The d-bar Neumann Problem and Schrödinger Operators 9783110315356, 9783110315301

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Friedrich Haslinger The ∂-Neumann Problem and Schrödinger Operators

De Gruyter Expositions in Mathematics

| Edited by Lev Birbrair, Fortaleza, Brazil Victor P. Maslov, Moscow, Russia Walter D. Neumann, New York City, New York, USA Markus J. Pflaum, Boulder, Colorado, USA Dierk Schleicher, Bremen, Germany Raymond O. Wells, Boulder, Colorado, USA

Volume 59

Friedrich Haslinger

The ∂-Neumann Problem and Schrödinger Operators |

Mathematics Subject Classification 2010 Primary: 32W05, 32A25, 32A36; Secondary: 35P05, 35J10 Author Prof. Dr. Friedrich Haslinger Universität Wien Fakultät für Mathematik Oskar-Morgenstern-Platz 1 1090 Wien [email protected]

ISBN 978-3-11-031530-1 e-ISBN 978-3-11-031535-6 Set-ISBN 978-3-11-031536-3 ISSN 0938-6572 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.dnb.de. © 2014 Walter de Gruyter GmbH, Berlin/Boston Typesetting: PTP-Berlin, Protago-TEX-Production GmbH, Berlin Printing and binding: CPI books GmbH, Leck ♾Printed on acid-free paper Printed in Germany www.degruyter.com

| To Hedi Enar and Philipp Kathi and Christopher

Preface The subject of this book is complex analysis in several variables and its connections to partial differential equations and to functional analysis. The first sections of each chapter contain prerequisites from functional analysis, Sobolev spaces, partial differential equations and spectral analysis, which are used in the following sections devoted to the main topic of the book. In this way the book becomes self-contained, with only one exception, where we do not provide all details in the proof of the general spectral theorem for unbounded self-adjoint operators. We concentrate on the Cauchy–Riemann equation (𝜕-equation) and investigate the properties of the canonical solution operator to 𝜕, the solution with minimal 𝐿2 norm and its relationship to the 𝜕-Neumann operator. The first chapter contains a discussion of Bergman spaces in one and several complex variables, including basic facts on Hilbert spaces. In the second chapter the solution operator to 𝜕 restricted to holomorphic 𝐿2 -functions in one complex variable is investigated, pointing out that the Bergman kernel of the associated Hilbert space of holomorphic functions plays an important role. We investigate operator properties like compactness and Schatten class membership, also for the solution operator on weighted spaces of entire functions (Fock spaces). In the third chapter we generalize the results to several complex variables and explain some new phenomena which do not appear in one variable. In the following we consider the general 𝜕-complex and derive properties of the complex Laplacian on 𝐿2 -spaces of bounded pseudoconvex domains and on weighted 𝐿2 -spaces. For this purpose we first concentrate on basic results about distributions, Sobolev spaces, and unbounded operators on Hilbert spaces. The key result in J. J. Kohn’s far-reaching method is the Kohn–Morrey formula, which is presented in different versions. Using this formula the basic properties of the 𝜕-Neumann operator – the bounded inverse of the complex Laplacian – are proved. In recent years it has turned out to be useful to investigate an even more general situation, namely the twisted 𝜕-complex, where 𝜕 is composed with a positive twist factor. In this way one obtains a rather general basic estimate, from which one gets Hörmander’s 𝐿2 -estimates for the solution of the Cauchy–Riemann equation together with results on related weighted spaces of entire functions, such as that these spaces are infinite-dimensional if the eigenvalues of the Levi matrix of the weight function show a certain behavior at infinity. In addition, it is pointed out that some 𝐿2 -estimates for 𝜕 can be interpreted in the sense of a general Brascamp–Lieb inequality. The next chapter contains a detailed account of the application of the 𝜕-methods to Schrödinger operators, Pauli and Dirac operators and to Witten–Laplacians. In this context, spectral analysis plays an important role. Therefore an extensive chapter on spectral analysis was inserted to provide a better understanding for the operator theoretic aspects in the 𝜕-Neumann problem, which, in particular, is used to exactly describe the spectrum of complex Laplacian on the Fock space. Returning to the

viii | Preface 𝜕-Neumann problem, we characterize compactness of the 𝜕-Neumann operator using a description of precompact subsets in 𝐿2 -spaces. Compactness of the 𝜕-Neumann operator is also related to properties of commutators of the Bergman projection and multiplication operators. In the last part we use the 𝜕-methods and some spectral theory to settle the question whether certain Schrödinger operators with a magnetic field have compact resolvent. It is also shown that a large class of Dirac operators fail to have compact resolvent. Finally we exhibit some situations where the 𝜕-Neumann operator is not compact. Numerous references for the topics of the text and for additional results are given in the notes at the end of each chapter. Most of the material of the book stems from various lectures of the author given at the University of Vienna, the Erwin Schrödinger International Institute for Mathematical Physics (ESI) in Vienna and at CIRM, Luminy, during programs on the 𝜕-Neumann operator in recent years. The author is indebted to both institutions, ESI and CIRM, for their help and hospitality. I would also like to thank my students Franz Berger, Damir Ferizović and Tobias Preinerstorfer for their constructive criticisms of the manuscript, and also for their help in eliminating a number of typos and minor errors. Vienna, March 2014

Friedrich Haslinger

Contents Preface | vii 1 1.1 1.2 1.3 1.4

Bergman spaces | 1 Elementary properties | 1 Examples | 8 Biholomorphic maps | 12 Notes | 15

2 2.1 2.2 2.3

The canonical solution operator to 𝜕 | 16 Compact operators on Hilbert spaces | 16 The canonical solution operator to 𝜕 restricted to 𝐴2 (𝔻) | 26 Notes | 35

3 3.1 3.2 3.3 3.4

Spectral properties of the canonical solution operator to 𝜕̄ | 36 Complex differential forms | 36 (0, 1)-forms with holomorphic coefficients | 37 Compactness and Schatten class membership | 39 Notes | 49

4 4.1 4.2 4.3 4.4 4.5

The 𝜕-complex | 50 Unbounded operators on Hilbert spaces | 50 Distributions | 65 A finite-dimensional analog | 71 The 𝜕-Neumann operator | 72 Notes | 86

5 5.1 5.2 5.3

Density of smooth forms | 87 Friedrichs’ Lemma and Sobolev spaces | 87 Density in the graph norm | 98 Notes | 103

6 6.1 6.2 6.3

The weighted 𝜕-complex | 104 The 𝜕-Neumann operator on (0, 1)-forms | 104 (0, 𝑞)-forms | 109 Notes | 112

7 7.1 7.2 7.3

The twisted 𝜕-complex | 114 An exact sequence of unbounded operators | 114 The twisted basic estimates | 115 Notes | 118

x | Contents 8 8.1 8.2 8.3

Applications | 119 Hörmander’s 𝐿2 -estimates | 119 Weighted spaces of entire functions | 122 Notes | 126

9 9.1 9.2 9.3 9.4 9.5 9.6

Spectral analysis | 127 Resolutions of the identity | 127 Spectral decomposition of bounded normal operators | 130 Spectral decomposition of unbounded self-adjoint operators | 136 Determination of the spectrum | 150 Variational characterization of the discrete spectrum | 160 Notes | 165

10 10.1 10.2 10.3 10.4 10.5 10.6

Schrödinger operators and Witten–Laplacians | 166 Difference quotients | 166 Interior regularity | 168 Schrödinger operators with magnetic field | 171 Witten–Laplacians | 178 Dirac and Pauli operators | 180 Notes | 182

11 11.1 11.2 11.3 11.4 11.5

Compactness | 183 Precompact sets in 𝐿2 -spaces | 183 Sobolev spaces and Gårding’s inequality | 186 Compactness in weighted spaces | 190 Bounded pseudoconvex domains | 202 Notes | 205

12 12.1 12.2 12.3

The 𝜕-Neumann operator and the Bergman projection | 207 The Stone–Weierstraß Theorem | 207 Commutators of the Bergman projection | 210 Notes | 215

13 13.1 13.2 13.3

Compact resolvents | 216 Schrödinger operators | 216 Dirac and Pauli operators | 218 Notes | 220

14 14.1 14.2 14.3

Spectrum of ◻ on the Fock space | 221 The general setting | 221 Determination of the spectrum | 223 Notes | 228

Contents

15 15.1 15.2 15.3

Obstructions to compactness | 229 The bidisc | 229 Weighted spaces | 230 Notes | 234

Bibliography | 235 Index | 239

| xi

1 Bergman spaces To investigate the solution to the inhomogeneous 𝜕-equation 𝜕𝑢 = 𝑔, we will first consider the case where the right-hand side 𝑔 is a holomorphic function. Therefore we need an appropriate Hilbert space of holomorphic functions – the Bergman space. We will use standard basic facts about Hilbert spaces, such as the Riesz representation theorem for continuous linear functionals, facts about orthogonal projections, and complete orthonormal bases. Let Ω ⊆ ℂ𝑛 be a domain and the Bergman space 𝐴2 (Ω) = {𝑓 : Ω 󳨀→ ℂ holomorphic : ‖𝑓‖2 = ∫ |𝑓(𝑧)|2 𝑑𝜆(𝑧) < ∞}, Ω 𝑛

where 𝜆 is the Lebesgue measure of ℂ . The inner product is given by (𝑓, 𝑔) = ∫ 𝑓(𝑧) 𝑔(𝑧) 𝑑𝜆(𝑧), Ω 2

for 𝑓, 𝑔 ∈ 𝐴 (Ω).

1.1 Elementary properties For sake of simplicity we first restrict ourselves to domains Ω ⊆ ℂ. We consider special continuous linear functionals on 𝐴2 (Ω) : the point evaluations. Let 𝑓 ∈ 𝐴2 (Ω) and fix 𝑧 ∈ Ω. By Cauchy’s integral theorem we have 𝑓(𝑧) =

𝑓(𝜁) 1 ∫ 𝑑𝜁, 2𝜋𝑖 𝜁 − 𝑧 𝛾𝑠

𝑖𝑡

where 𝛾𝑠 (𝑡) = 𝑧 + 𝑠𝑒 , 𝑡 ∈ [0, 2𝜋], 0 < 𝑠 ≤ 𝑟 and 𝐷(𝑧, 𝑟) = {𝑤 : |𝑤 − 𝑧| < 𝑟} ⊂ Ω. Using polar coordinates and integrating the above equality with respect to 𝑠 between 0 and 𝑟 we get 1 𝑓(𝑧) = 2 ∫ 𝑓(𝑤) 𝑑𝜆(𝑤). (1.1) 𝜋𝑟 𝐷(𝑧,𝑟)

Then, by Cauchy–Schwarz, |𝑓(𝑧)| ≤

1 𝜋𝑟2

∫ 1 . |𝑓(𝑤)| 𝑑𝜆(𝑤) 𝐷(𝑧,𝑟) 1/2

≤

1 ( ∫ 12 𝑑𝜆(𝑤)) 𝜋𝑟2

1/2

( ∫ |𝑓(𝑤)|2 𝑑𝜆(𝑤))

𝐷(𝑧,𝑟)

𝐷(𝑧,𝑟) 1/2

≤

1 ( ∫ |𝑓(𝑤)|2 𝑑𝜆(𝑤)) 𝜋1/2 𝑟 Ω

≤

1 ‖𝑓‖. 𝜋1/2 𝑟

2 | 1 Bergman spaces If 𝐾 is a compact subset of Ω, there is an 𝑟(𝐾) > 0 such that for any 𝑧 ∈ 𝐾 we have 𝐷(𝑧, 𝑟(𝐾)) ⊂ Ω and we get sup |𝑓(𝑧)| ≤ 𝑧∈𝐾

1 𝜋1/2 𝑟(𝐾)

‖𝑓‖.

If 𝐾 ⊂ Ω ⊂ ℂ𝑛 we can find a polycylinder 𝑃(𝑧, 𝑟(𝐾)) = {𝑤 ∈ ℂ𝑛 : |𝑤𝑗 − 𝑧𝑗 | < 𝑟(𝐾), 𝑗 = 1, . . . , 𝑛} such that for any 𝑧 ∈ 𝐾 we have 𝑃(𝑧, 𝑟(𝐾)) ⊂ Ω. Hence by iterating the above Cauchy integrals we get Proposition 1.1. Let 𝐾 ⊂ Ω be a compact set. Then there exists a constant 𝐶(𝐾), only depending on 𝐾 such that sup |𝑓(𝑧)| ≤ 𝐶(𝐾) ‖𝑓‖, (1.2) 𝑧∈𝐾

2

for any 𝑓 ∈ 𝐴 (Ω). Proposition 1.2. 𝐴2 (Ω) is a Hilbert space. Proof. If (𝑓𝑘 )𝑘 is a Cauchy sequence in 𝐴2 (Ω), by (1.2), it is also a Cauchy sequence with respect to uniform convergence on compact subsets of Ω. Hence the sequence (𝑓𝑘 )𝑘 has a holomorphic limit 𝑓 with respect to uniform convergence on compact subsets of Ω. On the other hand, the original 𝐿2 -Cauchy sequence has a subsequence, which converges pointwise almost everywhere to the 𝐿2 -limit of the original 𝐿2 -Cauchy sequence (see for instance [63]), and so the 𝐿2 -limit coincides with the holomorphic function 𝑓. Therefore 𝐴2 (Ω) is a closed subspace of 𝐿2 (Ω) and itself a Hilbert space. In the sequel we present basic facts about Hilbert spaces and their consequences for the Bergman spaces. Proposition 1.3. Let 𝐸 be a nonempty, convex, closed subset of the Hilbert space 𝐻, i.e. for 𝑥, 𝑦 ∈ 𝐸 one has 𝑡𝑥 + (1 − 𝑡)𝑦 ∈ 𝐸, for each 𝑡 ∈ [0, 1]. Then 𝐸 contains a uniquely determined element of minimal norm. Proof. The parallelogram rule says that ‖𝑥 + 𝑦‖2 + ‖𝑥 − 𝑦‖2 = 2 ‖𝑥‖2 + 2 ‖𝑦‖2 ,

𝑥, 𝑦 ∈ 𝐻.

Let 𝛿 = inf{‖𝑥‖ : 𝑥 ∈ 𝐸}. For 𝑥, 𝑦 ∈ 𝐸 we have 12 (𝑥 + 𝑦) ∈ 𝐸, hence 1/4 ‖𝑥 − 𝑦‖2 = 1/2 ‖𝑥‖2 + 1/2 ‖𝑦‖2 − ‖1/2(𝑥 + 𝑦)‖2 , implies that ‖𝑥 − 𝑦‖2 ≤ 2 ‖𝑥‖2 + 2 ‖𝑦‖2 − 4𝛿2 . So, if ‖𝑥‖ = ‖𝑦‖ = 𝛿, then 𝑥 = 𝑦 (uniqueness).

1.1 Elementary properties

|

3

By the definition of 𝛿 there exists a sequence (𝑦𝑘 )𝑘 in 𝐸 such that ‖𝑦𝑘 ‖ → 𝛿 if 𝑘 → ∞. The estimate ‖𝑦𝑘 − 𝑦𝑚 ‖2 ≤ 2 ‖𝑦𝑘 ‖2 + 2 ‖𝑦𝑚 ‖2 − 4𝛿2 implies that (𝑦𝑘 )𝑘 is a Cauchy sequence in 𝐻. Since 𝐻 is complete there exists 𝑥0 ∈ 𝐻 with ‖𝑦𝑘 − 𝑥0 ‖ → 0 and, as 𝐸 is closed, we have 𝑥0 ∈ 𝐸; the mapping 𝑥 󳨃→ ‖𝑥‖ is continuous and therefore ‖𝑥0 ‖ = lim𝑘→∞ ‖𝑦𝑘 ‖ = 𝛿. Theorem 1.4. Let 𝑀 be a closed subspace of the Hilbert space 𝐻. Then there exist uniquely determined mappings 𝑃 : 𝐻 󳨀→ 𝑀,

𝑄 : 𝐻 󳨀→ 𝑀⊥

such that (1) 𝑥 = 𝑃𝑥 + 𝑄𝑥 , ∀𝑥 ∈ 𝐻 (2) for 𝑥 ∈ 𝑀 we have 𝑃𝑥 = 𝑥, hence 𝑃2 = 𝑃 and 𝑄𝑥 = 0; for 𝑥 ∈ 𝑀⊥ we have 𝑃𝑥 = 0, 𝑄𝑥 = 𝑥, and 𝑄2 = 𝑄. (3) The distance of 𝑥 ∈ 𝐻 to 𝑀 is given by inf{‖𝑥 − 𝑦‖ : 𝑦 ∈ 𝑀} = ‖𝑥 − 𝑃𝑥‖. (4) For each 𝑥 ∈ 𝐻 we have

‖𝑥‖2 = ‖𝑃𝑥‖2 + ‖𝑄𝑥‖2 .

(5) 𝑃 and 𝑄 are continuous, linear, self-adjoint operators. 𝑃 and 𝑄 are the orthogonal projections of 𝐻 onto 𝑀 and 𝑀⊥ . Proof. For each 𝑥 ∈ 𝐻, the set 𝑥 + 𝑀 = {𝑥 + 𝑦 : 𝑦 ∈ 𝑀} is convex. Hence, by Proposition 1.3, there exists a uniquely determined element of minimal norm in 𝑥 + 𝑀, which is denoted by 𝑄𝑥. We set 𝑃𝑥 = 𝑥 − 𝑄𝑥 and see that 𝑃𝑥 ∈ 𝑀, since 𝑄𝑥 ∈ 𝑥 + 𝑀. Now we claim that 𝑄𝑥 ∈ 𝑀⊥ . We have to show that(𝑄𝑥, 𝑦) = 0 , ∀𝑦 ∈ 𝑀 : we can suppose that ‖𝑦‖ = 1, then we have (𝑄𝑥, 𝑄𝑥) = ‖𝑄𝑥‖2 ≤ ‖𝑄𝑥 − 𝛼𝑦‖2 = (𝑄𝑥 − 𝛼𝑦, 𝑄𝑥 − 𝛼𝑦),

∀𝛼 ∈ ℂ

by the minimality of 𝑄𝑥. Therefore we get 0 ≤ −𝛼(𝑦, 𝑄𝑥) − 𝛼(𝑄𝑥, 𝑦) + |𝛼|2 , setting 𝛼 = (𝑄𝑥, 𝑦), we obtain 0 ≤ −|(𝑄𝑥, 𝑦)|2 and (𝑄𝑥, 𝑦) = 0; hence 𝑄 : 𝐻 󳨀→ 𝑀⊥ . If 𝑥 = 𝑥0 +𝑥1 with 𝑥0 ∈ 𝑀 and 𝑥1 ∈ 𝑀⊥ , then 𝑥0 −𝑃𝑥 = 𝑄𝑥−𝑥1 , and since 𝑀∩𝑀⊥ = {0} we obtain 𝑥0 = 𝑃𝑥 and 𝑥1 = 𝑄𝑥, therefore 𝑃 and 𝑄 are uniquely determined. In a similar way, we get that 𝑃(𝛼𝑥 + 𝛽𝑦) − 𝛼𝑃𝑥 − 𝛽𝑃𝑦 = 𝛼𝑄𝑥 + 𝛽𝑄𝑦 − 𝑄(𝛼𝑥 + 𝛽𝑦).

4 | 1 Bergman spaces The left side belongs to 𝑀, the right side belongs to 𝑀⊥ , hence both sides are 0, which proves that 𝑃 and 𝑄 are linear. Property 3 follows by the definition of 𝑄, property 4 by the fact that (𝑃𝑥, 𝑄𝑥) = 0,

∀𝑥 ∈ 𝐻.

In addition we have ‖𝑄(𝑥 − 𝑦)‖ = inf{‖𝑥 − 𝑦 + 𝑚‖ : 𝑚 ∈ 𝑀} ≤ ‖𝑥 − 𝑦‖, hence 𝑄 and 𝑃 = 𝐼 − 𝑄 are continuous. For 𝑥, 𝑦 ∈ 𝐻 we have (𝑃𝑥, 𝑦) = (𝑃𝑥, 𝑃𝑦 + 𝑄𝑦) = (𝑃𝑥, 𝑃𝑦) and (𝑥, 𝑃𝑦) = (𝑃𝑥 + 𝑄𝑥, 𝑃𝑦) = (𝑃𝑥, 𝑃𝑦) hence (𝑃𝑥, 𝑦) = (𝑥, 𝑃𝑦), and so 𝑃 is self-adjoint. Corollary 1.5. If 𝑀 ≠ 𝐻 is a closed, proper subspace of the Hilbert space 𝐻, then there exists an element 𝑦 ≠ 0 with 𝑦 ⊥ 𝑀. Proof. Let 𝑥 ∈ 𝐻 such that 𝑥 ∉ 𝑀. Set 𝑦 = 𝑄𝑥 : then 𝑥 ≠ 𝑃𝑥 implies 𝑦 ≠ 0. The next result is the Riesz representation theorem. Theorem 1.6. Let 𝐿 be a continuous linear functional on the Hilbert space 𝐻. Then there exists a uniquely determined element 𝑦 ∈ 𝐻 such that 𝐿𝑥 = (𝑥, 𝑦), ∀𝑥 ∈ 𝐻. Proof. If 𝐿(𝑥) = 0, ∀𝑥 ∈ 𝐻, then we set 𝑦 = 0. Otherwise we define 𝑀 = {𝑥 : 𝐿𝑥 = 0}. Then, by the continuity of 𝐿, the subspace 𝑀 of 𝐻 is closed. By Corollary 1.5 we have 𝐿𝑧 𝑀⊥ ≠ 0. Let 𝑧 ∈ 𝑀⊥ with 𝑧 ≠ 0. Then 𝐿𝑧 ≠ 0. Now set 𝑦 = 𝛼𝑧, where 𝛼 = ‖𝑧‖ 2 . Then ⊥ 𝑦 ∈ 𝑀 and 𝐿𝑧 |𝐿𝑧|2 𝐿𝑦 = 𝐿(𝛼𝑧) = 𝐿𝑧 = = (𝑦, 𝑦) = |𝛼|2 (𝑧, 𝑧). 2 ‖𝑧‖ ‖𝑧‖2 For 𝑥 ∈ 𝐻 we define 𝑥󸀠 = 𝑥 −

𝐿𝑥 𝑦 (𝑦, 𝑦)

and 𝑥󸀠󸀠 =

𝐿𝑥 𝑦. (𝑦, 𝑦)

Then we obtain 𝐿𝑥󸀠 = 0 and 𝑥󸀠 ∈ 𝑀, hence (𝑥󸀠 , 𝑦) = 0 and (𝑥, 𝑦) = (𝑥󸀠󸀠 , 𝑦) = (

𝐿𝑥 𝑦, 𝑦) = 𝐿𝑥. (𝑦, 𝑦)

If (𝑥, 𝑦) = (𝑥, 𝑦󸀠 ), ∀𝑥 ∈ 𝐻, then we get (𝑥, 𝑦 − 𝑦󸀠 ) = 0, ∀𝑥 ∈ 𝐻, in particular (𝑦 − 𝑦󸀠 , 𝑦 − 𝑦󸀠 ) = 0. Therefore 𝑦 = 𝑦󸀠 , which shows that 𝑦 is uniquely determined. Corollary 1.7. Let 𝐻 be a Hilbert space and 𝐿 ∈ 𝐻󸀠 a continuous linear functional. Then the dual norm ‖𝐿‖ = sup{|𝐿𝑥| : ‖𝑥‖ ≤ 1}

1.1 Elementary properties

|

5

can be expressed in the form ‖𝐿‖ = sup{|(𝑥, 𝑦)| : ‖𝑥‖ ≤ 1} = ‖𝑦‖, where 𝑦 ∈ 𝐻 corresponds to 𝐿. For fixed 𝑧 ∈ Ω, (1.2) also implies that the point evaluation 𝑓 󳨃→ 𝑓(𝑧) is a continuous linear functional on 𝐴2 (Ω), hence, by the Riesz representation Theorem 1.6, there is a uniquely determined function 𝑘𝑧 ∈ 𝐴2 (Ω) such that 𝑓(𝑧) = (𝑓, 𝑘𝑧 ) = ∫ 𝑓(𝑤) 𝑘𝑧 (𝑤) 𝑑𝜆(𝑤).

(1.3)

Ω

We set 𝐾(𝑧, 𝑤) = 𝑘𝑧 (𝑤). Then 𝑤 󳨃→ 𝐾(𝑧, 𝑤) = 𝑘𝑧 (𝑤) is an element of 𝐴2 (Ω), hence the function 𝑤 󳨃→ 𝐾(𝑧, 𝑤) is anti-holomorphic on Ω and we have 𝑓(𝑧) = ∫ 𝐾(𝑧, 𝑤)𝑓(𝑤) 𝑑𝜆(𝑤),

𝑓 ∈ 𝐴2 (Ω).

Ω

The function of two complex variables (𝑧, 𝑤) 󳨃→ 𝐾(𝑧, 𝑤) is called the Bergman kernel of Ω and the above identity represents the reproducing property of the Bergman kernel. Now we use the reproducing property for the holomorphic function 𝑧 󳨃→ 𝑘𝑢 (𝑧), where 𝑢 ∈ Ω is fixed: 𝑘𝑢 (𝑧) = ∫ 𝐾(𝑧, 𝑤)𝑘𝑢 (𝑤) 𝑑𝜆(𝑤) = ∫ 𝑘𝑧 (𝑤) 𝐾(𝑢, 𝑤) 𝑑𝜆(𝑤) Ω

Ω −

= ( ∫ 𝐾(𝑢, 𝑤)𝑘𝑧 (𝑤) 𝑑𝜆(𝑤)) = 𝑘𝑧 (𝑢), Ω

hence we have 𝑘𝑢 (𝑧) = 𝑘𝑧 (𝑢), or 𝐾(𝑧, 𝑢) = 𝐾(𝑢, 𝑧). It follows that the Bergman kernel is holomorphic in the first variable and antiholomorphic in the second variable. Proposition 1.8. The Bergman kernel is uniquely determined by the properties that it is an element of 𝐴2 (Ω) in 𝑧 and that it is conjugate symmetric and reproduces 𝐴2 (Ω). Proof. To see this let 𝐾󸀠 (𝑧, 𝑤) be another kernel with these properties: Then we have 𝐾(𝑧, 𝑤) = ∫ 𝐾󸀠 (𝑧, 𝑢)𝐾(𝑢, 𝑤) 𝑑𝜆(𝑢) Ω

−

= ( ∫ 𝐾(𝑤, 𝑢)𝐾󸀠 (𝑢, 𝑧) 𝑑𝜆(𝑢)) Ω

= 𝐾󸀠 (𝑤, 𝑧) = 𝐾󸀠 (𝑧, 𝑤).

6 | 1 Bergman spaces Now let 𝜙 ∈ 𝐿2 (Ω). Since 𝐴2 (Ω) is a closed subspace of 𝐿2 (Ω) there exists a uniquely determined orthogonal projection 𝑃 : 𝐿2 (Ω) 󳨀→ 𝐴2 (Ω), see Theorem 1.4. For the function 𝑃𝜙 ∈ 𝐴2 (Ω) we use the reproducing property and obtain 𝑃𝜙(𝑧) = ∫ 𝐾(𝑧, 𝑤)𝑃𝜙(𝑤) 𝑑𝜆(𝑤) = (𝑃𝜙, 𝑘𝑧 ) = (𝜙, 𝑃𝑘𝑧 ) = (𝜙, 𝑘𝑧 );

(1.4)

Ω

where we still have used that 𝑃 is a self-adjoint operator and that 𝑃𝑘𝑧 = 𝑘𝑧 . Hence 𝑃𝜙(𝑧) = ∫ 𝐾(𝑧, 𝑤)𝜙(𝑤) 𝑑𝜆(𝑤).

(1.5)

Ω

𝑃 is called the Bergman projection. In order to compute the Bergman kernel for some special cases we need the concept of a complete orthonormal basis. A subset {𝑢𝛼 : 𝛼 ∈ 𝐴} of a Hilbert space is called orthonormal, if (𝑢𝛼 , 𝑢𝛽 ) = 𝛿𝛼𝛽 for each 𝛼, 𝛽 ∈ 𝐴. If (𝑥𝑘 )𝑘 is a linearly independent sequence in 𝐻, there is a standard procedure, called the Gram–Schmidt process, for converting (𝑥𝑘)𝑘 into an orthonormal 𝑁 sequence (𝑢𝑘 )𝑘 such that the linear span of (𝑢𝑘 )𝑁 𝑘=1 equals the linear span of (𝑥𝑘 )𝑘=1 for all 𝑁 ∈ ℕ. We start with 𝑢1 = 𝑥1 /‖𝑥1 ‖. Having defined 𝑢1 , . . . , 𝑢𝑁−1 , we set 𝑁−1

𝑣𝑁 = 𝑥𝑁 − ∑ (𝑥𝑁 , 𝑢𝑗 )𝑢𝑗 . 𝑗=1

The element 𝑣𝑁 is nonzero because 𝑥𝑁 is not in the linear span of 𝑥1 , . . . , 𝑥𝑁−1 and hence not in the span of 𝑢1 , . . . , 𝑢𝑁−1 . So we can set 𝑢𝑁 = 𝑣𝑁 /‖𝑣𝑁 ‖. It is now clear that (𝑢𝑘 )𝑁 𝑘=1 has the desired properties. Next we prove Bessel’s inequality. Proposition 1.9. If {𝑢𝛼 : 𝛼 ∈ 𝐴} is an orthonormal set in the Hilbert space 𝐻, then for any 𝑢 ∈ 𝐻 ∑ |(𝑢, 𝑢𝛼 )|2 ≤ ‖𝑢‖2 . (1.6) 𝛼∈𝐴

In particular, the set {𝛼 : (𝑢, 𝑢𝛼 ) ≠ 0} is countable. Proof. It suffices to show that ∑𝛼∈𝐹 |(𝑢, 𝑢𝛼 )|2 ≤ ‖𝑢‖2 , for any finite set 𝐹 ⊂ 𝐴. We use the property ‖𝑢𝛼 + 𝑢𝛽 ‖2 = ‖𝑢𝛼 ‖2 + ‖𝑢𝛽 ‖2 , for 𝛼, 𝛽 ∈ 𝐴, 𝛼 ≠ 𝛽. 0 ≤ ‖𝑢 − ∑ (𝑢, 𝑢𝛼 )𝑢𝛼 ‖2 𝛼∈𝐹

= ‖𝑢‖2 − 2ℜ(𝑢, ∑ (𝑢, 𝑢𝛼 )𝑢𝛼 ) + ‖ ∑ (𝑢, 𝑢𝛼 )𝑢𝛼 ‖2 𝛼∈𝐹

𝛼∈𝐹

= ‖𝑢‖2 − 2 ∑ |(𝑢, 𝑢𝛼 )|2 + ∑ |(𝑢, 𝑢𝛼 )|2 𝛼∈𝐹

= ‖𝑢‖2 − ∑ |(𝑢, 𝑢𝛼 )|2 . 𝛼∈𝐹

𝛼∈𝐹

1.1 Elementary properties

|

7

Proposition 1.10. If {𝑢𝛼 : 𝛼 ∈ 𝐴} is an orthonormal set in the Hilbert space 𝐻, then the following conditions are equivalent: (1) Completeness: if (𝑢, 𝑢𝛼 ) = 0 for all 𝛼 ∈ 𝐴, then 𝑢 = 0. (2) Parseval’s equation: ‖𝑢‖2 = ∑𝛼∈𝐴 |(𝑢, 𝑢𝛼 )|2 for all 𝑢 ∈ 𝐻. (3) 𝑢 = ∑𝛼∈𝐴 (𝑢, 𝑢𝛼 )𝑢𝛼 for each 𝑢 ∈ 𝐻, where the sum has only countably many nonzero terms and converges in norm to 𝑢 no matter how these terms are ordered. Proof. (1) implies (3): If 𝑢 ∈ 𝐻, let 𝛼1 , 𝛼2 , . . . be any enumeration of those 𝛼’s for which 2 (𝑢, 𝑢𝛼 ) ≠ 0. By Bessel’s inequality the series ∑∞ 𝑗=1 |(𝑢, 𝑢𝛼𝑗 )| converges, so 𝑀 󵄩󵄩 𝑀 󵄩2 󵄩󵄩 ∑ (𝑢, 𝑢𝛼 )𝑢𝛼 󵄩󵄩󵄩 = ∑ |(𝑢, 𝑢𝛼 )|2 → 0 as 𝑚, 𝑀 → ∞. 󵄩󵄩 󵄩󵄩 𝑗=𝑚 𝑗=𝑚 ∞ Hence the series ∑∞ 𝑗=1 (𝑢, 𝑢𝛼𝑗 )𝑢𝛼𝑗 converges in 𝐻. If 𝑣 = 𝑢−∑𝑗=1 (𝑢, 𝑢𝛼𝑗 )𝑢𝛼𝑗 , then (𝑣, 𝑢𝛼 ) = 0 for all 𝛼 ∈ 𝐴, so by (1), 𝑣 = 0. (3) implies (2): As in the proof of Bessel’s inequality we have 𝑚

𝑚

𝑗=1

𝑗=1

‖𝑢‖2 − ∑ |(𝑢, 𝑢𝛼𝑗 )|2 = ‖𝑢 − ∑ (𝑢, 𝑢𝛼𝑗 )𝑢𝛼𝑗 ‖2 → 0 as 𝑚 → ∞. Finally, that (2) implies (1) is obvious. An orthonormal set having the properties of Proposition 1.10 is called an orthonormal basis of 𝐻. Remark 1.11. An application of Zorn’s lemma shows that the collection of orthonormal sets in a Hilbert space, ordered by inclusion, has a maximal element. Maximality is equivalent to (1) of Proposition 1.10, hence for each orthonormal set there exists an orthonormal basis, which contains the given orthonormal set. If 𝐻 is a separable Hilbert space, it has a countable orthonormal basis. Proposition 1.12. Let 𝐾 ⊂ Ω be a compact subset and {𝜙𝑗 } be an orthonormal basis of 𝐴2 (Ω). Then the series ∞

∑ 𝜙𝑗 (𝑧) 𝜙𝑗 (𝑤)

𝑗=1

sums uniformly on 𝐾 × 𝐾 to the Bergman kernel 𝐾(𝑧, 𝑤). Proof. For the proof of this statement we use the duality for the sequence space 𝑙2 to get ∞

1/2

sup ( ∑ |𝜙𝑗 (𝑧)|2 ) 𝑧∈𝐾

𝑗=1

󵄨󵄨 ∞ 󵄨󵄨 ∞ = sup {󵄨󵄨󵄨 ∑ 𝑎𝑗 𝜙𝑗 (𝑧)󵄨󵄨󵄨 : ∑ |𝑎𝑗 |2 = 1, 𝑧 ∈ 𝐾} 󵄨 𝑗=1 󵄨 𝑗=1 = sup{|𝑓(𝑧)| : ‖𝑓‖ = 1, 𝑧 ∈ 𝐾} ≤ 𝐶(𝐾),

(1.7)

8 | 1 Bergman spaces where we have used (1.2) in the last inequality. Now Cauchy–Schwarz gives ∞

∞

𝑗=1

𝑗=1

∑ |𝜙𝑗 (𝑧) 𝜙𝑗 (𝑤)| ≤ ( ∑ |𝜙𝑗 (𝑧)|2 )

1/2

∞

( ∑ |𝜙𝑗 (𝑤)|2 )

1/2

𝑗=1

with uniform convergence in 𝑧, 𝑤 ∈ 𝐾. In addition it follows that (𝜙𝑗 (𝑧))𝑗 ∈ 𝑙2 and the function ∞

𝑤 󳨃→ ∑ 𝜙𝑗 (𝑧) 𝜙𝑗 (𝑤) 𝑗=1

belongs to 𝐴2 (Ω). Let the sum of the series be denoted by 𝐾󸀠 (𝑧, 𝑤). Notice that 𝐾󸀠 (𝑧, 𝑤) is conjugate symmetric and that for 𝑓 ∈ 𝐴2 (Ω) we get ∞

∫ 𝐾󸀠 (𝑧, 𝑤)𝑓(𝑤) 𝑑𝜆(𝑤) = ∑ ∫ 𝑓(𝑤)𝜙𝑗 (𝑤) 𝑑𝜆(𝑤) 𝜙𝑗 (𝑧) = 𝑓(𝑧) 𝑗=1

Ω

Ω

with convergence in the Hilbert space 𝐴2 (Ω). But (1.7) implies uniform convergence on compact subsets of Ω, hence 𝑓(𝑧) = ∫ 𝐾󸀠 (𝑧, 𝑤)𝑓(𝑤) 𝑑𝜆(𝑤), Ω

for all 𝑓 ∈ 𝐴2 (Ω), so 𝐾󸀠 (𝑧, 𝑤) is a reproducing kernel. By the uniqueness of the Bergman kernel we obtain 𝐾󸀠 (𝑧, 𝑤) = 𝐾(𝑧, 𝑤). We notice that (1.7) implies 𝐾(𝑧, 𝑧) = sup{|𝑓(𝑧)|2 : 𝑓 ∈ 𝐴2 (Ω) , ‖𝑓‖ = 1}.

(1.8)

1.2 Examples (a) 𝑧𝑛 , 𝑛 = 0, 1, 2, . . . constitute an orthonormal basis in The functions 𝜙𝑛 (𝑧) = √ 𝑛+1 𝜋

𝐴2 (𝔻) , 𝔻 = {𝑧 ∈ ℂ : |𝑧| < 1}. This follows from

2𝜋 1

∫𝑧

𝑛

𝑧𝑚

𝑑𝜆(𝑧) = ∫ ∫ 𝑟𝑛 𝑒𝑖𝑛𝜃 𝑟𝑚 𝑒−im𝜃 𝑟 𝑑𝑟 𝑑𝜃 = 0 0

𝔻

2𝜋 𝛿 . 𝑛 + 𝑚 + 2 𝑛,𝑚

𝑛 For each 𝑓 ∈ 𝐴2 (𝔻) with Taylor series expansion 𝑓(𝑧) = ∑∞ 𝑛=0 𝑎𝑛 𝑧 we get 1 2𝜋

(𝑓, 𝑧𝑛 ) = ∫ 𝑓(𝑧)𝑧𝑛 𝑑𝜆(𝑧) = ∫ ∫ 𝑓(𝑟𝑒𝑖𝜃 )𝑟𝑛 𝑒−𝑖𝑛𝜃 𝑟 𝑑𝑟 𝑑𝜃 𝔻

0 0

1.2 Examples 1 2𝜋

= ∫∫ 0 0

|

9

1

𝑓(𝑟𝑒𝑖𝜃 ) 𝑎 𝑟𝑒𝑖𝜃 𝑑𝜃 𝑟2𝑛+1 𝑑𝑟 = 2𝜋𝑎𝑛 ∫ 𝑟2𝑛+1 𝑑𝑟 = 𝜋 𝑛 , 𝑛+1 𝑛+1 𝑟 𝑒𝑖(𝑛+1)𝜃 0

where we used the fact that 𝑎𝑛 =

𝑓(𝑧) 1 𝑑𝑧, ∫ 2𝜋𝑖 𝑧𝑛+1 𝛾𝑟

for 𝛾𝑟 (𝜃) = 𝑟𝑒𝑖𝜃 . Hence, by the uniqueness of the Taylor series expansion, we obtain that (𝑓, 𝜙𝑛 ) = 0, for each 𝑛 = 0, 1, 2, . . . implies 𝑓 ≡ 0. This means that (𝜙𝑛 )∞ 𝑛=0 constitutes an orthonormal basis for 𝐴2 (𝔻) and we get ∞

‖𝑓‖2 = ∑ |(𝑓, 𝜙𝑛 )|2 , 𝑛=0

which is equivalent to ∞

|𝑎𝑛 |2 , 𝑛=0 𝑛 + 1

‖𝑓‖2 = 𝜋 ∑

∞

𝑓(𝑧) = ∑ 𝑎𝑛 𝑧𝑛 . 𝑛=0

Hence each 𝑓 ∈ 𝐴 (𝔻) can be written in the form 𝑓 = ∑∞ 𝑛=0 𝑐𝑛 𝜙𝑛 , where the sum 2 converges in 𝐴 (𝔻), but also uniformly on compact subsets of 𝔻. For the coefficients 𝑐𝑛 we have : 𝑐𝑛 = (𝑓, 𝜙𝑛 ). Now we compute an explicit formula for the Bergman kernel 𝐾(𝑧, 𝑤) of 𝔻. The function 𝑧 󳨃→ 𝐾(𝑧, 𝑤), with 𝑤 ∈ 𝔻 fixed, belongs to 𝐴2 (𝔻). Hence we get from the above formula that 2

∞

𝐾(𝑧, 𝑤) = ∑ 𝑐𝑛 𝜙𝑛 (𝑧), 𝑛=0

where 𝑐𝑛 = (𝐾(., 𝑤), 𝜙𝑛 ), in other words 𝑐𝑛 = (𝜙𝑛 , 𝐾(., 𝑤)) = ∫ 𝜙𝑛 (𝑧)𝐾(𝑤, 𝑧) 𝑑𝜆(𝑧) = 𝜙𝑛 (𝑤), 𝔻

by the reproducing property of the Bergman kernel. This implies that the Bergman kernel is of the form ∞

𝐾(𝑧, 𝑤) = ∑ 𝜙𝑛 (𝑧) 𝜙𝑛 (𝑤),

(1.9)

𝑛=0

where the sum converges uniformly in 𝑧 on all compact subsets of 𝔻. (This is true for any complete orthonormal system, as is shown above.) A simple computation now gives ∞ 1 1 ∞ 1 ∑ (𝑛 + 1)(𝑧𝑤)𝑛 = 𝐾(𝑧, 𝑤) = ∑ 𝜙𝑛 (𝑧) 𝜙𝑛 (𝑤) = . (1.10) 𝜋 𝜋 (1 − 𝑧𝑤)2 𝑛=0 𝑛=0 Hence for each 𝑓 ∈ 𝐴2 (𝔻) we have 𝑓(𝑧) =

1 1 𝑓(𝑤) 𝑑𝜆(𝑤). ∫ 𝜋 (1 − 𝑧𝑤)2 𝔻

10 | 1 Bergman spaces If we fix 𝑧 ∈ 𝔻 and set 𝑓(𝑤) = 1/(1 − 𝑤𝑧)2 , then we get the interesting formula 1 1 1 𝑑𝜆(𝑤) = . ∫ 𝜋 |1 − 𝑧𝑤|4 (1 − |𝑧|2 )2 𝔻

(b) Next we determine the Bergman kernel of the polycylinder. Proposition 1.13. Let Ω𝑗 ⊂ ℂ𝑛𝑗 , 𝑗 = 1, 2 be two bounded domains with Bergman kernels 𝐾Ω1 and 𝐾Ω2 . Then the Bergman kernel 𝐾Ω of the product domain Ω = Ω1 × Ω2 is given by 𝐾Ω ((𝑧1 , 𝑧2 ), (𝑤1 , 𝑤2 )) = 𝐾Ω1 (𝑧1 , 𝑤1 ) 𝐾Ω2 (𝑧2 , 𝑤2 ) (1.11) for (𝑧1 , 𝑧2 ), (𝑤1 , 𝑤2 ) ∈ Ω1 × Ω2 . Proof. In order to show this, let 𝐹 denote the function on the right-hand side of (1.11). It is clear that (𝑧1 , 𝑧2 ) 󳨃→ 𝐹((𝑧1 , 𝑧2 ), (𝑤1 , 𝑤2 )) belongs to 𝐴2 (Ω) for each fixed (𝑤1 , 𝑤2 ) ∈ Ω and that 𝐹 is anti-holomorphic in the second variable. The reproducing property 𝑓(𝑧1 , 𝑧2 ) =

∫ 𝐹((𝑧1 , 𝑧2 ), (𝑤1 , 𝑤2 ))𝑓(𝑤1 , 𝑤2 ) 𝑑𝜆(𝑤1 , 𝑤2 ) Ω1 ×Ω2

is a consequence of Fubini’s theorem and the corresponding reproducing properties of 𝐾Ω1 and 𝐾Ω2 . Hence, by the uniqueness property of the Bergman kernel, Proposition 1.8, we obtain 𝐹 = 𝐾Ω . From this we get that the Bergman kernel of the polycylinder 𝔻𝑛 is given by 𝐾𝔻𝑛 (𝑧, 𝑤) =

1 𝑛 1 ∏ . 𝜋𝑛 𝑗=1 (1 − 𝑧𝑗 𝑤𝑗 )2

(1.12)

(c) For the computation of the Bergman kernel 𝐾𝔹𝑛 of the unit ball in ℂ𝑛 we use the beta and gamma function 1

∫ 𝑥𝑘 (1 − 𝑥)𝑚 𝑑𝑥 = 𝐵(𝑘 + 1, 𝑚 + 1) = 0

Γ(𝑘 + 1)Γ(𝑚 + 1) , Γ(𝑘 + 𝑚 + 2)

where 𝑘, 𝑚 ∈ ℕ and that for 0 ≤ 𝑎 < 1, √1−𝑎2

∫ 𝑥2𝑘+1 (1 − 0

1

𝑚+1

𝑥2 ) 1 − 𝑎2

𝑑𝑥 =

1 (1 − 𝑎2 )𝑘+1 ∫ 𝑦𝑘 (1 − 𝑦)𝑚+1 𝑑𝑦 2 0

1 = (1 − 𝑎2 )𝑘+1 𝐵(𝑘 + 1, 𝑚 + 2) 2 Γ(𝑘 + 1)Γ(𝑚 + 2) 1 . = (1 − 𝑎2 )𝑘+1 2 Γ(𝑘 + 𝑚 + 3)

1.2 Examples

| 11

𝛼

Now we can normalize the orthogonal basis {𝑧𝛼 = 𝑧1 1 . . . 𝑧𝑛𝛼𝑛 } in 𝐴2 (𝔹𝑛 ) and obtain ‖𝑧𝛼 ‖2 = ∫ |𝑧1 |2𝛼1 . . . |𝑧𝑛 |2𝛼𝑛 𝑑𝜆(𝑧) 𝔹𝑛

=

𝜋 ∫ |𝑧1 |2𝛼1 . . . |𝑧𝑛−1 |2𝛼𝑛−1 (1 − |𝑧1 |2 − ⋅ ⋅ ⋅ − |𝑧𝑛−1 |2 )𝛼𝑛 +1 𝑑𝜆 𝛼𝑛 + 1 𝔹𝑛−1

𝜋 = ∫ |𝑧1 |2𝛼1 . . . |𝑧𝑛−2 |2𝛼𝑛−2 (1 − |𝑧1 |2 − ⋅ ⋅ ⋅ − |𝑧𝑛−2 |2 )𝛼𝑛 +1 𝛼𝑛 + 1 𝔹𝑛−1

|𝑧𝑛−1 |2 (1 − ) .|𝑧𝑛−1 | 1 − |𝑧1 |2 − ⋅ ⋅ ⋅ − |𝑧𝑛−2 |2 𝜋 𝜋Γ(𝛼𝑛−1 + 1)Γ(𝛼𝑛 + 2) = 𝛼𝑛 + 1 Γ(𝛼𝑛 + 𝛼𝑛−1 + 3) 2𝛼𝑛−1

𝛼𝑛 +1

𝑑𝜆

. ∫ |𝑧1 |2𝛼1 . . . |𝑧𝑛−2 |2𝛼𝑛−2 (1 − |𝑧1 |2 − ⋅ ⋅ ⋅ − |𝑧𝑛−2 |2 )𝛼𝑛 +𝛼𝑛−1 +2 𝑑𝜆 𝔹𝑛−2

𝜋Γ(𝛼1 + 1)Γ(𝛼𝑛 + ⋅ ⋅ ⋅ + 𝛼2 + 𝑛) 𝜋 𝜋Γ(𝛼𝑛−1 + 1)Γ(𝛼𝑛 + 2) ... 𝛼𝑛 + 1 Γ(𝛼𝑛 + 𝛼𝑛−1 + 3) Γ(𝛼𝑛 + ⋅ ⋅ ⋅ + 𝛼1 + 𝑛 + 1) 𝜋𝑛 𝛼1 ! . . . 𝛼𝑛 ! = . (𝛼𝑛 + ⋅ ⋅ ⋅ + 𝛼1 + 𝑛)! =

Hence the Bergman kernel of the unit ball is given by 𝐾𝔹𝑛 (𝑧, 𝑤) = ∑ 𝛼

(𝛼𝑛 + ⋅ ⋅ ⋅ + 𝛼1 + 𝑛)! 𝛼 𝛼 𝑧 𝑤 𝜋𝑛 𝛼1 ! . . . 𝛼𝑛 !

=

(𝛼𝑛 + ⋅ ⋅ ⋅ + 𝛼1 + 𝑛)! 𝛼 𝛼 1 ∞ ∑ ∑ 𝑧 𝑤 𝜋𝑛 𝑘=0 |𝛼|=𝑘 𝛼1 ! . . . 𝛼𝑛 !

=

1 ∞ ∑ (𝑘 + 𝑛)(𝑘 + 𝑛 − 1) . . . (𝑘 + 1)(𝑧1 𝑤1 + ⋅ ⋅ ⋅ + 𝑧𝑛 𝑤𝑛 )𝑘 𝜋𝑛 𝑘=0

=

1 𝑛! . 𝜋𝑛 (1 − (𝑧1 𝑤1 + ⋅ ⋅ ⋅ + 𝑧𝑛 𝑤𝑛 ))𝑛+1

(d) 2 In the sequel we will also consider the Fock space 𝐴2 (ℂ𝑛 , 𝑒−|𝑧| ) consisting of all entire functions 𝑓 such that 2

∫ |𝑓(𝑧)|2 𝑒−|𝑧| 𝑑𝜆(𝑧) < ∞. ℂ𝑛

It is clear that the Fock space is a Hilbert space with the inner product 2

(𝑓, 𝑔) = ∫ 𝑓(𝑧) 𝑔(𝑧) 𝑒−|𝑧| 𝑑𝜆(𝑧). ℂ𝑛

12 | 1 Bergman spaces 2

Similar to at the start of this chapter, setting 𝑛 = 1, we obtain for 𝑓 ∈ 𝐴2 (ℂ, 𝑒−|𝑧| ) that |𝑓(𝑧)| ≤

1 𝜋𝑟2

2

2

∫ 𝑒|𝑤| /2 |𝑓(𝑤)| 𝑒−|𝑤| /2 𝑑𝜆(𝑤) 𝐷(𝑧,𝑟) 1/2

≤

2 1 ( ∫ 𝑒|𝑤| 𝑑𝜆(𝑤)) 2 𝜋𝑟

2

1/2

( ∫ |𝑓(𝑤)|2 𝑒−|𝑤| 𝑑𝜆(𝑤))

𝐷(𝑧,𝑟)

𝐷(𝑧,𝑟) 1/2

2

≤ 𝐶( ∫ |𝑓(𝑤)|2 𝑒−|𝑤| 𝑑𝜆(𝑤)) ℂ

≤ 𝐶‖𝑓‖, where 𝐶 is a constant only depending on 𝑧. This implies that the Fock space 2 𝐴2 (ℂ𝑛 , 𝑒−|𝑧| ) has the reproducing property. The monomials {𝑧𝛼 } constitute an orthogonal basis and the norms of the monomials are 2

2

‖𝑧𝛼 ‖2 = ∫ |𝑧1 |2𝛼1 𝑒−|𝑧1 | 𝑑𝜆(𝑧1 ) ⋅ ⋅ ⋅ ∫ |𝑧𝑛 |2𝛼𝑛 𝑒−|𝑧𝑛 | 𝑑𝜆(𝑧𝑛 ) ℂ ∞

2

ℂ ∞

2

= (2𝜋)𝑛 ∫ 𝑟2𝛼1 +1 𝑒−𝑟 𝑑𝑟 ⋅ ⋅ ⋅ ∫ 𝑟2𝛼𝑛 +1 𝑒−𝑟 𝑑𝑟 𝑛

0

0

= 𝜋 𝛼1 ! ⋅ ⋅ ⋅ 𝛼𝑛 !. 2

Hence the Bergman kernel of 𝐴2 (ℂ𝑛 , 𝑒−|𝑧| ) is of the form 𝐾(𝑧, 𝑤) = ∑ 𝛼

𝑧𝛼 𝑤𝛼 𝑧𝛼 𝑤𝛼 1 ∞ 1 = 𝑛 ∑ ∑ exp(𝑧1 𝑤1 + ⋅ ⋅ ⋅ + 𝑧𝑛 𝑤𝑛 ). = 𝛼 2 ‖𝑧 ‖ 𝜋 𝑘=0 |𝛼|=𝑘 𝛼1 ! ⋅ ⋅ ⋅ 𝛼𝑛 ! 𝜋𝑛

(1.13)

1.3 Biholomorphic maps Here we describe the behavior of the Bergman kernel under biholomorphic maps. Proposition 1.14. Let 𝐹 : Ω1 󳨀→ Ω2 be a biholomorphic map between bounded do𝜕𝑓 (𝑧) mains in ℂ𝑛 . Let 𝑓1 , . . . , 𝑓𝑛 be the components of 𝐹 and 𝐹󸀠 (𝑧) = ( 𝜕𝑧𝑗 )𝑛𝑗,𝑘=1 . 𝑘 Then 𝐾Ω1 (𝑧, 𝑤) = det𝐹󸀠 (𝑧) 𝐾Ω2 (𝐹(𝑧), 𝐹(𝑤)) det𝐹󸀠 (𝑤), (1.14) for all 𝑧, 𝑤 ∈ Ω1 . Proof. Notice that, by the Cauchy–Riemann equations for 𝑓𝑗 = 𝑢𝑗 +𝑖𝑣𝑗 , 𝑗 = 1, . . . , 𝑛, the determinant of the Jacobian of the mapping 𝐹 equals to |det𝐹󸀠 (𝑧)|2 . The substitution formula for integrals implies that for 𝑔 ∈ 𝐿2 (Ω2 ) we have ∫ |𝑔(𝜁)|2 𝑑𝜆(𝜁) = ∫ |𝑔(𝐹(𝑧))|2 |det𝐹󸀠 (𝑧)|2 𝑑𝜆(𝑧). Ω2

Ω1

(1.15)

1.3 Biholomorphic maps

|

13

Hence the map 𝑇𝐹 : 𝑔 󳨃→ (𝑔 ∘ 𝐹) det𝐹󸀠 establishes an isometric isomorphism from 𝐿2 (Ω2 ) to 𝐿2 (Ω1 ), with inverse map 𝑇𝐹−1 , which restricts to an isomorphism between 𝐴2 (Ω1 ) and 𝐴2 (Ω2 ). Now let 𝑓 ∈ 𝐴2 (Ω1 ) and apply the reproducing property of 𝐾Ω2 to the function 𝑇𝐹−1 𝑓 = (𝑓 ∘ 𝐹−1 ) det(𝐹−1 )󸀠 , setting 𝐹(𝑧) = 𝑢 we get ∫ 𝐾Ω2 (𝑢, 𝑣)𝑇𝐹−1 𝑓(𝑣) 𝑑𝜆(𝑣) = 𝑇𝐹−1 𝑓(𝑢) = 𝑓(𝑧)(det𝐹󸀠 (𝑧))−1 .

(1.16)

Ω2

Since 𝑇𝐹 is an isometry, ∫ 𝑇𝐹−1 𝑓(𝑣)[𝐾Ω2 (𝑣, 𝑢)]− 𝑑𝜆(𝑣) = ∫ 𝑓(𝑤)[𝑇𝐹 𝐾Ω2 (., 𝑢)(𝑤)]− 𝑑𝜆(𝑤). Ω2

(1.17)

Ω1

From (1.16) and (1.17) we obtain 𝑓(𝑧) = ∫ det𝐹󸀠 (𝑧) 𝐾Ω2 (𝐹(𝑧), 𝐹(𝑤)) det𝐹󸀠 (𝑤) 𝑓(𝑤) 𝑑𝜆(𝑤), Ω1

which means that the right-hand side of (1.14) has the required reproducing property, belongs to 𝐴2 (Ω1 ) in the variable 𝑧 and is anti-holomorphic in the variable 𝑤, and hence must agree with 𝐾Ω1 (𝑧, 𝑤). Finally we derive a useful formula for the corresponding orthogonal projections 𝑃𝑗 : 𝐿2 (Ω𝑗 ) 󳨀→ 𝐴2 (Ω𝑗 ) , 𝑗 = 1, 2. Proposition 1.15. For all 𝑔 ∈ 𝐿2 (Ω2 ) one has 𝑃1 (det𝐹󸀠 𝑔 ∘ 𝐹) = det𝐹󸀠 (𝑃2 (𝑔) ∘ 𝐹).

(1.18)

Proof. The left-hand side of (1.18) can be written in the form 𝑃1 (𝑇𝐹 (𝑔)), hence, by (1.5), we obtain for 𝑃1 (𝑇𝐹 (𝑔))(𝑧) = ∫ 𝐾Ω1 (𝑧, 𝑤)𝑇𝐹 (𝑔)(𝑤) 𝑑𝜆(𝑤),

𝑧 ∈ Ω1 .

Ω1

Now (1.14), together with (1.17), implies that 𝐾Ω1 (𝑤, 𝑧) = [𝑇𝐹 (𝐾Ω2 (., 𝐹(𝑧)))(𝑤)] det𝐹󸀠 (𝑧), so, since 𝑇𝐹 is an isometric isomorphism, we get 𝑃1 (𝑇𝐹 (𝑔))(𝑧) = det𝐹󸀠 (𝑧) ∫ 𝑇𝐹 (𝑔)(𝑤) [𝑇𝐹 (𝐾Ω2 (., 𝐹(𝑧)))(𝑤)]− 𝑑𝜆(𝑤) Ω1

= det𝐹󸀠 (𝑧) ∫ 𝑔(𝑣) [𝐾Ω2 (𝑣, 𝐹(𝑧)))]− 𝑑𝜆(𝑣) Ω2 󸀠

= det𝐹 (𝑧) (𝑃2 (𝑔))(𝐹(𝑧)), which proves (1.18).

14 | 1 Bergman spaces Remark 1.16. If 𝑛 = 1 and Ω ⊊ ℂ is a simply connected domain, there is an interesting connection between the Bergman kernel 𝐾Ω of Ω and the Riemann mapping 𝐹 : Ω 󳨀→ 𝔻 with the uniqueness properties 𝐹(𝑎) = 0, 𝐹󸀠 (𝑎) > 0 for some 𝑎 ∈ Ω : 𝐹󸀠 (𝑧) = √

𝜋 𝐾 (𝑧, 𝑎), 𝐾Ω (𝑎, 𝑎) Ω

(1.19)

𝑧 ∈ Ω.

By (1.15), the transformation 𝑇𝐹 establishes an isometry between 𝐿2 (Ω) and 𝐿2 (𝔻) which restricts to be an isometry between 𝐴2 (Ω) and 𝐴2 (𝔻). Therefore we have (𝑇𝐹 𝑢, 𝑇𝐹 𝑢)Ω = (𝑢, 𝑢)𝔻 ,

𝑢 ∈ 𝐿2 (𝔻),

where (. , .)Ω denotes the inner product in 𝐿2 (Ω) and (. , .)𝔻 denotes the inner product of 𝐿2 (𝔻). Similarly, for 𝑣 ∈ 𝐿2 (Ω) and 𝐺 = 𝐹−1 we have (𝑇𝐺 𝑣, 𝑇𝐺 𝑣)𝔻 = (𝑣, 𝑣)Ω . The polarization identity (𝑢1 , 𝑢2 ) =

1 𝑖 (‖𝑢 + 𝑢2 ‖2 − ‖𝑢1 − 𝑢2 ‖2 ) − (‖𝑢1 + 𝑖𝑢2 ‖2 − ‖𝑢1 − 𝑖𝑢2 ‖2 ) 4 1 4

in an inner product space over ℂ yields (𝑇𝐹 𝑢1 , 𝑇𝐹 𝑢2 )Ω = (𝑢1 , 𝑢2 )𝔻 , 𝑢1 , 𝑢2 ∈ 𝐿2 (𝔻),

(1.20)

(𝑇𝐺 𝑣1 , 𝑇𝐺 𝑣2 )𝔻 = (𝑣1 , 𝑣2 )Ω , 𝑣1 , 𝑣2 ∈ 𝐿2 (Ω).

(1.21)

and Since 𝑇𝐹 𝑇𝐺 is the identity operator, we obtain from (1.20) and (1.21) (1.22)

(𝑇𝐹 𝑢, 𝑣)Ω = (𝑇𝐹 𝑢, 𝑇𝐹 (𝑇𝐺 𝑣))Ω = (𝑢, 𝑇𝐺 𝑣)𝔻 . Now let ℎ ∈ 𝐴2 (𝔻) and observe that, by (1.1) (ℎ, 1)𝔻 = 𝜋ℎ(0). By (1.22) we get for 𝑓 ∈ 𝐴2 (Ω) (𝑓, 𝐹󸀠 )Ω = (𝐺󸀠 (𝑓 ∘ 𝐺), 1)𝔻 = 𝜋𝐺󸀠 (0)𝑓(𝐺(0)) =

𝜋 𝐹󸀠 (𝑎)

󸀠

𝑓(𝑎).

It follows that the function 𝑘(𝑧, 𝑎) = 𝐹 𝜋(𝑎) 𝐹󸀠 (𝑧) has the reproducing property and belongs to 𝐴2 (Ω), so, by Proposition 1.8, 𝑘(𝑧, 𝑎) = 𝐾Ω (𝑧, 𝑎) and we get 𝐹󸀠 (𝑧) =

𝜋 𝐾 (𝑧, 𝑎), 𝐹󸀠 (𝑎) Ω

setting 𝑧 = 𝑎 we obtain 𝐹󸀠 (𝑎)2 = 𝜋𝐾Ω (𝑎, 𝑎) which proves (1.19).

1.4 Notes

| 15

1.4 Notes The basics on Hilbert spaces can be found for instance in [73]. There are numerous good texts on different aspects of Bergman spaces and their reproducing kernels, the reader may consult the books by S.-C. Chen and M.-C. Shaw [12], S. Krantz [53], and M. Jarnicki and P. Pflug [45]. The remark about the Riemann mapping and the Bergman kernel is contained in S. Bell’s book [3]. Further results regarding the Bergman kernel and the solution to 𝜕 will be discussed in the next chapters.

2 The canonical solution operator to 𝜕 In this chapter we will use properties of the Bergman kernel to solve the inhomogeneous Cauchy–Riemann equation 𝜕𝑢 =𝑔 𝜕𝑧 where

or

𝜕𝑢 = 𝑔,

1 𝜕 𝜕 𝜕 = ( + 𝑖 ), 𝜕𝑧 2 𝜕𝑥 𝜕𝑦

𝑧 = 𝑥 + 𝑖𝑦

(2.1)

and 𝑔 ∈ 𝐴2 (𝔻). Before we proceed we recall some basic facts from operator theory.

2.1 Compact operators on Hilbert spaces Let 𝐻1 and 𝐻2 be separable Hilbert spaces and 𝐴 : 𝐻1 󳨀→ 𝐻2 a bounded linear operator. The operator 𝐴 is compact if the image 𝐴(𝐵1 ) of the unit ball 𝐵1 in 𝐻1 is a relatively compact subset of 𝐻2 , since 𝐻2 is complete this is equivalent to the concept of a totally bounded set, i.e. for each 𝜖 > 0 there exists a finite number of elements 𝑣1 , . . . , 𝑣𝑚 ∈ 𝐻2 such that 𝑚

𝐴(𝐵1 ) ⊂ ⋃ 𝐵(𝑣𝑗 , 𝜖), 𝑗=1

where 𝐵(𝑣𝑗 , 𝜖) = {𝑣 ∈ 𝐻2 : ‖𝑣 − 𝑣𝑗 ‖ < 𝜖}. Another equivalent definition of compactness is: for each bounded sequence (𝑢𝑘 )𝑘 in 𝐻1 the image sequence (𝐴(𝑢𝑘 ))𝑘 has a convergent subsequence in 𝐻2 . Let L(𝐻1 , 𝐻2 ) denote the space of all bounded linear operators from 𝐻1 to 𝐻2 endowed with the topology generated by the operator norm ‖𝐴‖ = sup{‖𝐴𝑢‖ : ‖𝑢‖ ≤ 1}. In this way L(𝐻1 , 𝐻2 ) becomes a Banach space. Let K(𝐻1 , 𝐻2 ) denote the subspace of all compact operators from 𝐻1 to 𝐻2 . The following characterization of compactness is useful for the special operators in the text, see for instance [17]: Proposition 2.1. Let 𝐻1 and 𝐻2 be Hilbert spaces, and assume that 𝑆 : 𝐻1 → 𝐻2 is a bounded linear operator. The following three statements are equivalent: – 𝑆 is compact. – For every 𝜖 > 0 there is a 𝐶 = 𝐶𝜖 > 0 and a compact operator 𝑇 = 𝑇𝜖 : 𝐻1 → 𝐻2 such that (2.2) ‖𝑆𝑣‖ ≤ 𝐶 ‖𝑇𝑣‖ + 𝜖 ‖𝑣‖ .

2.1 Compact operators on Hilbert spaces

–

| 17

For every 𝜖 > 0 there is a 𝐶 = 𝐶𝜖 > 0 and a compact operator 𝑇 = 𝑇𝜖 : 𝐻1 → 𝐻2 such that (2.3) ‖𝑆𝑣‖2 ≤ 𝐶 ‖𝑇𝑣‖2 + 𝜖 ‖𝑣‖2 .

Proof. First we show that (2.2) and (2.3) are equivalent. Suppose that (2.3) holds. Write (2.3) with 𝜖 and 𝐶 replaced by their squares to obtain ‖𝑆𝑣‖2 ≤ 𝐶2 ‖𝑇𝑣‖2 + 𝜖2 ‖𝑣‖2 ≤ (𝐶 ‖𝑇𝑣‖ + 𝜖 ‖𝑣‖)2 , which implies (2.2). Now suppose that (2.2) holds. Choose 𝜂 with 𝜖 = 2𝜂2 and apply (2.2) with 𝜖 replaced by 𝜂 to get ‖𝑆𝑣‖2 ≤ 𝐶2 ‖𝑇𝑣‖2 + 2𝜂𝐶 ‖𝑣‖ ‖𝑇𝑣‖ + 𝜂2 ‖𝑣‖2 . It is easily seen (small constant – large constant trick) that there is 𝐶󸀠 > 0 such that 2𝜂𝐶 ‖𝑣‖ ‖𝑇𝑣‖ ≤ 𝜂2 ‖𝑣‖2 + 𝐶󸀠 ‖𝑇𝑣‖2 , hence ‖𝑆𝑣‖2 ≤ (𝐶2 + 𝐶󸀠 ) ‖𝑇𝑣‖2 + 2𝜂2 ‖𝑣‖2 = 𝐶󸀠󸀠 ‖𝑇𝑣‖2 + 𝜖 ‖𝑣‖2 . To prove the proposition it therefore suffices to show that (2.2) is equivalent to compactness. When 𝑆 is known to be compact, we choose 𝑇 = 𝑆 and 𝐶 = 1, and (2.2) holds for every positive 𝜖. For the converse let (𝑣𝑛 )𝑛 be a bounded sequence in 𝐻1 . We want to extract a Cauchy subsequence from (𝑆𝑣𝑛 )𝑛 . From (2.2) we have 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩𝑆𝑣𝑛 − 𝑆𝑣𝑚 󵄩󵄩󵄩 ≤ 𝐶 󵄩󵄩󵄩𝑇𝑣𝑛 − 𝑇𝑣𝑚 󵄩󵄩󵄩 + 𝜖 󵄩󵄩󵄩𝑣𝑛 − 𝑣𝑚 󵄩󵄩󵄩

(2.4)

Given a positive integer 𝑁, we may choose 𝜖 sufficiently small in (2.4) so that the second term on the right-hand side is at most 1/(2𝑁). The first term can be made smaller than 1/(2𝑁) by extracting a subsequence of (𝑣𝑛 )𝑛 (still labeled the same) for which (𝑇𝑣𝑛 )𝑛 converges, and then choosing 𝑛 and 𝑚 large enough. Let (𝑣𝑛(0) )𝑛 denote the original bounded sequence. The above argument shows that, for each positive integer 𝑁, there is a sequence (𝑣𝑛(𝑁) )𝑛 satisfying: (𝑣𝑛(𝑁) )𝑛 is a subsequence of (𝑣𝑛(𝑁−1) )𝑛 , and for any pair 𝑣 and 𝑤 in (𝑣𝑛(𝑁) )𝑛 we have ‖𝑆𝑣 − 𝑆𝑤‖ ≤ 1/𝑁. Let (𝑤𝑘 )𝑘 be the diagonal sequence defined by 𝑤𝑘 = 𝑣𝑘(𝑘) . Then (𝑤𝑘 )𝑘 is a subsequence of (𝑣𝑛(0) )𝑛 and the image sequence under 𝑆 of (𝑤𝑘 )𝑘 is a Cauchy sequence. Since 𝐻2 is complete, the image sequence converges and 𝑆 is compact. Proposition 2.2. K(𝐻1 , 𝐻2 ) is a closed subspace of L(𝐻1 , 𝐻2 ) endowed with the operator norm. Proof. Let 𝐴 ∈ L(𝐻1 , 𝐻2 ). Suppose, for each 𝜖 > 0, there is a compact operator 𝐴 𝜖 such that ‖𝐴 − 𝐴 𝜖 ‖ ≤ 𝜖. Then for each 𝑢 ∈ 𝐻1 we have ‖𝐴𝑢 − 𝐴 𝜖 𝑢‖ ≤ 𝜖‖𝑢‖.

18 | 2 The canonical solution operator to 𝜕 Now we get ‖𝐴𝑢‖ = ‖𝐴𝑢 − 𝐴 𝜖 𝑢 + 𝐴 𝜖 𝑢‖ ≤ ‖𝐴𝑢 − 𝐴 𝜖 𝑢‖ + ‖𝐴 𝜖 𝑢‖ ≤ 𝜖‖𝑢‖ + ‖𝐴 𝜖 𝑢‖. Proposition 2.1 implies that 𝐴 is compact. Proposition 2.3. Suppose that 𝐴 ∈ K(𝐻1 , 𝐻2 ), and that 𝑆 ∈ L(𝐻1 , 𝐻1 ) and 𝑇 ∈ L(𝐻2 , 𝐻2 ) is a bounded operator on 𝐻2 . Then both 𝐴𝑆 and 𝑇𝐴 are compact. Proof. If (𝑢𝑘 )𝑘 is a bounded sequence in 𝐻1 , then (𝑆(𝑢𝑘 ))𝑘 is also bounded, because 𝑆 is a bounded operator. 𝐴 is compact, so (𝐴(𝑆(𝑢𝑘 )))𝑘 has a convergent subsequence. Thus 𝐴𝑆 is compact. To show that 𝑇𝐴 is compact we use Proposition 2.1: ‖𝑇𝐴𝑢‖ ≤ ‖𝑇‖ ‖𝐴𝑢‖ ≤ ‖𝑇‖(𝜖‖𝑢‖ + 𝐶‖𝐴𝑢‖) ≤ 𝜖 ‖𝑇‖ ‖𝑢‖ + 𝐶 ‖𝑇‖ ‖𝐴𝑢‖. Corollary 2.4. Let 𝐻 be a Hilbert space. K(𝐻, 𝐻) forms a two-sided, closed ideal in L(𝐻, 𝐻). Theorem 2.5. Let 𝐴 : 𝐻1 󳨀→ 𝐻2 be a bounded linear operator. The following properties are equivalent: (i) 𝐴 is compact; (ii) the adjoint operator 𝐴∗ : 𝐻2 󳨀→ 𝐻1 is compact; (iii) 𝐴∗ 𝐴 : 𝐻1 󳨀→ 𝐻1 is compact. Proof. Suppose that 𝐴 is compact, then, by Proposition 2.3, 𝐴𝐴∗ is also compact. Given 𝜖 > 0, it follows that there is a constant 𝐶 such that ‖𝐴∗ 𝑢‖2 = (𝐴∗ 𝑢, 𝐴∗ 𝑢) = (𝑢, 𝐴𝐴∗ 𝑢) ≤ ‖𝑢‖ ‖𝐴𝐴∗ 𝑢‖ ≤ 𝜖‖𝑢‖2 + 𝐶‖𝐴𝐴∗ 𝑢‖2 . Therefore, by Proposition 2.1, 𝐴∗ is compact. Since 𝐴∗∗ = 𝐴, (i) and (ii) are equivalent. (i) implies (iii) (Proposition 2.3), so it remains to show that (iii) implies (i). Let 𝐴∗ 𝐴 be compact. Given 𝜖 > 0 there is again a constant 𝐶 such that ‖𝐴𝑢‖2 = (𝐴𝑢, 𝐴𝑢) = (𝑢, 𝐴∗ 𝐴𝑢) ≤ 𝜖‖𝑢‖2 + 𝐶‖𝐴∗ 𝐴𝑢‖2 , so Proposition 2.1 implies that 𝐴 is compact. Proposition 2.6. A bounded operator 𝐴 : 𝐻 󳨀→ 𝐻 is compact if and only if there exists a sequence (𝐴 𝑘 )𝑘 of linear operators with finite-dimensional range such that ‖𝐴 − 𝐴 𝑘 ‖ → 0 as 𝑘 → ∞.

2.1 Compact operators on Hilbert spaces

| 19

Proof. An operator with finite-dimensional range is compact, because any bounded sequence in a finite-dimensional Hilbert space has a convergent subsequence. The limit of compact operators in the operator norm is again compact, so we proved one direction. The converse will follow from the next two results. The following theorem is the spectral theorem for compact, self-adjoint operators: Theorem 2.7. Let 𝐴 : 𝐻 󳨀→ 𝐻 be a compact, self-adjoint operator on a separable Hilbert space 𝐻. Then there exists a real zero sequence (𝜇𝑛 )𝑛 and an orthonormal system (𝑒𝑛 )𝑛 in 𝐻 such that for 𝑥 ∈ 𝐻 ∞

𝐴𝑥 = ∑ 𝜇𝑛 (𝑥, 𝑒𝑛 )𝑒𝑛 , 𝑛=0

where the sum converges in the operator norm, i.e. 𝑁

sup ‖𝐴𝑥 − ∑ 𝜇𝑛 (𝑥, 𝑒𝑛 )𝑒𝑛 ‖ → 0,

‖𝑥‖≤1

𝑛=0

as 𝑁 → ∞. Proof. First we collect some elementary properties of self-adjoint operators: Claim (a): Let 𝐴 ∈ L(𝐻, 𝐻) = L(𝐻) and suppose that (𝐴𝑥, 𝑥) = 0 for all 𝑥 ∈ 𝐻. Then 𝐴 = 0. We have 0 = (𝐴(𝑥 + 𝑦), 𝑥 + 𝑦) − (𝐴(𝑥 − 𝑦), 𝑥 − 𝑦) = 2[(𝐴𝑥, 𝑦) + (𝐴𝑦, 𝑥)], hence (𝐴𝑥, 𝑦) + (𝐴𝑦, 𝑥) = 0, now replace 𝑥 by 𝑖𝑥, then 𝑖(𝐴𝑥, 𝑦) − 𝑖(𝐴𝑦, 𝑥) = 0, therefore (𝐴𝑥, 𝑦) = 0 for all 𝑥, 𝑦 ∈ 𝐻, which implies 𝐴 = 0. Claim (b): 𝐴 = 𝐴∗ , if and only if (𝐴𝑥, 𝑥) ∈ ℝ for all 𝑥 ∈ 𝐻. 𝐴 = 𝐴∗ implies (𝐴𝑥, 𝑥)− = (𝑥, 𝐴𝑥) = (𝐴𝑥, 𝑥) ∈ ℝ. If (𝐴𝑥, 𝑥) = (𝐴𝑥, 𝑥)− for all 𝑥 ∈ 𝐻, then (𝐴𝑥, 𝑥) = (𝑥, 𝐴∗ 𝑥)− = (𝐴∗ 𝑥, 𝑥), and so ((𝐴 − 𝐴∗ )𝑥, 𝑥) = 0 for all 𝑥 ∈ 𝐻. By (a) we have 𝐴 = 𝐴∗ . Now we show that for a self-adjoint operator 𝐴 ∈ L(𝐻) the norm of 𝐴 is given by ‖𝐴‖ = sup |(𝐴𝑥, 𝑥)|. ‖𝑥‖=1

For this aim let 𝑁𝐴 = sup‖𝑥‖=1 |(𝐴𝑥, 𝑥)|. Then 𝑁𝐴 ≤ ‖𝐴‖. We also have 2[(𝐴𝑥, 𝑦) + (𝐴𝑦, 𝑥)] = (𝐴(𝑥 + 𝑦), 𝑥 + 𝑦) − (𝐴(𝑥 − 𝑦), 𝑥 − 𝑦), as 𝐴∗ = 𝐴 and by the parallelogram rule |4ℜ(𝐴𝑥, 𝑦)| ≤ 𝑁𝐴 (‖𝑥 + 𝑦‖2 + ‖𝑥 − 𝑦‖2 ) = 2𝑁𝐴 (‖𝑥‖2 + ‖𝑦‖2 ). There is a 𝜃 ∈ ℝ such that 𝑒−𝑖𝜃 (𝐴𝑥, 𝑦) = |(𝐴𝑥, 𝑦)|. Now replace 𝑦 by 𝑒−𝑖𝜃 𝑦, then 2|(𝐴𝑥, 𝑦)| ≤ 𝑁𝐴 (‖𝑥‖2 + ‖𝑦‖2 ).

(2.5)

20 | 2 The canonical solution operator to 𝜕 For a 𝑡 > 0 we replace 𝑥 by 𝑡𝑥 and 𝑦 by 𝑦/𝑡. Then one obtains 2|(𝐴𝑥, 𝑦)| ≤ 𝑁𝐴 (𝑡2 ‖𝑥‖2 +

1 ‖𝑦‖2 ) . 𝑡2

We consider the right side of this inequality as a function in 𝑡 and get after differentiation with respect to 𝑡 2 2𝑡‖𝑥‖2 − 3 ‖𝑦‖2 , 𝑡 so the right side of the inequality will be minimal if 𝑡2 = ‖𝑦‖/‖𝑥‖. Hence we obtain 2|(𝐴𝑥, 𝑦)| ≤ 2𝑁𝐴 ‖𝑥‖ ‖𝑦‖, and ‖𝐴‖ ≤ 𝑁𝐴 , which proves (2.5). Next we show that, if 𝐴 ≠ 0, there exists an eigenvector 𝑥0 ∈ 𝐻 of 𝐴 such that ‖𝑥0 ‖ = 1 and |(𝐴𝑥0 , 𝑥0 )| = ‖𝐴‖ = sup |(𝐴𝑥, 𝑥)|, (2.6) ‖𝑥‖=1

and we show that the corresponding eigenvalue 𝜆 0 is real and satisfies |𝜆 0 | = ‖𝐴‖. By (2.5) we have ‖𝐴‖ = sup‖𝑥‖=1 |(𝐴𝑥, 𝑥)|. Hence there is a sequence (𝑥𝑛 )𝑛 in 𝐻 with ‖𝑥𝑛 ‖ = 1 and lim𝑛→∞ |(𝐴𝑥𝑛 , 𝑥𝑛 )| = ‖𝐴‖. The inner products (𝐴𝑥𝑛 , 𝑥𝑛 ) are real, so there exists a subsequence, which is again denoted by (𝑥𝑛)𝑛 , such that lim𝑛→∞ (𝐴𝑥𝑛 , 𝑥𝑛 ) = 𝜆 0 . Then we have 𝜆 0 = ‖𝐴‖ or 𝜆 0 = −‖𝐴‖. Now we get 0 ≤ ‖𝐴𝑥𝑛 − 𝜆 0 𝑥𝑛 ‖2 = ‖𝐴𝑥𝑛 ‖2 − 2𝜆 0 (𝐴𝑥𝑛 , 𝑥𝑛 ) + 𝜆20 ‖𝑥𝑛 ‖2 ≤ ‖𝐴‖2 − 2𝜆 0 (𝐴𝑥𝑛 , 𝑥𝑛 ) + ‖𝐴‖2 ,

and letting 𝑛 → ∞ lim ‖𝐴𝑥𝑛 − 𝜆 0 𝑥𝑛 ‖ = 0.

𝑛→∞

Since 𝐴 is compact and (𝑥𝑛 )𝑛 is a bounded sequence, there exists a subsequence (𝑥𝑛𝑘 )𝑘 such that lim𝑘→∞ 𝐴𝑥𝑛𝑘 = 𝑥. Hence lim𝑘→∞ 𝜆 0 𝑥𝑛𝑘 = 𝑥, and as 𝜆 0 ≠ 0, we also have lim𝑘→∞ 𝑥𝑛𝑘 = 𝑥0 with 𝜆 0 𝑥0 = 𝑥. Since 𝐴 is continuous, this implies lim 𝐴𝑥𝑛𝑘 = 𝐴𝑥0 ,

𝑘→∞

and hence 𝑥 = 𝜆 0 𝑥0 = 𝐴𝑥0 . So 𝑥0 is an eigenvector of 𝐴 with eigenvalue 𝜆 0 . This proves (2.6). Now, each eigenvalue of 𝐴 is real: 𝐴𝑥 = 𝜆𝑥 and 𝑥 ≠ 0, imply 𝜆(𝑥, 𝑥) = (𝐴𝑥, 𝑥) = (𝑥, 𝐴𝑥) = (𝑥, 𝜆𝑥) = 𝜆(𝑥, 𝑥). Therefore 𝜆 ∈ ℝ. Let 𝐻𝜆 = {𝑥 ∈ 𝐻 : 𝐴𝑥 = 𝜆𝑥} be the eigenspace of the eigenvalue 𝜆. Since 𝐴 is continuous, 𝐻𝜆 is closed. There exists an orthonormal basis in 𝐻𝜆 consisting of eigenvectors of 𝐴. If 𝜆 1 ≠ 𝜆 2 are eigenvalues, then 𝐻𝜆 1 ⊥𝐻𝜆 2 , because for 𝑥 ∈ 𝐻𝜆 1 and 𝑦 ∈ 𝐻𝜆 2 we have: 𝜆 1 (𝑥, 𝑦) = (𝐴𝑥, 𝑦) = (𝑥, 𝐴𝑦) = (𝑥, 𝜆 2 𝑦) = 𝜆 2 (𝑥, 𝑦), and hence (𝑥, 𝑦) = 0.

2.1 Compact operators on Hilbert spaces

| 21

We claim that the eigenspace 𝐻𝜆 of an eigenvalue 𝜆 ≠ 0 is always finitedimensional: if 𝐻𝜆 is of infinite dimension, we would have an infinite orthonormal system {𝑥𝛼 }𝛼∈𝐴 with 𝐴𝑥𝛼 = 𝜆𝑥𝛼 and since the restriction of 𝐴 to 𝐻𝜆 is also compact, we could find a convergent subsequence of (𝐴𝑥𝛼 )𝛼 , ending up in a contradiction to ‖𝜆𝑥𝛼1 − 𝜆𝑥𝛼2 ‖2 = 2|𝜆|2 for 𝛼1 ≠ 𝛼2 . Now we have two possible cases: (i) there are finitely many eigenvalues ≠ 0; (ii) there are infinitely many eigenvalues ≠ 0. (i) In this case we have 𝐻 = 𝐻𝜆 0 ⊕ ⋅ ⋅ ⋅ ⊕ 𝐻𝜆 𝑘−1 ⊕ 𝑀, where 𝑀 = (𝐻𝜆 0 ⊕ ⋅ ⋅ ⋅ ⊕ 𝐻𝜆 𝑘−1 )⊥ . We claim that 𝐴(𝑀) ⊆ 𝑀: for 𝑦 ∈ 𝑀 we have (𝐴𝑦, 𝑥) = (𝑦, 𝐴𝑥) = 0, for all 𝑥 ∈ 𝐻𝜆 0 ⊕ ⋅ ⋅ ⋅ ⊕ 𝐻𝜆 𝑘−1 , because 𝐴𝑥 ∈ 𝐻𝜆 0 ⊕ ⋅ ⋅ ⋅ ⊕ 𝐻𝜆 𝑘−1 , hence 𝐴𝑦 ∈ 𝑀. The restriction 𝐴 𝑀 of 𝐴 to 𝑀 is again compact and self-adjoint: for 𝑥, 𝑦 ∈ 𝑀 we have (𝐴 𝑀 𝑥, 𝑦) = (𝐴𝑥, 𝑦) = (𝑥, 𝐴𝑦) = (𝑥, 𝐴 𝑀 𝑦), because 𝐴(𝑀) ⊆ 𝑀. We claim that 𝐴 𝑀 = 0. Suppose 𝐴 𝑀 ≠ 0, then, by (2.5), there is a 𝜆 ≠ 0 with 𝐴 𝑀 𝑥 = 𝜆𝑥 for some 𝑥 ≠ 0, hence we would get 𝑥 ∈ 𝐻𝜆 , which is a contradiction. Let 𝑃𝑗 : 𝐻 󳨀→ 𝐻𝜆 𝑗 be the orthogonal projection. Then one obtains 𝑥 = 𝑃0 𝑥 + 𝑃1 𝑥 + ⋅ ⋅ ⋅ + 𝑃𝑘−1 𝑥 + 𝑦, where 𝐴𝑦 = 0, therefore 𝐴𝑥 = 𝜆 0 𝑃0 𝑥 + 𝜆 1 𝑃1 𝑥 + ⋅ ⋅ ⋅ + 𝜆 𝑘−1 𝑃𝑘−1 𝑥. (ii) For each 𝜖 > 0, the set {𝜆 𝑖 : 𝜆 𝑖 eigenvalue of 𝐴 with |𝜆 𝑖 | ≥ 𝜖} is a finite set, otherwise we would get an infinite orthonormal system {𝑥𝑖 } with 𝐴𝑥𝑖 = 𝜆 𝑖 𝑥𝑖 and the sequence (𝐴𝑥𝑖 )𝑖 , would have to contain a convergent subsequence. But ‖𝐴𝑥𝑘 − 𝐴𝑥𝑙 ‖2 = ‖𝜆 𝑘 𝑥𝑘 − 𝜆 𝑙 𝑥𝑙 ‖2 = |𝜆 𝑘 |2 + |𝜆 𝑙 |2 ≥ 2𝜖2 , which is a contradiction. Let 𝜆 0 , . . . , 𝜆 𝑘−1 be the eigenvalues with |𝜆 𝑖 | ≥ 𝜖 and let 𝑀 = (𝐻𝜆 0 ⊕ ⋅ ⋅ ⋅ ⊕ 𝐻𝜆 𝑘−1 )⊥ . The restriction 𝐴 𝑀 of 𝐴 to 𝑀 is self-adjoint and compact, hence, by (2.5), there is an eigenvector 𝑥 with 𝐴 𝑀 𝑥 = 𝜆𝑥 and |𝜆| = ‖𝐴 𝑀 ‖, and hence 𝐴𝑥 = 𝜆𝑥. Therefore 𝑥 ∈ (𝐻𝜆 𝑖 )⊥ and |𝜆| = ‖𝐴 𝑀 ‖ < 𝜖. So we have 𝐴𝑥 = 𝐴𝑥1 + 𝐴 𝑀 𝑥2 , 𝑥1 ∈ 𝑀⊥ ,

𝑥2 ∈ 𝑀.

Since 𝐴𝑥1 = 𝜆 0 𝑃0 𝑥 + ⋅ ⋅ ⋅ + 𝜆 𝑘−1 𝑃𝑘−1 𝑥 we obtain ‖𝐴 𝑀 𝑥2 ‖ = ‖𝐴𝑥 − 𝜆 0 𝑃0 𝑥 − ⋅ ⋅ ⋅ − 𝜆 𝑘−1 𝑃𝑘−1 𝑥‖ < 𝜖‖𝑥‖. Letting 𝜖 → 0 we get

∞

𝐴𝑥 = ∑ 𝜆 𝑗 𝑃𝑗 𝑥. 𝑗=0

22 | 2 The canonical solution operator to 𝜕 Now let {𝜓1 , . . . , 𝜓𝑘 } be an orthonormal basis of 𝐻𝜆 𝑖 . Then 𝑘

𝐴𝑃𝑖 (𝑥) = 𝜆 𝑖 𝑃𝑖 (𝑥) = 𝜆 𝑖 ∑ (𝑥, 𝜓𝑗 )𝜓𝑗 . 𝑗=1

The sequence of the eigenvalues can be ordered in the following way |𝜆 0 | ≥ |𝜆 1 | ≥ . . . , because for each 𝜖 > 0 there are only finitely many 𝜆 𝑖 with |𝜆 𝑖 | ≥ 𝜖. Finally, the orthogonality of the eigenvectors 𝑥𝑘 and Bessel’s inequality (1.6) give 𝑛 ∞ 2 󵄩2 󵄩󵄩 󵄩󵄩𝐴𝑥 − ∑ 𝜆 𝑘 (𝑥, 𝑥𝑘 )𝑥𝑘 󵄩󵄩󵄩 = ∑ |𝜆 𝑘 (𝑥, 𝑥𝑘 )|2 ≤ (‖𝑥‖ sup |𝜆 𝑘 |) , 󵄩󵄩 󵄩󵄩 𝑘>𝑛 𝑘=1 𝑘=𝑛+1

and since (𝜆 𝑘 )𝑘≥1 is a zero sequence, the above series converges to 𝐴 in the operator norm. Now we drop the assumption of self-adjointness and obtain Proposition 2.8. Let 𝐴 : 𝐻1 󳨀→ 𝐻2 be a compact operator. There exists a decreasing zero sequence (𝑠𝑛 )𝑛 in ℝ+ and orthonormal systems (𝑒𝑛 )𝑛≥0 in 𝐻1 and (𝑓𝑛 )𝑛≥0 in 𝐻2 , such that ∞

𝐴𝑥 = ∑ 𝑠𝑛 (𝑥, 𝑒𝑛 )𝑓𝑛 , 𝑛=0

∀𝑥 ∈ 𝐻1 ,

where the sum converges again in the operator norm. Proof. In order to show this, one applies the spectral theorem for the self-adjoint, compact operator 𝐴∗ 𝐴 : 𝐻1 󳨀→ 𝐻1 and gets ∞

𝐴∗ 𝐴𝑥 = ∑ 𝑠𝑛2 (𝑥, 𝑒𝑛 )𝑒𝑛 , 𝑛=0

where 𝑠𝑛2 are the eigenvalues of 𝐴∗ 𝐴. If 𝑠𝑛 > 0, we set 𝑓𝑛 = 𝑠𝑛−1 𝐴𝑒𝑛 and get (𝑓𝑛 , 𝑓𝑚 ) =

𝑠2 1 1 (𝐴𝑒𝑛 , 𝐴𝑒𝑚 ) = (𝐴∗ 𝐴𝑒𝑛 , 𝑒𝑚 ) = 𝑛 (𝑒𝑛 , 𝑒𝑚 ) = 𝛿𝑛,𝑚 . 𝑠𝑛 𝑠𝑚 𝑠𝑛 𝑠𝑚 𝑠𝑛 𝑠𝑚

For 𝑦 ∈ 𝐻1 with 𝑦 ⊥ 𝑒𝑛 for each 𝑛 ∈ ℕ0 we have by (2.7) that ‖𝐴𝑦‖2 = (𝐴𝑦, 𝐴𝑦) = (𝐴∗ 𝐴𝑦, 𝑦) = 0. Hence we have

∞

∞

𝐴𝑥 = 𝐴(𝑥 − ∑ (𝑥, 𝑒𝑛 )𝑒𝑛 ) + 𝐴( ∑ (𝑥, 𝑒𝑛 )𝑒𝑛 ) 𝑛=0

∞

𝑛=0

∞

= ∑ (𝑥, 𝑒𝑛 )𝐴𝑒𝑛 = ∑ 𝑠𝑛 (𝑥, 𝑒𝑛 )𝑓𝑛 . 𝑛=0

𝑛=0

(2.7)

2.1 Compact operators on Hilbert spaces

| 23

Similar to the last theorem, we get 𝑛 ∞ 2 󵄩2 󵄩󵄩 󵄩󵄩𝐴𝑥 − ∑ 𝑠𝑘 (𝑥, 𝑒𝑘 )𝑓𝑘 󵄩󵄩󵄩 = ∑ |𝑠𝑘 (𝑥, 𝑒𝑘 )|2 ≤ (‖𝑥‖ sup |𝑠𝑘 |) , 󵄩󵄩 󵄩󵄩 𝑘>𝑛 𝑘=1 𝑘=𝑛+1

which implies that the series converges in the operator norm. Remark 2.9. From the last statement we get the missing direction in the proof of Proposition 2.6. The numbers 𝑠𝑛 are called the 𝑠-numbers of 𝐴. They are uniquely determined by the operator 𝐴, since they are the square roots of the eigenvalues of 𝐴∗ 𝐴. Let 0 < 𝑝 < ∞. The operator 𝐴 belongs to the Schatten class S𝑝 , if its sequence (𝑠𝑛 )𝑛 of 𝑠-numbers belongs to 𝑙𝑝 . The elements of the Schatten class S2 are called Hilbert– Schmidt operators. Proposition 2.10. Let 𝐴 : 𝐻1 󳨀→ 𝐻2 be a bounded linear operator between Hilbert spaces. The following conditions are equivalent: (i) there is an orthonormal basis (𝑒𝑖 )𝑖∈𝐼 of 𝐻1 , such that ∑𝑖∈𝐼 ‖𝐴𝑒𝑖 ‖2 < ∞; (ii) for each orthonormal basis (𝑓𝑗 )𝑗∈𝐽 of 𝐻1 one has ∑𝑗∈𝐽 ‖𝐴𝑓𝑗 ‖2 < ∞; (iii) 𝐴 is a Hilbert–Schmidt operator. Proof. (i) implies (ii): Let (𝑒𝑖 )𝑖∈𝐼 be as in (i) and (𝑔𝑙 )𝑙∈𝐿 an orthonormal basis of 𝐻2 . Then, by Parseval’s equality (Proposition 1.10) ∑ ‖𝐴∗ 𝑔𝑙 ‖2 = ∑ ∑ |(𝐴∗ 𝑔𝑙 , 𝑒𝑖 )|2 = ∑ ∑ |(𝑔𝑙 , 𝐴𝑒𝑖 )|2 = ∑ ‖𝐴𝑒𝑖 ‖2 < ∞. 𝑙∈𝐿 𝑖∈𝐼

𝑙∈𝐿

𝑖∈𝐼 𝑙∈𝐿

𝑖∈𝐼

If (𝑓𝑗 )𝑗∈𝐽 is an arbitrary orthonormal basis of 𝐻1 , we get again ∑ ‖𝐴𝑓𝑗 ‖2 = ∑ ‖𝐴∗ 𝑔𝑙 ‖2 = ∑ ‖𝐴𝑒𝑖 ‖2 < ∞.

𝑗∈𝐽

𝑖∈𝐼

𝑙∈𝐿

(ii) implies (iii): Let (𝑒𝑖 )𝑖∈𝐼 be an orthonormal basis of 𝐻1 and 𝑀 a finite subset of 𝐼. We set 𝑃𝑀 𝑥 = ∑𝑖∈𝑀 (𝑥, 𝑒𝑖 )𝑒𝑖 . Then 󵄩󵄩 󵄩󵄩 ‖(𝐴 − 𝐴𝑃𝑀 )𝑥‖ = ‖𝐴(𝐼 − 𝑃𝑀 )𝑥‖ = 󵄩󵄩󵄩 ∑ (𝑥, 𝑒𝑖 )𝐴𝑒𝑖 󵄩󵄩󵄩 󵄩 󵄩 𝑖∈𝐼\𝑀 ≤ ( ∑ ‖𝐴𝑒𝑖 ‖2 ) 𝑖∈𝐼\𝑀

1/2

1/2

( ∑ |(𝑥, 𝑒𝑖 )|2 ) 𝑖∈𝐼\𝑀

≤ ( ∑ ‖𝐴𝑒𝑖 ‖2 )

1/2

‖𝑥‖,

𝑖∈𝐼\𝑀

where we used Bessel’s inequality (1.6). By assumption, ∑𝑖∈𝐼 ‖𝐴𝑒𝑖 ‖2 < ∞, hence we can approximate 𝐴 in the operator norm by operators with finite range. Therefore 𝐴 is compact and, by Proposition 2.8, can be written as ∞

𝐴𝑥 = ∑ 𝑠𝑛 (𝑥, 𝑥𝑛 )𝑓𝑛 . 𝑛=0

24 | 2 The canonical solution operator to 𝜕 By Remark 1.11, there exists an orthonormal basis (𝜉𝑗 )𝑗∈𝐽 , which contains the orthonormal system (𝑥𝑛 )𝑛≥0 . Then we have 󵄩󵄩 ∞ 󵄩󵄩2 ∞ ∑ ‖𝐴𝜉𝑗 ‖2 = ∑ 󵄩󵄩󵄩 ∑ 𝑠𝑛 (𝜉𝑗 , 𝑥𝑛 )𝑓𝑛 󵄩󵄩󵄩 = ∑ 𝑠𝑛2 < ∞, 󵄩 󵄩 𝑛=0 𝑗∈𝐽 𝑗∈𝐽 𝑛=0 so 𝐴 is a Hilbert–Schmidt operator. (iii) implies (i): If 𝐴 is a Hilbert–Schmidt operator, 𝐴 can be written as above and we obtain (i). Corollary 2.11. Let 𝐴 : 𝐻1 󳨀→ 𝐻2 be a Hilbert–Schmidt operator. Then for each orthonormal basis (𝑒𝑖 )𝑖∈𝐼 of 𝐻1 we have ∞

𝜈2 (𝐴)2 := ∑ 𝑠𝑛2 = ∑ ‖𝐴𝑒𝑖 ‖2 ≥ ‖𝐴‖2 . 𝑛=0

𝑖∈𝐼

In particular, 𝜈2 is a norm in the space 𝑆2 (𝐻1 , 𝐻2 ) of all Hilbert–Schmidt operators between 𝐻1 and 𝐻2 . Proof. We have

1/2

2

‖𝐴𝑥‖ = ‖ ∑(𝑥, 𝑒𝑖 )𝐴𝑒𝑖 ‖ ≤ ( ∑ ‖𝐴𝑒𝑖 ‖ ) 𝑖∈𝐼

‖𝑥‖,

𝑖∈𝐼

1/2

hence 𝜈2 (𝐴) ≥ ‖𝐴‖. It is easily seen that 𝐴 󳨃→ (∑𝑖∈𝐼 ‖𝐴𝑒𝑖 ‖2 )

is a norm.

If 𝐻1 and 𝐻2 are separable Hilbert spaces, 𝐴 ∈ L(𝐻1 , 𝐻2 ), and (𝑒𝑛 )𝑛≥1 an orthonormal basis of 𝐻1 and (𝑓𝑛 )𝑛≥1 an orthonormal basis of 𝐻2 , then for each 𝑥 ∈ 𝐻1 we have ∞

𝐴𝑥 = ∑ (𝑥, 𝑒𝑛 )𝐴𝑒𝑛 𝑛=1

and hence for all 𝑗 ∈ ℕ

∞

(𝐴𝑥, 𝑓𝑗 ) = ∑ (𝑥, 𝑒𝑛 )(𝐴𝑒𝑛 , 𝑓𝑗 ). 𝑛=1

The coefficients of 𝐴𝑥 with respect to (𝑓𝑗 )𝑗≥1 can be computed from the coefficients of 𝑥 with respect to (𝑒𝑛 )𝑛≥1 by means of the infinite matrix 𝑎𝑗,𝑛 = (𝐴𝑒𝑛 , 𝑓𝑗 ),

𝑗, 𝑛 ∈ ℕ.

Parseval’s equation implies ∞

∞

∞

𝑗,𝑛=1

𝑗,𝑛=1

𝑛=1

∑ |𝑎𝑗,𝑛 |2 = ∑ |(𝐴𝑒𝑛 , 𝑓𝑗 )|2 = ∑ ‖𝐴𝑒𝑛 ‖2 ,

and we obtain

2.1 Compact operators on Hilbert spaces

| 25

Corollary 2.12. 𝐴 ∈ L(𝐻1 , 𝐻2 ) is a Hilbert–Schmidt operator if and only if the matrix (𝑎𝑗,𝑛 )𝑗,𝑛≥1 of 𝐴 with respect to arbitrary orthonormal bases satisfies ∞

∑ |𝑎𝑗,𝑛 |2 < ∞.

𝑗,𝑛=1

In this case

∞

𝜈2 (𝐴)2 = ∑ |𝑎𝑗,𝑛 |2 . 𝑗,𝑛=1

2

Corollary 2.13. A linear operator 𝐴 : 𝑙 󳨀→ 𝑙2 is a Hilbert–Schmidt operator, if and only 2 2 if there exists a matrix (𝑎𝑗,𝑛 )𝑗,𝑛≥1 with ∑∞ 𝑗,𝑛=1 |𝑎𝑗,𝑛 | < ∞, such that for all 𝑥 = (𝑥𝑛 )𝑛≥1 ∈ 𝑙 one has ∞

𝐴𝑥 = ( ∑ 𝑎𝑗,𝑛 𝑥𝑛 ) 𝑛=1

. 𝑗≥1

Proof. If 𝐴 has the given form, one can apply Cauchy–Schwarz to get ∞

‖𝐴𝑥‖22 ≤ ( ∑ |𝑎𝑗,𝑛 |2 )‖𝑥‖22 , 𝑗,𝑛=1

hence everything follows from Corollary 2.12. Proposition 2.14. Let 𝑆 ⊆ ℝ𝑛 and 𝑇 ⊆ ℝ𝑚 be open sets and 𝐴 : 𝐿2 (𝑇) 󳨀→ 𝐿2 (𝑆) a bounded linear operator. 𝐴 is a Hilbert–Schmidt operator, if and only if there exists 𝐾 ∈ 𝐿2 (𝑆 × 𝑇), such that 𝐴𝑓(𝑠) = ∫ 𝐾(𝑠, 𝑡)𝑓(𝑡) 𝑑𝑡,

𝑓 ∈ 𝐿2 (𝑇).

𝑇

In this case, one has

1/2

𝜈2 (𝐴) = ( ∫ |𝐾(𝑠, 𝑡)|2 𝑑𝑠 𝑑𝑡)

.

𝑆×𝑇

Proof. Let (𝑔𝑘 )𝑘≥1 and (𝑓𝑗 )𝑗≥1 be orthonormal bases of 𝐿2 (𝑇) and 𝐿2 (𝑆) respectively. Then (ℎ𝑗,𝑘 )𝑗,𝑘≥1 , defined by ℎ𝑗,𝑘 (𝑠, 𝑡) := 𝑓𝑗 (𝑠)𝑔𝑘 (𝑡),

(𝑠, 𝑡) ∈ 𝑆 × 𝑇

2

is an orthonormal basis of 𝐿 (𝑆×𝑇). This can be shown as follows: if 𝐹 ∈ 𝐿2 (𝑆×𝑇), then 𝑡 󳨃→ 𝐹(𝑠, 𝑡)𝑓𝑗 (𝑠) is for almost every 𝑠 ∈ 𝑆 in 𝐿2 (𝑇). If 𝐹 ⊥ ℎ𝑗,𝑘 for all 𝑗, 𝑘 ∈ ℕ, it follows that the functions 𝑡 󳨃→ 𝐹(𝑠, 𝑡)𝑓𝑗 (𝑠) vanish in 𝐿2 (𝑇) for all 𝑗 ∈ ℕ almost everywhere with respect to 𝑠. This implies 𝐹 = 0 and (ℎ𝑗,𝑘 )𝑗,𝑘≥1 is an orthonormal basis. If 𝐴 is a Hilbert–Schmidt operator and (𝑎𝑗,𝑘 )𝑗,𝑘≥1 is its matrix with respect to (𝑔𝑘 )𝑘≥1 and (𝑓𝑗 )𝑗≥1 , then, by Corollary 2.12 𝐾(𝑠, 𝑡) := ∑ 𝑎𝑗,𝑘 ℎ𝑗,𝑘 (𝑠, 𝑡) = ∑ 𝑎𝑗,𝑘 𝑓𝑗 (𝑠)𝑔𝑘 (𝑡) 𝑗,𝑘≥1

𝑗,𝑘≥1

belongs to 𝐿2 (𝑆 × 𝑇) and ‖𝐾‖22 = ∑𝑗,𝑘≥1 |𝑎𝑗,𝑘 |2 = 𝜈2 (𝐴)2 .

26 | 2 The canonical solution operator to 𝜕 For 𝑓 ∈ 𝐿2 (𝑇) we have: 𝐴𝑓(𝑠) = ∑ 𝑎𝑗,𝑘 (𝑓, 𝑔𝑘 )𝑓𝑗 (𝑠) = ∫ ∑ 𝑎𝑗,𝑘 𝑓𝑗 (𝑠)𝑔𝑘 (𝑡)𝑓(𝑡) 𝑑𝑡 = ∫ 𝐾(𝑠, 𝑡)𝑓(𝑡) 𝑑𝑡. 𝑗,𝑘≥1

𝑇 𝑗,𝑘≥1

𝑇

If 𝐴 is given by the kernel function 𝐾 ∈ 𝐿2 (𝑆 × 𝑇), then 𝐴 is continuous, because ‖𝐴𝑓‖2 ≤ ( ∫ |𝐾(𝑠, 𝑡)|2 𝑑𝑠 𝑑𝑡) ∫ |𝑓(𝑡)|2 𝑑𝑡. 𝑆×𝑇

𝑇

For the matrix (𝑎𝑗,𝑘 )𝑗,𝑘≥1 of 𝐴 we get 𝑎𝑗,𝑘 = (𝐴𝑔𝑘 , 𝑓𝑗 ) = ∫ ( ∫ 𝐾(𝑠, 𝑡)𝑔𝑘 (𝑡) 𝑑𝑡)𝑓𝑗 (𝑠) 𝑑𝑠 = (𝐾, ℎ𝑗,𝑘 ), 𝑆

𝑇

for all 𝑗, 𝑘 ∈ ℕ. Bessel’s inequality and Corollary 2.12 imply that 𝐴 is a Hilbert–Schmidt operator.

2.2 The canonical solution operator to 𝜕 restricted to 𝐴2 (𝔻) We return to the inhomogeneous Cauchy–Riemann equation on the disc 𝔻 and use the notations of Chapter 1. Let 𝑆(𝑔)(𝑧) = ∫ 𝐾(𝑧, 𝑤)𝑔(𝑤)(𝑧 − 𝑤)− 𝑑𝜆(𝑤).

(2.8)

𝔻

Using the Bergman projection 𝑃 : 𝐿2 (𝔻) 󳨀→ 𝐴2 (𝔻) and (1.5) we get ̃ 𝑆(𝑔)(𝑧) = 𝑧𝑔(𝑧) − 𝑃(𝑔)(𝑧), ̃ where 𝑔(𝑤) = 𝑤𝑔(𝑤). We claim that 𝑆(𝑔) is a solution of the inhomogeneous Cauchy– Riemann equation: 𝜕𝑔 𝜕𝑃(𝑔)̃ 𝜕 𝜕𝑧 𝑆(𝑔)(𝑧) = 𝑔(𝑧) + 𝑧 + = 𝑔(𝑧), 𝜕𝑧 𝜕𝑧 𝜕𝑧 𝜕𝑧 because 𝑔 and 𝑃(𝑔)̃ are holomorphic functions, therefore 𝜕𝑆(𝑔) = 𝑔. In addition we have 𝑆(𝑔) ⊥ 𝐴2 (𝔻), because for arbitrary 𝑓 ∈ 𝐴2 (𝔻) we get ̃ 𝑓) = (𝑔,̃ 𝑓) − (𝑃(𝑔), ̃ 𝑓) = (𝑔,̃ 𝑓) − (𝑔,̃ 𝑃𝑓) = (𝑔,̃ 𝑓) − (𝑔,̃ 𝑓) = 0. (𝑆𝑔, 𝑓) = (𝑔̃ − 𝑃(𝑔), The operator 𝑆 : 𝐴2 (𝔻) 󳨀→ 𝐿2 (𝔻) is called the canonical solution operator to 𝜕.

2.2 The canonical solution operator to 𝜕 restricted to 𝐴2 (𝔻)

| 27

Now we want to show that 𝑆 is a compact operator. For this purpose we consider the adjoint operator 𝑆∗ and prove that 𝑆∗ 𝑆 is compact, which implies that 𝑆 is compact (Theorem 2.5). For 𝑔 ∈ 𝐴2 (𝔻) and 𝑓 ∈ 𝐿2 (𝔻) we have (𝑆𝑔, 𝑓) = ∫ ( ∫ 𝐾(𝑧, 𝑤)𝑔(𝑤)(𝑧 − 𝑤)− 𝑑𝜆(𝑤) )𝑓(𝑧) 𝑑𝜆(𝑧) 𝔻

𝔻

−

= ∫ ( ∫ 𝐾(𝑤, 𝑧)(𝑧 − 𝑤)𝑓(𝑧) 𝑑𝜆(𝑧)) 𝑔(𝑤) 𝑑𝜆(𝑤) = (𝑔, 𝑆∗ 𝑓), 𝔻

𝔻

hence 𝑆∗ (𝑓)(𝑤) = ∫ 𝐾(𝑤, 𝑧)(𝑧 − 𝑤)𝑓(𝑧) 𝑑𝜆(𝑧).

(2.9)

𝔻

Now set 𝑐𝑛2 = ∫ |𝑧|2𝑛 𝑑𝜆(𝑧) = 𝔻

𝜋 , 𝑛+1

𝑛

and 𝜙𝑛 (𝑧) = 𝑧 /𝑐𝑛 , 𝑛 ∈ ℕ0 , then the Bergman kernel 𝐾(𝑧, 𝑤) can be expressed in the form ∞ 𝑘 𝑘 𝑧 𝑤 𝐾(𝑧, 𝑤) = ∑ 2 . 𝑐𝑘 𝑘=0 Next we compute ∞

∞ 𝑘−1 𝑐𝑛 𝑧𝑛−1 𝑧𝑘 𝑤𝑘 𝑤𝑛 𝑧 𝑤𝑘 𝑤𝑛 ∑ ∫ 𝑤 𝑑𝜆(𝑤) = 𝑑𝜆(𝑤) = , 2 𝑐𝑛 𝑐𝑛 𝑐𝑘2 𝑐2 𝑐𝑛−1 𝑘=0 𝑘=1 𝑘−1

𝑃(𝜙𝑛̃ )(𝑧) = ∫ ∑ 𝔻

𝔻

hence we have 𝑆(𝜙𝑛 )(𝑧) = 𝑧 𝜙𝑛 (𝑧) −

𝑐𝑛 𝑧𝑛−1 , 2 𝑐𝑛−1

𝑛 ∈ ℕ.

Now we apply 𝑆∗ and get ∞

𝑤𝑘 𝑧𝑘 𝑧𝑧𝑛 𝑐𝑛 𝑧𝑛−1 ( (𝑧 − 𝑤) − 2 ) 𝑑𝜆(𝑧). 𝑐𝑛 𝑐𝑘2 𝑐𝑛−1 𝑘=0

𝑆∗ 𝑆(𝜙𝑛 )(𝑤) = ∫ ∑ 𝔻

The last integral is computed in two steps: first the multiplication by 𝑧 ∞ 𝑐𝑛 𝑤𝑘 𝑧𝑘 𝑧𝑧𝑛+1 𝑐𝑛 𝑧𝑛 𝑧𝑛+1 ∞ 𝑤𝑘 𝑧𝑘+1 𝑤𝑘 𝑧𝑘 𝑛 ∑ ∑ ( − ) 𝑑𝜆(𝑧) = ∫ 𝑑𝜆(𝑧) − ∫ 𝑧 𝑑𝜆(𝑧) 2 2 𝑐𝑛 𝑐𝑛 𝑘=0 𝑐𝑘2 𝑐𝑘2 𝑐𝑛−1 𝑐𝑛−1 𝑐𝑘2 𝑘=0 𝑘=0 ∞

∫∑ 𝔻

𝔻

𝔻

𝑤𝑛 𝑤𝑛 = 3 ∫ |𝑧|2𝑛+2 𝑑𝜆(𝑧) − 2 ∫ |𝑧|2𝑛 𝑑𝜆(𝑧) 𝑐𝑛 𝑐𝑛−1 𝑐𝑛 =

𝔻 2 𝑐 ( 𝑛+1 𝑐𝑛3

𝔻

−

𝑐𝑛 ) 𝑤𝑛 . 2 𝑐𝑛−1

28 | 2 The canonical solution operator to 𝜕 Next the multiplication by 𝑤 ∞

𝑤𝑘 𝑧𝑘 𝑧𝑧𝑛 𝑐𝑛 𝑧𝑛−1 𝑧𝑛 ∞ 𝑤𝑘 𝑧𝑘+1 ( ∑ − 2 ) 𝑑𝜆(𝑧) = 𝑤 ∫ 𝑑𝜆(𝑧) 2 𝑐𝑛 𝑐𝑛 𝑘=0 𝑐𝑘2 𝑐𝑘 𝑐𝑛−1 𝑘=0

𝑤∫ ∑ 𝔻

𝔻

− 𝑤∫ 𝔻

= 𝑤(

𝑐𝑛 𝑧𝑛−1 ∞ 𝑤𝑘 𝑧𝑘 ∑ 2 𝑑𝜆(𝑧) 2 𝑐𝑛−1 𝑐𝑘 𝑘=0

𝑐𝑛 𝑤𝑛−1 𝑐𝑛 𝑤𝑛−1 − 2 ) 2 𝑐𝑛−1 𝑐𝑛−1

= 0, it follows that 𝑆∗ 𝑆(𝜙𝑛 )(𝑤) = (

2 𝑐𝑛+1 𝑐𝑛2 − ) 𝜙𝑛 (𝑤), 2 𝑐𝑛2 𝑐𝑛−1

𝑛 = 1, 2, . . . ,

for 𝑛 = 0 an analogous computation shows 𝑆∗ 𝑆(𝜙0 )(𝑤) =

𝑐12 𝜙 (𝑤). 𝑐02 0

Finally we get Proposition 2.15. Let 𝑆 : 𝐴2 (𝔻) 󳨀→ 𝐿2 (𝔻) be the canonical solution operator for 𝜕 and (𝜙𝑘 )𝑘 the normalized monomials. Then 𝑆∗ 𝑆𝜙 =

2 ∞ 𝑐𝑛+1 𝑐𝑛2 𝑐12 ∑ (𝜙, 𝜙 )𝜙 + ( − ) (𝜙, 𝜙𝑛 )𝜙𝑛 0 0 2 𝑐𝑛2 𝑐02 𝑐𝑛−1 𝑛=1

(2.10)

for each 𝜙 ∈ 𝐴2 (𝔻). Since

2 𝑐𝑛+1 𝑐2 1 − 2𝑛 = → 0 as 𝑛 → ∞, 2 𝑐𝑛 𝑐𝑛−1 (𝑛 + 2)(𝑛 + 1)

it follows that 𝑆∗ 𝑆 is compact and 𝑆 too. We have also shown that the s-numbers of 𝑆 are ( ∞

∑( 𝑛=1

2 𝑐𝑛+1 𝑐𝑛2

−

𝑐𝑛2 2 ) 𝑐𝑛−1

1/2

and since

2 𝑐𝑛+1 𝑐2 − 2𝑛 ) < ∞ 2 𝑐𝑛 𝑐𝑛−1

it follows that 𝑆 is Hilbert–Schmidt. This can also be shown directly using Proposition 2.14. For this purpose we claim that the function (𝑧, 𝑤) 󳨃→ 𝐾(𝑧, 𝑤)(𝑧 − 𝑤)− belongs to 𝐿2 (𝔻 × 𝔻). We have to prove that ∫∫ 𝔻𝔻

|𝑧 − 𝑤|2 𝑑𝜆(𝑧) 𝑑𝜆(𝑤) < ∞. |1 − 𝑧𝑤|4

2.2 The canonical solution operator to 𝜕 restricted to 𝐴2 (𝔻)

| 29

An easy estimate gives |𝑧 − 𝑤| ≤ |1 − 𝑧𝑤|, for 𝑧, 𝑤 ∈ 𝔻. Hence ∫∫ 𝔻𝔻

|𝑧 − 𝑤|2 1 𝑑𝜆(𝑧) 𝑑𝜆(𝑤) ≤ ∫ ∫ 𝑑𝜆(𝑧) 𝑑𝜆(𝑤). |1 − 𝑧𝑤|4 |1 − 𝑧𝑤|2 𝔻𝔻

Introducing polar coordinates 𝑧 = 𝑟 𝑒𝑖𝜃 and 𝑤 = 𝑠 𝑒𝑖𝜙 we can write the last integral in the following form 1 1 2𝜋 2𝜋

𝑟 𝑠 𝑑𝜃 𝑑𝜙 𝑑𝑟 𝑑𝑠 1 ∫∫ 𝑑𝜆(𝑧) 𝑑𝜆(𝑤) = ∫ ∫ ∫ ∫ 2 |1 − 𝑧𝑤| 1 − 2 𝑟 𝑠 cos(𝜃 − 𝜙) + 𝑟2 𝑠2 0 0 0 0

𝔻𝔻

1 1 2𝜋 2𝜋

= ∫∫∫ ∫ 0 0 0 0

1 − 𝑟2 𝑠2 𝑟𝑠 𝑑𝜃 𝑑𝜙 𝑑𝑟 𝑑𝑠. 1 − 2 𝑟 𝑠 cos(𝜃 − 𝜙) + 𝑟2 𝑠2 1 − 𝑟2 𝑠2

Integration of the Poisson kernel with respect to 𝜃 yields 2𝜋

∫ 0

1 − 𝜌2 𝑑𝜃 = 2𝜋, 1 − 2𝜌 cos(𝜃 − 𝜙) + 𝜌2

0 < 𝜌 < 1.

Therefore we have 1 1 2𝜋 2𝜋

∫∫∫ ∫ 0 0 0 0

1 − 𝑟2 𝑠2 𝑟𝑠 𝑑𝜃 𝑑𝜙 𝑑𝑟 𝑑𝑠 1 − 2 𝑟 𝑠 cos(𝜃 − 𝜙) + 𝑟2 𝑠2 1 − 𝑟2 𝑠2 1 1

1

0 0

0

log(1 − 𝑠2 ) 𝑟𝑠 2 ∫ 𝑑𝑟 𝑑𝑠 = − (2𝜋) = (2𝜋) ∫ ∫ 𝑑𝑠 < ∞. 1 − 𝑟2 𝑠2 2𝑠 2

In the following we consider weighted spaces of entire functions 𝑚

𝑚

𝐴2 (ℂ, 𝑒−|𝑧| ) = {𝑓 : ℂ 󳨀→ ℂ : ‖𝑓‖2𝑚 := ∫ |𝑓(𝑧)|2 𝑒−|𝑧| 𝑑𝜆(𝑧) < ∞}, ℂ

where 𝑚 > 0. Let

𝑚

𝑐𝑘2 = ∫ |𝑧|2𝑘 𝑒−|𝑧| 𝑑𝜆(𝑧). ℂ

Then

∞

𝑧𝑘 𝑤𝑘 𝑐𝑘2 𝑘=0

𝐾𝑚 (𝑧, 𝑤) = ∑ 𝑚

is the reproducing kernel for 𝐴2 (ℂ, 𝑒−|𝑧| ). In the sequel the expression 2 𝑐𝑘+1 𝑐𝑘2 − 2 𝑐𝑘2 𝑐𝑘−1

30 | 2 The canonical solution operator to 𝜕 will become important. Using the integral representation of the Γ-function one easily sees that the above expression is equal to ) Γ ( 2𝑘+4 𝑚 Γ ( 2𝑘+2 ) 𝑚

−

) Γ ( 2𝑘+2 𝑚 Γ ( 2𝑘 ) 𝑚

.

For 𝑚 = 2 this expression equals to 1 for each 𝑘 = 1, 2, . . . . We will be interested in the limit behavior for 𝑘 → ∞. By Stirling’s formula the limit behavior is equivalent to the limit behavior of the expression 2𝑘 + 2 2/𝑚 2𝑘 2/𝑚 ) −( ) , 𝑚 𝑚

(

as 𝑘 → ∞. Hence we have shown the following Lemma 2.16. The expression ) Γ ( 2𝑘+4 𝑚 Γ ( 2𝑘+2 ) 𝑚

−

) Γ ( 2𝑘+2 𝑚 Γ ( 2𝑘 ) 𝑚

tends to ∞ for 0 < 𝑚 < 2, is equal to 1 for 𝑚 = 2 and tends to zero for 𝑚 > 2 as 𝑘 tends to ∞. 𝑚 Let 0 < 𝜌 < 1, define 𝑓𝜌 (𝑧) := 𝑓(𝜌𝑧) and 𝑓𝜌̃ (𝑧) = 𝑧𝑓𝜌 (𝑧), for 𝑓 ∈ 𝐴2 (ℂ, 𝑒−|𝑧| ). Then it 𝑚 𝑚 is easily seen that 𝑓 ̃ ∈ 𝐿2 (ℂ, 𝑒−|𝑧| ), but there are functions 𝑔 ∈ 𝐴2 (ℂ, 𝑒−|𝑧| ) such that 𝑚

𝜌

𝑧𝑔 ∈ ̸ 𝐿2 (ℂ, 𝑒−|𝑧| ). 𝑚 𝑚 Let 𝑃𝑚 : 𝐿2 (ℂ, 𝑒−|𝑧| ) 󳨀→ 𝐴2 (ℂ, 𝑒−|𝑧| ) denote the orthogonal projection. Then 𝑃𝑚 can be written in the form 𝑚

𝑃𝑚 (𝑓)(𝑧) = ∫ 𝐾𝑚 (𝑧, 𝑤)𝑓(𝑤)𝑒−|𝑤| 𝑑𝜆(𝑤),

𝑚

𝑓 ∈ 𝐿2 (ℂ, 𝑒−|𝑧| ).

ℂ

Proposition 2.17. Let 𝑚 ≥ 2. Then there is a constant 𝐶𝑚 > 0 depending only on 𝑚 such that 𝑚 𝑚 󵄨2 󵄨 󵄨 󵄨2 ∫ 󵄨󵄨󵄨𝑓𝜌̃ (𝑧) − 𝑃𝑚 (𝑓𝜌̃ )(𝑧)󵄨󵄨󵄨 𝑒−|𝑧| 𝑑𝜆(𝑧) ≤ 𝐶𝑚 ∫ 󵄨󵄨󵄨𝑓(𝑧)󵄨󵄨󵄨 𝑒−|𝑧| 𝑑𝜆(𝑧), ℂ

ℂ 2

for each 0 < 𝜌 < 1 and for each 𝑓 ∈ 𝐴 (ℂ, 𝑒

−|𝑧|𝑚

).

𝑘 Proof. First we observe that for the Taylor expansion of 𝑓(𝑧) = ∑∞ 𝑘=0 𝑎𝑘 𝑧 we have ∞

∞ 𝑚 𝑧𝑘 𝑤𝑘 (𝑤 ∑ 𝑎𝑗 𝜌𝑗 𝑤𝑗 ) 𝑒−|𝑤| 𝑑𝜆(𝑤) 2 𝑐𝑘 𝑗=0 𝑘=0

𝑃𝑚 (𝑓𝜌̃ )(𝑧) = ∫ ∑ ℂ ∞

= ∑ 𝑎𝑘 𝑘=1

𝑐𝑘2 𝑘 𝑘−1 𝜌 𝑧 . 2 𝑐𝑘−1

2.2 The canonical solution operator to 𝜕 restricted to 𝐴2 (𝔻)

| 31

Now we obtain ∞ ∞ 𝑚 𝑐2 󵄨 󵄨2 ∫ 󵄨󵄨󵄨󵄨𝑓𝜌̃ (𝑧) − 𝑃𝑚 (𝑓𝜌̃ )(𝑧)󵄨󵄨󵄨󵄨 𝑒−|𝑧| 𝑑𝜆(𝑧) = ∫ (𝑧 ∑ 𝑎𝑘 𝜌𝑘 𝑧𝑘 − ∑ 𝑎𝑘 2𝑘 𝜌𝑘 𝑧𝑘−1 ) 𝑐𝑘−1 𝑘=0 𝑘=1

ℂ

ℂ

∞

∞

𝑘=0 ∞

𝑘=1

× (𝑧 ∑ 𝑎𝑘 𝜌𝑘 𝑧𝑘 − ∑ 𝑎𝑘

𝑐𝑘2 𝑘 𝑘−1 −|𝑧|𝑚 𝜌 𝑧 )𝑒 𝑑𝜆(𝑧) 2 𝑐𝑘−1 ∞

= ∫ ( ∑ |𝑎𝑘 |2 𝜌2𝑘 |𝑧|2𝑘+2 − 2 ∑ |𝑎𝑘 |2 𝑘=0

ℂ

𝑘=1

𝑐4 ∑ |𝑎𝑘 |2 4𝑘 𝑐𝑘−1 𝑘=1 ∞

+

𝑐𝑘2 2𝑘 2𝑘 𝜌 |𝑧| 2 𝑐𝑘−1

𝑚

𝜌2𝑘 |𝑧|2𝑘−2 ) 𝑒−|𝑧| 𝑑𝜆(𝑧)

∞

= |𝑎0 |2 𝑐12 + ∑ |𝑎𝑘 |2 𝑐𝑘2 𝜌2𝑘 ( 𝑘=1

2 𝑐𝑘+1 𝑐𝑘2 ). − 2 𝑐𝑘2 𝑐𝑘−1

Now the result follows from the fact that ∞

𝑚

∫ |𝑓(𝑧)|2 𝑒−|𝑧| 𝑑𝜆(𝑧) = ∑ |𝑎𝑘 |2 𝑐𝑘2 , 𝑘=0

ℂ

and that the sequence (

2 𝑐𝑘+1 𝑐𝑘2

−

𝑐𝑘2 2 )𝑘 𝑐𝑘−1

is bounded. 𝑐2

𝑐2

Remark 2.18. Already in the last proposition, the sequence ( 𝑘+1 − 𝑐2𝑘 )𝑘 plays an im𝑐𝑘2 𝑘−1 portant role and it will turn out that this sequence is the sequence of eigenvalues of ∗ the operator 𝑆𝑚 𝑆𝑚 (see below). 𝑚

Proposition 2.19. Let 𝑚 ≥ 2 and consider an entire function 𝑓 ∈ 𝐴2 (ℂ, 𝑒−|𝑧| ) with 𝑘 Taylor series expansion 𝑓(𝑧) = ∑∞ 𝑘=0 𝑎𝑘 𝑧 . Let ∞

∞

𝐹(𝑧) := 𝑧 ∑ 𝑎𝑘 𝑧𝑘 − ∑ 𝑎𝑘 𝑘=0

𝑘=1

𝑐𝑘2 𝑘−1 𝑧 2 𝑐𝑘−1

𝑚

𝑚

and define 𝑆𝑚 (𝑓) := 𝐹. Then 𝑆𝑚 : 𝐴2 (ℂ, 𝑒−|𝑧| ) 󳨀→ 𝐿2 (ℂ, 𝑒−|𝑧| ) is a continuous linear 𝑚 operator, representing the canonical solution operator to 𝜕 restricted to 𝐴2 (ℂ, 𝑒−|𝑧| ), i.e. 𝑚 𝜕𝑆𝑚 (𝑓) = 𝑓 and 𝑆𝑚 (𝑓) ⊥ 𝐴2 (ℂ, 𝑒−|𝑧| ). Proof. By the proof of Proposition 2.17, by Abel’s Theorem and by Fatou’s Theorem (see for instance [19]) we have 𝑚 𝑚 󵄨2 󵄨 ∫ |𝐹(𝑧)|2 𝑒−|𝑧| 𝑑𝜆(𝑧) = ∫ lim 󵄨󵄨󵄨𝑓𝜌̃ (𝑧) − 𝑃𝑚 (𝑓𝜌̃ )(𝑧)󵄨󵄨󵄨 𝑒−|𝑧| 𝑑𝜆(𝑧)

ℂ

ℂ

𝜌→1

𝑚 󵄨2 󵄨 ≤ sup ∫ 󵄨󵄨󵄨𝑓𝜌̃ (𝑧) − 𝑃𝑚 (𝑓𝜌̃ )(𝑧)󵄨󵄨󵄨 𝑒−|𝑧| 𝑑𝜆(𝑧)

0 𝑑𝜆(𝑤),

(3.1)

Ω

where 𝐾 denotes the Bergman kernel of Ω, where 𝑛

𝑔(𝑧) = ∑ 𝑔𝑗 (𝑧) 𝑑𝑧𝑗 ∈ 𝐴2(0,1) (Ω) 𝑗=1

and

𝑛

< 𝑔(𝑤), 𝑧 − 𝑤 >= ∑ 𝑔𝑗 (𝑤)(𝑧𝑗 − 𝑤𝑗 ), 𝑗=1

for 𝑧 = (𝑧1 , . . . , 𝑧𝑛 ) and 𝑤 = (𝑤1 , . . . , 𝑤𝑛 ). Let 𝑣(𝑧) = ∑𝑛𝑗=1 𝑧𝑗 𝑔𝑗 (𝑧). Then it follows that 𝑛

𝑛 𝜕𝑣 𝑑𝑧𝑗 = ∑ 𝑔𝑗 𝑑𝑧𝑗 = 𝑔. 𝑗=1 𝜕𝑧𝑗 𝑗=1

𝜕𝑣 = ∑

Hence the canonical solution operator 𝑆 can be written in the form 𝑆(𝑔) = 𝑣 − 𝑃(𝑣), where 𝑃 : 𝐿2 (Ω) 󳨀→ 𝐴2 (Ω) is the Bergman projection. If 𝑣̃ is another solution to 𝜕𝑢 = 𝑔, then 𝑣 − 𝑣̃ ∈ 𝐴2 (Ω) hence 𝑣 = 𝑣̃ + ℎ, where ℎ ∈ 𝐴2 (Ω). Therefore ̃ 𝑣 − 𝑃(𝑣) = 𝑣̃ + ℎ − 𝑃(𝑣)̃ − 𝑃(ℎ) = 𝑣̃ − 𝑃(𝑣). Since 𝑔𝑗 ∈ 𝐴2 (Ω), 𝑗 = 1, . . . , 𝑛, we have 𝑔𝑗 (𝑧) = ∫ 𝐾(𝑧, 𝑤)𝑔𝑗 (𝑤) 𝑑𝜆(𝑤). Ω

38 | 3 Spectral properties of the canonical solution operator to 𝜕̄ Now we get 𝑛

𝑛

𝑆(𝑔)(𝑧) = ∑ 𝑧𝑗 𝑔𝑗 (𝑧) − ∫ 𝐾(𝑧, 𝑤)( ∑ 𝑤𝑗 𝑔𝑗 (𝑤)) 𝑑𝜆(𝑤) 𝑗=1

𝑗=1

Ω 𝑛

𝑛

= ∫ [( ∑ 𝑧𝑗 𝑔𝑗 (𝑤))𝐾(𝑧, 𝑤) − ( ∑ 𝑤𝑗 𝑔𝑗 (𝑤))𝐾(𝑧, 𝑤)] 𝑑𝜆(𝑤) 𝑗=1

Ω

𝑗=1

= ∫ 𝐾(𝑧, 𝑤) < 𝑔(𝑤), 𝑧 − 𝑤 > 𝑑𝜆(𝑤), Ω

and (3.1) is proved. Remark 3.1. It is pointed out that a (0, 1)-form 𝑔 = ∑𝑛𝑗=1 𝑔𝑗 𝑑𝑧𝑗 with holomorphic coefficients is not invariant under the pull back by a holomorphic map 𝐹 = (𝐹1 , . . . , 𝐹𝑛 ) : Ω1 󳨀→ Ω. Then 𝑛 𝑛 𝑛 𝜕𝐹 𝐹∗ 𝑔 = ∑ 𝑔𝑙 𝑑𝐹𝑙 = ∑ ( ∑ 𝑔𝑙 𝑙 ) 𝑑𝑧𝑗 , 𝜕𝑧𝑗 𝑗=1 𝑙=1 𝑙=1 where we used the fact that 𝑛

𝑑𝐹𝑙 = 𝜕𝐹𝑙 + 𝜕 𝐹𝑙 = ∑

𝑗=1

The expressions

𝜕𝐹𝑙 𝜕𝑧𝑗

𝑛 𝑛 𝜕𝐹𝑙 𝜕𝐹 𝜕𝐹 𝑑𝑧𝑗 + ∑ 𝑙 𝑑𝑧𝑗 = ∑ 𝑙 𝑑𝑧𝑗 . 𝜕𝑧𝑗 𝑗=1 𝜕𝑧𝑗 𝑗=1 𝜕𝑧𝑗

are not holomorphic.

Nevertheless it is true that 𝜕𝑢 = 𝑔 implies 𝜕(𝑢 ∘ 𝐹) = 𝐹∗ 𝑔, which follows from the fact that for a general differential form 𝜔 and a holomorphic map 𝐹 we have 𝜕(𝐹∗ 𝜔) = 𝐹∗ (𝜕𝜔) and 𝜕(𝐹∗ 𝜔) = 𝐹∗ (𝜕𝜔). For holomorphic (𝑛, 𝑛)-forms we have the following result Proposition 3.2. Let 𝜔 be a holomorphic (𝑛, 𝑛)-form, i.e. 𝜔 = 𝜔̃ 𝑑𝑧1 ∧ ⋅ ⋅ ⋅ ∧ 𝑑𝑧𝑛 ∧ 𝑑𝑧1 ∧ ⋅ ⋅ ⋅ ∧ 𝑑𝑧𝑛 , where 𝜔̃ ∈ 𝐴2 (Ω). Let 𝑢 be the following (𝑛, 𝑛 − 1)-form 𝑛

𝑢 = ∑ 𝑢𝑗 𝑑𝑧1 ∧ ⋅ ⋅ ⋅ ∧ 𝑑𝑧𝑛 ∧ 𝑑𝑧1 ∧ ⋅ ⋅ ⋅ ∧ [𝑑𝑧𝑗 ] ∧ ⋅ ⋅ ⋅ ∧ 𝑑𝑧𝑛 , 𝑗=1

where 𝑢𝑗 (𝑧) =

(−1)𝑛+𝑗−1 ̃ ∫(𝑧𝑗 − 𝑤𝑗 )𝐾(𝑧, 𝑤)𝜔(𝑤) 𝑑𝜆(𝑤) 𝑛 Ω

and [𝑑𝑧𝑗 ] means that 𝑑𝑧𝑗 is omitted. Then 𝑢𝑗 ⊥ 𝐴2 (Ω) , 𝑗 = 1, . . . , 𝑛 and 𝜕𝑢 = 𝜔.

3.3 Compactness and Schatten class membership

|

39

Proof. It follows that 𝑢𝑗 (𝑧) =

(−1)𝑛+𝑗−1 ̃ − 𝑃(𝑤𝑗 𝜔)(𝑧)) ̃ (𝑧𝑗 𝜔(𝑧) , 𝑛

from this we obtain 𝜕𝑢𝑗 𝜕𝑧𝑘

=

𝜕𝜔̃ (−1)𝑛+𝑗−1 𝜕𝑧𝑗 (−1)𝑛+𝑗−1 )= 𝜔̃ + 𝑧𝑗 ( 𝛿𝑗𝑘 𝜔,̃ 𝑛 𝜕𝑧𝑘 𝜕𝑧𝑘 𝑛

where 𝛿𝑗𝑘 is the Kronecker delta symbol. Hence 𝑛

𝑛

𝜕𝑢 = ∑ ∑ 𝑘=1 𝑗=1 𝑛

𝜕𝑢𝑗 𝜕𝑧𝑘

𝑑𝑧𝑘 ∧ 𝑑𝑧1 ∧ ⋅ ⋅ ⋅ ∧ 𝑑𝑧𝑛 ∧ 𝑑𝑧1 ∧ ⋅ ⋅ ⋅ ∧ [𝑑𝑧𝑗 ] ∧ ⋅ ⋅ ⋅ ∧ 𝑑𝑧𝑛

𝑛

= ∑ ∑ ((−1)𝑛+𝑗−1 /𝑛) 𝛿𝑗𝑘 𝜔̃ 𝑑𝑧𝑘 𝑘=1 𝑗=1

∧ 𝑑𝑧1 ∧ ⋅ ⋅ ⋅ ∧ 𝑑𝑧𝑛 ∧ 𝑑𝑧1 ∧ ⋅ ⋅ ⋅ ∧ [𝑑𝑧𝑗 ] ∧ ⋅ ⋅ ⋅ ∧ 𝑑𝑧𝑛 = 𝜔̃ 𝑑𝑧1 ∧ ⋅ ⋅ ⋅ ∧ 𝑑𝑧𝑛 ∧ 𝑑𝑧1 ∧ ⋅ ⋅ ⋅ ∧ 𝑑𝑧𝑛 . Remark 3.3. The pull back by a holomorphic map 𝐹 has in this case the form 󵄨󵄨 𝜕𝐹𝑗 󵄨󵄨󵄨󵄨2 󵄨󵄨 󵄨󵄨 𝜔̃ 𝑑𝑧1 ∧ ⋅ ⋅ ⋅ ∧ 𝑑𝑧𝑛 ∧ 𝑑𝑧1 ∧ ⋅ ⋅ ⋅ ∧ 𝑑𝑧𝑛 . 󵄨 𝐹 𝜔 = 󵄨󵄨det 󵄨󵄨 𝜕𝑧𝑘 󵄨󵄨󵄨 ∗

3.3 Compactness and Schatten class membership Now we will study boundedness, compactness, and Schatten class membership of the canonical solution operator to 𝜕, restricted to (0, 1)-forms with holomorphic coefficients, on 𝐿2 (𝑑𝜇) where 𝜇 is a Borel measure on ℂ𝑛 with the property that the monomials form an orthogonal family in 𝐿2 (𝑑𝜇). The characterizations are formulated in terms of moment properties of 𝜇. This situation covers a number of basic examples: – Lebesgue measure on bounded domains in ℂ𝑛 which are invariant under the torus action (𝜃1 , . . . , 𝜃𝑛 )(𝑧1 , . . . , 𝑧𝑛 ) 󳨃→ (𝑒𝑖𝜃1 𝑧1 , . . . , 𝑒𝑖𝜃𝑛 𝑧𝑛 ) – –

(i.e. Reinhardt domains). Weighted 𝐿2 spaces with radially symmetric weights (e.g., generalized Fock spaces). Weighted 𝐿2 spaces with decoupled radial weights, that is, 2

𝑑𝜇 = 𝑒∑𝑗 𝜑𝑗 (|𝑧𝑗 | ) 𝑑𝜆, where 𝜑𝑗 : ℝ → ℝ is a weight function.

40 | 3 Spectral properties of the canonical solution operator to 𝜕̄ We denote by 𝐴2 (𝑑𝜇) = {𝑧𝛼 : 𝛼 ∈ ℕ𝑛 }, the closure of the monomials in 𝐿2 (𝑑𝜇), and write 𝑚𝛼 = 𝑐𝛼−1 = ∫ |𝑧𝛼 |2 𝑑𝜇. We will give necessary and sufficient conditions in terms of these multimoments of the measure 𝜇 for the canonical solution operator to 𝜕̄ – when restricted to (0, 1)-forms with coefficients in 𝐴2 (𝑑𝜇) – to be bounded, compact, and to belong to the Schatten class S𝑝 . This is accomplished by presenting a complete diagonalization of the solution operator by orthonormal bases with corresponding estimates. In the following we will investigate the canonical solution operator 𝑆 to 𝜕 which assigns to each 𝑛

𝑓𝑗 ∈ 𝐴2 (𝑑𝜇)

𝜔 = ∑ 𝑓𝑗 𝑑𝑧̄𝑗 , 𝑗=0

̄ = 𝜔 which is orthogonal to 𝐴2 (𝑑𝜇). the solution to the equation 𝜕𝑢 We will frequently encounter multi-indices 𝛾 which might have one (but not more than one) entry equal to −1: in that case, we define 𝑐𝛾 = 0. We will denote the set of these multi-indices by Γ. We let 𝑒𝑗 = (0, ⋅ ⋅ ⋅ , 1, ⋅ ⋅ ⋅ , 0) be the multi-index with a 1 in the 𝑗-th spot and 0 elsewhere. Theorem 3.4. 𝑆 : 𝐴2(0,1) (𝑑𝜇) → 𝐿2 (𝑑𝜇) is bounded if and only if there exists a constant 𝐶 such that 𝑐𝛾+𝑒𝑗 𝑐𝛾 − 0. Then 𝑆 : 𝐴2(0,1) (𝑑𝜇) → 𝐿2 (𝑑𝜇) is in the Schatten-𝑝-class S𝑝 if and only if 𝑝 2 𝑐𝛾+𝑒𝑗 𝑐𝛾 ∑ (∑( )) < ∞. − (3.3) 𝑐𝛾+2𝑒𝑗 𝑐𝛾+𝑒𝑗 𝛾∈Γ 𝑗 The condition above is substantially easier to check if 𝑝 = 2 (we will show that the sum is actually a telescoping sum then), i.e. for the case of the Hilbert–Schmidt class; we state this as a theorem: Theorem 3.7. The canonical solution operator 𝑆 is in the Hilbert–Schmidt class if and only if 𝑐𝛾 ∑ < ∞. (3.4) lim 𝑘→∞ 𝑐 𝛾∈ℕ𝑛 ,|𝛾|=𝑘 𝛾+𝑒𝑗 1≤𝑗≤𝑛

Before we prove the theorems we will derive some consequences. Let us apply Theorem 3.4 to the case of decoupled weights, or more generally, of product measures 𝑑𝜇 = 𝑑𝜇1 × ⋅ ⋅ ⋅ × 𝑑𝜇𝑛 , where each 𝑑𝜇𝑘 is a (circle-invariant) measure on ℂ. Note that for such measures, there is definitely no compactness by Theorem 3.5. If we denote by −1

𝑐𝑗𝑘 = ( ∫ |𝑧|2𝑘 𝑑𝜇𝑗 ) , ℂ

we have that

𝑛

𝛾

𝑐(𝛾1 ,...,𝛾𝑛 ) = ∏ 𝑐𝑗 𝑗 . 𝑗=1

We thus obtain the following corollary. Corollary 3.8. For a product measure 𝑑𝜇 = 𝑑𝜇1 × ⋅ ⋅ ⋅ × 𝑑𝜇𝑛 as above, the canonical solution operator 𝑆 : 𝐴2(0,1) (𝑑𝜇) → 𝐿2 (𝑑𝜇) is bounded if and only if there exists a constant 𝐶 such that 𝑐𝑗𝑘+1 𝑐𝑗𝑘 − 𝛿 > 0,

42 | 3 Spectral properties of the canonical solution operator to 𝜕̄ In the case of a rotation-invariant measure 𝜇, we write 𝑚𝑑 = ∫ |𝑧|2𝑑 𝑑𝜇; ℂ𝑛

and claim that 𝑐𝛾 =

(𝑛 − 1 + |𝛾|)! , (𝑛 − 1)! 𝛾! 𝑚|𝛾|

(3.5)

where |𝛾| = 𝛾1 + ⋅ ⋅ ⋅ + 𝛾𝑛 and 𝛾! = 𝛾1 ! . . . 𝛾𝑛 !. To show this, let U be the unitary group consisting of all 𝑛 × 𝑛 unitary matrices and let 𝑑𝑈 denote the Haar probability measure on U. Let 𝜎 be the rotation-invariant probability measure on the unit sphere 𝕊 in ℂ𝑛 . For a multi-index 𝛼 = (𝛼1 , . . . , 𝛼𝑛 ) we 𝛼 define 𝑧𝛼 = 𝑧1 1 . . . 𝑧𝑛𝛼𝑛 and 𝛼! = 𝛼1 ! . . . 𝛼𝑛 ! and |𝛼| = 𝛼1 + ⋅ ⋅ ⋅ + 𝛼𝑛 . Due to the invariance of 𝜇 it follows by Fubini’s theorem that ∫ 𝑧𝛼 𝑧𝛽 𝑑𝜇(𝑧) = ∫ ∫ (𝑈𝑧)𝛼 (𝑈𝑧)𝛽 𝑑𝜇(𝑧) 𝑑𝑈 ℂ𝑛

U ℂ𝑛

= ∫ ∫(𝑈𝑧)𝛼 (𝑈𝑧)𝛽 𝑑𝑈 𝑑𝜇(𝑧) ℂ𝑛 U 𝛽

= ∫ |𝑧||𝛼|+|𝛽| ∫ 𝜁𝛼 𝜁 𝑑𝜎(𝜁) 𝑑𝜇(𝑧), ℂ𝑛

(3.6)

𝕊

where we used the fact that for a continuous function 𝑓 ∈ C(𝕊) we have ∫ 𝑓(𝜁) 𝑑𝜎(𝜁) = ∫ 𝑓(𝑈𝜂) 𝑑𝑈, 𝕊

U

for any 𝜂 ∈ 𝕊 (see [64, Proposition 1.4.7]). It is clear that for 𝛼 ≠ 𝛽 we have 𝛽

∫ 𝜁𝛼 𝜁 𝑑𝜎(𝜁) = 0. 𝕊

Next we claim that for any multi-index 𝛾 ∫ |𝜁𝛾 |2 𝑑𝜎(𝜁) = 𝕊

(𝑛 − 1)! 𝛾! (𝑛 − 1 + |𝛾|)!

(3.7)

To prove (3.7) we use the integral 𝑛

𝐼 = ∫ |𝑧𝛾 |2 exp(−|𝑧|2 ) 𝑑𝜆 2𝑛 (𝑧) = ∏ ∫ |𝑤|2𝛾𝑗 exp(−|𝑤|2 ) 𝑑𝜆 2 (𝑤), 𝑗=1

ℂ𝑛

ℂ

2𝑛

where 𝜆 2𝑛 is the Lebesgue measure on ℝ . It follows easily that 𝐼 = 𝜋𝑛 𝛾!. Now we apply integration in polar coordinates to 𝐼 and get ∞

2

𝜋𝑛 𝛾! = 2𝑛 𝑐𝑛 ∫ 𝑟2|𝛾|+2𝑛−1 𝑒−𝑟 𝑑𝑟 ∫ |𝜁𝛾 |2 𝑑𝜎(𝜁), 0

where 𝑐𝑛 is the volume of the unit ball in ℂ𝑛 .

𝕊

3.3 Compactness and Schatten class membership

Hence ∫ |𝜁𝛾 |2 𝑑𝜎(𝜁) = 𝕊

|

43

𝜋𝑛 𝛾! , (𝑛 − 1 + |𝛾|)! 𝑛𝑐𝑛

𝑛

taking 𝛾 = 0 we get 𝑐𝑛 = 𝜋 /𝑛!, which proves (3.7). For 𝑑 ∈ ℕ we set 𝑚𝑑 = ∫ |𝑧|2𝑑 𝑑𝜇 and 𝑐𝛾−1 = ∫ |𝑧𝛾 |2 𝑑𝜇 ℂ𝑛

ℂ𝑛

and obtain from (3.6) and (3.7) 𝑐𝛾 =

(𝑛 − 1 + |𝛾|)! . (𝑛 − 1)! 𝛾! 𝑚|𝛾|

In order to express the conditions of our theorems, we compute (setting 𝑑 = |𝛾|+1) ∑( 𝑗

𝑐𝛾+𝑒𝑗 𝑐𝛾+2𝑒𝑗

𝑚

{ 𝑑+2𝑛−1 𝑑+1 − ) = { 1𝑑+𝑛𝑚 𝑚𝑑 − 𝑑+1 𝑐𝛾+𝑒𝑗 { 𝑑+𝑛 𝑚𝑑 𝑐𝛾

𝑚𝑑 𝑚𝑑−1

𝛾𝑗 ≠ −1 for all 𝑗 else.

(3.8)

Note that the Cauchy–Schwarz inequality implies that the first case in (3.8) always dominates the second case for 𝑛 ≥ 2; for 𝑛 = 1 we observe that the second case in (3.8) 𝑚 reduces to 𝑚1 , compare with Proposition 2.15. 0 Using this observation and some trivial inequalities, we get the following corollaries. Corollary 3.9. Let 𝜇 be a rotation invariant measure on ℂ𝑛 . Then the canonical solution operator to 𝜕̄ is bounded on 𝐴2(0,1) (𝑑𝜇) if and only if sup (

𝑑∈ℕ

(2𝑛 + 𝑑 − 1)𝑚𝑑+1 𝑚𝑑 − ) < ∞. (𝑛 + 𝑑)𝑚𝑑 𝑚𝑑−1

(3.9)

Corollary 3.10. Let 𝜇 be a rotation invariant measure on ℂ𝑛 . Then the canonical solution operator to 𝜕̄ is compact on 𝐴2(0,1) (𝑑𝜇) if and only if lim (

𝑑→∞

(2𝑛 + 𝑑 − 1)𝑚𝑑+1 𝑚𝑑 ) = 0. − (𝑛 + 𝑑)𝑚𝑑 𝑚𝑑−1

(3.10)

Corollary 3.11. Let 𝜇 be a rotation invariant measure on ℂ𝑛 . Then the canonical solution operator to 𝜕̄ is a Hilbert–Schmidt operator on 𝐴2(0,1) (𝑑𝜇) if and only if 𝑛 + 𝑑 − 2 𝑚𝑑+1 ) < ∞. lim ( 𝑚𝑑 𝑛−1

𝑑→∞

(3.11)

Remark 3.12. It follows that the canonical solution operator to 𝜕̄ is a Hilbert–Schmidt operator on 𝐴2 (𝔻), but fails to be Hilbert–Schmidt on 𝐴2 (𝔹𝑛 ), where 𝔹𝑛 is the unit ball in ℂ𝑛 , for 𝑛 ≥ 2.

44 | 3 Spectral properties of the canonical solution operator to 𝜕̄ Corollary 3.13. Let 𝜇 be a rotation invariant measure on ℂ𝑛 , 𝑝 > 0. Then the canonical solution operator to 𝜕̄ is in the Schatten class S𝑝 , as an operator from 𝐴2(0,1) (𝑑𝜇) to 𝐿2 (𝑑𝜇) if and only if 𝑝 ∞ 𝑚𝑑 2 𝑛 + 𝑑 − 2 (2𝑛 + 𝑑 − 1)𝑚𝑑+1 ∑( ) < ∞. )( − (3.12) (𝑛 + 𝑑)𝑚𝑑 𝑚𝑑−1 𝑛−1 𝑑=1 Now we prove the theorems of this section. In what follows, we will denote by 𝑢𝛼 = √𝑐𝛼 𝑧𝛼 the orthonormal basis of monomials for the space 𝐴2 (𝑑𝜇), and by 𝑈𝛼,𝑗 = 𝑢𝛼 𝑑𝑧̄𝑗

(3.13)

the corresponding basis of 𝐴2(0,1) (𝑑𝜇). We first note that it is always possible to solve the ̄ 𝜕-equation for the elements of this basis; indeed, 𝜕̄ 𝑧̄𝑗 𝑢𝛼 = 𝑈𝛼,𝑗 . The canonical solution operator is also easily determined for forms with monomial coefficients: Lemma 3.14. The canonical solution 𝑆𝑧𝛼 𝑑𝑧̄𝑗 for monomial forms is given by 𝑆𝑧𝛼 𝑑𝑧̄𝑗 = 𝑧̄𝑗 𝑧𝛼 −

𝑐𝛼−𝑒𝑗 𝑐𝛼

𝑧𝛼−𝑒𝑗 ,

𝛼 ∈ ℕ𝑛0 .

(3.14)

Proof. We have ⟨𝑧̄𝑗 𝑧𝛼 , 𝑧𝛽 ⟩ = ⟨𝑧𝛼 , 𝑧𝛽+𝑒𝑗 ⟩; so this expression is nonzero only if 𝛽 = 𝛼 − 𝑒𝑗 (in particular, this implies (3.14) for multi-indices 𝛼 with 𝛼𝑗 = 0; recall our convention that 𝑐𝛾 = 0 if one of the entries of 𝛾 is negative). Thus 𝑆𝑧𝛼 𝑑𝑧̄𝑗 = 𝑧̄𝑗 𝑧𝛼 + 𝑐𝑧𝛼−𝑒𝑗 , and 𝑐 is computed by −1 , 0 = ⟨𝑧̄𝑗 𝑧𝛼 + 𝑐𝑧𝛼−𝑒𝑗 , 𝑧𝛼−𝑒𝑗 ⟩ = 𝑐𝛼−1 + 𝑐𝑐𝛼−𝑒 𝑗 which gives 𝑐 = −𝑐𝛼−𝑒𝑗 /𝑐𝛼 . We are going to introduce an orthogonal decomposition 𝐴2(0,1) (𝑑𝜇) = ⨁ 𝐸𝛾 𝛾∈Γ

of 𝐴2(0,1) (𝑑𝜇) into at most 𝑛-dimensional subspaces 𝐸𝛾 indexed by multi-indices 𝛾 ∈ Γ (we will describe the index set below), and a corresponding sequence of mutually orthogonal finite-dimensional subspaces 𝐹𝛾 ⊂ 𝐿2 (𝑑𝜇) which diagonalizes 𝑆 (by this we mean that 𝑆𝐸𝛾 = 𝐹𝛾 ). To motivate the definition of 𝐸𝛾 , note that {0 ⟨𝑆𝑧𝛼 𝑑𝑧̄𝑘 , 𝑆𝑧𝛽 𝑑𝑧̄ℓ ⟩ = { 1 𝑐 ( 𝛼 − { 𝑐𝛼 𝑐𝛼+𝑒ℓ

𝑐𝛼−𝑒𝑘

𝑐𝛼+𝑒ℓ −𝑒𝑘

𝛽 ≠ 𝛼 + 𝑒ℓ − 𝑒𝑘 , ) 𝛽 = 𝛼 + 𝑒ℓ − 𝑒𝑘 ,

(3.15)

so that ⟨𝑆𝑧𝛼 𝑑𝑧̄𝑘 , 𝑆𝑧𝛽 𝑑𝑧̄ℓ ⟩ ≠ 0 if and only if there exists a multi-index 𝛾 such that 𝛼 = 𝛾 + 𝑒𝑘 and 𝛽 = 𝛾 + 𝑒ℓ . Recall (3.13) and define 𝐸𝛾 = span {𝑈𝛾+𝑒𝑗 ,𝑗 : 1 ≤ 𝑗 ≤ 𝑛} = span {𝑧𝛾+𝑒𝑗 𝑑𝑧̄𝑗 : 1 ≤ 𝑗 ≤ 𝑛},

3.3 Compactness and Schatten class membership

|

45

and likewise 𝐹𝛾 = 𝑆𝐸𝛾 . Recall that Γ is defined to be the set of all multi-indices whose entries are greater than or equal to −1 and has at most one negative entry. Note that 𝐸𝛾 is 1-dimensional if exactly one entry in 𝛾 equals −1, and 𝑛-dimensional otherwise. We have already observed that 𝐹𝛾 are mutually orthogonal subspaces of 𝐿2 (𝑑𝜇) (see (3.15)). Whenever we use multi-indices 𝛾 and integers 𝑝 ∈ {1, . . . , 𝑛} as indices, we use the convention that the 𝑝 runs over all 𝑝 such that 𝛾+𝑒𝑝 ≥ 0; that is, for a fixed multi-index 𝛾 ∈ Γ, either the indices are all 𝑝 ∈ {1, . . . , 𝑛} or there is exactly one 𝑝 such that 𝛾𝑝 = −1, in which case the index is exactly this one 𝑝. We next observe that we can find an orthonormal basis of 𝐸𝛾 and an orthonormal basis of 𝐹𝛾 such that in these bases 𝑆𝛾 = 𝑆|𝐸𝛾 : 𝐸𝛾 → 𝐹𝛾 acts diagonally. First note that it is enough to do this if dim 𝐸𝛾 = 𝑛 (since an operator between one-dimensional spaces is automatically diagonal). Fixing 𝛾, the functions 𝑈𝑗 := 𝑈𝛾+𝑒𝑗 ,𝑗 are an orthonormal basis of 𝐸𝛾 . The operator 𝑆𝛾 is clearly nonsingular on this space, so the functions 𝑆𝑈𝑗 = 𝑗

Ψ𝑗 constitute a basis of 𝐹𝛾 . For a basis 𝐵 of vectors 𝑣𝑗 = (𝑣1 , . . . , 𝑣𝑛𝑗 ), 𝑗 = 1, . . . , 𝑛 of ℂ𝑛 we consider the new basis 𝑛

𝑗

𝑉𝑘 = ∑ 𝑣𝑘 𝑈𝑗 ; 𝑗=1

since the basis given by the 𝑈𝑗 is orthonormal, the basis given by the 𝑉𝑘 is also orthonormal provided that the vectors 𝑣𝑘 = (𝑣𝑘1 , . . . , 𝑣𝑘𝑛 ) constitute an orthonormal basis for ℂ𝑛 with the standard Hermitian product. Let us write 𝑗

Φ𝑘 = 𝑆𝑉𝑘 = ∑ 𝑣𝑘 𝑆𝑈𝑗 . 𝑗

𝑗

The inner product ⟨Φ𝑝 , Φ𝑞 ⟩ is then given by ∑𝑗,𝑘 𝑣𝑝 𝑣𝑞̄𝑘 ⟨𝑆𝑈𝑗 , 𝑆𝑈𝑘 ⟩. We therefore have ⟨Φ1 , Φ1 ⟩ .. ( . ⟨Φ𝑛 , Φ1 ⟩

⋅⋅⋅ ⋅⋅⋅ 𝑣11 ( ... 𝑣𝑛1

⟨Φ1 , Φ𝑛 ⟩ .. )= . ⟨Φ𝑛 , Φ𝑛 ⟩ ⋅⋅⋅ ⋅⋅⋅

⟨Ψ1 , Ψ1 ⟩ 𝑣1𝑛 .. ) ( .. . . ⟨Ψ𝑛 , Ψ1 ⟩ 𝑣𝑛𝑛

⟨Ψ1 , Ψ𝑛 ⟩ 𝑣11̄ .. ) ( ... . ⟨Ψ𝑛 , Ψ𝑛 ⟩ 𝑣1̄𝑛

⋅⋅⋅ ⋅⋅⋅

⋅⋅⋅ ⋅⋅⋅

𝑣𝑛̄1 .. ) . . 𝑣𝑛̄𝑛

(3.16)

Since the matrix (⟨Ψ𝑗 , Ψ𝑘 ⟩)𝑗,𝑘 is Hermitian, we can unitarily diagonalize it; that is, we can choose an orthonormal basis 𝐵 of ℂ𝑛 such that with this choice of 𝐵 the vectors 𝑗

𝜑𝛾,𝑘 = 𝑉𝑘 = ∑ 𝑣𝑘 𝑈𝛾+𝑒𝑗 ,𝑗

(3.17)

𝑗

of 𝐸𝛾 are orthonormal, and their images Φ𝑘 = 𝑆𝑉𝑘 are orthogonal in 𝐹𝛾 . Therefore, Φ𝑘 /‖Φ𝑘 ‖ is an orthonormal basis of 𝐹𝛾 such that 𝑆𝛾 : 𝐸𝛾 → 𝐹𝛾 is diagonal when expressed in terms of the bases {𝑉1 , . . . , 𝑉𝑛 } ⊂ 𝐸𝛾 and {Φ1 , . . . , Φ𝑛 } ⊂ 𝐹𝛾 , with entries ‖Φ𝑘 ‖.

46 | 3 Spectral properties of the canonical solution operator to 𝜕̄ Furthermore, the ‖Φ𝑘 ‖ are exactly the square roots of the eigenvalues of the matrix (⟨Ψ𝑝 , Ψ𝑞 ⟩)𝑝,𝑞 which by (3.15) is given by ⟨Ψ𝑝 , Ψ𝑞 ⟩ = ⟨𝑆𝑈𝛾+𝑒𝑝 ,𝑝 , 𝑆𝑈𝛾+𝑒𝑞 ,𝑞 ⟩ = √𝑐𝛾+𝑒𝑝 √𝑐𝛾+𝑒𝑞 ⟨𝑆 𝑧𝛾+𝑒𝑝 𝑑𝑧̄𝑝 , 𝑆 𝑧𝛾+𝑒𝑞 𝑑𝑧̄𝑞 ⟩ = √𝑐𝛾+𝑒𝑝 𝑐𝛾+𝑒𝑞

1

𝑐𝛾+𝑒𝑝

(

−

𝑐𝛾+𝑒𝑝 𝑐𝛾+𝑒𝑝 +𝑒𝑞 𝑐𝛾+𝑒𝑝 𝑐𝛾+𝑒𝑞 − 𝑐𝛾 𝑐𝛾+𝑒𝑝 +𝑒𝑞 = . 𝑐𝛾+𝑒𝑝 +𝑒𝑞 √𝑐𝛾+𝑒𝑝 𝑐𝛾+𝑒𝑞

𝑐𝛾 𝑐𝛾+𝑒𝑞

)

(3.18)

Summarizing, we have the following proposition. Proposition 3.15. With 𝜇 as above, the canonical solution operator 𝑆 : 𝐴2(0,1) (𝑑𝜇) → 𝐿2(0,1) (𝑑𝜇) admits a diagonalization by orthonormal bases. In fact, we have a decomposition 𝐴2(0,1) = ⨁𝛾 𝐸𝛾 into mutually orthogonal finite-dimensional subspaces 𝐸𝛾 , indexed by the multi-indices 𝛾 with at most one negative entry (equal to −1), which are of dimension 1 or 𝑛, and orthonormal bases 𝜑𝛾,𝑗 of 𝐸𝛾 , such that 𝑆𝜑𝛾,𝑗 is a set of mutually orthogonal vectors in 𝐿2 (𝑑𝜇). For fixed 𝛾, the norms ‖𝑆𝜑𝛾,𝑗 ‖ are the square roots of the eigenvalues of the matrix 𝐶𝛾 = (𝐶𝛾,𝑝,𝑞 )𝑝,𝑞 given by 𝑐𝛾+𝑒𝑝 𝑐𝛾+𝑒𝑞 − 𝑐𝛾 𝑐𝛾+𝑒𝑝 +𝑒𝑞 . (3.19) 𝐶𝛾,𝑝,𝑞 = 𝑐𝛾+𝑒𝑝 +𝑒𝑞 √𝑐𝛾+𝑒𝑝 𝑐𝛾+𝑒𝑞 In particular, we have that 𝑛 𝑛 𝑐𝛾+𝑒𝑝 𝑐𝛾 󵄩 󵄩2 ∑ 󵄩󵄩󵄩󵄩𝑆𝜑𝛾,𝑗 󵄩󵄩󵄩󵄩 = tr(𝐶𝛾,𝑝,𝑞 )𝑝,𝑞 = ∑ ( − ). (3.20) 𝑐𝛾+2𝑒 𝑐𝛾+𝑒 𝑝=1

𝑗=1

𝑝

𝑝

In order to prove Theorem 3.4, we use Proposition 3.15. We have seen that we have an orthonormal basis 𝜑𝛾,𝑗 , 𝛾 ∈ Γ, 𝑗 ∈ {1, . . . , 𝑛}, such that the images 𝑆𝜑𝛾,𝑗 are mutually orthogonal. Thus, 𝑆 is bounded if and only if there exists a constant 𝐶 such that 󵄩󵄩 󵄩2 󵄩󵄩𝑆𝜑𝛾,𝑗 󵄩󵄩󵄩 ≤ 𝐶 󵄩 󵄩 for all 𝛾 ∈ Γ and 𝑗 ∈ {1, . . . , dim 𝐸𝛾 }. If dim 𝐸𝛾 = 1, then 𝛾 has exactly one entry (say the 𝑗-th one) equal to −1; in that case, let us write 𝜑𝛾 = 𝑈𝛾+𝑒𝑗 𝑑𝑧̄𝑗 . We have 𝑆𝜑𝛾 = √𝑐𝛾+𝑒𝑗 𝑧̄𝑗 𝑧𝛾+𝑒𝑗 , and so 𝑐𝛾+𝑒𝑗 󵄩󵄩 󵄩󵄩2 󵄩󵄩𝑆𝜑𝛾 󵄩󵄩 = . 󵄩 󵄩 𝑐𝛾+2𝑒 𝑗

On the other hand, if dim 𝐸𝛾 = 𝑛, we argue as follows: Writing ‖𝑆𝜑𝛾,𝑗 ‖2 = 𝜆2𝛾,𝑗 with 𝜆 𝛾,𝑗 > 0, from (3.20) we find that 𝑛

𝑛

∑ 𝜆2𝛾,𝑗 = ∑ (

𝑗=1

𝑗=1

𝑐𝛾+𝑒𝑗 𝑐𝛾+2𝑒𝑗

−

𝑐𝛾 𝑐𝛾+𝑒𝑗

).

The last two equations complete the proof of Theorem 3.4.

3.3 Compactness and Schatten class membership

|

47

In order to prove Theorem 3.5, we use Proposition 2.1. Proof of Theorem 3.5. We first show that (3.2) implies compactness. We will use the notation which was already used in the proof of Theorem 3.4; that is, we write ‖𝑆𝜑𝛾,𝑗 ‖2 = 𝜆2𝛾,𝑗 . Let 𝜀 > 0. There exists a finite set 𝐴 𝜀 of multi-indices 𝛾 ∈ Γ such that for all 𝛾 ∉ 𝐴 𝜀 , 𝑛

𝑛

𝑐𝛾+𝑒𝑗

𝑗=1

𝑐𝛾+2𝑒𝑗

∑ 𝜆2𝛾,𝑗 = ∑ (

𝑗=1

−

𝑐𝛾 𝑐𝛾+𝑒𝑗

) < 𝜀.

Hence, if we consider the finite-dimensional (and thus, compact) operator 𝑇𝜀 defined by 𝑇𝜀 ( ∑ 𝑎𝛾,𝑗 𝜑𝛾,𝑗 ) = ∑ 𝑎𝛾,𝑗 𝑆𝜑𝛾,𝑗 , 𝛾∈𝐴 𝜀

for any 𝑣 = ∑ 𝑎𝛾,𝑗 𝜑𝛾,𝑗 ∈ 𝐴2(0,1) (𝑑𝜇) we obtain 󵄩󵄩2 󵄩 󵄩2 󵄩󵄩 ‖𝑆𝑣‖2 = 󵄩󵄩󵄩𝑇𝜀 𝑣󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑆 ∑ 𝑎𝛾,𝑗 𝜑𝛾,𝑗 󵄩󵄩󵄩󵄩 󵄩 󵄩 𝛾∉𝐴 𝜀

󵄩 󵄩2 󵄩 󵄩2 = 󵄩󵄩󵄩𝑇𝜀 𝑣󵄩󵄩󵄩 + ∑ |𝑎𝛾,𝑗 |2 󵄩󵄩󵄩𝑆𝜑𝛾,𝑗 󵄩󵄩󵄩 𝛾∉𝐴 𝜀

󵄩 󵄩2 = 󵄩󵄩󵄩𝑇𝜀 𝑣󵄩󵄩󵄩 + ∑ |𝑎𝛾,𝑗 |2 𝜆2𝛾,𝑗 𝛾∉𝐴 𝜀

󵄩 󵄩2 ≤ 󵄩󵄩󵄩𝑇𝜀 𝑣󵄩󵄩󵄩 + 𝜀 ∑ |𝑎𝛾,𝑗 |2 𝛾∉𝐴 𝜀

󵄩 󵄩2 󵄩 󵄩2 ≤ 󵄩󵄩󵄩𝑇𝜀 𝑣󵄩󵄩󵄩 + 𝜀󵄩󵄩󵄩𝑣󵄩󵄩󵄩 . Hence, (2.2) holds and we have proved the first implication in Theorem 3.5. We now turn to the other direction. Assume that (3.2) is not satisfied. Then there exists a 𝐾 > 0 and an infinite family 𝐴 of multi-indices 𝛾 such that for all 𝛾 ∈ 𝐴, 𝑛

𝑛

∑ 𝜆2𝛾,𝑗 = ∑ (

𝑗=1

𝑗=1

𝑐𝛾+𝑒𝑗 𝑐𝛾+2𝑒𝑗

−

𝑐𝛾 𝑐𝛾+𝑒𝑗

) > 𝑛𝐾.

In particular, for each 𝛾 ∈ 𝐴, there exists a 𝑗𝛾 such that 𝜆2𝛾,𝑗𝛾 > 𝐾. Thus, we have an infinite orthonormal family {𝜑𝛾,𝑗𝛾 : 𝛾 ∈ 𝐴} of vectors such that their images 𝑆𝜑𝛾,𝑗𝛾 are orthogonal and have norm bounded from below by √𝐾, which contradicts compactness. In the following we will also need to introduce the usual grading on the index set Γ, that is, we write Γ𝑘 = {𝛾 ∈ Γ : |𝛾| = 𝑘} , 𝑘 ≥ −1. (3.21) In order to study the membership in the Schatten class, we need the following elementary Lemma:

48 | 3 Spectral properties of the canonical solution operator to 𝜕̄ Lemma 3.16. Assume that 𝑝(𝑥) and 𝑞(𝑥) are continuous, real-valued functions on ℝ𝑁 which are homogeneous of degree 1 (i.e. 𝑝(𝑡𝑥) = 𝑡𝑝(𝑥) and 𝑞(𝑡𝑥) = 𝑡𝑞(𝑥) for 𝑡 ∈ ℝ), and 𝑞(𝑥) = 0 as well as 𝑝(𝑥) = 0 implies 𝑥 = 0. Then there exists a constant 𝐶 such that 1 |𝑞(𝑥)| ≤ |𝑝(𝑥)| ≤ 𝐶|𝑞(𝑥)|. 𝐶

(3.22)

Proof. Note that the set 𝐵𝑞 = {𝑥 : |𝑞(𝑥)| = 1} is compact: it is closed since 𝑞 is continuous, and since |𝑞| is bounded from below on the unit sphere of ℝ𝑁 by some 𝑚 > 0, it is necessarily contained in the closed ball of radius 1/𝑚. Now, the function |𝑝| is bounded on the compact set 𝐵𝑞 ; say, by 1/𝐶 from below and 𝐶 from above. Thus for all 𝑥 ∈ ℝ𝑁 , 1 󵄨󵄨󵄨󵄨 𝑥 󵄨󵄨󵄨󵄨 )󵄨 ≤ 𝐶, ≤ 󵄨󵄨𝑝 ( 𝐶 󵄨󵄨 𝑞(𝑥) 󵄨󵄨󵄨 which proves (3.22). Proof of Theorem 3.6. Note that 𝑆 is in the Schatten class S𝑝 if and only if 𝑝

(3.23)

∑ 𝜆 𝛾,𝑗 < ∞.

𝛾∈Γ, 𝑗

We rewrite this sum as 𝑝

∑ ( ∑ 𝜆 𝛾,𝑗 ) =: 𝑀 ∈ ℝ ∪ {∞} .

𝛾∈Γ

𝑗

Lemma 3.16 implies that there exists a constant 𝐶 such that for every 𝛾 ∈ Γ, 1 ( ∑ 𝜆2 ) 𝐶 𝑗 𝛾,𝑗

𝑝/2

≤

𝑝 ∑ 𝜆 𝛾,𝑗 𝑗

𝑝/2 2 𝐶( ∑ 𝜆 𝛾,𝑗 ) . 𝑗

≤

Hence, 𝑀 < ∞ if and only if 𝑝/2 2 ∑ ( ∑ 𝜆 𝛾,𝑗 ) 𝛾 𝑗

< ∞,

which after applying (3.20) becomes the condition (3.6) claimed in Theorem 3.6. Proof of Theorem 3.7. 𝑆 is in the Hilbert–Schmidt class if and only if ∑ 𝜆2𝛾,𝑗 < ∞.

(3.24)

𝛾∈Γ,𝑗

We will prove that 𝑘

∑ ∑ 𝜆2𝛾,𝑗 =

ℓ=−1 𝛾∈Γℓ ,𝑗

∑

𝑐𝛼

𝑐 𝛼∈ℕ𝑛 ,|𝛼|=𝑘+1 𝛼+𝑒𝑝 1≤𝑝≤𝑛

,

(3.25)

3.4 Notes

| 49

which immediately implies Theorem 3.7. The proof is by induction over 𝑘. For 𝑘 = −1, the left-hand side of (3.25) is 𝑛 𝑛 𝑛 𝑐 󵄩 󵄩2 ∑ 𝜆2−𝑒𝑗 ,𝑗 = ∑ 󵄩󵄩󵄩󵄩𝑧𝑗 󵄩󵄩󵄩󵄩 𝑐0 = ∑ 0 , 𝑐 𝑗=1 𝑗=1 𝑗=1 𝑒𝑝

which is equal to the right-hand side of (3.25). Now assume that (3.25) holds for 𝑘 = 𝐾 − 1; we will show that this implies that it holds for 𝑘 = 𝐾. We write 𝐾

∑ ∑ 𝜆2𝛾,𝑗 =

ℓ=−1 𝛾∈Γℓ ,𝑗

=

𝑐𝛼

∑ 𝛼∈ℕ𝑛 ,|𝛼|=𝐾−1 1≤𝑝≤𝑛

∑ 𝑛

𝛼∈ℕ ,|𝛼|=𝐾 1≤𝑝≤𝑛

𝑐𝛼+𝑒𝑝

+ ∑ ( 𝛾∈Γ𝐾 ,𝑗

𝑐𝛾+𝑒𝑗 𝑐𝛾+2𝑒𝑗

−

𝑐𝛾 𝑐𝛾+𝑒𝑗

)

𝑐𝛼 . 𝑐𝛼+𝑒𝑝

This finishes the proof of Theorem 3.7.

3.4 Notes Most of the material in this chapter is taken from [35], which can be viewed as a generalization of results from [29, 30] and [55]. Corollary 3.13 improves Theorem C of [55] in the sense that it also covers the case 0 < 𝑝 < 2. We would like to note that our techniques can be adapted to the setting of [55] by considering the canonical solution operator on a Hilbert space H of holomorphic functions endowed with a norm which is comparable to the 𝐿2 -norm on each subspace generated by monomials of a fixed degree 𝑑, if in addition to the requirements in [55] we also assume that the monomials belong to H; this introduces the additional weights found by [55] in the formulas, as the reader can check. In our setting, the formulas are somewhat “cleaner” by working with 𝐴2 (𝑑𝜇) (in particular, Corollary 3.11 only holds in this setting). The restriction of the canonical solution operator to 𝜕 can be seen as a Hankel operator. Additional contributions were made by Knirsch and Schneider [48, 68]. Later, in Chapter 12, we will investigate more general properties of this restriction, which are related to commutators between multiplication operators and the Bergman projection. In Chapter 13 we will study the canonical solution operator to 𝜕 on 𝐿2 -spaces without restriction.

4 The 𝜕-complex Our main task will be to solve the inhomogeneous Cauchy–Riemann equation 𝜕𝑢 = 𝑓, where the right-hand side 𝑓 is given and satisfies the necessary condition 𝜕𝑓 = 0. For 𝑛 > 1 this is an overdetermined system of partial differential equations, which will be reduced to a system with equal numbers of unknowns and equations. But we also need some background from the theory of unbounded operators on Hilbert spaces and the basics of the theory of distributions. Afterwards we introduce the complex Laplacian (box operator). We show that, under suitable assumptions, this operator has a bounded inverse, the 𝜕-Neumann operator and we discuss important properties of the 𝜕-Neumann operator.

4.1 Unbounded operators on Hilbert spaces In the sequel we develop elements of unbounded self-adjoint operators which are used for the 𝜕-complex. Definition 4.1. Let 𝐻1 , 𝐻2 be Hilbert spaces and 𝑇 : dom(𝑇) 󳨀→ 𝐻2 be a densely defined linear operator, i.e. dom(𝑇) is a dense linear subspace of 𝐻1 . Let dom(𝑇∗ ) be the space of all 𝑦 ∈ 𝐻2 such that 𝑥 󳨃→ (𝑇𝑥, 𝑦)2 defines a continuous linear functional on dom(𝑇). Since dom(𝑇) is dense in 𝐻1 there exists a uniquely determined element 𝑇∗ 𝑦 ∈ 𝐻1 such that (𝑇𝑥, 𝑦)2 = (𝑥, 𝑇∗ 𝑦)1 (Riesz representation Theorem 1.6). The map 𝑦 󳨃→ 𝑇∗ 𝑦 is linear and 𝑇∗ : dom(𝑇∗ ) 󳨀→ 𝐻1 is the adjoint operator to 𝑇. 𝑇 is called a closed operator, if the graph G(𝑇) = {(𝑓, 𝑇𝑓) ∈ 𝐻1 × 𝐻2 : 𝑓 ∈ dom(𝑇)}

is a closed subspace of 𝐻1 × 𝐻2 . The inner product in 𝐻1 × 𝐻2 is ((𝑥, 𝑦), (𝑢, 𝑣)) = (𝑥, 𝑢)1 + (𝑦, 𝑣)2 . If 𝑉̃ is a linear subspace of 𝐻1 which contains dom(𝑇) and the operator 𝑇̃ is defined ̃ = 𝑇𝑥 for all 𝑥 ∈ dom(𝑇), then we say that 𝑇̃ is an extension of 𝑇. on 𝑉̃ such that 𝑇𝑥 An operator 𝑇 with domain dom(𝑇) is said to be closable if it has a closed extension 𝑇.̃ Lemma 4.2. Let 𝑇 be a densely defined closable operator. Then there is a closed extension 𝑇, called its closure, whose domain is smallest among all closed extensions. Proof. Let V be the set of 𝑥 ∈ 𝐻1 for which there exist 𝑥𝑘 ∈ dom(𝑇) and 𝑦 ∈ 𝐻2 such that lim𝑘→∞ 𝑥𝑘 = 𝑥 and lim𝑘→∞ 𝑇𝑥𝑘 = 𝑦. Since 𝑇̃ is a closed extension of 𝑇 it follows ̃ = 𝑦. Therefore 𝑦 is uniquely determined by 𝑥. We define that 𝑥 ∈ dom(𝑇)̃ and 𝑇𝑥

4.1 Unbounded operators on Hilbert spaces

|

51

𝑇𝑥 = 𝑦 with dom(𝑇) = V. Then 𝑇 is an extension of 𝑇 and every closed extension of 𝑇 is also an extension of 𝑇. The graph of 𝑇 is the closure of the graph of 𝑇 in 𝐻1 × 𝐻2 . Hence 𝑇 is a closed operator. Lemma 4.3. Let 𝑇1 : dom(𝑇1 ) 󳨀→ 𝐻2 be a densely defined operator and 𝑇2 : 𝐻2 󳨀→ 𝐻3 be a bounded operator. Then (𝑇2 𝑇1 )∗ = 𝑇1∗ 𝑇2∗ , which includes that dom((𝑇2 𝑇1 )∗ ) = dom(𝑇1∗ 𝑇2∗ ). Proof. Note that dom(𝑇1∗ 𝑇2∗ ) = {𝑓 ∈ dom(𝑇2∗ ) : 𝑇2∗ (𝑓) ∈ dom(𝑇1∗ )}. Let 𝑓 ∈ dom(𝑇1∗ 𝑇2∗ ) and 𝑔 ∈ dom(𝑇2 𝑇1 ). Then (𝑇1∗ 𝑇2∗ 𝑓, 𝑔) = (𝑇2∗ 𝑓, 𝑇1 𝑔) = (𝑓, 𝑇2 𝑇1 𝑔), hence dom(𝑇1∗ 𝑇2∗ ) ⊆ dom((𝑇2 𝑇1 )∗ ). Now let 𝑓 ∈ dom((𝑇2 𝑇1 )∗ ). As 𝑇2∗ is bounded and everywhere defined on 𝐻3 , we have for all 𝑔 ∈ dom(𝑇2 𝑇1 ) = dom(𝑇1 ) that ((𝑇2 𝑇1 )∗ 𝑓, 𝑔) = (𝑓, 𝑇2 𝑇1 𝑔) = (𝑇2∗ 𝑓, 𝑇1 𝑔). Hence 𝑇2∗ 𝑓 ∈ dom(𝑇1∗ ) and 𝑓 ∈ dom(𝑇1∗ 𝑇2∗ ). Lemma 4.4. Let 𝑇 be a densely defined operator on 𝐻 and let 𝑆 be a bounded operator on 𝐻. Then (𝑇 + 𝑆)∗ = 𝑇∗ + 𝑆∗ . Proof. Let 𝑓 ∈ dom(𝑇∗ + 𝑆∗ ) = dom(𝑇∗ ). Then for all 𝑔 ∈ dom(𝑇 + 𝑆) = dom(𝑇) we have ((𝑇∗ + 𝑆∗ )𝑓, 𝑔) = (𝑇∗ 𝑓, 𝑔) + (𝑆∗ 𝑓, 𝑔) = (𝑓, 𝑇𝑔) + (𝑓, 𝑆𝑔) = (𝑓, (𝑇 + 𝑆)𝑔), hence 𝑓 ∈ dom((𝑇 + 𝑆)∗ ) and (𝑇 + 𝑆)∗ 𝑓 = 𝑇∗ 𝑓 + 𝑆∗ 𝑓. If 𝑓 ∈ dom((𝑇 + 𝑆)∗ ), then for all 𝑔 ∈ dom(𝑇 + 𝑆) = dom(𝑇) we have ([(𝑇 + 𝑆)∗ − 𝑆∗ ]𝑓, 𝑔) = (𝑓, (𝑇 + 𝑆)𝑔) − (𝑓, 𝑆𝑔) = (𝑓, 𝑇𝑔), therefore 𝑓 ∈ dom(𝑇∗ ) and dom((𝑇 + 𝑆)∗ ) = dom(𝑇∗ + 𝑆∗ ) = dom(𝑇∗ ). Lemma 4.5. Let 𝑇 : dom(𝑇) 󳨀→ 𝐻2 be a densely defined linear operator and define 𝑉 : 𝐻1 × 𝐻2 󳨀→ 𝐻2 × 𝐻1 by 𝑉((𝑥, 𝑦)) = (𝑦, −𝑥). Then ∗

⊥

⊥

G(𝑇 ) = [𝑉(G(𝑇))] = 𝑉(G(𝑇) );

in particular 𝑇∗ is always closed. Proof. (𝑦, 𝑧) ∈ G(𝑇∗ ) ⇔ (𝑇𝑥, 𝑦)2 = (𝑥, 𝑧)1 for each 𝑥 ∈ dom(𝑇) ⇔ ((𝑥, 𝑇𝑥), (−𝑧, 𝑦)) = 0 for each 𝑥 ∈ dom(𝑇) ⇔ 𝑉−1 ((𝑦, 𝑧)) = (−𝑧, 𝑦) ∈ G(𝑇)⊥ . Hence G(𝑇∗ ) = 𝑉(G(𝑇)⊥ ) and since 𝑉 is unitary we have 𝑉∗ = 𝑉−1 and [𝑉(G(𝑇))]⊥ = 𝑉(G(𝑇)⊥ ).

52 | 4 The 𝜕-complex Lemma 4.6. Let 𝑇 : dom(𝑇) 󳨀→ 𝐻2 be a densely defined, closed linear operator. Then 𝐻2 × 𝐻1 = 𝑉(G(𝑇)) ⊕ G(𝑇∗ ). Proof. G(𝑇) is closed, therefore, by Lemma 4.5: G(𝑇∗ )⊥ = 𝑉(G(𝑇)). Lemma 4.7. Let 𝑇 : dom(𝑇) 󳨀→ 𝐻2 be a densely defined, closed linear operator. Then dom(𝑇∗ ) is dense in 𝐻2 and 𝑇∗∗ = 𝑇. Proof. Let 𝑧⊥dom(𝑇∗ ). Hence (𝑧, 𝑦)2 = 0 for each 𝑦 ∈ dom(𝑇∗ ). We have 𝑉−1 : 𝐻2 × 𝐻1 󳨀→ 𝐻1 × 𝐻2 where 𝑉−1 ((𝑦, 𝑥)) = (−𝑥, 𝑦), and 𝑉−1 𝑉 = Id. Now, by Lemma 4.6, we have 𝐻1 × 𝐻2 ≅ 𝑉−1 (𝐻2 × 𝐻1 ) = 𝑉−1 (𝑉(G(𝑇)) ⊕ G(𝑇∗ )) ≅ G(𝑇) ⊕ 𝑉−1 (G(𝑇∗ )). Hence (𝑧, 𝑦)2 = 0 ⇔ ((0, 𝑧), (−𝑇∗ 𝑦, 𝑦)) = 0 for each 𝑦 ∈ dom(𝑇∗ ). This implies (0, 𝑧) ∈ G(𝑇) and therefore 𝑧 = 𝑇(0) = 0, which means that dom(𝑇∗ ) is dense in 𝐻2 . Since 𝑇 and 𝑇∗ are densely defined and closed we have by Lemma 4.5 G(𝑇) = G(𝑇)

⊥⊥

= [𝑉−1 G(𝑇∗ )]⊥ = G(𝑇∗∗ ),

where −𝑉−1 corresponds to 𝑉 in considering operators from 𝐻2 to 𝐻1 . Lemma 4.8. Let 𝑇 : dom(𝑇) 󳨀→ 𝐻2 be a densely defined linear operator. Then ker𝑇∗ = (im𝑇)⊥ , which means that ker𝑇∗ is closed. Proof. Let 𝑣 ∈ ker𝑇∗ and 𝑦 ∈ im𝑇, which means that there exists 𝑢 ∈ dom(𝑇) such that 𝑇𝑢 = 𝑦. Hence (𝑣, 𝑦)2 = (𝑣, 𝑇𝑢)2 = (𝑇∗ 𝑣, 𝑢)1 = 0, and ker𝑇∗ ⊆ (im𝑇)⊥ . And if 𝑦 ∈ (im𝑇)⊥ , then (𝑦, 𝑇𝑢)2 = 0 for each 𝑢 ∈ dom(𝑇), which implies that 𝑦 ∈ dom(𝑇∗ ) and (𝑦, 𝑇𝑢)2 = (𝑇∗ 𝑦, 𝑢)1 for each 𝑢 ∈ dom(𝑇). Since dom(𝑇) is dense in 𝐻1 we obtain 𝑇∗ 𝑦 = 0 and (im𝑇)⊥ ⊆ ker𝑇∗ . Lemma 4.9. Let 𝑇 : dom(𝑇) 󳨀→ 𝐻2 be a densely defined, closed linear operator. Then ker𝑇 is a closed linear subspace of 𝐻1 . Proof. We use Lemma 4.8 for 𝑇∗ and get ker𝑇∗∗ = (im𝑇∗ )⊥ . Since, by Lemma 4.7, 𝑇∗∗ = 𝑇 we obtain ker𝑇 = (im𝑇∗ )⊥ and that ker𝑇 is a closed linear subspace of 𝐻1 . For our applications to the 𝜕-equation it will be important to know whether the differential operators involved have closed range or are even surjective. In the following propositions we will explain the functional analysis background and show how inequalities correspond to the closed range property. First we collect some corresponding results about bounded operators and begin with the Baire category theorem and the open-mapping theorem.

4.1 Unbounded operators on Hilbert spaces

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53

Theorem 4.10. Let 𝑋 be a complete metric space. (i) If {𝑈𝑛 }𝑛 is a sequence of open dense subsets of 𝑋, then ⋂∞ 𝑛=1 𝑈𝑛 is dense in 𝑋. (ii) 𝑋 is not a countable union of nowhere dense sets, i.e. of sets 𝐸𝑛 such that 𝐸𝑛 has empty interior. Proof. (i) Let 𝑊 be a nonempty open set in 𝑋. We have to show that ∞

( ⋂ 𝑈𝑛 ) ∩ 𝑊 ≠ 0. 𝑛=1

Since 𝑈1 ∩𝑊 is open and nonempty, it contains a ball 𝐵(𝑟1 , 𝑥1 ), and we can assume that 0 < 𝑟1 < 1. We choose sequences (𝑥𝑛 )𝑛 in 𝑋 and (𝑟𝑛 )𝑛 in ℝ+ inductively as follows: having chosen 𝑥𝑗 and 𝑟𝑗 for 𝑗 < 𝑛, we observe that 𝑈𝑛 ∩ 𝐵(𝑟𝑛−1 , 𝑥𝑛−1 ) is open and nonempty, so we can choose 𝑥𝑛 and 𝑟𝑛 such that 0 < 𝑟𝑛 < 2−𝑛 and 𝐵(𝑟𝑛 , 𝑥𝑛 ) ⊂ 𝑈𝑛 ∩ 𝐵(𝑟𝑛−1 , 𝑥𝑛−1 ). Then, for 𝑛, 𝑚 ≥ 𝑁, we see that 𝑥𝑛 , 𝑥𝑚 ∈ 𝐵(𝑟𝑁 , 𝑥𝑁 ), and since 𝑟𝑛 → 0, the sequence (𝑥𝑛 )𝑛 is a Cauchy sequence. As 𝑋 is complete, the limit 𝑥 = lim𝑛→∞ 𝑥𝑛 exists. Since 𝑥𝑛 ∈ 𝐵(𝑟𝑁 , 𝑥𝑁 ) for all 𝑛 ≥ 𝑁 we have 𝑥 ∈ 𝐵(𝑟𝑁 , 𝑥𝑁 ) ⊂ 𝑈𝑁 ∩ 𝐵(𝑟1 , 𝑥1 ) ⊂ 𝑈𝑁 ∩ 𝑊, for all 𝑁, which proves (i). (ii) If {𝐸𝑛 }𝑛 is a sequence of nowhere dense sets, then {(𝐸𝑛 )𝑐 }𝑛 is a sequence of open 𝑐 dense sets. Since, by (i), ⋂∞ 𝑛=1 (𝐸𝑛 ) ≠ 0, we have ∞

∞

⋃ 𝐸𝑛 ⊂ ⋃ 𝐸𝑛 ≠ 𝑋.

𝑛=1

𝑛=1

A set 𝐸 ⊂ 𝑋 is called meager (or of the first category), if it is a countable union of nowhere dense sets, otherwise 𝐸 is of the second category. Now we apply the Baire category theorem to linear maps between Banach spaces, to prove the open-mapping theorem and the closed graph theorem. Theorem 4.11. Let 𝑋 and 𝑌 be Banach spaces and 𝑇 : 𝑋 󳨀→ 𝑌 be a bounded linear operator which is surjective. Then 𝑇 is open. Proof. Let 𝐵𝑟 denote the open ball of radius 𝑟 and center 0 in 𝑋. We have to show that 𝑇(𝐵𝑟 ) contains an open ball about 0 in 𝑌. Since 𝑋 = ⋃∞ 𝑛=1 𝐵𝑛 and 𝑇 is surjective, we have 𝑌 = ⋃∞ 𝑇(𝐵 ). As 𝑌 is complete and the map 𝑦 󳨃→ 𝑛𝑦 is a homeomorphism 𝑛 𝑛=1 of 𝑌 which maps 𝑇(𝐵1 ) to 𝑇(𝐵𝑛 ), we can apply Theorem 4.10 and obtain that 𝑇(𝐵1 ) cannot be nowhere dense in 𝑌. Hence there exists 𝑦0 ∈ 𝑌 and 𝑟 > 0 such that the ball 𝐵(4𝑟, 𝑦0 ) = {𝑦 ∈ 𝑌 : ‖𝑦 − 𝑦0 ‖ < 4𝑟} is contained in 𝑇(𝐵1 ). Pick 𝑦1 = 𝑇𝑥1 ∈ 𝑇(𝐵1 ) such that ‖𝑦1 − 𝑦0 ‖ < 2𝑟 : then 𝐵(2𝑟, 𝑦1 ) ⊂ 𝑇(𝐵1 ), so if ‖𝑦‖ < 2𝑟, we have 𝑦 = 𝑇𝑥1 + (𝑦 − 𝑦1 ) ∈ 𝑇(𝑥1 + 𝐵1 ) ⊂ 𝑇(𝐵2 ). Dividing both sides by 2, we obtain that there exists 𝑟 > 0 such that if ‖𝑦‖ < 𝑟 then 𝑦 ∈ 𝑇(𝐵1 ). If we could replace 𝑇(𝐵1 ) by 𝑇(𝐵1 ) the proof would be complete.

54 | 4 The 𝜕-complex Dilation invariance implies that if ‖𝑦‖ < 𝑟2−𝑛 then 𝑦 ∈ 𝑇(𝐵2−𝑛 ). Suppose that ‖𝑦‖ < 𝑟/2; we can find 𝑥1 ∈ 𝐵1/2 such that ‖𝑦 − 𝑇𝑥1 ‖ < 𝑟/4, and proceeding inductively we can find 𝑥𝑛 ∈ 𝐵2−𝑛 such that 𝑛

‖𝑦 − ∑ 𝑇𝑥𝑗 ‖ < 𝑟2−𝑛−1 .

(4.1)

𝑗=1

∞ −𝑛 We have ∑∞ = 1 and, since 𝑋 is complete, the absolutely convergent 𝑛=1 ‖𝑥𝑛 ‖ < ∑𝑛=1 2 ∞ series ∑𝑛=1 𝑥𝑛 is also convergent say to 𝑥 = ∑∞ 𝑛=1 𝑥𝑛 ∈ 𝑋 and, by (4.1), we have 𝑦 = 𝑇𝑥. This means that 𝑇(𝐵1 ) contains all 𝑦 with ‖𝑦‖ < 𝑟/2.

Corollary 4.12. Let 𝑇 : 𝑋 󳨀→ 𝑌 be a bijective bounded linear operator between Banach spaces. Then 𝑇−1 is also bounded. Proof. By Theorem 4.11, 𝑇 is open, hence 𝑇−1 is bounded. Now it is easy to prove the closed graph theorem. Theorem 4.13. Let 𝑋 and 𝑌 be Banach spaces. Let 𝑇 : 𝑋 󳨀→ 𝑌 be a closed operator with dom(𝑇) = 𝑋. Then 𝑇 is bounded. Proof. Let 𝜋1 and 𝜋2 be the projections of G(𝑇) onto 𝑋 and 𝑌. They are both continuous. 𝑋 × 𝑌 is also complete, G(𝑇) is a closed subspace of 𝑋 × 𝑌 and therefore also complete. 𝜋1 is a bijection from G(𝑇) to 𝑋. By Corollary 4.12, 𝜋1−1 is bounded. But 𝑇 = 𝜋2 ∘ 𝜋1−1 , therefore 𝑇 is bounded. Before we proceed in the theory for unbounded operators on Hilbert spaces we still prove the uniform boundedness principle again as an application of the Baire category theorem. Theorem 4.14. Suppose 𝑋 and 𝑌 are normed spaces and A is a subset of L(𝑋, 𝑌). (i) If sup𝑇∈A ‖𝑇𝑥‖ < ∞ for all 𝑥 in a nonmeager subset of 𝑋, then sup ‖𝑇‖ < ∞.

𝑇∈A

(ii) If 𝑋 is a Banach space and sup𝑇∈A ‖𝑇𝑥‖ < ∞ for all 𝑥 ∈ 𝑋, then sup ‖𝑇‖ < ∞.

𝑇∈A

Proof. Let 𝐸𝑛 = {𝑥 ∈ 𝑋 : sup ‖𝑇𝑥‖ ≤ 𝑛} = ⋂ {𝑥 ∈ 𝑋 : ‖𝑇𝑥‖ ≤ 𝑛}. 𝑇∈A

𝑇∈A

Then the sets 𝐸𝑛 are closed, so under the hypothesis of (i) some 𝐸𝑛 must contain a nontrivial closed ball 𝐵(𝑟, 𝑥0 ). Then 𝐸2𝑛 ⊃ 𝐵(𝑟, 0), for if ‖𝑥‖ ≤ 𝑟 then 𝑥 − 𝑥0 ∈ 𝐸𝑛 and ‖𝑇𝑥‖ ≤ ‖𝑇(𝑥 − 𝑥0 )‖ + ‖𝑇𝑥0 ‖ ≤ 2𝑛.

4.1 Unbounded operators on Hilbert spaces

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55

In other words, ‖𝑇𝑥‖ ≤ 2𝑛 whenever 𝑇 ∈ A and ‖𝑥‖ ≤ 𝑟, so sup ‖𝑇‖ ≤ 2𝑛/𝑟.

𝑇∈A

(ii) follows from (i) by Theorem 4.10. The following characterization of compactness uses the uniform boundedness principle and the concept of weak convergence. Definition 4.15. A sequence (𝑥𝑘 )𝑘 in a Hilbert space 𝐻 is a weak null-sequence, if (𝑥𝑘 , 𝑥) → 0 for each 𝑥 ∈ 𝐻. A sequence (𝑥𝑘 )𝑘 converges weakly to 𝑥0 , if (𝑥𝑘 − 𝑥0 )𝑘 is a weak null sequence. Remark 4.16. A weakly convergent sequence (𝑥𝑘 )𝑘 in a Hilbert space is always bounded: we have sup |(𝑥𝑘 − 𝑥0 , 𝑥)| < ∞, 𝑘

for all 𝑥 ∈ 𝐻, then, by Theorem 4.14, sup ‖𝑥𝑘 − 𝑥0 ‖ = sup sup |(𝑥𝑘 − 𝑥0 , 𝑥)| < ∞ 𝑘

𝑘 ‖𝑥‖≤1

and therefore ‖𝑥𝑘 ‖ ≤ ‖𝑥𝑘 − 𝑥0 ‖ + ‖𝑥0 ‖ < ∞, for all 𝑘 ∈ ℕ. In the same way we can show that each weak Cauchy sequence is bounded. If 𝐴 ∈ L(𝐻1 , 𝐻2 ) and (𝑥𝑘 )𝑘 is a weakly convergent sequence in 𝐻1 , then (𝐴𝑥𝑘 )𝑘 converges weakly in 𝐻2 , which follows from (𝐴𝑥𝑘 − 𝐴𝑥0 , 𝑦)2 = (𝑥𝑘 − 𝑥0 , 𝐴∗ 𝑦)1 , where 𝑦 ∈ 𝐻2 . Proposition 4.17. Let 𝐴 ∈ L(𝐻1 , 𝐻2 ) be a bounded operator between Hilbert spaces. 𝐴 is compact if and only if (𝐴𝑥𝑘 )𝑘 converges to 0 in 𝐻2 for each weak null sequence (𝑥𝑘 )𝑘 in 𝐻1 . Proof. Let 𝐴 be compact and (𝑥𝑘 )𝑘 be a weak null sequence in 𝐻1 . Then (𝐴𝑥𝑘 )𝑘 is a weak null sequence in 𝐻2 . Suppose that (‖𝐴𝑥𝑘 ‖)𝑘 does not converge to 0. Then there exists 𝜖 > 0 and a subsequence (𝑦𝑘 )𝑘 of (𝑥𝑘 )𝑘 such that ‖𝐴𝑦𝑘 ‖ ≥ 𝜖 for each 𝑘 ∈ ℕ. By Remark 4.16, the sequence (‖𝑦𝑘 ‖)𝑘 is bounded. As 𝐴 is compact, there exists a subsequence (𝑧𝑘 )𝑘 of (𝑦𝑘 )𝑘 such that (𝐴𝑧𝑘 )𝑘 is convergent in 𝐻2 . Since (𝐴𝑧𝑘 )𝑘 converges weakly to 0, we would have ‖𝐴𝑧𝑘 ‖ = sup |(𝐴𝑧𝑘 , 𝑦)| → 0, ‖𝑦‖≤1

contradicting ‖𝐴𝑦𝑘 ‖ ≥ 𝜖. For the opposite direction we need a number of prerequisites: first we claim the following assertion: if (𝑥𝑘 )𝑘 is a bounded sequence in a Hilbert space 𝐻 and if ((𝑥𝑘 , 𝑦))𝑘

56 | 4 The 𝜕-complex is a Cauchy sequence in ℂ for each 𝑦 in a dense subset 𝑀 of 𝐻, then (𝑥𝑘 )𝑘 is a weak Cauchy sequence in 𝐻. To show this, let 𝑥 ∈ 𝐻 be an arbitrary element and choose 𝑦 ∈ 𝑀 such that ‖𝑥 − 𝑦‖ < 𝜖. Then |(𝑥𝑘 − 𝑥𝑚 , 𝑥)| ≤ |(𝑥𝑘 , 𝑥 − 𝑦)| + |(𝑥𝑘 − 𝑥𝑚 , 𝑦)| + |(𝑥𝑚 , 𝑦 − 𝑥)|. Since (𝑥𝑘 )𝑘 is a bounded sequence, we get that ((𝑥𝑘 , 𝑥))𝑘 is a Cauchy sequence in ℂ for each 𝑥 ∈ 𝐻. Next we claim that each weak Cauchy sequence in 𝐻 is also weakly convergent: a weak Cauchy sequence is bounded, therefore we have |(𝑥𝑘 , 𝑥)| ≤ 𝐶‖𝑥‖, for some constant 𝐶 > 0 and for each 𝑘 ∈ ℕ. Hence 𝐹(𝑥) := lim (𝑥, 𝑥𝑘 ) 𝑘→∞

defines a continuous linear functional, for which there exists 𝑥0 ∈ 𝐻 such that (𝑥, 𝑥0 ) = lim (𝑥, 𝑥𝑘 ), 𝑘→∞

which means that (𝑥𝑘 )𝑘 converges weakly to 𝑥0 . Next we show that each bounded sequence (𝑥𝑘 )𝑘 contains a weakly convergent subsequence: the sequence ((𝑥𝑘 , 𝑥))𝑘 is bounded in ℂ, hence we can find subsequences (𝑥𝑘(𝑚) )𝑘 of (𝑥𝑘(𝑚−1) )𝑘 for 𝑚 ∈ ℕ with (𝑥𝑘 )𝑘 = (𝑥𝑘(0) )𝑘 such that ((𝑥𝑘(𝑚) , 𝑥𝑚 ))𝑘 converges. The diagonal sequence (𝑥𝑘(𝑘) )𝑘 has the property that the sequence ((𝑥𝑘(𝑘) , 𝑥𝑚 ))𝑘 is convergent for each 𝑚 ∈ ℕ. This implies that ((𝑥𝑘(𝑘) , 𝑥))𝑘 converges for each 𝑥 in the linear span 𝐿 of the elements {𝑥ℓ : ℓ ∈ ℕ}. Since the sequence (𝑥𝑘(𝑘) )𝑘 is bounded, we have from above that ((𝑥𝑘(𝑘) , 𝑥))𝑘 is convergent for each 𝑥 ∈ 𝐿. It is clear that ((𝑥𝑘(𝑘) , 𝑥)) = 0 for each 𝑥 ∈ 𝐿⊥ , and therefore we get that (𝑥𝑘(𝑘) )𝑘 is a weak Cauchy sequence, which converges weakly, by our prerequisites. Now we can finish the proof of the opposite direction: let (𝑥𝑘 )𝑘 be a bounded sequence in 𝐻1 . We know that it contains a weakly convergent subsequence 𝑥𝑘ℓ → 𝑥0 , hence the sequence (𝑥𝑘ℓ − 𝑥0 )ℓ is a weak null sequence. By assumption we obtain that 𝐴𝑥𝑘ℓ → 𝐴𝑥0 in 𝐻2 . So 𝐴 is a compact operator. Now we continue to derive further important properties of unbounded operators on Hilbert spaces which will later be used for the 𝜕-operator. Lemma 4.18. Let 𝑇 : 𝐻1 󳨀→ 𝐻2 be a bounded linear operator. 𝑇(𝐻1 ) is closed if and only if 𝑇|(ker𝑇)⊥ is bounded from below, i.e. ‖𝑇𝑓‖ ≥ 𝐶‖𝑓‖ , ∀𝑓 ∈ (ker𝑇)⊥ . Proof. If 𝑇(𝐻1 ) is closed, then the mapping 𝑇 : (ker𝑇)⊥ 󳨀→ 𝑇(𝐻1 ) is bijective and continuous and, by the open-mapping theorem 4.11, also open. This implies the desired inequality.

4.1 Unbounded operators on Hilbert spaces

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57

To prove the other direction, let (𝑓𝑛 )𝑛 be a sequence in 𝐻1 with 𝑇𝑓𝑛 → 𝑦 in 𝐻2 . We have to show that there exists ℎ ∈ 𝐻1 with 𝑇ℎ = 𝑦. Decompose 𝑓𝑛 = 𝑔𝑛 + ℎ𝑛 , where 𝑔𝑛 ∈ ker𝑇 and ℎ𝑛 ∈ (ker𝑇)⊥ . By assumption we have ‖ℎ𝑛 − ℎ𝑚 ‖ ≤ 𝐶‖𝑇ℎ𝑛 − 𝑇ℎ𝑚 ‖ = 𝐶‖𝑇𝑓𝑛 − 𝑇𝑓𝑚 ‖ < 𝜖, for all sufficiently large 𝑛 and 𝑚. Hence (ℎ𝑛 )𝑛 is a Cauchy sequence. Let ℎ = lim𝑛→∞ ℎ𝑛 . Then we have ‖𝑇𝑓𝑛 − 𝑇ℎ‖ = ‖𝑇ℎ𝑛 − 𝑇ℎ‖ ≤ ‖𝑇‖ ‖ℎ𝑛 − ℎ‖, and therefore 𝑦 = lim 𝑇𝑓𝑛 = 𝑇ℎ. 𝑛→∞

Lemma 4.19. Let 𝑇 be as before. 𝑇(𝐻1 ) is closed if and only if 𝑇∗ (𝐻2 ) is closed. Proof. Since 𝑇∗∗ = 𝑇, it suffices to show one direction. We will show that the closedness of 𝑇(𝐻1 ) implies that (ker𝑇)⊥ = im𝑇∗ ; since (ker𝑇)⊥ is closed, we will be finished. Let 𝑥 ∈ im𝑇∗ . Then there exists 𝑦 ∈ 𝐻2 with 𝑥 = 𝑇∗ 𝑦. Now we get for 𝑥󸀠 ∈ ker𝑇 that (𝑥, 𝑥󸀠 ) = (𝑇∗ 𝑦, 𝑥󸀠 ) = (𝑦, 𝑇𝑥󸀠 ) = 0, hence im𝑇∗ ⊆ (ker𝑇)⊥ . For 𝑥󸀠 ∈ (ker𝑇)⊥ we define a linear functional 𝜆(𝑇𝑥) = (𝑥, 𝑥󸀠 ) on the closed subspace 𝑇(𝐻1 ) of 𝐻2 . We remark that 𝜆 is well defined, since 𝑇𝑥 = 𝑇𝑥̃ implies that 𝑥 − 𝑥̃ ∈ ker𝑇, hence (𝑥 − 𝑥,̃ 𝑥󸀠 ) = 0 and (𝑥, 𝑥󸀠 ) = (𝑥,̃ 𝑥󸀠 ). The operator 𝑇 : 𝐻1 󳨀→ 𝑇(𝐻1 ) is continuous and surjective. Since 𝑇(𝐻1 ) is closed, the open-mapping theorem implies ‖𝑣‖ ≤ 𝐶‖𝑇𝑣‖, for all 𝑣 ∈ (ker𝑇)⊥ where 𝐶 > 0 is a constant. Set 𝑦 = 𝑇𝑥 and write 𝑥 = 𝑢 + 𝑣, where 𝑢 ∈ ker𝑇 and 𝑣 ∈ (ker𝑇)⊥ . Then we obtain |𝜆(𝑦)| = |(𝑥, 𝑥󸀠 )| = |(𝑣, 𝑥󸀠 )| ≤ ‖𝑣‖‖𝑥󸀠 ‖ ≤ 𝐶‖𝑇𝑣‖‖𝑥󸀠 ‖ = 𝐶‖𝑇𝑥‖‖𝑥󸀠 ‖ = 𝐶‖𝑦‖‖𝑥󸀠 ‖. Hence 𝜆 is continuous on im𝑇. By Theorem 1.6, there exists a uniquely determined element 𝑧 ∈ im𝑇 with 𝜆(𝑦) = (𝑦, 𝑧)2 = (𝑥, 𝑥󸀠 )1 . This implies (𝑦, 𝑧)2 = (𝑇𝑥, 𝑧)2 = (𝑥, 𝑇∗ 𝑧)1 = (𝑥, 𝑥󸀠 )1 , for all 𝑥 ∈ 𝐻1 , and hence 𝑥󸀠 = 𝑇∗ 𝑧 ∈ im𝑇∗ .

58 | 4 The 𝜕-complex Lemma 4.20. Let 𝑇 : 𝐻1 󳨀→ 𝐻2 be a densely defined closed operator. im𝑇 is closed in 𝐻2 if and only if 𝑇|dom(𝑇)∩(ker𝑇)⊥ is bounded from below, i.e. ‖𝑇𝑓‖ ≥ 𝐶‖𝑓‖ , ∀𝑓 ∈ dom(𝑇) ∩ (ker𝑇)⊥ . ̃ 𝑇𝑓)) = 𝑇𝑓 and get a bounded Proof. On the graph G(𝑇) we define the operator 𝑇((𝑓, linear operator 𝑇̃ : G(𝑇) 󳨀→ 𝐻2 , since ̃ ‖𝑇((𝑓, 𝑇𝑓))‖ = ‖𝑇𝑓‖ ≤ (‖𝑓‖2 + ‖𝑇𝑓‖2 )1/2 = ‖(𝑓, 𝑇𝑓)‖; and im𝑇̃ = im𝑇. 󵄨 By Lemma 4.18, im𝑇 is closed if and only if 𝑇̃ 󵄨󵄨󵄨(ker𝑇)̃ ⊥ is bounded from below. 󵄨 ̃ We have ker𝑇 = ker𝑇 ⊕ {0}, and it remains to show that 𝑇̃ 󵄨󵄨󵄨(ker𝑇)̃ ⊥ is bounded from 󵄨󵄨 below, if and only if 𝑇󵄨󵄨dom(𝑇)∩(ker𝑇)⊥ is bounded from below. But this follows from ̃ ‖𝑇((𝑓, 𝑇𝑓))‖ = ‖𝑇𝑓‖ ≥ 𝐶(‖𝑓‖2 + ‖𝑇𝑓‖2 )1/2 , and hence, for 0 < 𝐶 < 1, ‖𝑇𝑓‖2 ≥

𝐶2 ‖𝑓‖2 . 1 − 𝐶2

Lemma 4.21. Let 𝑃, 𝑄 : 𝐻 󳨀→ 𝐻 be orthogonal projections on the Hilbert space 𝐻. Then the following assertions are equivalent (i) im(𝑃𝑄) is closed; (ii) im(𝑄𝑃) is closed; (iii) im(𝐼 − 𝑃)(𝐼 − 𝑄) is closed; (iv) 𝑃(𝐻) + (𝐼 − 𝑄)(𝐻) is closed. Proof. (i) and (ii) are equivalent by Lemma 4.19 and the fact that 𝑄𝑃 = 𝑄∗ 𝑃∗ = (𝑃𝑄)∗ . Suppose (ii) holds and let (𝑓𝑛 )𝑛 and (𝑔𝑛 )𝑛 be sequences in 𝐻 with 𝑃𝑓𝑛 + (𝐼 − 𝑄)𝑔𝑛 → ℎ. Then 𝑄(𝑃𝑓𝑛 + (𝐼 − 𝑄)𝑔𝑛 ) = 𝑄𝑃𝑓𝑛 → 𝑄ℎ. By assumption, im(𝑄𝑃) is closed, hence there exists 𝑓 ∈ 𝐻 with 𝑄𝑃𝑓 = 𝑄ℎ; it follows that 𝑄ℎ = 𝑃𝑓 − (𝐼 − 𝑄)(𝑃𝑓) and ℎ = 𝑄ℎ + (𝐼 − 𝑄)ℎ = 𝑃𝑓 − (𝐼 − 𝑄)(𝑃𝑓) + (𝐼 − 𝑄)ℎ = 𝑃𝑓 + (𝐼 − 𝑄)(ℎ − 𝑃𝑓) ∈ 𝑃(𝐻) + (𝐼 − 𝑄)(𝐻), which yields (iv). If (iv) holds and (𝑓𝑛 )𝑛 is a sequence in 𝐻 with 𝑄𝑃𝑓𝑛 → ℎ, we get 𝑄𝑃𝑓𝑛 = 𝑃𝑓𝑛 − (𝐼 − 𝑄)𝑃𝑓𝑛 ∈ 𝑃(𝐻) + (𝐼 − 𝑄)(𝐻).

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59

Hence there exist 𝑓, 𝑔 ∈ 𝐻 with ℎ = 𝑃𝑓 + (𝐼 − 𝑄)𝑔; and it follows that 𝑄ℎ = 𝑄( lim 𝑄𝑃𝑓𝑛 ) = lim 𝑄2 𝑃𝑓𝑛 = ℎ, 𝑛→∞

𝑛→∞

and ℎ = 𝑃𝑓 + (𝐼 − 𝑄)𝑔 = 𝑄ℎ = 𝑄𝑃𝑓 ∈ im(𝑄𝑃), therefore (ii) holds. Finally, replace 𝑃 by 𝐼 − 𝑃 and 𝑄 by 𝐼 − 𝑄. Then, using the assertions proved so far, we obtain the equivalence im(𝐼 − 𝑃)(𝐼 − 𝑄) closed ⇔ (𝐼 − 𝑃)(𝐻) + 𝑄(𝐻) closed, which proves the remaining assertions. At this point, we are able to prove Lemma 4.19 for densely defined closed operators. Proposition 4.22. Let 𝑇 : 𝐻1 󳨀→ 𝐻2 be a densely defined closed operator. im𝑇 is closed if and only if im𝑇∗ is closed. Proof. Let 𝑃 : 𝐻1 × 𝐻2 󳨀→ G(𝑇) be the orthogonal projection of 𝐻1 × 𝐻2 on the closed subspace G(𝑇) of 𝐻1 × 𝐻2 , and let 𝑄 : 𝐻1 × 𝐻2 󳨀→ {0} × 𝐻2 be the canonical orthogonal projection. Then im𝑇 ≅ im𝑄𝑃 and since 𝐼 − 𝑄 : 𝐻1 × 𝐻2 󳨀→ 𝐻1 × {0} and 𝐼 − 𝑃 : 𝐻1 × 𝐻2 󳨀→ G(𝑇)⊥ = 𝑉(G(𝑇∗ )) ≅ G(𝑇∗ ) we obtain the desired result from Lemma 4.21. Proposition 4.23. Let 𝑇 : 𝐻1 󳨀→ 𝐻2 be a densely defined closed operator and 𝐺 a 󵄨 closed subspace of 𝐻2 with 𝐺 ⊇ im𝑇. Suppose that 𝑇∗ 󵄨󵄨󵄨dom(𝑇∗ )∩𝐺 is bounded from below, i.e. ‖𝑓‖ ≤ 𝐶‖𝑇∗ 𝑓‖ for all 𝑓 ∈ dom(𝑇∗ ) ∩ 𝐺, where 𝐶 > 0 is a constant. Then 𝐺 = im𝑇. Proof. We have ker𝑇∗ = (im𝑇)⊥ . Since im𝑇 ⊆ 𝐺, it follows that ker𝑇∗ ⊇ 𝐺⊥ . If 𝐺⊥ is a proper subspace of ker𝑇∗ , then 𝐺 ∩ ker𝑇∗ ≠ {0}, which is a contradiction to the 󵄨 assumption that 𝑇∗ 󵄨󵄨󵄨dom(𝑇∗ )∩𝐺 is bounded from below. Hence ker𝑇∗ = 𝐺⊥ and 𝐺 = 𝐺⊥⊥ = (ker𝑇∗ )⊥ = im𝑇⊥⊥ = (im𝑇). In addition we have

󵄨 󵄨 𝑇∗ 󵄨󵄨󵄨dom(𝑇∗ )∩𝐺 = 𝑇∗ 󵄨󵄨󵄨dom(𝑇∗ )∩(ker𝑇∗ )⊥

and, by Lemma 4.20 we obtain, that im𝑇∗ is closed. By Proposition 4.22, im𝑇 is also closed and we get that 𝐺 = im𝑇. Remark 4.24. The last proposition also holds in the other direction: if 𝑇 : 𝐻1 󳨀→ 𝐻2 is a densely defined closed operator and 𝐺 is a closed subspace of 𝐻2 with 𝐺 = im𝑇, 󵄨 then 𝑇∗ 󵄨󵄨󵄨dom(𝑇∗ )∩𝐺 is bounded from below. Since in this case 𝐺 = im𝑇, we have that im𝑇 is closed and hence, by Lemma 4.22, im𝑇∗ is also closed. Therefore, Lemma 4.20 and 󵄨 the fact that 𝐺 = (ker𝑇∗ )⊥ implies that 𝑇∗ 󵄨󵄨󵄨dom(𝑇∗ )∩𝐺 is bounded from below.

60 | 4 The 𝜕-complex Proposition 4.25. Let 𝑇 : 𝐻1 󳨀→ 𝐻2 be a densely defined closed operator and let 𝐺 󵄨 be a closed subspace of 𝐻2 with 𝐺 ⊇ im𝑇. Suppose that 𝑇∗ 󵄨󵄨󵄨dom(𝑇∗ )∩𝐺 is bounded from below. Then for each 𝑣 ∈ 𝐻1 with 𝑣 ⊥ ker𝑇 there exists 𝑓 ∈ dom(𝑇∗ ) ∩ 𝐺 with 𝑇∗ 𝑓 = 𝑣 and ‖𝑓‖ ≤ 𝐶‖𝑣‖. Proof. We have ker𝑇 = (im𝑇∗ )⊥ , hence 𝑣 ∈ (ker𝑇)⊥ = im𝑇∗ . In addition 𝐺⊥ ⊆ (im𝑇)⊥ = ker𝑇∗ and therefore 󵄨 im𝑇∗ 󵄨󵄨󵄨dom(𝑇∗ )∩𝐺 = im𝑇∗ , this means that im𝑇∗ is closed and that for 𝑣 ∈ (ker𝑇)⊥ = im𝑇∗ there exists 𝑓 ∈ dom(𝑇∗ ) ∩ 𝐺 with 𝑇∗ 𝑓 = 𝑣. The desired norm-inequality follows from the assumption 󵄨 that 𝑇∗ 󵄨󵄨󵄨dom(𝑇∗ )∩𝐺 is bounded from below. In the following we introduce the fundamental concept of an unbounded self-adjoint operator, which will be crucial for both spectral theory and its applications to complex analysis. Definition 4.26. Let 𝑇 : dom(𝑇) 󳨀→ 𝐻 be a densely defined linear operator. 𝑇 is symmetric if (𝑇𝑥, 𝑦) = (𝑥, 𝑇𝑦) for all 𝑥, 𝑦 ∈ dom(𝑇). We say that 𝑇 is self-adjoint if 𝑇 is symmetric and dom(𝑇) = dom(𝑇∗ ). This is equivalent to requiring that 𝑇 = 𝑇∗ and implies that 𝑇 is closed. We say that 𝑇 is essentially self-adjoint if it is symmetric and its closure 𝑇 is self-adjoint. Remark 4.27. (a) If 𝑇 is a symmetric operator, there are two natural closed extensions. We have dom(𝑇) ⊆ dom(𝑇∗ ) and 𝑇∗ = 𝑇 on dom(𝑇). Since 𝑇∗ is closed (Lemma 4.5), 𝑇∗ is a closed extension of 𝑇, it is the maximal closed extension. 𝑇 is also closable, by Lemma 4.2, therefore 𝑇 exists, it is the minimal closed extension. (b) If 𝑇 is essentially self-adjoint, then its self-adjoint extension is unique. To prove this, let 𝑆 be a self-adjoint extension of 𝑇. Then 𝑆 is closed and, being an extension of 𝑇, it is also an extension of its smallest extension 𝑇. Hence 𝑇 ⊂ 𝑆 = 𝑆∗ ⊂ (𝑇)∗ = 𝑇, and 𝑆 = 𝑇. Lemma 4.28. Let 𝑇 be a densely defined, symmetric operator. (i) If dom(𝑇) = 𝐻, then 𝑇 is self-adjoint and 𝑇 is bounded. (ii) If 𝑇 is self-adjoint and injective, then im𝑇 is dense in 𝐻, and 𝑇−1 is self-adjoint. (iii) If im𝑇 is dense in 𝐻, then 𝑇 is injective. (iv) If im𝑇 = 𝐻, then 𝑇 is self-adjoint, and 𝑇−1 is bounded. Proof. (i) By assumption dom(𝑇) ⊆ dom(𝑇∗ ). If dom(𝑇) = 𝐻, it follows that 𝑇 is selfadjoint, therefore also closed (Lemma 4.5) and continuous by the closed graph theorem. (ii) Suppose 𝑦⊥im𝑇. Then 𝑥 󳨃→ (𝑇𝑥, 𝑦) = 0 is continuous on dom(𝑇), hence 𝑦 ∈ dom(𝑇∗ ) = dom(𝑇), and (𝑥, 𝑇𝑦) = (𝑇𝑥, 𝑦) = 0 for all 𝑥 ∈ dom(𝑇). Thus 𝑇𝑦 = 0 and

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61

since 𝑇 is assumed to be injective, it follows that 𝑦 = 0. This proves that im𝑇 is dense in 𝐻. 𝑇−1 is therefore densely defined, with dom(𝑇−1 ) = im𝑇, and (𝑇−1 )∗ exists. Now let 𝑈 : 𝐻 × 𝐻 󳨀→ 𝐻 × 𝐻 be defined by 𝑈((𝑥, 𝑦)) = (−𝑦, 𝑥). It easily follows that 𝑈2 = −𝐼 and 𝑈2 (𝑀) = 𝑀 for any subspace 𝑀 of 𝐻 × 𝐻, and we get G(𝑇−1 ) = 𝑈(G(−𝑇)) and 𝑈(G(𝑇−1 )) = G(−𝑇). Being self-adjoint, 𝑇 is closed; hence −𝑇 is closed and 𝑇−1 is closed. By Lemma 4.6 applied to 𝑇−1 and to −𝑇 we get the orthogonal decompositions 𝐻 × 𝐻 = 𝑈(G(𝑇−1 )) ⊕ G((𝑇−1 )∗ ) and 𝐻 × 𝐻 = 𝑈(G(−𝑇)) ⊕ G(−𝑇)) = G(𝑇−1 ) ⊕ 𝑈(G(𝑇−1 )). Consequently −1 ∗

−1

⊥

−1

G((𝑇 ) ) = [𝑈(G(𝑇 ))] = G(𝑇 ),

which shows that (𝑇−1 )∗ = 𝑇−1 . (iii) Suppose 𝑇𝑥 = 0. Then (𝑥, 𝑇𝑦) = (𝑇𝑥, 𝑦) = 0 for each 𝑦 ∈ dom(𝑇). Thus 𝑥⊥im(𝑇), and therefore 𝑥 = 0. (iv) Since im(𝑇) = 𝐻, (iii) implies that 𝑇 is injective, dom(𝑇−1 ) = 𝐻. If 𝑥, 𝑦 ∈ 𝐻, then 𝑥 = 𝑇𝑧 and 𝑦 = 𝑇𝑤, for some 𝑧 ∈ dom(𝑇) and 𝑤 ∈ dom(𝑇), so that (𝑇−1 𝑥, 𝑦) = (𝑧, 𝑇𝑤) = (𝑇𝑧, 𝑤) = (𝑥, 𝑇−1 𝑦). Hence 𝑇−1 is symmetric. (i) implies that 𝑇−1 is self-adjoint (and bounded), and now it follows from (ii) that 𝑇 = (𝑇−1 )−1 is also self-adjoint. Lemma 4.29. Let 𝑇 be a densely defined closed operator, dom(𝑇) ⊆ 𝐻1 and 𝑇 : dom(𝑇) 󳨀→ 𝐻2 . Then 𝐵 = (𝐼 + 𝑇∗ 𝑇)−1 and 𝐶 = 𝑇(𝐼 + 𝑇∗ 𝑇)−1 are everywhere defined and bounded, ‖𝐵‖ ≤ 1, ‖𝐶‖ ≤ 1; in addition 𝐵 is self-adjoint and positive. Proof. Let ℎ ∈ 𝐻1 be an arbitrary element and consider (ℎ, 0) ∈ 𝐻1 ×𝐻2 . From the proof of Lemma 4.7 we get 𝐻1 × 𝐻2 = G(𝑇) ⊕ 𝑉−1 (G(𝑇∗ )), (4.2) which implies that (ℎ, 0) can be written in a unique way as (ℎ, 0) = (𝑓, 𝑇𝑓) + (−𝑇∗ (−𝑔), −𝑔), for 𝑓 ∈ dom(𝑇) and 𝑔 ∈ dom(𝑇∗ ), which gives ℎ = 𝑓 + 𝑇∗ 𝑔 and 0 = 𝑇𝑓 − 𝑔. We set 𝐵ℎ := 𝑓 and 𝐶ℎ := 𝑔. In this way we get two linear operators 𝐵 and 𝐶 everywhere defined on 𝐻1 . The two equations from above can now be written as 𝐼 = 𝐵 + 𝑇∗ 𝐶, 0 = 𝑇𝐵 − 𝐶, which gives 𝐶 = 𝑇𝐵 and 𝐼 = 𝐵 + 𝑇∗ 𝑇𝐵 = (𝐼 + 𝑇∗ 𝑇)𝐵.

(4.3)

62 | 4 The 𝜕-complex The decomposition in (4.2) is orthogonal, therefore we obtain ‖ℎ‖2 = ‖(ℎ, 0)‖2 = ‖(𝑓, 𝑇𝑓)‖2 + ‖(𝑇∗ 𝑔, −𝑔)‖2 = ‖𝑓‖2 + ‖𝑇𝑓‖2 + ‖𝑇∗ 𝑔‖2 + ‖𝑔‖2 , and hence ‖𝐵ℎ‖2 + ‖𝐶ℎ‖2 = ‖𝑓‖2 + ‖𝑔‖2 ≤ ‖ℎ‖2 , which implies ‖𝐵‖ ≤ 1 and ‖𝐶‖ ≤ 1. For each 𝑢 ∈ dom(𝑇∗ 𝑇) we get ((𝐼 + 𝑇∗ 𝑇)𝑢, 𝑢) = (𝑢, 𝑢) + (𝑇𝑢, 𝑇𝑢) ≥ (𝑢, 𝑢) hence, if (𝐼 + 𝑇∗ 𝑇)𝑢 = 0 we get 𝑢 = 0. Therefore (𝐼 + 𝑇∗ 𝑇)−1 exists and (4.3) implies that (𝐼 + 𝑇∗ 𝑇)−1 is defined everywhere and 𝐵 = (𝐼 + 𝑇∗ 𝑇)−1 . Finally let 𝑢, 𝑣 ∈ 𝐻1 . Then (𝐵𝑢, 𝑣) = (𝐵𝑢, (𝐼 + 𝑇∗ 𝑇)𝐵𝑣) = (𝐵𝑢, 𝐵𝑣) + (𝐵𝑢, 𝑇∗ 𝑇𝐵𝑣) = (𝐵𝑢, 𝐵𝑣) + (𝑇∗ 𝑇𝐵𝑢, 𝐵𝑣) = ((𝐼 + 𝑇∗ 𝑇)𝐵𝑢, 𝐵𝑣) = (𝑢, 𝐵𝑣) and (𝐵𝑢, 𝑢) = (𝐵𝑢, (𝐼 + 𝑇∗ 𝑇)𝐵𝑢) = (𝐵𝑢, 𝐵𝑢) + (𝑇𝐵𝑢, 𝑇𝐵𝑢) ≥ 0, which proves the lemma. At this point we can describe the concept of the core of an operator, which will be very useful later for spectral analysis. Definition 4.30. Let 𝑇 be a closable operator with domain dom(𝑇). A subspace 𝐷 ⊂ dom(𝑇) is called a core of the operator 𝑇 if the closure of the restriction 𝑇 |𝐷 is an extension of 𝑇. Remark 4.31. If 𝑇 is a closed operator, then 𝑇 |𝐷 = 𝑇. Lemma 4.32. Let 𝑇 be a densely defined closed operator, dom(𝑇) ⊆ 𝐻1 and 𝑇 : dom(𝑇) 󳨀→ 𝐻2 . Then dom(𝑇∗ 𝑇) is a core of the operator 𝑇. Proof. We have to show that G(𝑇) = G(𝑇 |dom(𝑇∗ 𝑇) ). For this purpose we consider elements (𝑥, 𝑇𝑥) in the graph of 𝑇. We suppose that (𝑥, 𝑇𝑥) ⊥ (𝑦, 𝑇𝑦) for each 𝑦 ∈ dom(𝑇∗ 𝑇). Then (𝑥, (𝐼 + 𝑇∗ 𝑇)𝑦) = (𝑥, 𝑦) + (𝑇𝑥, 𝑇𝑦) = ((𝑥, 𝑇𝑥), (𝑦, 𝑇𝑦)) = 0, and as im(𝐼 + 𝑇∗ 𝑇) = 𝐻1 (Lemma 4.29) we conclude that 𝑥 = 0, which means that G(𝑇 |dom(𝑇∗ 𝑇) ) is dense in G(𝑇). Finally we describe a general method to construct self-adjoint operators associated with Hermitian sesquilinear forms. This leads to a self-adjoint extension of an unbounded operator, which is known as the Friedrichs extension.

4.1 Unbounded operators on Hilbert spaces

|

63

Definition 4.33. Let (V, ‖.‖V ) and (𝐻, ‖.‖𝐻 ) be Hilbert spaces such that V ⊂ 𝐻,

(4.4)

and suppose that there exists a constant 𝐶 > 0 such that for all 𝑢 ∈ V we have ‖𝑢‖𝐻 ≤ 𝐶 ‖𝑢‖V .

(4.5)

We also assume that V is dense in 𝐻. In this situation the space 𝐻 can be imbedded into the dual space V󸀠 : for ℎ ∈ 𝐻 the mapping 𝐿(𝑢) = (𝑢, ℎ)𝐻 , for 𝑢 ∈ V is continuous on V, this follows from (4.5): |𝐿(𝑢)| ≤ ‖𝑢‖𝐻 ‖ℎ‖𝐻 ≤ 𝐶‖ℎ‖𝐻 ‖𝑢‖V . Hence there exists a uniquely determined 𝑣ℎ ∈ V󸀠 such that 𝑣ℎ (𝑢) = (𝑢, ℎ)𝐻 ,

for 𝑢 ∈ V,

and the mapping ℎ 󳨃→ 𝑣ℎ is injective, as V is dense in 𝐻. Definition 4.34. A form 𝑎 : V × V 󳨀→ ℂ is sesquilinear, if it is linear in the first component and anti-linear in the second component. The form 𝑎 is continuous if there exists a constant 𝐶 > 0 such that |𝑎(𝑢, 𝑣)| ≤ 𝐶‖𝑢‖V ‖𝑣‖V (4.6) for all 𝑢, 𝑣 ∈ V and it is Hermitian if 𝑎(𝑢, 𝑣) = 𝑎(𝑣, 𝑢) for all 𝑢, 𝑣 ∈ V. The form 𝑎 is called V-elliptic if there exists a constant 𝛼 > 0 such that |𝑎(𝑢, 𝑢)| ≥ 𝛼‖𝑢‖2V

(4.7)

for all 𝑢 ∈ V. Proposition 4.35. Let 𝑎 be a continuous, V-elliptic form on V × V. Using (4.6) and the Riesz representation Theorem 1.6 we can define a linear operator 𝐴 : V 󳨀→ V such that 𝑎(𝑢, 𝑣) = (𝐴𝑢, 𝑣)V . This operator 𝐴 is a topological isomorphism from V onto V.

(4.8)

64 | 4 The 𝜕-complex Proof. First we show that 𝐴 is injective: (4.8) and (4.7) imply that for 𝑢 ∈ V we have ‖𝐴𝑢‖V ‖𝑢‖V ≥ |(𝐴𝑢, 𝑢)V | ≥ 𝛼‖𝑢‖2V , hence ‖𝐴𝑢‖V ≥ 𝛼‖𝑢‖V ,

(4.9)

which implies that 𝐴 is injective. Now we claim that 𝐴(V) is dense in V. Let 𝑢 ∈ V be such that (𝐴𝑣, 𝑢)V = 0 for each 𝑣 ∈ V. Taking 𝑣 = 𝑢 we get 𝑎(𝑢, 𝑢) = 0 and, by (4.7), 𝑢 = 0, which proves the claim. Next we observe that (4.8) implies 𝑎(𝑢, 𝐴𝑢) = ‖𝐴𝑢‖2V , therefore, using (4.6), we obtain ‖𝐴(𝑢)‖V ≤ 𝐶‖𝑢‖V , hence 𝐴 ∈ L(V). If (𝑣𝑛 )𝑛 is a Cauchy sequence in 𝐴(V) and 𝐴𝑢𝑛 = 𝑣𝑛 , we derive from (4.9) that (𝑢𝑛 )𝑛 is also a Cauchy sequence. Let 𝑢 = lim𝑛→∞ 𝑢𝑛 . We know already that 𝐴 is continuous, therefore lim𝑛→∞ 𝐴𝑢𝑛 = 𝐴𝑢, which shows that lim𝑛→∞ 𝑣𝑛 = 𝑣 = 𝐴𝑢 and 𝐴(V) is closed. As we have already shown that 𝐴(V) is dense in V, we conclude that 𝐴 is surjective. Finally (4.9) yields that 𝐴−1 is continuous. Proposition 4.36. Let 𝑎 be a Hermitian, continuous, V-elliptic form on V × V and suppose that (4.4) and (4.5) hold. Let dom(𝑆) be the set of all 𝑢 ∈ V such that the mapping 𝑣 󳨃→ 𝑎(𝑢, 𝑣) is continuous on V for the topology induced by 𝐻. For each 𝑢 ∈ dom(𝑆) there exists a uniquely determined element 𝑆𝑢 ∈ 𝐻 such that 𝑎(𝑢, 𝑣) = (𝑆𝑢, 𝑣)𝐻

(4.10)

for each 𝑣 ∈ V (by the Riesz representation theorem 1.6). Then 𝑆 : dom(𝑆) 󳨀→ 𝐻 is a bijective densely defined self-adjoint operator and −1 𝑆 ∈ L(𝐻). Moreover, dom(𝑆) is also dense in V. Proof. First we show that 𝑆 is injective. For each 𝑢 ∈ dom(𝑆) we get from (4.7) and (4.5) that 𝛼‖𝑢‖2𝐻 ≤ 𝐶𝛼‖𝑢‖2V ≤ 𝐶|𝑎(𝑢, 𝑢)| = 𝐶|(𝑆𝑢, 𝑢)𝐻 | ≤ 𝐶‖𝑆𝑢‖𝐻 ‖𝑢‖𝐻 , which implies that 𝛼‖𝑢‖𝐻 ≤ 𝐶‖𝑆𝑢‖𝐻 ,

(4.11)

for all 𝑢 ∈ dom(𝑆), therefore 𝑆 is injective. Now let ℎ ∈ 𝐻 and consider the mapping 𝑣 󳨃→ (ℎ, 𝑣)𝐻 for 𝑣 ∈ V. Then, by (4.5), we obtain |(ℎ, 𝑣)𝐻 | ≤ ‖ℎ‖𝐻 ‖𝑣‖𝐻 ≤ 𝐶‖ℎ‖𝐻 ‖𝑣‖V , which implies that there exists a uniquely determined 𝑤 ∈ V such that (ℎ, 𝑣)𝐻 = (𝑤, 𝑣)V for all 𝑣 ∈ V. Now we apply Proposition 4.35 and get from (4.8) that 𝑎(𝑢, 𝑣) = (𝑤, 𝑣)V , where 𝑢 = 𝐴−1 𝑤. Since 𝑎(𝑢, 𝑣) = (ℎ, 𝑣)𝐻 for each 𝑣 ∈ V, we conclude that 𝑢 ∈ dom(𝑆) and that 𝑆𝑢 = ℎ, which shows that 𝑆 is surjective.

4.2 Distributions

|

65

Suppose that (𝑢, ℎ)𝐻 = 0 for each 𝑢 ∈ dom(𝑆). As 𝑆 is surjective, there is 𝑣 ∈ dom(𝑆) such that 𝑆𝑣 = ℎ and we get that (𝑢, 𝑆𝑣)𝐻 = 0 for each 𝑢 ∈ dom(𝑆). Using the Vellipticity (4.7) we get for 𝑢 = 𝑣 that 0 = (𝑆𝑣, 𝑣)𝐻 = 𝑎(𝑣, 𝑣) ≥ 𝛼‖𝑣‖2V , which implies that 𝑣 = 0 and consequently ℎ = 0. Therefore we have shown that dom(𝑆) is dense in 𝐻. As 𝑎(𝑢, 𝑣) is Hermitian, we get for 𝑢, 𝑣 ∈ dom(𝑆) that (𝑆𝑢, 𝑣)𝐻 = 𝑎(𝑢, 𝑣) = 𝑎(𝑣, 𝑢) = (𝑆𝑣, 𝑢)𝐻 = (𝑢, 𝑆𝑣)𝐻 . Hence 𝑆 is symmetric and dom(𝑆) ⊂ dom(𝑆∗ ). Let 𝑣 ∈ dom(𝑆∗ ). Since 𝑆 is surjective, there exists 𝑣0 ∈ dom(𝑆) such that 𝑆𝑣0 = 𝑆∗ 𝑣. This implies (𝑆𝑢, 𝑣0 )𝐻 = (𝑢, 𝑆𝑣0 )𝐻 = (𝑢, 𝑆∗ 𝑣)𝐻 = (𝑆𝑢, 𝑣)𝐻 , for all 𝑢 ∈ dom(𝑆). Using again the surjectivity of 𝑆, we derive that 𝑣 = 𝑣0 ∈ dom(𝑆). This implies that dom(𝑆) = dom(𝑆∗ ) and that 𝑆 is self-adjoint. Finally we show that dom(𝑆) is dense in V. Let ℎ ∈ V be such that (𝑢, ℎ)V = 0, for all 𝑢 ∈ dom(𝑆). By Proposition 4.35 there exists 𝑓 ∈ V such that 𝐴𝑓 = ℎ. Then 0 = (𝑢, ℎ)V = (𝑢, 𝐴𝑓)V = (𝐴𝑓, 𝑢)V = 𝑎(𝑓, 𝑢) = 𝑎(𝑢, 𝑓) = (𝑆𝑢, 𝑓)𝐻 . 𝑆 is surjective, therefore we obtain 𝑓 = 0 and ℎ = 𝐴𝑓 = 0.

4.2 Distributions In our context the unbounded operator 𝑇 will mainly be the 𝜕-operator. In order to achieve an appropriate closed extension we will have to consider the derivatives 𝜕𝑧𝜕 𝑘 in the sense of distributions. Therefore we will now briefly summarize elementary definitions and results of distribution theory. ∞ Definition 4.37. Let Ω ⊆ ℝ𝑛 be an open subset and D(Ω) = C∞ 0 (Ω) the space of C functions with compact support (test functions). A sequence (𝜙𝑗 )𝑗 tends to 0 in D(Ω) if there exists a compact set 𝐾 ⊂ Ω such that supp(𝜙𝑗 ) ⊂ 𝐾 for every 𝑗 and 𝜕|𝛼| 𝜙𝑗 𝛼 → 0 𝛼 𝜕𝑥1 1 . . . 𝜕𝑥𝑛𝑛

uniformly on 𝐾 for each 𝛼 = (𝛼1 , . . . , 𝛼𝑛 ). A distribution is a linear functional 𝑢 on D(Ω) such that for every compact subset 𝐾 ⊂ Ω there exists 𝑘 ∈ ℕ0 = ℕ ∪ {0} and a constant 𝐶 > 0 with 󵄨󵄨 󵄨 󵄨 𝜕|𝛼| 𝜙(𝑥) 󵄨󵄨󵄨 |𝑢(𝜙)| ≤ 𝐶 ∑ sup 󵄨󵄨󵄨󵄨 𝛼1 𝛼𝑛 󵄨󵄨󵄨, 󵄨 󵄨 |𝛼|≤𝑘 𝑥∈𝐾 󵄨 𝜕𝑥1 . . . 𝜕𝑥𝑛 󵄨

66 | 4 The 𝜕-complex for each 𝜙 ∈ D(Ω) with support in 𝐾. We denote the space of distributions on Ω by D󸀠 (Ω). It is easily seen that 𝑢 ∈ D󸀠 (Ω) if and only if 𝑢(𝜙𝑗 ) → 0 for every sequence (𝜙𝑗 )𝑗 in D(Ω) converging to 0 in D(Ω). Example 4.38. (1) Let 𝑓 ∈ 𝐿1𝑙𝑜𝑐 (Ω), where 𝐿1𝑙𝑜𝑐 (Ω) = {𝑓 : Ω 󳨀→ ℂ measurable : 𝑓 |𝐾 ∈ 𝐿1 (𝐾) ∀𝐾 ⊂ Ω, 𝐾 compact}. The mapping 𝑇𝑓 (𝜙) = ∫Ω 𝑓(𝑥)𝜙(𝑥) 𝑑𝜆(𝑥) , 𝜙 ∈ D(Ω), is a distribution. (2) Let 𝑎 ∈ Ω and 𝛿𝑎 (𝜙) := 𝜙(𝑎), which is the point evaluation in 𝑎. The distribution 𝛿𝑎 is called Dirac delta distribution. In the sequel, certain operations for ordinary functions, such as multiplication of functions and differentiation, are generalized to distributions. Definition 4.39. Let 𝑓 ∈ C∞ (Ω) and 𝑢 ∈ D󸀠 (Ω). The multiplication of 𝑢 with 𝑓 is defined by (𝑓𝑢)(𝜙) := 𝑢(𝑓𝜙) for 𝜙 ∈ D(Ω). Notice that 𝑓𝜙 ∈ D(Ω). For 𝑢 ∈ D󸀠 (ℝ𝑛 ) and 𝑓 ∈ D(ℝ𝑛 ) the convolution of 𝑢 and 𝑓 is defined by (𝑢 ∗ 𝑓)(𝑥) := 𝑢(𝑦 󳨃→ 𝑓(𝑥 − 𝑦)). If 𝑢 = 𝑇𝑔 for some locally integrable function 𝑔, it is the usual convolution of functions (𝑇𝑔 ∗ 𝑓)(𝑥) = ∫ 𝑔(𝑦)𝑓(𝑥 − 𝑦) 𝑑𝜆(𝑦) = (𝑔 ∗ 𝑓)(𝑥). Ω

Let 𝐷𝑘 =

𝜕 𝜕𝑥𝑘

and 𝐷𝛼 =

𝛼 𝜕𝑥1 1

𝜕|𝛼| 𝛼 , . . . 𝜕𝑥𝑛𝑛

where 𝛼 = (𝛼1 , . . . , 𝛼𝑛 ) is a multi-index. The partial derivative of a distribution 𝑢 ∈

D󸀠 (Ω) is defined by

(𝐷𝑘 𝑢)(𝜙) := −𝑢(𝐷𝑘 𝜙),

𝜙 ∈ D(Ω);

higher order mixed derivatives are defined as (𝐷𝛼 𝑢)(𝜙) := (−1)|𝛼| 𝑢(𝐷𝛼 𝜙),

𝜙 ∈ D(Ω).

This definition stems from integrating by parts: ∫(𝐷𝑘 𝑓)𝜙 𝑑𝜆 = − ∫ 𝑓(𝐷𝑘 𝜙) 𝑑𝜆, Ω

where 𝑓 ∈ C1 (Ω) and 𝜙 ∈ D(Ω).

Ω

4.2 Distributions

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67

Definition 4.40. Let Ω1 ⊆ Ω be an open subset and 𝑢 ∈ D󸀠 (Ω). The restriction of 𝑢 to Ω1 is the distribution 𝑢 |Ω1 defined by for 𝜙 ∈ D(Ω1 ) ⊆ D(Ω).

𝑢 |Ω1 (𝜙) = 𝑢(𝜙),

The support supp(𝑢) of 𝑢 ∈ D󸀠 (Ω) consists of 𝑥 ∈ Ω such that there is no open neighborhood 𝑈 of 𝑥 with 𝑢 |𝑈 = 0. We denote the space of distributions with compact support by E󸀠 (Ω). A fundamental solution of a differential operator 𝑃(𝐷) is a distribution 𝐸 ∈ E󸀠 (ℝ𝑛 ) with 𝑃(𝐷)𝐸 = 𝛿0 . In what follows we will find a fundamental solution for 𝜕 and we will investigate regularity properties of the 𝜕-operator. For this purpose we recall Stokes’ theorem in the following way: let Ω ⊂ ℂ be a bounded domain with piecewise C1 -boundary 𝛾 (positively oriented), let 𝑓 ∈ C1 (Ω) and 𝜁 ∈ Ω; for 0 < 𝜖 < dist(𝜁, Ω𝑐 ), let Ω𝜖 := {𝑧 ∈ Ω : |𝑧 − 𝜁| > 𝜖}, where the boundary of Ω𝜖 consists of 𝛾 and the negatively oriented circle 𝑓(𝑧) 𝜌𝜖 = {𝑧 : |𝑧 − 𝜁| = 𝜖}. Consider the differential 𝜔 = 𝑧−𝜁 𝑑𝑧, then by Stokes’ theorem 2𝜋

𝑓(𝑧) ∫ 𝑑𝜔 = ∫ 𝜔 + ∫ 𝜔 = ∫ 𝑑𝑧 − ∫ 𝑖𝑓(𝜁 + 𝜖𝑒𝑖𝜃 ) 𝑑𝜃. 𝑧−𝜁 𝛾

Ω𝜖

We have 𝑑𝜔 =

𝜌𝜖

𝛾

(4.12)

0

𝜕𝑓 1 𝜕 𝑓(𝑧) ( ) 𝑑𝑧 ∧ 𝑑𝑧 = − 𝑑𝑧 ∧ 𝑑𝑧, 𝜕𝑧 𝑧 − 𝜁 𝜕𝑧 𝑧 − 𝜁

2𝜋

and by continuity ∫0 𝑖𝑓(𝜁 + 𝜖𝑒𝑖𝜃 ) 𝑑𝜃 → 2𝜋𝑖𝑓(𝜁) as 𝜖 → 0. Hence we obtain 𝑓(𝜁) =

𝑓(𝑧) 𝜕𝑓 1 1 1 ∫ 𝑑𝑧 + ∫ 𝑑𝑧 ∧ 𝑑𝑧. 2𝜋𝑖 𝑧 − 𝜁 2𝜋𝑖 𝜕𝑧 𝑧 − 𝜁 𝛾

(4.13)

Ω

In particular, if 𝜙 ∈ D(Ω), it follows that 𝜙(𝜁) = −

1 𝜕𝜙 1 ∫ 𝑑𝜆(𝑧), 𝜋 𝜕𝑧 𝑧 − 𝜁

(4.14)

Ω

where we used the fact that 𝑑𝑧 ∧ 𝑑𝑧 = (−2𝑖) 𝑑𝑥 ∧ 𝑑𝑦 = (−2𝑖) 𝑑𝜆(𝑧). This can be interpreted in the sense of distributions as 𝛿𝜁 (𝜙) = 𝜙(𝜁) = −

𝜕 1 𝜕𝜙 1 1 ∫ 𝑑𝜆(𝑧) = ∫ (𝑧 󳨃→ ) 𝜙(𝑧) 𝑑𝜆(𝑧), 𝜋 𝜕𝑧 𝑧 − 𝜁 𝜕𝑧 𝜋(𝑧 − 𝜁) Ω

which means that 𝑧 󳨃→

1 𝜋𝑧

(4.15)

Ω

is a fundamental solution to 𝜕.

Proposition 4.41. Let Ω ⊆ ℂ be a domain and 𝑢 ∈ D󸀠 (Ω) a distribution satisfying 𝜕𝑢 = 0. Then there exists a holomorphic function 𝑓 on Ω such that 𝑢 = 𝑇𝑓 .

68 | 4 The 𝜕-complex Proof. Let 𝐸 be the fundamental solution to 𝜕 given in (4.15). We have 𝜕𝐸 = 𝛿0 . Now let Ω󸀠 ⊂⊂ Ω be a relatively compact subset and choose 𝑔 ∈ D(Ω) such that 𝑔 = 1 in some neighborhood of Ω󸀠 . Let 𝑆 := 𝜕(𝑔𝑢). Then 𝑆 ∈ E󸀠 (Ω) with supp(𝑆) ⊂ ℂ \ Ω󸀠 , hence 𝑔𝑢 = 𝛿0 ∗ (𝑔𝑢) = (𝜕𝐸) ∗ (𝑔𝑢) = 𝐸 ∗ 𝜕(𝑔𝑢) = 𝐸 ∗ 𝑆.

(4.16)

Now pick 𝑧0 ∈ Ω󸀠 and 𝜖 > 0 such that 𝑉𝜖 = {𝑧 : dist(𝑧, ℂ \ Ω󸀠 ) > 𝜖} is a neighborhood of 𝑧0 . Let 𝜓𝜖 ∈ D(ℂ) such that 𝜓𝜖 (𝑧) = 1 for |𝑧| ≤ 𝜖/2 and 𝜙𝜖 (𝑧) = 0 for |𝑧| ≥ 𝜖. We can write (4.16) in the form 𝑔𝑢 = (𝜓𝜖 𝐸) ∗ 𝑆 + ((1 − 𝜓𝜖 )𝐸) ∗ 𝑆.

(4.17)

Since supp((𝜓𝜖 𝐸) ∗ 𝑆) ⊆ supp(𝜓𝜖 𝐸) + supp(𝑆) ⊆ {𝑧 : dist(𝑧, supp(𝑆) ≤ 𝜖}, we have that (𝜓𝜖 𝐸) ∗ 𝑆 has to vanish in 𝑉𝜖 . This implies 𝑢 |𝑉𝜖 = (𝑔𝑢) |𝑉𝜖 = (((1 − 𝜓𝜖 )𝐸) ∗ 𝑆) |𝑉𝜖 .

(4.18)

The function (1 − 𝜓𝜖 )𝐸 is in C∞ (ℂ) and supp(𝑆) is compact, therefore ((1 − 𝜓𝜖 )𝐸) ∗ 𝑆 is smooth. Hence, using (4.18), we obtain that 𝑢 is smooth in a neighborhood of 𝑧0 , and 𝑧0 being arbitrary, 𝑢 is smooth in Ω󸀠 and therefore also in Ω. This implies that 𝑢 is holomorphic, since it also solves the Cauchy–Riemann equation. For an appropriate description of the appearing phenomena we will use further Hilbert spaces of differentiable functions – the Sobolev spaces. Definition 4.42. If Ω is a bounded open set in ℝ𝑛 , and 𝑘 is a nonnegative integer we define the Sobolev space 𝑊𝑘 (Ω) = {𝑓 ∈ 𝐿2 (Ω) : 𝜕𝛼 𝑓 ∈ 𝐿2 (Ω), |𝛼| ≤ 𝑘}, where the derivatives are taken in the sense of distributions and endow the space with the norm 1/2

‖𝑓‖𝑘,Ω = [ ∑ ∫ |𝜕𝛼 𝑓|2 𝑑𝜆]

,

(4.19)

|𝛼|≤𝑘 Ω

where 𝛼 = (𝛼1 , . . . , 𝛼𝑛 ) is a multi-index , |𝛼| = ∑𝑛𝑗=1 𝛼𝑗 and 𝜕𝛼 𝑓 =

𝜕|𝛼| 𝑓 𝛼 . . . . 𝜕𝑥𝑛𝑛

𝛼 𝜕𝑥1 1

𝑘 ∞ 𝑊0𝑘 (Ω) denotes the completion of C∞ 0 (Ω) under 𝑊 (Ω)-norm. Since C0 (Ω) is dense 0 0 2 in 𝐿 (Ω), it follows that 𝑊0 (Ω) = 𝑊 (Ω) = 𝐿 (Ω). Using the Fourier transform it is also possible to introduce Sobolev spaces of non-integer exponent. (See [1, 23].) 2

In general a function can belong to a Sobolev space, and yet be discontinuous and unbounded.

4.2 Distributions

| 69

Example 4.43. Take Ω = 𝔹 the open unit ball in ℝ𝑛 , and 𝑢(𝑥) = |𝑥|−𝛼 ,

𝑥 ∈ 𝔹, 𝑥 ≠ 0.

. We claim that 𝑢 ∈ 𝑊1 (𝔹) if and only if 𝛼 < 𝑛−2 2 First note that 𝑢 is smooth away from 0, and that −𝛼𝑥𝑗

𝑢𝑥𝑗 (𝑥) =

|𝑥|𝛼+2

,

𝑥 ≠ 0.

|∇𝑢(𝑥)| =

|𝛼| , |𝑥|𝛼+1

𝑥 ≠ 0.

Hence

Now, recall the Gauß–Green Theorem: for a smoothly bounded 𝜔 ⊆ ℝ𝑛 we have ∫ ∇ . 𝐹(𝑥) 𝑑𝜆(𝑥) = ∫ (𝐹(𝑥), 𝜈(𝑥)) 𝑑𝜎(𝑥), 𝜔

𝑏𝜔

where 𝜈(𝑥) = ∇𝑟(𝑥) is the normal to 𝑏𝜔 at 𝑥, and 𝐹 is a C1 vector field on 𝜔, and 𝑛

∇ . 𝐹(𝑥) = ∑ 𝑗=1

𝜕𝐹𝑗 𝜕𝑥𝑗

,

(see [22]). Let 𝜙 ∈ C∞ 0 (𝔹) and let 𝔹𝜖 be the open ball around 0 with radius 𝜖 > 0. Take 𝜔 = 𝔹 \ 𝔹𝜖 and 𝐹(𝑥) = (0, . . . , 0, 𝑢𝜙, 0, . . . , 0), where 𝑢𝜙 appears at the 𝑗-th component. Then ∫ 𝑢(𝑥) 𝜙𝑥𝑗 (𝑥) 𝑑𝜆(𝑥) = − ∫ 𝑢𝑥𝑗 (𝑥) 𝜙(𝑥) 𝑑𝜆(𝑥) + ∫ 𝑢(𝑥)𝜙(𝑥)𝜈𝑗 (𝑥) 𝑑𝜎(𝑥), 𝔹\𝔹𝜖

𝔹\𝔹𝜖

𝑏𝔹𝜖

where 𝜈(𝑥) = (𝜈1 (𝑥), . . . , 𝜈𝑛 (𝑥)) denotes the inward pointing normal on 𝑏𝔹𝜖 . If 𝛼 < 𝑛− 1, then |∇𝑢(𝑥)| ∈ 𝐿1 (𝔹), and we obtain 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 ∫ 𝑢(𝑥)𝜙(𝑥)𝜈𝑗 (𝑥) 𝑑𝜎(𝑥)󵄨󵄨󵄨 ≤ ‖𝜙‖∞ ∫ 𝜖−𝛼 𝑑𝜎(𝑥) 󵄨󵄨 󵄨󵄨 󵄨 󵄨 𝑏𝔹 𝑏𝔹 𝜖

𝜖

≤ 𝐶 𝜖𝑛−1−𝛼 → 0, as 𝜖 → 0. Thus ∫ 𝑢(𝑥) 𝜙𝑥𝑗 (𝑥) 𝑑𝜆(𝑥) = − ∫ 𝑢𝑥𝑗 (𝑥) 𝜙(𝑥) 𝑑𝜆(𝑥) 𝔹

for all 𝜙 ∈

C∞ 0 (𝔹).

𝔹

As |∇𝑢(𝑥)| =

𝛼 ∈ 𝐿2 (𝔹) |𝑥|𝛼+1

if and only if 2(𝛼 + 1) < 𝑛 we get that 𝑢 ∈ 𝑊1 (𝔹) if and only if 𝛼
0. Let 𝑓 ∈ 𝐿2 (ℝ𝑛 ) and define for 𝑥 ∈ ℝ𝑛 𝑓𝜖 (𝑥) = (𝑓 ∗ 𝜒𝜖 )(𝑥) = ∫ 𝑓(𝑥󸀠 )𝜒𝜖 (𝑥 − 𝑥󸀠 ) 𝑑𝜆(𝑥󸀠 ) ℝ𝑛

= ∫ 𝑓(𝑥 − 𝑥󸀠 )𝜒𝜖 (𝑥󸀠 ) 𝑑𝜆(𝑥󸀠 ) ℝ𝑛

= ∫ 𝑓(𝑥 − 𝜖𝑥󸀠 )𝜒(𝑥󸀠 ) 𝑑𝜆(𝑥󸀠 ). ℝ𝑛

In the first integral we can differentiate under the integral sign to show that 𝑓𝜖 ∈ C∞ (ℝ𝑛 ). The family of functions (𝜒𝜖 )𝜖 is called an approximation to the identity. Lemma 5.3. ‖𝑓𝜖 − 𝑓‖2 → 0 as 𝜖 → 0. Proof. 𝑓𝜖 (𝑥) − 𝑓(𝑥) = ∫ [𝑓(𝑥 − 𝜖𝑥󸀠 ) − 𝑓(𝑥)] 𝜒(𝑥󸀠 ) 𝑑𝜆(𝑥󸀠 ). ℝ𝑛

We use Minkowski’s inequality (5.1) to get ‖𝑓𝜖 − 𝑓‖2 ≤ ∫ ‖𝑓−𝜖𝑥󸀠 − 𝑓‖2 |𝜒(𝑥󸀠 )| 𝑑𝜆(𝑥󸀠 ). ℝ𝑛

But ‖𝑓−𝜖𝑥󸀠 − 𝑓‖2 is bounded by 2‖𝑓‖2 and tends to 0 as 𝜖 → 0 by Lemma 5.3. Now set 𝐹𝜖 (𝑥󸀠 ) = ‖𝑓−𝜖𝑥󸀠 − 𝑓‖2 𝜒(𝑥󸀠 ).

5.1 Friedrichs’ Lemma and Sobolev spaces

|

89

Then 𝐹𝜖 (𝑥󸀠 ) → 0 as 𝜖 → 0 and |𝐹𝜖 (𝑥󸀠 )| ≤ 2‖𝑓‖2 𝜒(𝑥󸀠 ), and we can apply the dominated convergence theorem to get the desired result. 𝑛 If 𝑢 ∈ C∞ 0 (ℝ ) we have

𝐷𝑗 (𝑢 ∗ 𝜒𝜖 ) = (𝐷𝑗 𝑢) ∗ 𝜒𝜖 , where 𝐷𝑗 = 𝜕/𝜕𝑥𝑗 . This is also true, if 𝑢 ∈ 𝐿2 (ℝ𝑛 ) and 𝐷𝑗 𝑢 is defined in the sense of distributions. We will show even more using these methods for approximating a function in a Sobolev space by smooth functions. Let Ω ⊆ ℝ𝑛 be an open subset and let Ω𝜖 = {𝑥 ∈ Ω : dist(𝑥, 𝑏Ω) > 𝜖}. Lemma 5.4. Let 𝑢 ∈ 𝑊𝑘 (Ω) and set 𝑢𝜖 = 𝑢 ∗ 𝜒𝜖 in Ω𝜖 . Then (i) 𝑢𝜖 ∈ C∞ (Ω𝜖 ), for each 𝜖 > 0, (ii) 𝐷𝛼 𝑢𝜖 = 𝐷𝛼 𝑢 ∗ 𝜒𝜖 in Ω𝜖 , for |𝛼| ≤ 𝑘. Proof. (i) has already been shown. (ii) means that the ordinary 𝛼-th partial derivative of the smooth functions 𝑢𝜖 is the 𝜖-mollification of the 𝛼-th weak partial derivative of 𝑢. To see this, we take 𝑥 ∈ Ω𝜖 and compute 𝐷𝛼 𝑢𝜖 (𝑥) = 𝐷𝛼 ∫ 𝑢(𝑦)𝜒𝜖 (𝑥 − 𝑦) 𝑑𝜆(𝑦) Ω

= ∫ 𝐷𝑥𝛼 𝜒𝜖 (𝑥 − 𝑦)𝑢(𝑦) 𝑑𝜆(𝑦) Ω

= (−1)|𝛼| ∫ 𝐷𝑦𝛼 𝜒𝜖 (𝑥 − 𝑦)𝑢(𝑦) 𝑑𝜆(𝑦). Ω

For a fixed 𝑥 ∈ Ω𝜖 the function 𝜙(𝑦) := 𝜒𝜖 (𝑥 − 𝑦) belongs to C∞ (Ω). The definition of the 𝛼-th weak partial derivative implies ∫ 𝐷𝑦𝛼 𝜒𝜖 (𝑥 − 𝑦)𝑢(𝑦) 𝑑𝜆(𝑦) = (−1)|𝛼| ∫ 𝜒𝜖 (𝑥 − 𝑦) 𝐷𝛼 𝑢(𝑦) 𝑑𝜆(𝑦). Ω

Ω

Thus 𝐷𝛼 𝑢𝜖 (𝑥) = (−1)|𝛼|+|𝛼| ∫ 𝜒𝜖 (𝑥 − 𝑦) 𝐷𝛼 𝑢(𝑦) 𝑑𝜆(𝑦) Ω

= (𝐷𝛼 𝑢 ∗ 𝜒𝜖 )(𝑥), which proves (ii).

90 | 5 Density of smooth forms We are now ready to prove Lemma 5.5 (Friedrichs’ Lemma). If 𝑣 ∈ 𝐿2 (ℝ𝑛 ) is a function with compact support and 𝑎 is a C1 -function in a neighborhood of the support of 𝑣, it follows that ‖𝑎𝐷𝑗 (𝑣 ∗ 𝜒𝜖 ) − (𝑎𝐷𝑗 𝑣) ∗ 𝜒𝜖 ‖2 → 0 as 𝜖 → 0, where 𝐷𝑗 = 𝜕/𝜕𝑥𝑗 and 𝑎𝐷𝑗 𝑣 is defined in the sense of distributions. 𝑛 Proof. If 𝑣 ∈ C∞ 0 (ℝ ), we have

𝐷𝑗 (𝑣 ∗ 𝜒𝜖 ) = (𝐷𝑗 𝑣) ∗ 𝜒𝜖 → 𝐷𝑗 𝑣,

(𝑎𝐷𝑗 𝑣) ∗ 𝜒𝜖 → 𝑎𝐷𝑗 𝑣,

with uniform convergence. We claim that ‖𝑎𝐷𝑗 (𝑣 ∗ 𝜒𝜖 ) − (𝑎𝐷𝑗 𝑣) ∗ 𝜒𝜖 ‖2 ≤ 𝐶‖𝑣‖2 ,

(5.3)

𝑛 where 𝑣 ∈ 𝐿2 (ℝ𝑛 ) and 𝐶 is some positive constant independent of 𝜖 and 𝑣. Since C∞ 0 (ℝ ) 2 𝑛 is dense in 𝐿 (ℝ ), the lemma will follow as in the proof of Lemma 5.3 from (5.3) and the dominated convergence theorem. To show (5.3) we may assume that 𝑎 ∈ C10 (ℝ𝑛 ), since 𝑣 has compact support. We 𝑛 have for 𝑣 ∈ C∞ 0 (ℝ ),

𝑎(𝑥)𝐷𝑗 (𝑣 ∗ 𝜒𝜖 )(𝑥) − ((𝑎𝐷𝑗 𝑣) ∗ 𝜒𝜖 )(𝑥) = 𝑎(𝑥) 𝐷𝑗 ∫ 𝑣(𝑥 − 𝑦)𝜒𝜖 (𝑦) 𝑑𝜆(𝑦) − ∫ 𝑎(𝑥 − 𝑦) = ∫(𝑎(𝑥) − 𝑎(𝑥 − 𝑦))

𝜕𝑣 (𝑥 − 𝑦)𝜒𝜖 (𝑦) 𝑑𝜆(𝑦) 𝜕𝑥𝑗

= − ∫(𝑎(𝑥) − 𝑎(𝑥 − 𝑦))

𝜕𝑣 (𝑥 − 𝑦)𝜒𝜖 (𝑦) 𝑑𝜆(𝑦) 𝜕𝑦𝑗

= ∫(𝑎(𝑥) − 𝑎(𝑥 − 𝑦))𝑣(𝑥 − 𝑦) − ∫(

𝜕𝑣 (𝑥 − 𝑦)𝜒𝜖 (𝑦) 𝑑𝜆(𝑦) 𝜕𝑥𝑗

𝜕 𝜒 (𝑦) 𝑑𝜆(𝑦) 𝜕𝑦𝑗 𝜖

𝜕 𝑎(𝑥 − 𝑦)) 𝑣(𝑥 − 𝑦)𝜒𝜖 (𝑦) 𝑑𝜆(𝑦). 𝜕𝑦𝑗

Let 𝑀 be the Lipschitz constant for 𝑎 such that |𝑎(𝑥) − 𝑎(𝑥 − 𝑦)| ≤ 𝑀|𝑦|, for all 𝑥, 𝑦 ∈ ℝ𝑛 . Then 󵄨󵄨 𝜕 󵄨󵄨 |𝑎(𝑥)𝐷𝑗 (𝑣 ∗ 𝜒𝜖 )(𝑥) − ((𝑎𝐷𝑗 𝑣) ∗ 𝜒𝜖 )(𝑥)| ≤ 𝑀 ∫ |𝑣(𝑥 − 𝑦)|(𝜒𝜖 (𝑦) + 󵄨󵄨󵄨𝑦 𝜒 (𝑦)󵄨󵄨) 𝑑𝜆(𝑦). 󵄨 𝜕𝑦𝑗 𝜖 󵄨󵄨 By Minkowski’s inequality (5.1) we obtain 󵄨󵄨 𝜕 󵄨󵄨 𝜒 (𝑦)󵄨󵄨) 𝑑𝜆(𝑦) = 𝑀(1 + 𝑚𝑗 )‖𝑣‖2 , ‖𝑎𝐷𝑗 (𝑣 ∗ 𝜒𝜖 ) − (𝑎𝐷𝑗 𝑣) ∗ 𝜒𝜖 ‖2 ≤ 𝑀 ‖𝑣‖2 ∫ (𝜒𝜖 (𝑦) + 󵄨󵄨󵄨𝑦 󵄨 𝜕𝑦𝑗 𝜖 󵄨󵄨 where

󵄨󵄨 𝜕 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝜕 𝑚𝑗 = ∫ 󵄨󵄨󵄨𝑦 𝜒 (𝑦)󵄨󵄨 𝑑𝜆(𝑦) = ∫ 󵄨󵄨󵄨𝑦 𝜒(𝑦)󵄨󵄨󵄨 𝑑𝜆(𝑦). 󵄨 𝜕𝑦𝑗 𝜖 󵄨󵄨 󵄨 󵄨 𝜕𝑦𝑗

5.1 Friedrichs’ Lemma and Sobolev spaces

| 91

𝑛 ∞ 𝑛 2 𝑛 This shows (5.3) when 𝑣 ∈ C∞ 0 (ℝ ). Since C0 (ℝ ) is dense in 𝐿 (ℝ ), we have proved (5.3) and the lemma.

Lemma 5.6. Let

𝑛

𝐿 = ∑ 𝑎𝑗 𝐷𝑗 + 𝑎0 𝑗=1

be a first order differential operator with variable coefficients where 𝑎𝑗 ∈ C1 (ℝ𝑛 ) and 𝑎0 ∈ C(ℝ𝑛 ). If 𝑣 ∈ 𝐿2 (ℝ𝑛 ) is a function with compact support and 𝐿𝑣 = 𝑓 ∈ 𝐿2 (ℝ𝑛 ) 𝑛 where 𝐿𝑣 is defined in the distribution sense, the convolution 𝑣𝜖 = 𝑣 ∗ 𝜒𝜖 is in C∞ 0 (ℝ ) and 𝑣𝜖 → 𝑣, 𝐿𝑣𝜖 → 𝑓 in 𝐿2 (ℝ𝑛 ) as 𝜖 → 0. Proof. Since 𝑎0 𝑣 ∈ 𝐿2 (ℝ𝑛 ), we have lim 𝑎0 (𝑣 ∗ 𝜒𝜖 ) = lim(𝑎0 𝑣 ∗ 𝜒𝜖 ) = 𝑎0 𝑣

𝜖→0

𝜖→0

in 𝐿2 (ℝ𝑛 ). Using Friedrichs’ Lemma 5.5, we have 𝐿𝑣𝜖 − 𝐿𝑣 ∗ 𝜒𝜖 = 𝐿𝑣𝜖 − 𝑓 ∗ 𝜒𝜖 → 0 in 𝐿2 (ℝ𝑛 ) as 𝜖 → 0. The lemma follows easily since 𝑓 ∗ 𝜒𝜖 → 𝑓 in 𝐿2 (ℝ𝑛 ). Before we proceed with results about Sobolev spaces we prove an important inequality for the sgn-function. Let 𝑧 ∈ ℂ. Define {𝑧/|𝑧| sgn𝑧 = { 0 {

𝑧 ≠ 0 𝑧 = 0.

Proposition 5.7. Suppose that 𝑓 ∈ 𝐿1loc (ℝ𝑛 ) with ∇𝑓 ∈ 𝐿1loc (ℝ𝑛 ). Then ∇|𝑓| ∈ 𝐿1loc (ℝ𝑛 ) and ∇|𝑓|(𝑥) = ℜ[sgn(𝑓(𝑥)) ∇𝑓(𝑥)]

(5.4)

almost everywhere. In particular, we have 󵄨󵄨 󵄨 󵄨󵄨∇|𝑓|󵄨󵄨󵄨 ≤ |∇𝑓|, almost everywhere. Proof. Let 𝑧 ∈ ℂ and 𝜖 > 0. We define |𝑧|𝜖 := √|𝑧|2 + 𝜖2 − 𝜖 and observe that 0 ≤ |𝑧|𝜖 ≤ |𝑧|

and

lim |𝑧|𝜖 = |𝑧|.

𝜖→0

(5.5)

92 | 5 Density of smooth forms If 𝑢 ∈ C∞ (ℝ𝑛 ), then |𝑢|𝜖 ∈ C∞ (ℝ𝑛 ) and as |𝑢|2 = 𝑢 𝑢 we get ∇|𝑢|𝜖 =

ℜ(𝑢 ∇𝑢) . √|𝑢|2 + 𝜖2

(5.6)

Now let 𝑓 be as assumed, take an approximation to the identity (𝜒𝛿 )𝛿 and define 𝑓𝛿 = 𝑓 ∗ 𝜒𝛿 . By Lemma 5.2, Lemma 5.3 and Lemma 5.4, we obtain that 𝑓𝛿 → 𝑓, |𝑓𝛿 | → |𝑓|, and ∇𝑓𝛿 → ∇𝑓 in 𝐿1loc (ℝ𝑛 ) as 𝛿 → 0. 𝑛 Let 𝜙 ∈ C∞ 0 (ℝ ) be a test function. There exists a subsequence 𝛿𝑘 → 0 such that 𝑓𝛿𝑘 (𝑥) → 𝑓(𝑥) for almost every 𝑥 ∈ supp(𝜙). For simplicity we omit the index 𝑘 now. Using the dominated convergence theorem and (5.6) we get ∫(∇𝜙) |𝑓| 𝑑𝜆 = lim ∫(∇𝜙) |𝑓|𝜖 𝑑𝜆 𝜖→0

= lim lim ∫(∇𝜙) |𝑓𝛿 |𝜖 𝑑𝜆 𝜖→0 𝛿→0

= − lim lim ∫ 𝜙 𝜖→0 𝛿→0

ℜ(𝑓𝛿 ∇𝑓𝛿 ) √|𝑓𝛿 |2 + 𝜖2

𝑑𝜆.

Since ∇𝑓𝛿 → ∇𝑓 in 𝐿1loc (ℝ𝑛 ), we get taking the limit 𝛿 → 0 that ∫(∇𝜙) |𝑓| 𝑑𝜆 = − lim ∫ 𝜙 𝜖→0

ℜ(𝑓∇𝑓) √|𝑓|2 + 𝜖2

𝑑𝜆,

and since 𝜙∇𝑓 ∈ 𝐿1 (ℝ𝑛 ) and 𝑓/√|𝑓|2 + 𝜖2 → sgn𝑓 as 𝜖 → 0 we get the desired result by applying once more dominated convergence. In the sequel we still use the methods from above for approximating a function in a Sobolev space by smooth functions. In a similar way as in the last lemma one gets 𝑘 Lemma 5.8. If 𝑢 ∈ 𝑊𝑘 (Ω), then 𝑢𝜖 → 𝑢 in 𝑊𝑙𝑜𝑐 (Ω), as 𝜖 → 0, this means that this 𝑘 happens in each space 𝑊 (𝜔), where 𝜔 is an open subset with 𝜔 ⊂⊂ Ω.

Using a smooth partition of unity we still show that one can find smooth functions 𝑘 which approximate in the 𝑊𝑘 (Ω)-norm, and not just in 𝑊𝑙𝑜𝑐 (Ω). Lemma 5.9. Let Ω ⊂ ℝ𝑛 be a bounded open set and let 𝑢 ∈ 𝑊𝑘 (Ω). Then there exist functions 𝑢𝑚 ∈ C∞ (Ω) ∩ 𝑊𝑘 (Ω) such that 𝑢𝑚 → 𝑢 in 𝑊𝑘 (Ω). Note that we do not assert that 𝑢𝑚 ∈ C∞ (Ω). Proof. We write Ω = ⋃∞ 𝑗=1 𝜔𝑗 , where 𝜔𝑗 := {𝑥 ∈ Ω : dist(𝑥, 𝑏Ω) > 1/𝑗} , 𝑗 = 1, 2, . . . .

5.1 Friedrichs’ Lemma and Sobolev spaces

|

93

Set 𝑈𝑗 := 𝜔𝑗+3 \ 𝜔𝑗+1 , and choose any open set 𝑈0 ⊂⊂ Ω so that Ω = ⋃∞ 𝑗=0 𝑈𝑗 . Let (𝜙𝑗 )𝑗 be a smooth partition of unity subordinate to the open sets (𝑈𝑗 )𝑗 : that is 0 ≤ 𝜙𝑗 ≤ 1 , 𝜙𝑗 ∈ ∞ C∞ 0 (𝑈𝑗 ) and ∑𝑗=0 𝜙𝑗 = 1 on Ω.

According to Proposition 4.44 𝜙𝑗 𝑢 ∈ 𝑊𝑘 (Ω) and the support of 𝜙𝑗 𝑢 is contained in 𝑈𝑗 . Now we use Lemma 5.8: fix 𝜖 > 0 and choose 𝜖𝑗 > 0 so small that 𝑢𝑗 := (𝜙𝑗 𝑢) ∗ 𝜒𝜖𝑗 satisfies 𝜖 ‖𝑢𝑗 − 𝜙𝑗 𝑢‖𝑊𝑘 (Ω) ≤ 𝑗+1 , 𝑗 = 0, 1, . . . , 2 and 𝑢𝑗 has support in 𝑉𝑗 := 𝜔𝑗+4 \ 𝜔𝑗 ⊃ 𝑈𝑗 for 𝑗 = 1, 2, . . . . ∞ Now define 𝑣 := ∑∞ 𝑗=0 𝑢𝑗 . This function belongs to C (Ω), since for each open set 𝜔 ⊂⊂ Ω there are at most finitely many nonzero terms in the sum. Since 𝑢 = ∑∞ 𝑗=0 𝜙𝑗 𝑢, we have for each 𝜔 ⊂⊂ Ω ∞

∞

‖𝑣 − 𝑢‖𝑊𝑘 (𝜔) ≤ ∑ ‖𝑢𝑗 − 𝜙𝑗 𝑢‖𝑊𝑘 (Ω) ≤ 𝜖 ∑ 𝑗=0

𝑗=0

1 2𝑗+1

= 𝜖.

Finally, take the supremum over all sets 𝜔 ⊂⊂ Ω, to conclude that ‖𝑣 − 𝑢‖𝑊𝑘 (Ω) ≤ 𝜖. Before we proceed to prove the density result for the 𝜕-setting we show that a function 𝑢 ∈ 𝑊𝑘 (Ω) can be approximated by functions in C∞ (Ω), where all derivatives extend continuously to Ω. This of course requires some conditions on the boundary 𝑏Ω. Proposition 5.10. Let Ω be a bounded open set in ℝ𝑛 and assume that 𝑏Ω is C1 . Let 𝑢 ∈ 𝑊𝑘 (Ω). Then there exist functions 𝑢𝑚 ∈ C∞ (Ω) such that 𝑢𝑚 → 𝑢 in 𝑊𝑘 (Ω). Proof. Let 𝑥0 ∈ 𝑏Ω. As 𝑏Ω is C1 , there exists a radius 𝑟 > 0 and a C1 -function 𝛾 : ℝ𝑛−1 󳨀→ ℝ such that Ω ∩ 𝐵(𝑥0 , 𝑟) = {𝑥 ∈ 𝐵(𝑥0 , 𝑟) : 𝑥𝑛 > 𝛾(𝑥1 , . . . , 𝑥𝑛−1 )}. We set 𝑉 := Ω ∩ 𝐵(𝑥0 , 𝑟/2) and define the shifted point 𝑥𝜖 := 𝑥 + 𝜇𝜖𝑒𝑛 for 𝑥 ∈ 𝑉 and 𝜖 > 0. We see that for some fixed, sufficiently large number 𝜇 > 0 the ball 𝐵(𝑥𝜖 , 𝜖) lies in Ω ∩ 𝐵(𝑥0 , 𝑟) for all 𝑥 ∈ 𝑉 and all small 𝜖 > 0. Now we define 𝑢𝜖 (𝑥) := 𝑢(𝑥𝜖 ) for 𝑥 ∈ 𝑉; this is the function 𝑢 translated a distance 𝜇𝜖 in the 𝑒𝑛 -direction. Next we write 𝑣𝜖 = 𝑢𝜖 ∗ 𝜒𝜖 . The idea is that we have moved up enough so that there is room to mollify within Ω. We have 𝑣𝜖 ∈ C∞ (𝑉). We now claim that 𝑣𝜖 → 𝑢 in 𝑊𝑘 (𝑉) as 𝜖 → 0. Let 𝛼 be a multi-index with |𝛼| ≤ 𝑘. Then ‖𝐷𝛼 𝑣𝜖 − 𝐷𝛼 𝑢‖𝐿2 (𝑉) ≤ ‖𝐷𝛼 𝑣𝜖 − 𝐷𝛼 𝑢𝜖 ‖𝐿2 (𝑉) + ‖𝐷𝛼 𝑢𝜖 − 𝐷𝛼 𝑢‖𝐿2 (𝑉) . The second term on the right-hand side goes to zero with 𝜖, since, by Lemma 5.2, translation is continuous in the 𝐿2 -norm. The first term also vanishes as 𝜖 → 0, by a similar reasoning as in Lemma 5.6.

94 | 5 Density of smooth forms Let 𝛿 > 0. Since 𝑏Ω is compact, one can find finitely many points 𝑥𝑗 ∈ 𝑏Ω, radii 𝑟𝑗 > 0, corresponding sets 𝑉𝑗 = Ω ∩ 𝐵(𝑥𝑗 , 𝑟𝑗 /2), and functions 𝑣𝑗 ∈ C∞ (𝑉𝑗 ) , 𝑗 = 1, . . . , 𝑁 such that 𝑁

𝑏Ω ⊂ ⋃ 𝐵(𝑥𝑗 , 𝑟𝑗 /2) 𝑗=1

and ‖𝑣𝑗 − 𝑢‖𝑊𝑘 (𝑉𝑗 ) ≤ 𝛿.

(5.7)

Now we take an open set 𝑉0 ⊂⊂ Ω such that 𝑁

Ω ⊂ ⋃ 𝑉𝑗 𝑗=0

and select, using Lemma 5.8, a function 𝑣0 ∈ C∞ (𝑉0 ) satisfying (5.8)

‖𝑣0 − 𝑢‖𝑊𝑘 (𝑉0 ) ≤ 𝛿.

Finally we take a smooth partition (𝜙𝑗 )𝑗 of unity subordinate to the open sets (𝑉𝑗 )𝑗 𝑁 ∞ in Ω for 𝑗 = 0, . . . , 𝑁. Define 𝑣 := ∑𝑁 𝑗=0 𝜙𝑗 𝑣𝑗 . Then 𝑣 ∈ C (Ω). Since 𝑢 = ∑𝑗=0 𝜙𝑗 𝑢 we see that for each |𝛼| ≤ 𝑘 : 𝑁

𝑁

𝑗=0

𝑗=0

‖𝐷𝛼 𝑣 − 𝐷𝛼 𝑢‖𝐿2 (Ω) ≤ ∑ ‖𝐷𝛼 (𝜙𝑗 𝑣𝑗 ) − 𝐷𝛼 (𝜙𝑗 𝑢)‖𝐿2 (𝑉𝑗 ) ≤ ∑ ‖𝑣𝑗 − 𝑢‖𝑊𝑘 (𝑉𝑗 ) ≤ (𝑁 + 1)𝛿, where we used (5.7) and (5.8). In the next step we show that functions in the Sobolev space 𝑊1 (Ω) can be extended to functions in 𝑊1 (ℝ𝑛 ), provided that Ω is a bounded domain with a C1 -boundary. Proposition 5.11. Assume that Ω is a bounded domain with a C1 -boundary. Select a bounded open set 𝑉 such that Ω ⊂⊂ 𝑉. Then there exists a bounded linear operator 𝐸 : 𝑊1 (Ω) 󳨀→ 𝑊1 (ℝ𝑛 ) such that for each 𝑢 ∈ 𝑊1 (Ω) : (i) 𝐸𝑢 = 𝑢 almost everywhere in Ω, (ii) 𝐸𝑢 has support within 𝑉, (iii) there exists a constant 𝐶 depending only on Ω and 𝑉 such that ‖𝐸𝑢‖𝑊1 (ℝ𝑛 ) ≤ 𝐶‖𝑢‖𝑊1 (Ω) . Proof. We use the method of a higher order reflection for the extension. Let 𝑥0 ∈ 𝑏Ω and suppose first that 𝑏Ω is flat near 𝑥0 , lying in the plane {𝑥𝑛 = 0}. Then we may assume there exists an open ball 𝐵 centered in 𝑥0 with radius 𝑟 such that 𝐵+ := 𝐵 ∩ {𝑥𝑛 ≥ 0} ⊂ Ω , 𝐵− := 𝐵 ∩ {𝑥𝑛 ≤ 0} ⊂ ℝ𝑛 \ Ω. Temporarily we suppose that 𝑢 ∈ C∞ (Ω). We define then {𝑢(𝑥) if 𝑥 ∈ 𝐵+ ̃ 𝑢(𝑥) ={ 𝑥 −3𝑢(𝑥1 , . . . , 𝑥𝑛−1 , −𝑥𝑛 ) + 4𝑢(𝑥1 , . . . , 𝑥𝑛−1 , − 2𝑛 ) if 𝑥 ∈ 𝐵− . {

5.1 Friedrichs’ Lemma and Sobolev spaces

|

95

This is called a higher order reflection of 𝑢 from 𝐵+ to 𝐵− . First we show: 𝑢̃ ∈ C1 (𝐵). To check this we write 𝑢− := 𝑢̃ |𝐵−

and 𝑢+ := 𝑢̃ |𝐵+ .

By definition, we have 𝑥 𝜕𝑢 𝜕𝑢 𝜕𝑢− (𝑥 , . . . , 𝑥𝑛−1 , − 𝑛 ) (𝑥) = 3 (𝑥 , . . . , 𝑥𝑛−1 , −𝑥𝑛 ) − 2 𝜕𝑥𝑛 𝜕𝑥𝑛 1 𝜕𝑥𝑛 1 2 and so 𝑢𝑥−𝑛 |{𝑥𝑛 =0} = 𝑢𝑥+𝑛 |{𝑥𝑛 =0} . Now since 𝑢+ = 𝑢− on {𝑥𝑛 = 0}, we see that also 𝑢𝑥−𝑗 |{𝑥𝑛 =0} = 𝑢𝑥+𝑗 |{𝑥𝑛 =0} for 𝑗 = 1, . . . , 𝑛 − 1. Hence we have 𝐷𝛼 𝑢− |{𝑥𝑛 =0} = 𝐷𝛼 𝑢+ |{𝑥𝑛 =0} , for each |𝛼| ≤ 1, which implies 𝑢̃ ∈ C1 (𝐵). Using these computations one readily sees that (5.9)

‖𝑢‖̃ 𝑊1 (𝐵) ≤ 𝐶‖𝑢‖𝑊1 (𝐵+ ) ,

for some constant 𝐶 > 0 which does not depend on 𝑢. If 𝑏Ω is not flat near 𝑥0 , we can find a C1 -mapping Φ, with inverse Ψ, which straightens out 𝑏Ω near 𝑥0 . We write 𝑦 = Φ(𝑥) and 𝑥 = Ψ(𝑦) and define 𝑢∗ (𝑦) := 𝑢(Ψ(𝑦)). We choose a small ball 𝐵 and use the same reasoning as before to extend 𝑢∗ from 𝐵+ to a function 𝑢̃∗ defined on all of 𝐵, such that 𝑢̃∗ ∈ C1 (𝐵) and as in 5.9 we get ‖𝑢̃∗ ‖𝑊1 (𝐵) ≤ 𝐶‖𝑢∗ ‖𝑊1 (𝐵+ ) .

(5.10)

Let 𝑊 := Ψ(𝐵). Then converting back to the 𝑥-variables, we obtain an extension 𝑢̃ of 𝑢 to 𝑊, with ‖𝑢‖̃ 𝑊1 (𝑊) ≤ 𝐶‖𝑢‖𝑊1 (Ω) . (5.11) 𝑗

Since 𝑏Ω is compact, there exist finitely many points 𝑥0 ∈ 𝑏Ω, open sets 𝑊𝑗 , and extensions 𝑢̃𝑗 of 𝑢 to 𝑊𝑗 for 𝑗 = 1, . . . , 𝑁, such that 𝑏Ω ⊂ ⋃𝑁 𝑗=1 𝑊𝑗 . Take 𝑊0 ⊂⊂ Ω with

Ω ⊂ ⋃𝑁 𝑗=0 𝑊𝑗 , and let (𝜙𝑗 )𝑗 be an associated partition of unity. Write 𝑁

𝑢̃ = ∑ 𝜙𝑗 𝑢̃𝑗 , 𝑗=0

where 𝑢̃0 = 𝑢. Then, by (5.11), we obtain the estimate ‖𝑢‖̃ 𝑊1 (ℝ𝑛 ) ≤ 𝐶‖𝑢‖𝑊1 (Ω)

(5.12)

96 | 5 Density of smooth forms for some constant 𝐶 > 0 independent of 𝑢. In addition we arrange for the support of 𝑢̃ to lie within 𝑉 ⊃⊃ Ω. We define 𝐸𝑢 := 𝑢̃ and observe that the mapping 𝑢 󳨃→ 𝐸𝑢 is linear. So far we have assumed that 𝑢 ∈ C∞ (Ω). Now take 𝑢 ∈ 𝑊1 (Ω), and choose a sequence 𝑢𝑚 ∈ C∞ (Ω) converging to 𝑢 in 𝑊1 (Ω)(see Proposition 5.10). Estimate (5.12) implies ‖𝐸𝑢𝑚 − 𝐸𝑢ℓ ‖𝑊1 (ℝ𝑛 ) ≤ 𝐶‖𝑢𝑚 − 𝑢ℓ ‖𝑊1 (Ω) . Hence (𝐸𝑢𝑚 )𝑚 is a Cauchy sequence and so converges to 𝑢̃ =: 𝐸𝑢. This extension does not depend on the particular choice of the approximating sequence (𝑢𝑚 )𝑚 . In a similar way we treat the problem how to assign boundary values along 𝑏Ω to a function 𝑢 ∈ 𝑊1 (Ω), assuming that 𝑏Ω is C1 . Proposition 5.12. Let Ω be a bounded domain with C1 -boundary. Then there exists a bounded linear operator 𝑇 : 𝑊1 (Ω) 󳨀→ 𝐿2 (𝑏Ω) such that (i) 𝑇𝑢 = 𝑢 |𝑏Ω , if 𝑢 ∈ 𝑊1 (Ω) ∩ C(Ω); and (ii) ‖𝑇𝑢‖𝐿2 (𝑏Ω) ≤ 𝐶‖𝑢‖𝑊1 (Ω) , for each 𝑢 ∈ 𝑊1 (Ω), with the constant 𝐶 depending only on Ω. We call 𝑇𝑢 the trace of 𝑢 on 𝑏Ω. Proof. First we assume that 𝑢 ∈ C1 (Ω) and proceed as in the proof of Proposition 5.11. We suppose that 𝑥0 ∈ 𝑏Ω and that 𝑏Ω is flat near 𝑥0 , lying in the plane {𝑥𝑛 = 0}. We choose an open ball 𝐵 as in the previous proof and let 𝐵̃ denote the concentric ball of ̃ radius 𝑟/2. Select a function 𝜒 ∈ C∞ 0 (𝐵) with 𝜒 ≥ 0 in 𝐵 and 𝜒 = 1 on 𝐵. Denote Γ the 󸀠 𝑛−1 ̃ portion of 𝑏Ω within 𝐵. Set 𝑥 = (𝑥1 , . . . , 𝑥𝑛−1 ) ∈ ℝ = {𝑥𝑛 = 0}. Then we have ∫ |𝑢|2 𝑑𝜆(𝑥󸀠 ) ≤ ∫ 𝜒|𝑢|2 𝑑𝜆(𝑥󸀠 ) = − ∫ (𝜒|𝑢|2 )𝑥𝑛 𝑑𝜆(𝑥) Γ

{𝑥𝑛 =0}

𝐵+

= − ∫ (|𝑢|2 𝜒𝑥𝑛 + |𝑢|(sgn𝑢) 𝑢𝑥𝑛 𝜒) 𝑑𝜆(𝑥) 𝐵+

≤ 𝐶 ∫ (|𝑢|2 + |∇𝑢|2 ) 𝑑𝜆(𝑥), 𝐵+

where we used Proposition 5.7 and the inequality 𝑎𝑏 ≤ 𝑎2 /2 + 𝑏2 /2, for 𝑎, 𝑏 ≥ 0.

5.1 Friedrichs’ Lemma and Sobolev spaces

|

97

After this we straighten out the boundary near 𝑥0 to get the estimate ∫ |𝑢|2 𝑑𝜆(𝑥󸀠 ) ≤ 𝐶 ∫(|𝑢|2 + |∇𝑢|2 ) 𝑑𝜆(𝑥), Γ

Ω

where Γ is some open subset of 𝑏Ω containing 𝑥0 . Since 𝑏Ω is compact, there exist finitely many points 𝑥0,𝑘 ∈ 𝑏Ω and open subsets Γ𝑘 ⊂ 𝑏Ω (𝑘 = 1, . . . , 𝑁) such that 𝑁

𝑏Ω = ⋃ Γ𝑘 𝑘=1

and ‖𝑢‖𝐿2 (Γ𝑘 ) ≤ 𝐶‖𝑢‖𝑊1 (Ω) , for 𝑘 = 1, . . . , 𝑁. Hence, if we write 𝑇𝑢 := 𝑢 |𝑏Ω , we get ‖𝑇𝑢‖𝐿2 (𝑏Ω) ≤ 𝐶‖𝑢‖𝑊1 (Ω) ,

(5.13)

for some constant 𝐶, which does not depend on 𝑢. Inequality (5.13) holds for 𝑢 ∈ C1 (Ω). Assume now that 𝑢 ∈ 𝑊1 (Ω). Then, by Proposition 5.10 there exist functions 𝑢𝑚 ∈ C∞ (Ω) converging to 𝑢 in 𝑊1 (Ω). By (5.13) we have ‖𝑇𝑢𝑚 − 𝑇𝑢ℓ ‖𝐿2 (𝑏Ω) ≤ 𝐶‖𝑢𝑚 − 𝑢ℓ ‖𝑊1 (Ω) ,

(5.14)

hence (𝑇𝑢𝑚 )𝑚 is a Cauchy sequence in 𝐿2 (𝑏Ω). Set 𝑇𝑢 := lim 𝑇𝑢𝑚 , 𝑚→∞

2

where the limit is taken in 𝐿 (𝑏Ω). By (5.14), this definition does not depend on the particular choice of the smooth functions approximating 𝑢. If 𝑢 ∈ 𝑊1 (Ω) ∩ C(Ω), one can use the fact that the functions 𝑢𝑚 ∈ C∞ (Ω) constructed in the proof of Proposition 5.10 converge uniformly to 𝑢 on Ω. This implies 𝑇𝑢 = 𝑢 |𝑏Ω . Proposition 5.13. Let Ω be a bounded domain with C1 -boundary. Then 𝑢 ∈ 𝑊01 (Ω), if and only if 𝑇𝑢 = 0 on 𝑏Ω. Proof. If 𝑢 ∈ 𝑊01 (Ω), then there exist functions 𝑢𝑚 ∈ C∞ 0 (Ω) such that 𝑢𝑚 → 𝑢 in 1 𝑊 (Ω). As 𝑇𝑢𝑚 = 0 on 𝑏Ω and 𝑇 : 𝑊1 (Ω) 󳨀→ 𝐿2 (𝑏Ω) is a continuous linear operator (Proposition 5.12), we obtain 𝑇𝑢 = 0 on 𝑏Ω. The converse statement is more difficult, it will not be important in the following, so we omit the proof, see for instance [20].

98 | 5 Density of smooth forms

5.2 Density in the graph norm We are now able to prove the density result for the basic estimates. ∗

∗

Proposition 5.14. If 𝑏Ω is C𝑘+1 , then C𝑘(0,𝑞) (Ω) ∩ dom(𝜕 ) is dense in dom(𝜕) ∩ dom(𝜕 ) ∗

in the graph norm 𝑢 󳨃→ (‖𝑢‖2 + ‖𝜕𝑢‖2 + ‖𝜕 𝑢‖2 )1/2 . The statement also holds with 𝑘 + 1 and 𝑘 replaced by ∞. Before we begin with the proof of this important approximation result we gather some ∗ observations about dom(𝜕) and dom(𝜕 ).

Remark 5.15. ∗ (a) It is useful to know that dom(𝜕 ) is preserved under multiplication by a function ∗ in C1 (Ω) : let 𝑢 ∈ dom(𝜕 ) , 𝑣 ∈ dom(𝜕) and 𝜓 ∈ C1 (Ω). Then ∗

(𝜕𝑣, 𝜓𝑢) = (𝜓 𝜕𝑣, 𝑢) = (𝜕(𝜓𝑣), 𝑢) − (𝜕 𝜓 ∧ 𝑣, 𝑢) = (𝜓𝑣, 𝜕 𝑢) − (𝜕 𝜓 ∧ 𝑣, 𝑢). ∗

The right-hand side is bounded by ‖𝑣‖, hence 𝜓𝑢 ∈ dom(𝜕 ), (see [71]). ∗ (b) Compactly supported forms are not dense in dom(𝜕) ∩ dom(𝜕 ) in the graph norm: for compactly supported forms Proposition 4.55 gives 󵄩󵄩 𝜕𝑢 󵄩󵄩2 󵄩 𝑗 󵄩󵄩 󵄩󵄩 , ‖𝜕𝑢‖ + ‖𝜕 𝑢‖ = ∑ 󵄩󵄩󵄩󵄩 󵄩 𝜕𝑧𝑘 󵄩󵄩󵄩 𝑗,𝑘=1 󵄩 ∗

2

2

𝑛

and integration by parts also shows that in this case 󵄩󵄩 𝜕𝑢 󵄩󵄩2 󵄩󵄩 𝜕𝑢 󵄩󵄩2 󵄩󵄩 𝑗 󵄩󵄩 󵄩 𝑗 󵄩󵄩 󵄩󵄩 󵄩󵄩 = 󵄩󵄩󵄩 󵄩󵄩 . 󵄩󵄩󵄩 𝜕𝑧𝑘 󵄩󵄩󵄩 󵄩󵄩󵄩 𝜕𝑧𝑘 󵄩󵄩󵄩 Hence

∗

‖𝑢‖21 ≤ 2 (‖𝜕𝑢‖2 + ‖𝜕 𝑢‖2 ), where ‖𝑢‖21 denotes the standard Sobolev-1 norm of 𝑢 on Ω. Therefore the closure ∗ of the compactly supported forms in dom(𝜕) ∩ dom(𝜕 ) in the graph norm is contained in the Sobolev space 𝑊01 (Ω) for forms that are C∞ on Ω, this means that they are zero on the boundary (Proposition 5.13), which is stronger than the condition 𝑛

∑ 𝑗=1

𝜕𝑟 𝑢 =0 𝜕𝑧𝑗 𝑗

on 𝑏Ω from Proposition 4.52. ∗ (c) If Ω is a smoothly bounded pseudoconvex domain, then dom(𝜕) ∩ dom(𝜕 ) is a ∗ Hilbert space in the graph norm 𝑢 󳨃→ (‖𝜕𝑢‖2 + ‖𝜕 𝑢‖2 )1/2 . This follows from (4.37). The proof of Proposition 5.14 will be carried out in several steps. Lemma 5.16. Let Ω be as in Proposition 5.14. Then C∞ (0,𝑞) (Ω) is dense in dom(𝜕) ∩ ∗

∗

dom(𝜕 ) in the graph norm 𝑢 󳨃→ (‖𝑢‖2 + ‖𝜕𝑢‖2 + ‖𝜕 𝑢‖2 )1/2 .

5.2 Density in the graph norm

| 99

∗

Proof. By this we mean that if 𝑢 ∈ dom(𝜕) ∩ dom(𝜕 ), one can construct a sequence 2 𝑢𝑚 ∈ C∞ (0,𝑞) (Ω) such that 𝑢𝑚 → 𝑢 , 𝜕𝑢𝑚 → 𝜕𝑢 and 𝜗𝑢𝑚 → 𝜗𝑢 in 𝐿 (Ω), recall (4.27). We use a method closely related to Friedrichs’ Lemma 5.5 and use the notation from there. Let (𝜒𝜖 )𝜖 be an approximation of the identity and (𝛿𝜈 )𝜈 a sequence of small positive numbers with 𝛿𝜈 → 0, and define Ω𝛿𝜈 = {𝑧 ∈ Ω : 𝑟(𝑧) < −𝛿𝜈 }. Then (Ω𝛿𝜈 )𝜈 is a sequence of relatively compact open subsets of Ω with union equal to Ω. The forms 𝑢𝜖 = 𝑢 ∗ 𝜒𝜖 belong to C∞ (0,𝑞) (Ω𝛿𝜈 ) and 𝑢𝜖 → 𝑢 , 𝜕𝑢𝜖 → 𝜕𝑢 and 𝜗𝑢𝜖 → 𝜗𝑢 in 𝐿2 (Ω𝛿𝜈 ), see Lemma 5.3 and Lemma 5.5. To see that this can be done up to the boundary, we first assume that the domain Ω is star-shaped and 0 ∈ Ω is a center. Let Ω𝜖 = {(1 + 𝜖)𝑧 : 𝑧 ∈ Ω} and 𝑢𝜖 (𝑧) = 𝑢 (

𝑧 ), 1+𝜖

where the dilation is performed for each coefficient of 𝑢. Then Ω ⊂⊂ Ω𝜖 and 𝑢𝜖 ∈ 𝐿2 (Ω𝜖 ). Also, by the dominated convergence theorem, 𝑢𝜖 → 𝑢 , 𝜕𝑢𝜖 → 𝜕𝑢 and 𝜗𝑢𝜖 → 𝜗𝑢 in 𝐿2 (Ω). Now we regularize 𝑢𝜖 defining 𝑢(𝜖) = 𝑢𝜖 ∗ 𝜒𝛿𝜖 ,

(5.15)

where 𝛿𝜖 → 0 as 𝜖 → 0 and 𝛿𝜖 is chosen sufficiently small. Then 𝑢(𝜖) ∈ C∞ (0,𝑞) (Ω) and

𝑢(𝜖) → 𝑢 , 𝜕𝑢(𝜖) → 𝜕𝑢 and 𝜗𝑢(𝜖) → 𝜗𝑢 in 𝐿2 (Ω). Thus, C∞ (0,𝑞) (Ω) is dense in the graph norm when Ω is star-shaped. The general case follows by using a partition of unity since we assume that our domain has at least C2 boundary.

Lemma 5.17. Let Ω be as in Proposition 5.14. Then compactly supported smooth forms ∗ ∗ are dense in dom(𝜕 ) in the graph norm 𝑢 󳨃→ (‖𝑢‖2 + ‖𝜕 𝑢‖2 )1/2 . ∗

Proof. We remark that if 𝑢 ∈ dom(𝜕 ) and if we extend 𝑢 to 𝑢̃ on the whole space ℂ𝑛 by setting 𝑢̃ to be zero outside of Ω, then 𝜗𝑢̃ ∈ 𝐿2 (ℂ𝑛 ) in the distribution sense: in fact ∗ for 𝑢 ∈ dom(𝜕 ) we have ̃ 𝜗𝑢̃ = 𝜗𝑢 ̃ = 𝜗𝑢 in Ω and 𝜗𝑢 ̃ = 0 outside of Ω. This can be checked from the definition where 𝜗𝑢 ∗ ∞ of 𝜕 , since for any 𝑣 ∈ C(0,𝑞−1) (ℂ𝑛 ), ̃ 𝑣) 2 𝑛 . (𝑢,̃ 𝜕𝑣)𝐿2 (ℂ𝑛 ) = (𝑢, 𝜕𝑣)𝐿2 (Ω) = (𝜗𝑢, 𝑣)𝐿2 (Ω) = (𝜗𝑢, 𝐿 (ℂ ) We assume again without loss of generality that Ω is star-shaped with 0 as a center. We first approximate 𝑢̃ by 𝑧 𝑢̃−𝜖 (𝑧) = 𝑢̃ ( ). 1−𝜖

100 | 5 Density of smooth forms Now we have forms 𝑢̃−𝜖 with compact support in Ω and 𝜗𝑢̃−𝜖 → 𝜗𝑢̃ in 𝐿2 (ℂ𝑛 ). Regularizing 𝑢̃−𝜖 as before, we define 𝑢(−𝜖) = 𝑢̃−𝜖 ∗ 𝜒𝛿𝜖 . (5.16) Then the 𝑢(−𝜖) are (0, 𝑞)-forms with coefficients in C∞ 0 (Ω) such that 𝑢(−𝜖) → 𝑢 and 𝜗𝑢(−𝜖) → 𝜗𝑢 in 𝐿2 (Ω). However, compactly supported smooth forms are not dense in dom(𝜕) in the graph norm 𝑢 󳨃→ (‖𝑢‖2 + ‖𝜕𝑢‖2 )1/2 . Nevertheless, we have ∗

Lemma 5.18. Let Ω be as in Proposition 5.14. Then C𝑘(0,𝑞) (Ω)∩dom(𝜕 ) is dense in dom(𝜕)

in the graph norm 𝑢 󳨃→ (‖𝑢‖2 + ‖𝜕𝑢‖2 )1/2 .

Proof. By Lemma 5.16 it suffices to show that for any 𝑢 ∈ C∞ (0,𝑞) (Ω) one can find a se∗

quence 𝑢𝑚 ∈ C𝑘(0,𝑞) (Ω) ∩ dom(𝜕 ) such that 𝑢𝑚 → 𝑢 and 𝜕𝑢𝑚 → 𝜕𝑢 in 𝐿2 (Ω).

Let 𝑟 be a C𝑘+1 defining function such that |𝑑𝑟| = 1 on 𝑏Ω. We now introduce some special vector fields and (1, 0)-forms associated with 𝑏Ω. Near a point 𝑝 ∈ 𝑏Ω we choose fields 𝐿 1 , 𝐿 2 , . . . , 𝐿 𝑛−1 of type (1, 0) that are orthonormal and span 𝑇𝑝1,0 (𝑏Ω). This can be done by choosing a basis, and then using the Gram–Schmidt process. To this collection add 𝐿 𝑛 , the complex normal, normalized to have length 1. So 𝐿 𝑛 is a smooth multiple of 𝑛 𝜕𝑟 𝜕 ∑ . 𝜕𝑧 𝑗 𝜕𝑧𝑗 𝑗=1 Now denote by 𝑤1 , 𝑤2 , . . . , 𝑤𝑛 the (1, 0)-forms such that 𝑤𝑗 (𝐿 𝑘 ) = 𝛿𝑗𝑘 . 𝐿 𝑛 is defined globally, in contrast to 𝐿 1 , . . . , 𝐿 𝑛−1 . The 𝑤𝑗 ’s then form an orthonormal basis for the 𝜕𝑟 (1, 0)-forms near 𝑝. The (1, 0)-form 𝑤𝑛 is a smooth multiple of ∑𝑛𝑗=1 𝜕𝑧 𝑑𝑧𝑗 , and is again 𝑗

globally defined. Taking wedge products of the 𝑤𝑗 ’s yields (local) orthonormal bases for the (1, 0)-forms. We will regularize near a boundary point 𝑝 ∈ 𝑏Ω. Let 𝑈 be a small neighborhood of 𝑝. By a partition of unity, we may assume that Ω ∩ 𝑈 is star-shaped and 𝑢 is supported in 𝑈 ∩ Ω. Shrinking 𝑈 if necessary, we can choose a special boundary chart (𝑡1 , 𝑡2 , . . . , 𝑡2𝑛−1 , 𝑟), where (𝑡1 , 𝑡2 , . . . , 𝑡2𝑛−1 , 0) are coordinates on 𝑏Ω near 𝑝. Let 𝑤1 , . . . , 𝑤𝑛 be an orthonormal basis for the (0, 1)-forms on 𝑈 such that 𝜕𝑟 = 𝑤𝑛 . Let 𝐿 𝑗 = ∑𝑛𝑠=1 𝑎𝑗𝑠 𝜕𝑧𝜕 , 𝑤𝑗 = ∑𝑛𝑠=1 𝑏𝑗𝑠 𝑑𝑧𝑠 , 1 ≤ 𝑗 ≤ 𝑛. Then 𝑠

𝑛

𝑛

𝛿𝑗𝑘 = 𝑤𝑗 (𝐿 𝑘 ) = ∑ 𝑏𝑗𝑠 𝑑𝑧𝑠 ( ∑ 𝑎𝑘ℓ 𝑠=1

ℓ=1

𝑛 𝜕 ) = ∑ 𝑏𝑗𝑠 𝑎𝑘𝑠 . 𝜕𝑧ℓ 𝑠=1

Consequently, if 𝑓 is a function, 𝑛

𝜕𝑓 = ∑ 𝑠=1

𝑛 𝑛 𝜕𝑓 𝑑𝑧𝑠 = ∑ 𝑎𝑠𝑘 (𝐿𝑘 𝑓) 𝑏𝑠𝑗 𝑤𝑗 = ∑ (𝐿𝑗 𝑓) 𝑤𝑗 , 𝜕𝑧𝑠 𝑗=1 𝑗,𝑘,𝑠=1

(5.17)

where the superscripts denote the entries of the inverses of the corresponding matrices ∗ with subscripts. Since multiplication by functions in C1 (Ω) preserves dom(𝜕 ), we may

| 101

5.2 Density in the graph norm

assume that the form 𝑢 is supported in a special boundary chart. So 𝑢 = ∑ 𝑘

󸀠 |𝐽|=𝑞

𝑢𝐽 𝑤𝐽 ,

where 𝑤𝐽 = 𝑤𝑗1 ∧ ⋅ ⋅ ⋅ ∧ 𝑤𝑗𝑞 and each 𝑢𝐽 is a function in C (Ω). Then, in view of (5.17) 󸀠

󸀠

|𝐽|=𝑞

|𝐽|=𝑞

(5.18)

𝜕𝑢 = 𝜕( ∑ 𝑢𝐽 𝑤𝐽 ) = ∑ (𝜕𝑢𝐽 ∧ 𝑤𝐽 + 𝑢𝐽 𝜕𝑤𝐽 ) 󸀠

𝑛

󸀠

(5.19)

= ∑ ∑ (𝐿𝑗 𝑢𝐽 ) 𝑤𝑗 ∧ 𝑤𝐽 + ∑ 𝑢𝐽 𝜕𝑤𝐽 . |𝐽|=𝑞 𝑗=1

|𝐽|=𝑞

Using the special boundary chart we get from (4.26) that ∗

𝑢 ∈ dom(𝜕 ) ⇐⇒ 𝑢𝐽 = 0 on 𝑏Ω when 𝑛 ∈ 𝐽.

(5.20)

Indeed, the only boundary terms that arise when proving (4.26) come from integrating ∫ 𝐿𝑛 𝛼𝐾 𝑢𝑛𝐾 𝑑𝜆 Ω

by parts and they equal ∫ 𝛼𝐾 𝑢𝑛𝐾 (𝐿𝑛 𝑟) 𝑑𝜎. 𝑏Ω

Since 𝛼𝐾 can be arbitrary on 𝑏Ω and 𝐿𝑛 𝑟 ≠ 0 on 𝑏Ω, we conclude that 𝑢𝑛𝐾 = 0 on 𝑏Ω for all 𝐾. To see that the condition is sufficient, note that the computation to prove (4.29) shows that (𝑢, 𝜕𝛼) = (𝜗𝑢, 𝛼) when (4.26) holds and 𝛼 ∈ C∞ (0,𝑞) (Ω). In view of

2 2 1/2 Lemma 5.16 C∞ (0,𝑞) (Ω) is dense in dom(𝜕) in the graph norm 𝑢 󳨃→ (‖𝑢‖ +‖𝜕𝑢‖ ) . Hence, ∗

∗

(𝑢, 𝜕𝛼) = (𝜗𝑢, 𝛼) for all 𝛼 ∈ dom(𝜕), which implies 𝑢 ∈ dom(𝜕 ) and 𝜕 𝑢 = 𝜗𝑢. ∗ These arguments also give a formula for 𝜗 and 𝜕 in special boundary frames: 󸀠

󸀠

𝜗𝑢 = 𝜗( ∑ 𝑢𝐽 𝑤𝐽 ) = − ∑ |𝐽|=𝑞

|𝐾|=𝑞−1

𝑛

( ∑ 𝐿 𝑗 𝑢𝑗𝐾 ) 𝑤𝐾 + 0-th order(𝑢).

(5.21)

𝑗=1

0-th order(u) indicates terms that contain no derivatives of the 𝑢𝐽 ’s. We note that both 𝜕 and 𝜗 are first order differential operators with variable coefficients in C𝑘 (Ω) when computed in the special frame 𝑤1 , . . . , 𝑤𝑛 . We write 𝑢 = 𝑢𝜏 + 𝑢𝜈 , where 𝑢𝜏 =

󸀠

∑

|𝐽|=𝑞,𝑛∉𝐽

𝑢𝐽 𝑤𝐽 , 𝑢𝜈 =

∑

󸀠

|𝐽|=𝑞,𝑛∈𝐽

𝑢𝐽 𝑤𝐽 .

𝑢𝜏 is the complex tangential part of 𝑢, and 𝑢𝜈 is the complex normal part of 𝑢. Our arguments from above imply that ∗

𝑢 ∈ C𝑘(0,𝑞) (Ω) ∩ dom(𝜕 ) ⇐⇒ 𝑢𝜈 = 0 on 𝑏Ω.

102 | 5 Density of smooth forms ∞ For 𝑢 ∈ C∞ (0,𝑞) (Ω) and 𝛼 ∈ C(0,𝑞+1) (Ω) we have by (4.30)

(𝜕𝑢, 𝛼) = (𝑢, 𝜗𝛼) + ∫ ⟨𝜕𝑟 ∧ 𝑢, 𝛼⟩ 𝑑𝜎 𝑏Ω

and 𝜕𝑟 ∧ 𝑢 = 𝜕𝑟 ∧ 𝑢𝜏 on 𝑏Ω, which follows from the representation in special boundary charts: 󸀠 𝜕𝑟 ∧ 𝑢𝜈 = 𝑐𝑤𝑛 ∧ ∑ 𝑢𝐽 𝑤𝐽 = 0. (5.22) |𝐽|=𝑞,𝑛∈𝐽

∗

𝑘 In order to approximate a form 𝑢 ∈ C∞ (0,𝑞) (Ω) by forms in C(0,𝑞) (Ω) ∩ dom(𝜕 ) we 𝜈 only change the complex normal part 𝑢 and leave the complex tangential part 𝑢𝜏 un∗ 𝜏 𝑘 𝜈 changed: for 𝑢 ∈ C∞ (0,𝑞) (Ω) it follows that 𝑢 ∈ C(0,𝑞) (Ω) ∩ dom(𝜕 ) and we denote by 𝑢̃ 𝜈 𝑛 𝜈 the extension of 𝑢 to ℂ by setting 𝑢̃ equal to zero outside of Ω. We approximate 𝑢̃𝜈 as in Lemma 5.17 by 𝜈 𝑢(−𝜖) = (𝑢̃ 𝜈 )−𝜖 ∗ 𝜒𝛿𝜖 . 𝜈 Then 𝑢(−𝜖) is smooth and supported in a compact subset of Ω ∩ 𝑈. By this, we approxi𝜈 𝜈 2 𝜈 mate 𝑢 by 𝑢(−𝜖) ∈ C∞ 0 (Ω ∩ 𝑈) in the 𝐿 -norm. Furthermore, by extending 𝜕𝑢 to be zero ̃ outside Ω ∩ 𝑈 and denoting the extension by 𝜕𝑢𝜈 , we have

̃ 𝜕𝑢̃𝜈 = 𝜕𝑢𝜈 in 𝐿2 (ℂ𝑛 ) in the sense of distributions. This follows from (4.30) and (5.22), since 𝑢𝜈 ∈ 𝑛 C𝑘(0,𝑞) (Ω) and for 𝛼 ∈ C∞ (0,𝑞+1) (ℂ ) we have ̃ (𝑢̃𝜈 , 𝜗𝛼)𝐿2 (ℂ𝑛 ) = (𝜕𝑢𝜈 , 𝛼)𝐿2 (Ω) − ∫ ⟨𝜕𝑟 ∧ 𝑢𝜈 , 𝛼⟩ 𝑑𝜎 = (𝜕𝑢𝜈 , 𝛼)𝐿2 (ℂ𝑛 ) . 𝑏Ω

Since 𝜕 is a first order differential operator with variable coefficients, we get from Friedrichs’ Lemma 5.6 𝜈 𝜕𝑢(−𝜖) → 𝜕𝑢̃𝜈 in 𝐿2 (ℂ𝑛 ). (5.23) We set 𝜈 𝑢(−𝜖) = 𝑢𝜏 + 𝑢(−𝜖) . ∗

𝜈 and 𝑤𝑗 is in It follows that 𝑢(−𝜖) ∈ C𝑘(0,𝑞) (Ω) ∩ dom(𝜕 ), since each coefficient of 𝑢𝜏 , 𝑢(−𝜖) ∗

C𝑘 (Ω ∩ 𝑈). Therefore we get 𝑢(−𝜖) ∈ C𝑘(0,𝑞) (Ω) ∩ dom(𝜕 ) and 𝑢(−𝜖) → 𝑢 in 𝐿2 (Ω).

To see that 𝜕𝑢(−𝜖) → 𝜕𝑢 in the 𝐿2 (Ω)-norm, using (5.23), we find that 𝜈 𝜕𝑢(−𝜖) = 𝜕𝑢𝜏 + 𝜕𝑢(−𝜖) → 𝜕𝑢

in 𝐿2 (Ω) as 𝜖 → 0.

5.3 Notes

| 103 ∗

To finish the proof of Proposition 5.14 we consider an arbitrary 𝑢 ∈ dom(𝜕) ∩ dom(𝜕 ) and use a partition of unity and the same notation as before to regularize 𝑢 in each small star-shaped neighborhood near the boundary. We regularize the complex tangential and normal part separately by setting 𝜏 𝜈 𝑢((𝜖)) = 𝑢(𝜖) + 𝑢(−𝜖) ,

this means that we first consider 𝑢(𝜖) as it was defined in (5.15) and take then the tan𝜏 gential components 𝑢(𝜖) , then we consider 𝑢(−𝜖) as it is defined in (5.16) and then take 𝜈 𝜈 the normal components 𝑢(−𝜖) . It follows that for sufficiently small 𝜖 > 0, 𝑢(−𝜖) has coef∞ 𝜏 ∞ ficients in C0 (Ω) and 𝑢(𝜖) has coefficients in C (Ω). ∗

Thus we see that 𝑢((𝜖)) ∈ C𝑘(0,𝑞) (Ω) ∩ dom(𝜕 ). We get from Lemma 5.16 that 𝑢(𝜖) → 𝑢 ∗

𝜏 → 𝑢𝜏 in the graph norm 𝑢 󳨃→ in the graph norm 𝑢 󳨃→ (‖𝑢‖2 +‖𝜕𝑢‖2 +‖𝜕 𝑢‖2 )1/2 , hence 𝑢(𝜖) ∗

(‖𝑢‖2 + ‖𝜕𝑢‖2 + ‖𝜕 𝑢‖2 )1/2 . From Lemma 5.17 we obtain 𝑢(−𝜖) → 𝑢 in the graph norm 𝑢 󳨃→ ∗

∗

𝜈 (‖𝑢‖2 + ‖𝜕 𝑢‖2 )1/2 , hence 𝑢(−𝜖) → 𝑢𝜈 in the graph norm 𝑢 󳨃→ (‖𝑢‖2 + ‖𝜕 𝑢‖2 )1/2 . Finally, 𝜈 we use Lemma 5.18, in particular formula (5.23), and see that 𝜕𝑢(−𝜖) → 𝜕𝑢̃𝜈 in 𝐿2 (ℂ𝑛 ), ∗

hence 𝑢((𝜖)) → 𝑢 in the graph norm 𝑢 󳨃→ (‖𝑢‖2 + ‖𝜕𝑢‖2 + ‖𝜕 𝑢‖2 )1/2 . ∗

∗

This shows that C𝑘(0,𝑞) (Ω) ∩ dom(𝜕 ) is dense in dom(𝜕) ∩ dom(𝜕 ) in the graph ∗

norm 𝑢 󳨃→ (‖𝑢‖2 + ‖𝜕𝑢‖2 + ‖𝜕 𝑢‖2 )1/2 .

5.3 Notes Friedrichs’ Lemma can be found in [12], it will also be important in the following chapters. In the proof of the density result we mainly follow the presentation in [12], E. Straube [71] uses a different approach for the same result, whereas L. Hörmander [43] proves a corresponding density result in weighted 𝐿2 -spaces.

6 The weighted 𝜕-complex In this chapter we investigate the basic properties of the 𝜕-operator on weighted 𝐿2 spaces on the whole of ℂ𝑛 . It will turn out that the behavior of the 𝜕-operator depends very much on properties of the Levi-matrix of the weight function. For 𝑛 = 1, there is an interesting connection between 𝜕 and the theory of Schrödinger operators with magnetic fields, for 𝑛 ≥ 2, the corresponding concept is the Witten Laplacian. The corresponding density result from the last chapter is much easier to handle in the weighted case, therefore also the basic estimate and existence of the 𝜕-Neumann operator.

6.1 The 𝜕-Neumann operator on (0, 1)-forms Let 𝜑 : ℂ𝑛 󳨀→ ℝ+ be a plurisubharmonic C2 -weight function and define the space 𝐿2 (ℂ𝑛 , 𝑒−𝜑 ) = {𝑓 : ℂ𝑛 󳨀→ ℂ : ∫ |𝑓|2 𝑒−𝜑 𝑑𝜆 < ∞}, ℂ𝑛

where 𝜆 denotes the Lebesgue measure, the space 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑 ) of (0, 1)-forms with coefficients in 𝐿2 (ℂ𝑛 , 𝑒−𝜑 ) and the space 𝐿2(0,2) (ℂ𝑛 , 𝑒−𝜑 ) of (0, 2)-forms with coefficients in 𝐿2 (ℂ𝑛 , 𝑒−𝜑 ). Let (𝑓, 𝑔)𝜑 = ∫ 𝑓 𝑔𝑒−𝜑 𝑑𝜆 ℂ𝑛

denote the inner product and ‖𝑓‖2𝜑 = ∫ |𝑓|2 𝑒−𝜑 𝑑𝜆 ℂ𝑛

the norm in 𝐿2 (ℂ𝑛 , 𝑒−𝜑 ). We define dom(𝜕) to be the space of all functions 𝑓 ∈ 𝐿2 (ℂ𝑛 , 𝑒−𝜑 ) such that 𝜕𝑓, in the sense of distributions, belongs to 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑 ), and consider the weighted 𝜕complex 𝜕

𝜕

←󳨀

←󳨀

𝐿2 (ℂ𝑛 , 𝑒−𝜑 ) 󳨀→ 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑 ) 󳨀→ 𝐿2(0,2) (ℂ𝑛 , 𝑒−𝜑 ), ∗ 𝜕𝜑

(6.1)

∗ 𝜕𝜑

∗

where 𝜕𝜑 is the adjoint operator to 𝜕 with respect to the weighted inner product. For a ∗

smooth (0, 1)-form 𝑢 = ∑𝑛𝑗=1 𝑢𝑗 𝑑𝑧𝑗 ∈ dom(𝜕𝜑 ) one has ∗

𝑛

𝜕𝜑 𝑢 = − ∑ ( 𝑗=1

𝜕𝜑 𝜕 ) 𝑢𝑗 . − 𝜕𝑧𝑗 𝜕𝑧𝑗

The complex Laplacian on (0, 1)-forms is defined as ∗

∗

◻𝜑 := 𝜕 𝜕𝜑 + 𝜕𝜑 𝜕,

(6.2)

6.1 The 𝜕-Neumann operator on (0, 1)-forms

| 105

and dom(◻𝜑 ) is the space of all 𝑓 ∈ 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑 ) such that ∗

𝑓 ∈ dom(𝜕) ∩ dom(𝜕𝜑 ) ∗

∗

and 𝜕𝑓 ∈ dom(𝜕𝜑 ) and 𝜕𝜑 𝑓 ∈ dom(𝜕). ◻𝜑 is a self-adjoint and positive operator (see Chapter 4), which means that (◻𝜑 𝑓, 𝑓)𝜑 ≥ 0 , for 𝑓 ∈ dom(◻𝜑 ). The associated Dirichlet form is denoted by ∗

∗

(6.3)

𝑄𝜑 (𝑓, 𝑔) = (𝜕𝑓, 𝜕𝑔)𝜑 + (𝜕𝜑 𝑓, 𝜕𝜑 𝑔)𝜑 , ∗

for 𝑓, 𝑔 ∈ dom(𝜕) ∩ dom(𝜕𝜑 ). The weighted 𝜕-Neumann operator 𝑁𝜑 is – if it exists the bounded inverse of ◻𝜑 . In the weighted space 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑 ) we can give a simple characterization of ∗

dom (𝜕𝜑 ) : ∗

Proposition 6.1. Let 𝑓 = ∑ 𝑓𝑗 𝑑𝑧𝑗 ∈ 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑 ). Then 𝑓 ∈ dom(𝜕𝜑 ) if and only if 𝑛

𝑒𝜑 ∑ 𝑗=1

𝜕 (𝑓 𝑒−𝜑 ) ∈ 𝐿2 (ℂ𝑛 , 𝑒−𝜑 ), 𝜕𝑧𝑗 𝑗

(6.4)

where the derivative is to be taken in the sense of distributions. Proof. Suppose first that 𝑒𝜑 ∑𝑛𝑗=1

𝜕 (𝑓 𝑒−𝜑 ) 𝜕𝑧𝑗 𝑗

∈ 𝐿2 (ℂ𝑛 , 𝑒−𝜑 ). We have to show that there

exists a constant 𝐶 such that |(𝜕𝑔, 𝑓)𝜑 | ≤ 𝐶‖𝑔‖𝜑 for all 𝑔 ∈ dom(𝜕). To this end let (𝜒𝑅 )𝑅∈ℕ be a family of radially symmetric smooth cut-off functions, which are identically one on 𝔹𝑅 , the ball with radius 𝑅, such that the support of 𝜒𝑅 is contained in 𝔹𝑅+1 , 𝑠𝑢𝑝𝑝(𝜒𝑅 ) ⊂ 𝔹𝑅+1 , and such that furthermore all first order derivatives of all functions 𝑛 in this family are uniformly bounded by a constant 𝑀. Then for all 𝑔 ∈ C∞ 0 (ℂ ): 𝑛

(𝜕𝑔, 𝜒𝑅 𝑓)𝜑 = ∑ ( 𝑗=1

𝑛 𝜕𝑔 𝜕 , 𝜒𝑅 𝑓𝑗 ) = − ∫ ∑ 𝑔 (𝜒𝑅 𝑓𝑗 𝑒−𝜑 )𝑑𝜆, 𝜕𝑧𝑗 𝜕𝑧 𝑗 𝜑 𝑛 𝑗=1 ℂ

by integration by parts, which in particular means 󵄨󵄨 󵄨󵄨 𝑛 𝜕 󵄨 󵄨 (𝜒𝑅 𝑓𝑗 𝑒−𝜑 ) 𝑑𝜆󵄨󵄨󵄨󵄨. |(𝜕𝑔, 𝑓)𝜑 | = lim |(𝜕𝑔, 𝜒𝑅 𝑓)𝜑 | = lim 󵄨󵄨󵄨󵄨 ∫ ∑ 𝑔 𝑅→∞ 𝑅→∞ 󵄨 󵄨󵄨 󵄨 ℂ𝑛 𝑗=1 𝜕𝑧𝑗 Now we use the triangle inequality, afterwards Cauchy–Schwarz, to get 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝑛 𝑛 𝑛 𝜕𝜒 𝜕 𝜕 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 −𝜑 −𝜑 󵄨󵄨 ≤󵄨󵄨 ∫ 𝜒𝑅 𝑔 ∑ 󵄨󵄨 + 󵄨󵄨 ∫ ∑ 𝑓𝑗 𝑔 𝑅 𝑒−𝜑 𝑑𝜆󵄨󵄨󵄨 󵄨󵄨 ∫ ∑ 𝑔 (𝜒 𝑓 𝑒 ) 𝑑𝜆 (𝑓 𝑒 ) 𝑑𝜆 𝑅 𝑗 𝑗 󵄨 󵄨󵄨 󵄨 𝜕𝑧 𝜕𝑧 󵄨󵄨󵄨 󵄨󵄨󵄨 𝑛 󵄨 󵄨 󵄨󵄨󵄨 𝑛 𝑗=1 𝜕𝑧𝑗 𝑗 𝑗 󵄨 󵄨 ℂ𝑛 𝑗=1 󵄨 𝑗=1 ℂ ℂ 󵄩󵄩 𝑛 󵄩󵄩 𝜕 󵄩 󵄩 (𝑓 𝑒−𝜑 )󵄩󵄩󵄩󵄩 + 𝑀‖𝑔‖𝜑 ‖𝑓‖𝜑 ≤‖𝜒𝑅 𝑔‖𝜑 󵄩󵄩󵄩󵄩𝑒𝜑 ∑ 󵄩󵄩 𝑗=1 𝜕𝑧𝑗 𝑗 󵄩󵄩𝜑 󵄩󵄩 𝑛 󵄩󵄩 𝜕 󵄩 󵄩 ≤‖𝑔‖𝜑 󵄩󵄩󵄩󵄩𝑒𝜑 ∑ (𝑓𝑗 𝑒−𝜑 )󵄩󵄩󵄩󵄩 + 𝑀‖𝑔‖𝜑 ‖𝑓‖𝜑 . 󵄩󵄩 𝑗=1 𝜕𝑧𝑗 󵄩󵄩𝜑

106 | 6 The weighted 𝜕-complex Hence by assumption, 󵄩󵄩 𝑛 󵄩󵄩 𝜕 󵄩 󵄩 |(𝜕𝑔, 𝑓)𝜑 | ≤ ‖𝑔‖𝜑 󵄩󵄩󵄩󵄩𝑒𝜑 ∑ (𝑓𝑗 𝑒−𝜑 )󵄩󵄩󵄩󵄩 + 𝑀‖𝑔‖𝜑 ‖𝑓‖𝜑 ≤ 𝐶‖𝑔‖𝜑 󵄩󵄩 𝑗=1 𝜕𝑧𝑗 󵄩󵄩𝜑 𝑛 ∞ 𝑛 for all 𝑔 ∈ C∞ 0 (ℂ ), and by density of C0 (ℂ ) this is true for all 𝑔 ∈ dom(𝜕). ∗ Conversely, let 𝑓 ∈ dom(𝜕𝜑 ), which means that there exists a uniquely determined ∗

element 𝜕𝜑 𝑓 ∈ 𝐿2 (ℂ𝑛 , 𝑒−𝜑 ) such that for each 𝑔 ∈ dom(𝜕) we have ∗

(𝜕𝑔, 𝑓)𝜑 = (𝑔, 𝜕𝜑 𝑓)𝜑 . 𝑛 Now take 𝑔 ∈ C∞ 0 (ℂ ). Then 𝑔 ∈ dom(𝜕) and ∗

(𝑔, 𝜕𝜑 𝑓)𝜑 = (𝜕𝑔, 𝑓)𝜑 𝑛

= ∑( 𝑗=1

𝜕𝑔 ,𝑓 ) 𝜕𝑧𝑗 𝑗 𝜑 𝑛

𝜕 (𝑓𝑗 𝑒−𝜑 )) 𝜕𝑧 𝑗 𝑗=1 𝐿2

= −(𝑔, ∑

𝑛

= −(𝑔, 𝑒𝜑 ∑ 𝑗=1

Since

𝑛 C∞ 0 (ℂ )

2

𝑛

𝜕 (𝑓 𝑒−𝜑 )) . 𝜕𝑧𝑗 𝑗 𝜑

−𝜑

is dense in 𝐿 (ℂ , 𝑒 ), we conclude that 𝑛

∗

𝜕 (𝑓𝑗 𝑒−𝜑 ), 𝜕𝑧 𝑗 𝑗=1

𝜕𝜑 𝑓 = −𝑒𝜑 ∑ which in particular implies that 𝑒𝜑 ∑𝑛𝑗=1

𝜕 (𝑓 𝑒−𝜑 ) 𝜕𝑧𝑗 𝑗

∈ 𝐿2 (ℂ𝑛 , 𝑒−𝜑 ).

The following lemma will be important for our considerations. ∗

𝑛 Lemma 6.2. Forms with coefficients in C∞ 0 (ℂ ) are dense in dom(𝜕) ∩ dom(𝜕𝜑 ) in the

graph norm 𝑓 󳨃→ (‖𝑓‖2𝜑 + ‖𝜕𝑓‖2𝜑 +

1 ∗ ‖𝜕𝜑 𝑓‖2𝜑 ) 2 .

Proof. First we show that compactly supported 𝐿2 -forms are dense in the graph norm. So let {𝜒𝑅 }𝑅∈ℕ be a family of smooth radially symmetric cut-offs identically one on 𝔹𝑅 and supported in 𝔹𝑅+1 , such that all first order derivatives of the functions in this family are uniformly bounded in 𝑅 by a constant 𝑀. ∗ ∗ Let 𝑓 ∈ dom(𝜕) ∩ dom(𝜕𝜑 ). Then, clearly, 𝜒𝑅 𝑓 ∈ dom(𝜕) ∩ dom(𝜕𝜑 ) and 𝜒𝑅 𝑓 → 𝑓 in 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑 ) as 𝑅 → ∞. As observed in Proposition 6.1, we have ∗

𝑛

𝜕 (𝑓𝑗 𝑒−𝜑 ), 𝜕𝑧 𝑗 𝑗=1

𝜕𝜑 𝑓 = −𝑒𝜑 ∑ hence ∗

𝑛

𝜕 (𝜒𝑅 𝑓𝑗 𝑒−𝜑 ). 𝜕𝑧 𝑗 𝑗=1

𝜕𝜑 (𝜒𝑅 𝑓) = −𝑒𝜑 ∑

6.1 The 𝜕-Neumann operator on (0, 1)-forms

| 107

We need to estimate the difference of these expressions ∗

∗

∗

∗

𝑛

𝜕𝜑 𝑓 − 𝜕𝜑 (𝜒𝑅 𝑓) = 𝜕𝜑 𝑓 − 𝜒𝑅 𝜕𝜑 𝑓 + ∑

𝑗=1

𝜕𝜒𝑅 𝑓, 𝜕𝑧𝑗 𝑗

which is by the triangle inequality ∗

∗

∗

∗

𝑛

‖𝜕𝜑 𝑓 − 𝜕𝜑 (𝜒𝑅 𝑓)‖𝜑 ≤‖𝜕𝜑 𝑓 − 𝜒𝑅 𝜕𝜑 𝑓‖𝜑 + 𝑀 ∑ ∫ |𝑓𝑗 |2 𝑒−𝜑 𝑑𝜆. 𝑗=1

ℂ𝑛 \𝔹𝑅

Now both terms tend to 0 as 𝑅 → ∞, and one can see similarly that also 𝜕(𝜒𝑅 𝑓) → 𝜕𝑓 as 𝑅 → ∞. So we have density of compactly supported forms in the graph norm, and density 𝑛 of forms with coefficients in C∞ 0 (ℂ ) will follow by applying Friedrichs’ Lemma 5.6. As in the case of bounded domains, the canonical solution operator to 𝜕, which we ∗ denote by 𝑆𝜑 , is given by 𝜕𝜑 𝑁𝜑 . Existence and compactness of 𝑁𝜑 and 𝑆𝜑 are closely related. At first, we notice that equivalent weight functions have the same properties in this regard. Lemma 6.3. Let 𝜑1 and 𝜑2 be two equivalent weights, i.e., 𝐶−1 ‖.‖𝜑1 ≤ ‖.‖𝜑2 ≤ 𝐶‖.‖𝜑1 for some 𝐶 > 0. Suppose that 𝑆𝜑2 exists. Then 𝑆𝜑1 also exists and 𝑆𝜑1 is compact if and only if 𝑆𝜑2 is compact. An analog statement is true for the weighted 𝜕-Neumann operator. Proof. Let 𝜄 be the identity 𝜄 : 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑1 ) → 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑2 ), 𝜄𝑓 = 𝑓, let 𝜅 be the identity 𝜅 : 𝐿2 (ℂ𝑛 , 𝑒−𝜑2 ) → 𝐿2 (ℂ𝑛 , 𝑒−𝜑1 ). Since the weights are equivalent, 𝜄 and 𝜅 are continuous, so if 𝑆𝜑2 is compact, 𝜅 ∘ 𝑆𝜑2 ∘ 𝜄 gives a solution operator on 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑1 ) that is compact. Therefore the canonical solution operator 𝑆𝜑1 = (𝐼 − 𝑃1 ) ∘ 𝜅 ∘ 𝑆𝜑2 ∘ 𝜄 is also compact, where 𝑃1 denotes the orthogonal projection of 𝐿2 (ℂ𝑛 , 𝑒−𝜑1 ) onto ker𝜕. Since the problem is symmetric in 𝜑1 and 𝜑2 , we are done. The assertion for the Neumann operator follows by the identity 𝑁𝜑 = 𝑆𝜑 𝑆𝜑∗ + 𝑆𝜑∗ 𝑆𝜑 . For the rest of this chapter we will suppose that 𝑛 ≥ 2. The case 𝑛 = 1 will be discussed separately in Chapter 10. Now we suppose that the lowest eigenvalue 𝜇𝜑 of the Levi matrix 𝑀𝜑 = (

𝜕2 𝜑 ) 𝜕𝑧𝑗 𝜕𝑧𝑘 𝑗𝑘

of 𝜑 satisfies lim inf 𝜇𝜑 (𝑧) > 0. |𝑧|→∞

(6.5)

Then, by Lemma 6.3, we may assume without loss of generality that 𝜇𝜑 (𝑧) > 𝜖 for some 𝜖 > 0 and all 𝑧 ∈ ℂ𝑛 , since changing the weight function on a compact set does not influence our considerations. So we have the following basic estimate

108 | 6 The weighted 𝜕-complex Proposition 6.4. For a plurisubharmonic weight function 𝜑 satisfying (6.5), there is a 𝐶 > 0 such that ∗ ‖𝑢‖2𝜑 ≤ 𝐶(‖𝜕𝑢‖2𝜑 + ‖𝜕𝜑 𝑢‖2𝜑 ) (6.6) ∗

for each (0, 1)-form 𝑢 ∈dom (𝜕) ∩ dom (𝜕𝜑 ). Proof. By Lemma 6.2 and the assumption on 𝜑 it suffices to show that ∗ 𝜕2 𝜑 𝑢𝑗 𝑢𝑘 𝑒−𝜑 𝑑𝜆 ≤ ‖𝜕𝑢‖2𝜑 + ‖𝜕𝜑 𝑢‖2𝜑 , 𝜕𝑧 𝜕𝑧 𝑗 𝑘 𝑗,𝑘=1 𝑛

∫ ∑ ℂ𝑛

(6.7)

𝑛 for each (0, 1)-form 𝑢 = ∑𝑛𝑘=1 𝑢𝑘 𝑑𝑧𝑘 with coefficients 𝑢𝑘 ∈ C∞ 0 (ℂ ), for 𝑘 = 1, . . . , 𝑛. 𝜕𝜑 𝜕 For this purpose we set 𝛿𝑘 = 𝜕𝑧 − 𝜕𝑧 and get since 𝑘

𝜕𝑢 = ∑ ( 𝑗 0. |𝑧|→∞

Then there exists a uniquely determined bounded linear operator 𝑁𝜑,𝑞 : 𝐿2(0,𝑞) (ℂ𝑛 , 𝑒−𝜑 ) 󳨀→ 𝐿2(0,𝑞) (ℂ𝑛 , 𝑒−𝜑 ), such that ◻𝜑 ∘ 𝑁𝜑,𝑞 𝑢 = 𝑢, for any 𝑢 ∈ 𝐿2(0,𝑞) (ℂ𝑛 , 𝑒−𝜑 ). Proof. Let 𝜇𝜑,1 ≤ 𝜇𝜑,2 ≤ ⋅ ⋅ ⋅ ≤ 𝜇𝜑,𝑛 denote the eigenvalues of 𝑀𝜑 and suppose that 𝑀𝜑 is diagonalized. Then, in a suitable basis, 󸀠

∑

|𝐾|=𝑞−1

𝑛 𝜕2 𝜑 󸀠 𝑢𝑗𝐾 𝑢𝑘𝐾 = ∑ ∑ 𝜇𝜑,𝑗 |𝑢𝑗𝐾 |2 𝜕𝑧𝑗 𝜕𝑧𝑘 𝑗,𝑘=1 |𝐾|=𝑞−1 𝑗=1 𝑛

∑

=

󸀠

∑

𝐽=(𝑗1 ,...,𝑗𝑞 )

(𝜇𝜑,𝑗1 + ⋅ ⋅ ⋅ + 𝜇𝜑,𝑗𝑞 )|𝑢𝐽 |2

≥ 𝑠𝑞 |𝑢|2 The last equality results as follows: for 𝐽 = (𝑗1 , . . . , 𝑗𝑞 ) fixed, |𝑢𝐽 |2 occurs precisely 𝑞 times in the second sum, once as |𝑢𝑗1 𝐾1 |2 , once as |𝑢𝑗2 𝐾2 |2 , etc. At each occurrence, it is multiplied by 𝜇𝜑,𝑗ℓ . For the rest of the proof we use (6.11) and proceed as in the proof of Proposition 6.6. ̄ Remark 6.9. For the 𝜕-Neumann operator 𝑁𝜑,𝑞 on (0, 𝑞)-forms one obtains in a similar ∗ way as in Proposition 6.4 that 𝑁𝜑,𝑞 = 𝑗𝜑,𝑞 ∘ 𝑗𝜑,𝑞 , where ∗

𝑗𝜑,𝑞 : dom (𝜕) ∩ dom (𝜕𝜑 ) 󳨀→ 𝐿2(0,𝑞) (ℂ𝑛 , 𝑒−𝜑 ), ∗

∗

and dom (𝜕) ∩ dom (𝜕𝜑 ) is endowed with the graph norm (‖𝜕𝑢‖2𝜑 + ‖𝜕𝜑 𝑢‖2𝜑 )1/2 . It also follows in this case that both 𝜕 and tion 4.57.

∗ 𝜕𝜑

have closed range, see Proposi-

112 | 6 The weighted 𝜕-complex

6.3 Notes Most of the material in this chapter is taken from [27, 28, 34]. It will be fundamental for a closer analysis of the 𝜕-Neumann operator and its connection to Schrödinger operators with magnetic fields and the Witten–Laplacian. We indicate that formula (6.11) has an interesting differential geometric interpretation: let (𝑀, 𝜔) be a Kähler manifold with fundamental form 𝜔 and (𝐸, 𝑀, 𝜋) a holomorphic vector bundle over 𝑀. Let ∇ : Γ(𝑇ℂ (𝑀)) × Γ(𝐸) 󳨀→ Γ(𝐸) be the uniquely determined connection on 𝐸 that is both holomorphic and compatible with the metric. The operator Θ := ∇2 is called the curvature of the connection ∇. We consider the weighted 𝜕-complex on ℂ𝑛 with fundamental form 𝑛

𝜔 = 𝑖 ∑ 𝑑𝑧𝑘 ∧ 𝑑𝑧𝑘 . 𝑘=1

The weight factor 𝑒 and

−𝜑

can be interpreted as a metric on the trivial line bundle over ℂ𝑛 𝜕2 𝜑 𝑑𝑧 ∧ 𝑑𝑧𝑘 , 𝜕𝑧𝑗 𝜕𝑧𝑘 𝑗 𝑗,𝑘=1 𝑛

Θ = 𝜕𝜕𝜑 = ∑

see [74] for the details. Let Λ denote the interior multiplication with the fundamental form 𝜔 : (Λ𝛼, 𝑤) = (𝛼, 𝜔 ∧ 𝑤), for suitable differential forms 𝛼 and 𝑤 and let 𝑢 = ∑󸀠|𝐽|=𝑞 𝑢𝐽 𝑑𝑧𝐽 be a (0, 𝑞)-form with 𝑛 coefficients in C∞ 0 (ℂ ). We want to interpret the term ∑

󸀠

𝑛

∑ ∫

|𝐾|=𝑞−1 𝑗,𝑘=1 ℂ𝑛

𝜕2 𝜑 𝑢 𝑢 𝑒−𝜑 𝑑𝜆 𝜕𝑧𝑗 𝜕𝑧𝑘 𝑗𝐾 𝑘𝐾

of (6.11) by the curvature Θ and the operator Λ. For this purpose we consider (𝑛, 𝑞)forms 󸀠 𝜉 = ∑ 𝜉𝐼 𝑑𝑧 ∧ 𝑑𝑧𝐼 , |𝐼|=𝑞

instead of (0, 𝑞)-forms, where 𝑑𝑧 = 𝑑𝑧1 ∧ ⋅ ⋅ ⋅ ∧ 𝑑𝑧𝑛 . We use the notation ̂ ∧ ⋅ ⋅ ⋅ ∧ 𝑑𝑧 , 𝑑𝑧̂𝑗 := 𝑑𝑧1 ∧ ⋅ ⋅ ⋅ ∧ 𝑑𝑧 𝑗 𝑛 which means that 𝑑𝑧𝑗 is excluded. It follows that 𝑛

󸀠

Λ𝜉 = 𝑖 ∑ ∑ 𝜉𝑗𝐽 𝑑𝑧̂𝑗 ∧ 𝑑𝑧𝐽 𝑗=1 |𝐽|=𝑞−1

6.3 Notes

| 113

and since Θ𝜉 = 0 we obtain for the commutator [Θ, Λ] that (𝑖[Θ, Λ]𝜉, 𝜉)𝜑 = ∑

󸀠

𝑛

∑ ∫

|𝐽|=𝑞−1 𝑗,𝑘=1 ℂ𝑛

𝜕2 𝜑 𝜉 𝜉 𝑒−𝜑 𝑑𝜆. 𝜕𝑧𝑗 𝜕𝑧𝑘 𝑗𝐽 𝑘𝐽

The commutator [Θ, Λ] appears in the Nakano vanishing theorem, see [74].

7 The twisted 𝜕-complex We will consider the twisted 𝜕-complex 𝑇

𝑆

𝐿2 (Ω) 󳨀→ 𝐿2(0,1) (Ω) 󳨀→ 𝐿2(0,2) (Ω)

(7.1)

for operators 𝑇 = 𝜕 ∘ √𝜏 and 𝑆 = √𝜏 ∘ 𝜕, where 𝜏 ∈ C2 (Ω) and 𝜏 > 0 on Ω. It will turn out that this approach leads to generalized basic estimates which are seminal for the applications which are explained in the next chapter. The key method consists of a keen use of integration by parts.

7.1 An exact sequence of unbounded operators First we prove a general result about operators like 𝑇 and 𝑆. Proposition 7.1. Let 𝐻1 , 𝐻2 , 𝐻3 be Hilbert spaces and 𝑇 : 𝐻1 󳨀→ 𝐻2 and 𝑆 : 𝐻2 󳨀→ 𝐻3 densely defined linear operators, such that 𝑆(𝑇(𝑓)) = 0, for each 𝑓 ∈ dom(𝑇), and let 𝑃 : 𝐻2 󳨀→ 𝐻2 be a positive invertible operator such that ‖𝑃𝑢‖22 ≤ ‖𝑇∗ 𝑢‖21 + ‖𝑆𝑢‖23 ,

(7.2)

for all 𝑢 ∈ dom(𝑆) ∩ dom(𝑇∗ ), where dom(𝑇∗ ) = {𝑢 ∈ 𝐻2 : |(𝑢, 𝑇𝑓)2 | ≤ 𝐶 ‖𝑓‖1 , for all 𝑓 ∈ dom(𝑇)}. Suppose (7.2) holds and let 𝛼 ∈ 𝐻2 , such that 𝑆𝛼 = 0. Then there exists 𝜎 ∈ 𝐻1 , such that (i) 𝑇(𝜎) = 𝛼 and (ii) ‖𝜎‖21 ≤ ‖𝑃−1 𝛼‖22 . Proof. Since 𝑃 is positive, it follows that 𝑃 = 𝑃∗ . Now let 𝛼 ∈ 𝐻2 be such that 𝑆𝛼 = 0. We consider the linear functional 𝑇∗ 𝑢 󳨃→ (𝑢, 𝛼)2 for 𝑢 ∈ dom(𝑇∗ ). We show that this linear functional is well defined and continuous. If 𝑢 ∈ ker𝑆, then |(𝑢, 𝛼)2 | = |(𝑃𝑢, 𝑃−1 𝛼)2 | ≤ ‖𝑃𝑢‖2 ‖𝑃−1 𝛼‖2 ≤ (‖𝑇∗ 𝑢‖21 + ‖𝑆𝑢‖23 )1/2 ‖𝑃−1 𝛼‖2 = ‖𝑇∗ 𝑢‖1 ‖𝑃−1 𝛼‖2 , if 𝑢⊥2 ker𝑆, then (𝑢, 𝛼)2 = 0. In addition we have that 𝑇∗ 𝑤 = 0 for all 𝑤⊥2 ker𝑆, this follows from the assumption that 𝑇𝑓 ∈ ker𝑆, so 0 = (𝑤, 𝑇𝑓)2 ≤ 𝐶‖𝑓‖1 , which means that 𝑤 ∈ dom(𝑇∗ ) and 𝑇∗ 𝑤 = 0, since (𝑇∗ 𝑤, 𝑓)1 = (𝑤, 𝑇𝑓)2 = 0 for all 𝑓 ∈ dom(𝑇). If 𝑇∗ 𝑢 = 0, it follows from the above estimate that (𝑢, 𝛼)2 = 0. We apply the Hahn–Banach Theorem, where we keep the constant for the estimate of the functional and the Riesz representation theorem to get 𝜎 ∈ 𝐻1 , such that (𝑇∗ 𝑢, 𝜎)1 = (𝑢, 𝛼)2 , which implies that (𝑢, 𝑇𝜎)2 = (𝑢, 𝛼)2 . Hence 𝑇𝜎 = 𝛼 and, again by the above estimate ‖𝜎‖1 ≤ ‖𝑃−1 𝛼‖2 .

7.2 The twisted basic estimates

|

115

7.2 The twisted basic estimates Let Ω be a smoothly bounded pseudoconvex domain in ℂ𝑛 , with defining function 𝑟 such that |∇𝑟(𝑧)| = 1 on 𝑏Ω. Let 𝜏 ∈ C2 (Ω) and 𝜏 > 0 on Ω. For 𝑓 ∈ C∞ (Ω) we define 𝑛

𝜕 (√𝜏𝑓) 𝑑𝑧𝑘 , 𝜕𝑧 𝑘 𝑘=1

𝑇𝑓 = (𝜕 ∘ √𝜏)𝑓 = ∑

(7.3)

and for 𝑢 = ∑𝑛𝑗=1 𝑢𝑗 𝑑𝑧𝑗 with coefficients 𝑢𝑗 in C∞ (Ω), we will write 𝑢 ∈ Λ0,1 (Ω), we define 𝜕𝑢𝑗 𝜕𝑢 𝑆𝑢 = ∑ √𝜏 ( 𝑘 − ) 𝑑𝑧𝑗 ∧ 𝑑𝑧𝑘 . (7.4) 𝜕𝑧𝑗 𝜕𝑧𝑘 𝑗 0 on Ω and let 𝑢 ∈ D0,1 . Then ∗

‖√𝜏 + 𝐴 𝜕𝜑 𝑢‖2𝜑 + ‖√𝜏 𝜕𝑢‖2𝜑 ≥ ‖√𝜏𝑢‖2𝜑,𝑧 + ∫ Θ(𝑢, 𝑢) 𝑒−𝜑 𝑑𝜆 + ∫ 𝜏𝑖𝜕𝜕𝑟(𝑢, 𝑢) 𝑒−𝜑 𝑑𝜎, Ω

where Θ(𝑢, 𝑢) = 𝜏𝑖𝜕𝜕𝜑(𝑢, 𝑢) − 𝑖𝜕𝜕𝜏(𝑢, 𝑢) −

𝑏Ω

|⟨𝜕𝜏, 𝑢⟩|2 . 𝐴

(7.8)

Proof. As in the case without twists we get ‖√𝜏 𝜕𝑢‖2𝜑 = ∑ ‖√𝜏 ( 𝑗 𝑘}, for 𝑘 = 0, 1, . . . , 𝑛. We shall prove inductively that for every 𝑘 there is a holomorphic function 𝑢𝑘 in Ω𝑘 with 𝑢𝑘 (𝑧0 ) = 1 and ∫ |𝑢𝑘 (𝑧)|2 𝑒−𝜑(𝑧) (1 + |𝑧|2 )−3𝑘 𝑑𝜆(𝑧) < ∞. Ω𝑘

When 𝑘 = 0 we can take 𝑢0 (𝑧) ≡ 1, and 𝑢𝑛 will have the desired properties. Assume that 0 < 𝑘 ≤ 𝑛 and that 𝑢𝑘−1 has already been constructed. Choose 𝜓 ∈ ∞ C0 (ℂ) so that 𝜓(𝑧𝑘 ) = 0 when |𝑧𝑘 | > 𝑟/2 and 𝜓(𝑧𝑘 ) = 1 when |𝑧𝑘 | < 𝑟/3, and set 𝑢𝑘 (𝑧) := 𝜓(𝑧𝑘 )𝑢𝑘−1 (𝑧) − 𝑧𝑘 𝑣(𝑧),

8.2 Weighted spaces of entire functions

|

123

notice that 𝜓(𝑧𝑘 )𝑢𝑘−1 (𝑧) = 0 in Ω𝑘 \ Ω𝑘−1 . To make 𝑢𝑘 holomorphic we must choose 𝑣 as a solution of the equation 𝜕𝑣 = 𝑧𝑘−1 𝑢𝑘−1 𝜕𝜓 = 𝑓. By the inductive hypothesis we have ∫ |𝑓(𝑧)|2 𝑒−𝜑(𝑧) (1 + |𝑧|2 )−3(𝑘−1) 𝑑𝜆(𝑧) < ∞. Ω𝑘

Hence it follows from Theorem 8.8 that 𝑣 can be found so that ∫ |𝑣(𝑧)|2 𝑒−𝜑(𝑧) (1 + |𝑧|2 )1−3𝑘 𝑑𝜆(𝑧) < ∞. Ω𝑘

Together with the inductive hypothesis on 𝑢𝑘−1 this implies that ∫ |𝑢𝑘 (𝑧)|2 𝑒−𝜑(𝑧) (1 + |𝑧|2 )−3𝑘 𝑑𝜆(𝑧) < ∞. Ω𝑘

Since 𝜕𝑣 = 0 in a neighborhood of 0, 𝑣 is a C∞ function there and we have 𝑢𝑘 (0) = 𝑢𝑘−1 (0) = 1 so 𝑢𝑘 has the required properties. Lemma 8.10. Let 𝜁 ∈ ℂ𝑛 and 𝐾 > 0 and define 𝑔(𝑧) = log(1 + 𝐾|𝑧 − 𝜁|2 ). Then for each 𝑤 ∈ ℂ𝑛 we have 𝐾 𝐾 |𝑤|2 ≤ 𝑖𝜕𝜕𝑔(𝑤, 𝑤)(𝑧) ≤ |𝑤|2 , (1 + 𝐾|𝑧 − 𝜁|2 )2 1 + 𝐾|𝑧 − 𝜁|2

(8.11)

where we use notation (7.5). Proof. An easy computation shows 𝐾2 (𝑧𝑗 − 𝜁𝑗 )(𝑧𝑘 − 𝜁𝑘 ) 𝐾𝛿𝑗𝑘 𝜕2 𝑔 (𝑧) = − + 𝜕𝑧𝑗 𝜕𝑧𝑘 (1 + 𝐾|𝑧 − 𝜁|2 )2 1 + 𝐾|𝑧 − 𝜁|2 𝐾 = [(1 + 𝐾|𝑧 − 𝜁|2 ) 𝛿𝑗𝑘 − 𝐾(𝑧𝑗 − 𝜁𝑗 )(𝑧𝑘 − 𝜁𝑘 )]. (1 + 𝐾|𝑧 − 𝜁|2 )2 This implies 𝑖𝜕𝜕𝑔(𝑤, 𝑤)(𝑧) =

𝐾 [(1 + 𝐾|𝑧 − 𝜁|2 ) |𝑤|2 − 𝐾|(𝑤, 𝑧 − 𝜁)|2 ] (1 + 𝐾|𝑧 − 𝜁|2 )2

and hence 𝐾 𝐾 [(1 + 𝐾|𝑧 − 𝜁|2 ) |𝑤|2 − 𝐾|𝑤|2 |𝑧 − 𝜁|2 ] ≤ 𝑖𝜕𝜕𝑔(𝑤, 𝑤)(𝑧) ≤ |𝑤|2 . (1 + 𝐾|𝑧 − 𝜁|2 )2 1 + 𝐾|𝑧 − 𝜁|2

We are now able to show that weighted spaces of entire functions are of infinite dimension if the weight function has an appropriate behavior at infinity. Theorem 8.11. Let 𝑊 : ℂ𝑛 󳨀→ ℝ be a C∞ -function and let 𝜇(𝑧) denote the lowest eigenvalue of the Levi matrix 𝑛 𝜕2 𝑊(𝑧) ) . 𝑖𝜕𝜕𝑊(𝑧) = ( 𝜕𝑧𝑗 𝜕𝑧𝑘 𝑗,𝑘=1

124 | 8 Applications Suppose that lim |𝑧|2 𝜇(𝑧) = ∞.

|𝑧|→∞

(8.12)

Then the Hilbert space 𝐴2 (ℂ𝑛 , 𝑒−𝑊 ) of all entire functions 𝑓 such that ∫ |𝑓(𝑧)|2 exp(−𝑊(𝑧)) 𝑑𝜆(𝑧) < ∞, ℂ𝑛

is of infinite dimension. Proof. Assumption (8.12) implies that there exists a constant 𝐾 > 0 such that 𝑖𝜕𝜕𝑊(𝑤, 𝑤)(𝑧) ≥ −𝐾|𝑤|2 , for all 𝑧, 𝑤 ∈ ℂ𝑛 , and that 𝑖𝜕𝜕𝑊(𝑧) is strictly positive for large |𝑧|. From Lemma 8.10 we have 𝑖𝜕𝜕𝑔(𝑤, 𝑤)(𝑧) ≥

4𝐾 |𝑤|2 , (1 + 4𝐾|𝑧 − 𝜁|2 )2

where 𝑔(𝑧) = log(1 + 4𝐾|𝑧 − 𝜁|2 ). Hence, for |𝑧 − 𝜁| ≤ 1/√4𝐾, we have 𝑖𝜕𝜕𝑔(𝑤, 𝑤)(𝑧) ≥ 𝐾|𝑤|2 . Since 𝑖𝜕𝜕𝑊(𝑤, 𝑤)(𝑧) is negative at most on a compact set in ℂ𝑛 there exist finitely many points 𝜁1 , . . . , 𝜁𝑀 ∈ ℂ𝑛 such that this compact set is covered by the balls {𝑧 : |𝑧 − 𝜁𝑙 | < 1/√4𝐾}. Hence 𝑀

̃ := 𝑊(𝑧) + ∑ 𝑔𝑙 (𝑧) 𝜑(𝑧) 𝑙=1

is strictly plurisubharmonic, where 𝑔𝑙 (𝑧) = log(1 + 4𝐾|𝑧 − 𝜁𝑙 |2 ) , 𝑙 = 1, . . . , 𝑀. ̃ be the least eigenvalue of 𝑖𝜕𝜕𝜑.̃ Then, by assumption (8.12), we have Let 𝜇(𝑧) ̃ = ∞. lim |𝑧|2 𝜇(𝑧)

|𝑧|→∞

For each 𝑁 ∈ ℕ there exists 𝑅 > 0 such that ̃ ≥ 𝜇(𝑧)

𝑁+𝑀+1 , |𝑧|2

for |𝑧| > 𝑅.

̃ Let 𝜇0̃ := inf{𝜇(𝑧) : |𝑧| ≤ 𝑅}. Then 𝜇0̃ > 0. Set 𝜅= and

𝑀

𝜇0̃ 2(𝑁 + 𝑀)

𝜑(𝑧) := 𝑊(𝑧) + ∑ 𝑔𝑙 (𝑧) − (𝑁 + 𝑀) log(1 + 𝜅|𝑧|2 ). 𝑙=1

It follows that 𝑒

−𝜑

is locally integrable.

8.2 Weighted spaces of entire functions

|

125

Next we claim that 𝜑 is strictly plurisubharmonic. Notice that ̃ − 𝑖𝜕𝜕𝜑(𝑤, 𝑤)(𝑧) ≥ |𝑤|2 (𝜇(𝑧)

(𝑁 + 𝑀)𝜅 ). 1 + 𝜅|𝑧|2

For |𝑧| ≤ 𝑅 we have ̃ − 𝜇(𝑧)

(𝑁 + 𝑀)𝜇0̃ 𝜇̃ (𝑁 + 𝑀)𝜅 ≥ 𝜇0̃ − (𝑁 + 𝑀)𝜅 = 𝜇0̃ − = 0 >0 1 + 𝜅|𝑧|2 2(𝑁 + 𝑀) 2

and for |𝑧| > 𝑅 we have ̃ − 𝜇(𝑧)

1 (𝑁 + 𝑀)𝜅 𝑁 + 𝑀 + 1 𝑁 + 𝑀 ≥ − = 2, 2 2 2 1 + 𝜅|𝑧| |𝑧| |𝑧| |𝑧|

which implies that 𝜑 is strictly plurisubharmonic. Therefore we can apply Theorem 8.9 and get an entire function 𝑓 with 𝑓(0) = 1 and ∫ |𝑓(𝑧)|2 (1 + |𝑧|2 )−3𝑛 𝑒−𝜑(𝑧) 𝑑𝜆(𝑧) < ∞. ℂ𝑛

Now we set 𝑁̃ = 𝑁 − 3𝑛 and we get ̃

∫ |𝑓(𝑧)|2 (1 + |𝑧|2 )𝑁 𝑒−𝑊(𝑧) 𝑑𝜆(𝑧) = ∫ ℂ𝑛

ℂ𝑛

2 ∏𝑀 ̃ 𝑙=1 (1 + 4𝐾|𝑧 − 𝜁𝑙 | ) |𝑓(𝑧)|2 (1 + |𝑧|2 )𝑁 𝑒−𝜑(𝑧) 𝑑𝜆(𝑧) (1 + 𝜅|𝑧|2 )𝑁+𝑀

≤ sup { 𝑧∈ℂ𝑛

2 (1 + |𝑧|2 )𝑁 ∏𝑀 𝑙=1 (1 + 4𝐾|𝑧 − 𝜁𝑙 | ) } 2 𝑁+𝑀 (1 + 𝜅|𝑧| )

× ∫ |𝑓(𝑧)|2 (1 + |𝑧|2 )−3𝑛 𝑒−𝜑(𝑧) 𝑑𝜆(𝑧) ℂ𝑛

< ∞. Hence 𝑓𝑝 ∈ 𝐴2 (ℂ𝑛 , 𝑒−𝑊 ) for any polynomial 𝑝 of degree < 𝑁,̃ and since 𝑁 = 𝑁̃ + 3𝑛 was arbitrary, we are done. The following example in ℂ2 shows that Theorem 8.11 is not sharp. Let 𝜑(𝑧, 𝑤) = |𝑧|2 |𝑤|2 + |𝑤|4 . In this case we have that 𝐴2 (ℂ2 , 𝑒−𝜑 ) contains all the functions 𝑓𝑘 (𝑧, 𝑤) = 𝑤𝑘 for 𝑘 ∈ ℕ, since ∞∞

∫

2 2 4 ∫ 𝑟22𝑘 𝑒−(𝑟1 𝑟2 +𝑟2 ) 𝑟1 𝑟2

0 0

∞

∞

2 2

4

𝑑𝑟1 𝑑𝑟2 = ∫ ( ∫ 𝑟1 𝑟22 𝑒−𝑟1 𝑟2 𝑑𝑟1 ) 𝑟22𝑘−1 𝑒−𝑟2 𝑑𝑟2 0 ∞

0 ∞

∞

0

0

4 4 1 1 = ∫ ( ∫ 𝑒−𝑠 𝑑𝑠) 𝑟22𝑘−1 𝑒−𝑟2 𝑑𝑟2 = ∫ 𝑟22𝑘−1 𝑒−𝑟2 𝑑𝑟2 < ∞. 2 2

0

The Levi matrix of 𝜑 has the form 𝑖𝜕𝜕𝜑 = (

|𝑤|2 𝑤𝑧

𝑧𝑤 ) |𝑧|2 + 4|𝑤|2

126 | 8 Applications hence 𝜑 is plurisubharmonic and the lowest eigenvalue has the form 1 (5|𝑤|2 + |𝑧|2 − (9|𝑤|4 + 10|𝑧|2 |𝑤|2 + |𝑧|4 )1/2 ) 2 16|𝑤|4 = , 2 2 2 (5|𝑤| + |𝑧| + (9|𝑤|4 + 10|𝑧|2 |𝑤|2 + |𝑧|4 )1/2 )

𝜇𝜑 (𝑧, 𝑤) =

hence lim |𝑧|2 𝜇𝜑 (𝑧, 0) = 0,

|𝑧|→∞

which implies that condition (8.12) of Theorem 8.11 is not satisfied.

8.3 Notes The original proof of Hörmander’s 𝐿2 -estimates uses a density lemma in weighted spaces. The 𝐿2 -estimates turned out to be an important method in different problems of analysis such as approximation and interpolation. The method is known as a real method in complex analysis (see [45]). It can also be used to prove the Levi problem, that domains of holomorphy coincide with pseudoconvex domains ([43]). The basic estimates (4.37) also hold for general bounded pseudoconvex domains (see [71]). We simplified the proof of Shikegawa’s result Theorem 8.11. In the following we will show that the infinite dimension of weighted spaces of entire functions is closely related to compactness of the 𝜕-Neumann operator on corresponding weighted 𝐿2 spaces – a phenomenon which also appears in applications to Dirac and Pauli operators. Rozenblum and Shirokov [62] showed that 𝐴2 (ℂ, 𝑒−𝜑 ) is of infinite dimension if 𝜑 is a subharmonic C2 -function such that ∫ℂ 󳵻𝜑(𝑧) 𝑑𝜆(𝑧) = ∞, see also Chapter 13.

9 Spectral analysis In this chapter we discuss the spectral theorem for self-adjoint, unbounded operators on a separable Hilbert space and important consequences of it. For the general approach basic concepts of measure theory are used, such as 𝜎-algebras and Borel sets. Moreover we prove Ruelle’s lemma and show how it can be used to interpret Hörmander’s 𝐿2 -estimates. In the following we will discuss applications of these results to the 𝜕-Neumann operator, to Schrödinger operators with magnetic fields, Dirac operators and to the Witten–Laplacian.

9.1 Resolutions of the identity We start with some definitions and basic facts. Definition 9.1. Let Ω be a subset of ℂ and M be a 𝜎-algebra in Ω and let 𝐻 be a Hilbert space. A resolution of the identity is a mapping 𝐸 : M 󳨀→ L(𝐻) of M to the algebra L(𝐻) of bounded linear operators on 𝐻 with the following properties (a) 𝐸(0) = 0 (zero operator), 𝐸(Ω) = 𝐼 (identity on 𝐻). (b) For each 𝜔 ∈ M the image 𝐸(𝜔) is an orthogonal projection on 𝐻. (c) 𝐸(𝜔󸀠 ∩ 𝜔󸀠󸀠 ) = 𝐸(𝜔󸀠 )𝐸(𝜔󸀠󸀠 ). (d) If 𝜔󸀠 ∩ 𝜔󸀠󸀠 = 0, then 𝐸(𝜔󸀠 ∪ 𝜔󸀠󸀠 ) = 𝐸(𝜔󸀠 ) + 𝐸(𝜔󸀠󸀠 ). (e) For every 𝑥 ∈ 𝐻 and 𝑦 ∈ 𝐻, the set function 𝐸𝑥,𝑦 defined by 𝐸𝑥,𝑦 (𝜔) = (𝐸(𝜔)𝑥, 𝑦) is a complex measure on M. We collect some immediate consequences of these properties. Since each 𝐸(𝜔) is an orthogonal (i.e. self-adjoint) projection, we have for 𝑥 ∈ 𝐻 𝐸𝑥,𝑥 (𝜔) = (𝐸(𝜔)𝑥, 𝑥) = (𝐸(𝜔)2 𝑥, 𝑥) = (𝐸(𝜔)𝑥, 𝐸(𝜔)𝑥) = ‖𝐸(𝜔)𝑥‖2 ,

(9.1)

hence each 𝐸𝑥,𝑥 is a positive measure on M with total variation 𝐸𝑥,𝑥 (Ω) = ‖𝑥‖2 .

(9.2)

By (c), any two of the projections 𝐸(𝜔) commute with each other; if 𝜔 ∩ 𝜔󸀠 = 0, (a) and (c) show that im(𝐸(𝜔)) ⊥ im(𝐸(𝜔󸀠 )), which follows from (𝐸(𝜔)𝑥, 𝐸(𝜔󸀠 )𝑦) = (𝐸(𝜔)2 𝑥, 𝐸(𝜔󸀠 )𝑦) = (𝐸(𝜔)𝑥, 𝐸(𝜔)𝐸(𝜔󸀠 )𝑦) = 0. By (d), 𝐸 is finitely additive. Concerning countable additivity we have the following result

128 | 9 Spectral analysis Proposition 9.2. If 𝐸 is a resolution of the identity, and if 𝑥 ∈ 𝐻, then 𝜔 󳨃→ 𝐸(𝜔)𝑥 is a countably additive 𝐻-valued measure on M. If 𝜔𝑛 ∈ M and 𝐸(𝜔𝑛 ) = 0 for 𝑛 ∈ ℕ, and if 𝜔 = ⋃∞ 𝑛=1 𝜔𝑛 , then 𝐸(𝜔) = 0. Proof. By (d), 𝜔 󳨃→ (𝐸(𝜔)𝑥, 𝑦) is a complex measure, hence ∞

∑ (𝐸(𝜔𝑛 )𝑥, 𝑦) = (𝐸(𝜔)𝑥, 𝑦),

(9.3)

𝑛=1

for every 𝑦 ∈ 𝐻. For 𝑛 ≠ 𝑚 we have 𝐸(𝜔𝑛 )𝑥 ⊥ 𝐸(𝜔𝑚 )𝑥. Let 𝑁

Λ 𝑁 (𝑦) = ∑ (𝑦, 𝐸(𝜔𝑛 )𝑥). 𝑛=1

By (9.3), the sequence (Λ 𝑁 (𝑦))𝑁 converges for every 𝑦 ∈ 𝐻. The uniform boundedness principle Theorem 4.14 implies that (‖Λ 𝑁 ‖)𝑁 is bounded, where ‖Λ 𝑁 ‖ = ‖𝐸(𝜔1 )𝑥 + ⋅ ⋅ ⋅ + 𝐸(𝜔𝑁 )𝑥‖ = (‖𝐸(𝜔1 )𝑥‖2 + ⋅ ⋅ ⋅ + ‖𝐸(𝜔𝑁 )𝑥‖2 )1/2 , hence, using the orthogonality, the partial sums 𝑁

∑ 𝐸(𝜔𝑛 )𝑥

𝑛=1

form a Cauchy sequence in 𝐻, so ∞

∑ 𝐸(𝜔𝑛 )𝑥 = 𝐸(𝜔)𝑥

𝑛=1

and 𝜔 󳨃→ 𝐸(𝜔)𝑥 is countably additive and therefore a complex measure on M. For the second claim, observe that 𝐸(𝜔𝑛 ) = 0 implies 𝐸𝑥,𝑥 (𝜔𝑛 ) = 0 for every 𝑥 ∈ 𝐻. Since 𝐸𝑥,𝑥 is countably additive, it follows that 𝐸𝑥,𝑥 (𝜔) = 0. But ‖𝐸(𝜔)𝑥‖2 = 𝐸𝑥,𝑥 (𝜔). Hence 𝐸(𝜔) = 0. Definition 9.3. Let 𝐸 be a resolution of the identity on M and let 𝑓 be a complex M-measurable function on Ω. There is a countable family (𝐷𝑘 )𝑘 of open discs forming a base for the topology of ℂ. Let 𝑉 be the union of those 𝐷𝑘 for which 𝐸(𝑓−1 (𝐷𝑘 )) = 0. By Proposition 9.2, 𝐸(𝑓−1 (𝑉)) = 0. Also, 𝑉 is the largest open subset of ℂ with this property. The essential range of 𝑓 is, by definition, the complement of 𝑉. It is the smallest closed subset of ℂ that contains 𝑓(𝑧) for all 𝑧 ∈ Ω except those that lie in some set 𝜔 ∈ M with 𝐸(𝜔) = 0. We say 𝑓 is essentially bounded if its essential range is bounded, hence compact. The largest value of |𝜆|, as 𝜆 runs through the essential range of 𝑓, is called the essential supremum ‖𝑓‖∞ of 𝑓.

9.1 Resolutions of the identity

|

129

Let 𝐵 be the algebra of all bounded complex M-measurable functions on Ω with the norm ‖𝑓‖ = sup |𝑓(𝑧)|, 𝑧∈Ω

and let 𝑁 = {𝑓 ∈ 𝐵 : ‖𝑓‖∞ = 0}, which, by Proposition 9.2, is a closed ideal. Hence 𝐵/𝑁 is a Banach algebra, which is denoted by 𝐿∞ (𝐸). The norm of a coset [𝑓] = 𝑓 + 𝑁 is ‖𝑓‖∞ , and the spectrum 𝜎([𝑓]) is the essential range of 𝑓, the spectrum of an element 𝑔 in a Banach algebra is the set of all complex numbers 𝜆 such that 𝜆𝑒 − 𝑔 is not invertible. In the next step we describe that a resolution of the identity induces an isometric isomorphism of the Banach algebra 𝐿∞ (𝐸) onto a closed normal subalgebra A of L(𝐻), the algebra of all bounded linear operators from 𝐻 to 𝐻, a normal subalgebra is a commutative one which contains 𝑇∗ for every 𝑇 ∈ A. For this purpose let {𝜔1 , . . . , 𝜔𝑛 } be a partition of Ω, with 𝜔𝑗 ∈ M and let 𝑠 be a simple function, such that 𝑠 = 𝛼𝑗 on 𝜔𝑗 . Define Ψ(𝑠) ∈ L(𝐻) by 𝑛

Ψ(𝑠) = ∑ 𝛼𝑗 𝐸(𝜔𝑗 ). 𝑗=1

Since each 𝐸(𝜔𝑗 ) is self-adjoint, Ψ(𝑠)∗ = Ψ(𝑠). If 𝑡 is another simple function and 𝛼, 𝛽 ∈ ℂ, we have Ψ(𝑠)Ψ(𝑡) = Ψ(𝑠𝑡) and Ψ(𝛼𝑠 + 𝛽𝑡) = 𝛼Ψ(𝑠) + 𝛽Ψ(𝑡). For 𝑥, 𝑦 ∈ 𝐻 we get 𝑛

𝑛

(Ψ(𝑠)𝑥, 𝑦) = ∑ 𝛼𝑗 (𝐸(𝜔𝑗 )𝑥, 𝑦) = ∑ 𝛼𝑗 𝐸𝑥,𝑦 (𝜔𝑗 ) = ∫ 𝑠 𝑑𝐸𝑥,𝑦 . 𝑗=1

𝑗=1

Ω

In addition we have Ψ(𝑠)∗ Ψ(𝑠) = Ψ(|𝑠|2 ) and ‖Ψ(𝑠)𝑥‖2 = ∫ |𝑠|2 𝑑𝐸𝑥,𝑥 . Ω

By (9.1) this implies ‖Ψ(𝑠)𝑥‖ ≤ ‖𝑠‖∞ ‖𝑥‖,

(9.4)

and if 𝑥 ∈ im(𝐸(𝜔𝑗 )), then Ψ(𝑠)𝑥 = 𝛼𝑗 𝐸(𝜔𝑗 )𝑥 = 𝛼𝑗 𝑥, since the projections 𝐸(𝜔𝑗 ) have mutually orthogonal ranges. If 𝑗 is chosen so that |𝛼𝑗 | = ‖𝑠‖∞ it follows by (9.4) that ‖Ψ(𝑠)‖ = sup ‖Ψ(𝑠)𝑥‖ = ‖𝑠‖∞ . ‖𝑥‖≤1

(9.5)

130 | 9 Spectral analysis Now suppose that 𝑓 ∈ 𝐿∞ (𝐸). There is a sequence of simple measurable functions 𝑠𝑘 that converges to 𝑓 in the norm of 𝐿∞ (𝐸). By (9.5), the corresponding operators Ψ(𝑠𝑘 ) form a Cauchy sequence in L(𝐻), which is therefore norm-convergent to an operator that we call Ψ(𝑓). By (9.5), we get (9.6)

‖Ψ(𝑓)‖ = ‖𝑓‖∞ . ∞

∞

Thus Ψ is an isometric isomorphism of 𝐿 (𝐸) into L(𝐻). Since 𝐿 (𝐸) is complete,

A = Ψ(𝐿∞ (𝐸)) is closed in L(𝐻). In addition we have

(Ψ(𝑓)𝑥, 𝑦) = ∫ 𝑓 𝑑𝐸𝑥,𝑦

and ‖Ψ(𝑓)𝑥‖2 = ∫ |𝑓|2 𝑑𝐸𝑥,𝑥 ,

Ω

Ω

which justifies the notation Ψ(𝑓) = ∫ 𝑓 𝑑𝐸. Ω

9.2 Spectral decomposition of bounded normal operators We will now relate a resolution of the identity to a certain operator in L(𝐻). Definition 9.4. The spectrum 𝜎(𝑇) of an operator 𝑇 ∈ L(𝐻) is the set of all 𝜆 ∈ ℂ, such that 𝜆𝐼 − 𝑇 has no inverse in L(𝐻). The complement 𝜌(𝑇) = ℂ \ 𝜎(𝑇) is called the resolvent set. If 𝜆 ∈ 𝜌(𝑇) the operator (𝜆𝐼 − 𝑇)−1 ∈ L(𝐻) is called the resolvent of 𝑇 at 𝜆 and is denoted by 𝑅𝑇 (𝜆). We have an operator-valued function 𝑅𝑇 : 𝜌(𝑇) 󳨀→ L(𝐻). An operator 𝑇 ∈ L(𝐻) is called normal if 𝑇𝑇∗ = 𝑇∗ 𝑇. Proposition 9.5. Let 𝑇 ∈ L(𝐻). Then the spectrum 𝜎(𝑇) of 𝑇 is a compact subset of ℂ and |𝜆| ≤ ‖𝑇‖, for every 𝜆 ∈ 𝜎(𝑇). Proof. First we show that 𝜌(𝑇) is open. Let 𝜆 ∈ 𝜌(𝑇). Then 𝛼 = ‖𝑅𝑇 (𝜆)‖−1 > 0. Let 𝜇 ∈ ℂ with |𝜇| < 𝛼. We will show that (𝜆 + 𝜇)𝐼 − 𝑇 has a bounded inverse. Then we prove that 𝜌(𝑇) is open. We have (𝜆 + 𝜇)𝐼 − 𝑇 = 𝜆𝐼 − 𝑇 + 𝜇𝐼 = (𝜆𝐼 − 𝑇)[𝐼 + 𝜇(𝜆𝐼 − 𝑇)−1 ] = (𝜆𝐼 − 𝑇)(𝐼 + 𝜇𝑅𝑇 (𝜆)). Formally

∞

(𝐼 + 𝜇𝑅𝑇 (𝜆))−1 = 𝐼 + ∑ (−1)𝑘 (𝜇𝑅𝑇 (𝜆))𝑘 , 𝑘=1

9.2 Spectral decomposition of bounded normal operators

|

131

but as |𝜇| < 𝛼 we have ∞ 󵄩 ∞ 󵄩󵄩 ∞ 󵄩󵄩 ∑ (−1)𝑘 (𝜇𝑅𝑇 (𝜆))𝑘 󵄩󵄩󵄩 ≤ ∑ |𝜇|𝑘 ‖𝑅𝑇 (𝜆)‖𝑘 = ∑ (|𝜇|/𝛼)𝑘 < ∞, 󵄩󵄩 󵄩󵄩 𝑘=1 𝑘=1 𝑘=1 𝑘 𝑘 therefore the partial sums of ∑∞ 𝑘=1 (−1) (𝜇𝑅𝑇 (𝜆)) form a Cauchy sequence in L(𝐻). Since L(𝐻) is complete, we obtain that ∞

∑ (−1)𝑘 (𝜇𝑅𝑇 (𝜆))𝑘 ∈ L(𝐻),

𝑘=1

and 𝜌(𝑇) is open. If 𝜂 ∈ ℂ with |𝜂| > ‖𝑇‖, then 𝐼 − 𝑇/𝜂 has a bounded inverse, since ∞

(𝐼 − 𝑇/𝜂)−1 = 𝐼 + ∑ (𝑇/𝜂)𝑘 . 𝑘=1

This implies that 𝜂𝐼 − 𝑇 has a bounded inverse. Hence, if 𝜆 ∈ 𝜎(𝑇), then |𝜆| ≤ ‖𝑇‖, and 𝜎(𝑇) is a bounded set. The resolvent has the following properties: Lemma 9.6. If 𝜆, 𝜇 ∈ 𝜌(𝑇), then 𝑅𝑇 (𝜆) − 𝑅𝑇 (𝜇) = (𝜇 − 𝜆)𝑅𝑇 (𝜆)𝑅𝑇 (𝜇).

(9.7)

If 𝜆 ∈ 𝜌(𝑇) and |𝜆 − 𝜇| < ‖𝑅𝑇 (𝜆)‖−1 , then ∞

𝑅𝑇 (𝜇) = ∑ (𝜆 − 𝜇)𝑘 [𝑅𝑇 (𝜆)]𝑘+1 ,

(9.8)

𝑘=0

therefore one says that 𝑅𝑇 is a holomorphic operator valued function. Proof. (9.7) follows from 𝑅𝑇 (𝜆) = 𝑅𝑇 (𝜆)(𝜇𝐼 − 𝑇)𝑅𝑇 (𝜇) = 𝑅𝑇 (𝜆)[(𝜆𝐼 − 𝑇) + (𝜇 − 𝜆)𝐼]𝑅𝑇 (𝜇) = 𝑅𝑇 (𝜇) + (𝜇 − 𝜆)𝑅𝑇 (𝜆)𝑅𝑇 (𝜇). (9.8) follows immediately from the proof of Proposition 9.5. The spectral theorem indicates that every bounded normal operator 𝑇 on a Hilbert space induces a resolution 𝐸 of the identity on the Borel subsets of its spectrum 𝜎(𝑇) and that 𝑇 can be reconstructed from 𝐸 by an integral of the type discussed before. Using Banach algebra techniques such as the Gelfand transform (see [65]) one obtains the spectral decomposition for a single normal operator.

132 | 9 Spectral analysis Proposition 9.7. If 𝑇 ∈ L(𝐻) and 𝑇 is normal, then there exists a uniquely determined resolution of the identity 𝐸 on the Borel subsets of the spectrum 𝜎(𝑇) which satisfies 𝑇 = ∫ 𝜆 𝑑𝐸(𝜆) and (𝑇𝑥, 𝑦) = ∫ 𝜆 𝑑𝐸𝑥,𝑦 (𝜆). 𝜎(𝑇)

(9.9)

𝜎(𝑇)

Furthermore, every projection 𝐸(𝜔) commutes with every 𝑆 ∈ L(𝐻) which commutes with 𝑇. We shall refer to this 𝐸 as the spectral decomposition of 𝑇. We list a few consequences of the spectral decomposition. – If 𝜔 ⊆ 𝜎(𝑇) is a nonempty open set, then 𝐸(𝜔) ≠ 0. – If 𝑓 is a bounded Borel function on 𝜎(𝑇), it is customary to denote the operator Ψ(𝑓) = ∫ 𝑓 𝑑𝐸 𝜎(𝑇)

by 𝑓(𝑇). The mapping 𝑓 󳨃→ 𝑓(𝑇) establishes a homomorphism of the algebra of all bounded Borel functions on 𝜎(𝑇) into L(𝐻), which carries the function 1 to 𝐼 and the identity function on 𝜎(𝑇) to 𝑇, and satisfies 𝑓(𝑇) = 𝑓(𝑇)∗ and ‖𝑓(𝑇)‖ ≤ sup{|𝑓(𝜆)| : 𝜆 ∈ 𝜎(𝑇)}. The procedure explained above is also called symbolic calculus. If 𝑓 ∈ C(𝜎(𝑇)), then 𝑓 󳨃→ 𝑓(𝑇) is an isomorphism on C(𝜎(𝑇)) satisfying ‖𝑓(𝑇)𝑥‖2 = ∫ |𝑓|2 𝑑𝐸𝑥,𝑥 . 𝜎(𝑇)

The eigenvalues of a normal operator can be characterized in terms of the spectral decomposition. For this purpose we mention the following applications of the symbolic calculus. Proposition 9.8. Let 𝑇 ∈ L(𝐻) be a normal operator and 𝐸 its spectral decomposition. If 𝑓 ∈ C(𝜎(𝑇)) and 𝜔0 = 𝑓−1 (0), then ker(𝑓(𝑇)) = im(𝐸(𝜔0 )). Proof. We set ℎ(𝜆) = 1 on 𝜔0 and ℎ(𝜆) = 0 on 𝜔̃ = 𝜎(𝑇) \ 𝜔0 . Then 𝑓ℎ = 0 and by the symbolic calculus 𝑓(𝑇)ℎ(𝑇) = 0. Since ℎ(𝑇) = 𝐸(𝜔0 ), it follows that im(𝐸(𝜔0 )) ⊆ ker(𝑓(𝑇)). For the opposite inclusion we define for 𝑛 ∈ ℕ the set 𝜔𝑛 = {𝜆 ∈ 𝜎(𝑇) : 1/𝑛 ≤ |𝑓(𝜆)| < 1/(𝑛 − 1)}.

9.2 Spectral decomposition of bounded normal operators

|

133

Then 𝜔̃ is the union of the disjoint Borel sets 𝜔𝑛 . We define 𝑓𝑛 (𝜆) = 1/𝑓(𝜆) on 𝜔𝑛 and 𝑓𝑛 (𝜆) = 0 on 𝜎(𝑇) \ 𝜔𝑛 . Then each 𝑓𝑛 is a bounded Borel function on 𝜎(𝑇) and 𝑓𝑛 (𝑇)𝑓(𝑇) = 𝐸(𝜔𝑛 ),

𝑛 ∈ ℕ.

If 𝑓(𝑇)𝑥 = 0, it follows that 𝐸(𝜔𝑛 )𝑥 = 0. Since the mapping 𝜔 󳨃→ 𝐸(𝜔)𝑥 is countably ̃ = 0. But we also have that additive (Proposition 9.2), we obtain 𝐸(𝜔)𝑥 𝐸(𝜔)̃ + 𝐸(𝜔0 ) = 𝐼. Hence 𝐸(𝜔0 )𝑥 = 𝑥 and therefore ker(𝑓(𝑇)) ⊆ im(𝐸(𝜔0 )). Proposition 9.9. Let 𝑇 ∈ L(𝐻) be a normal operator and 𝐸 its spectral decomposition. Let 𝜆 0 ∈ 𝜎(𝑇) and 𝐸0 = 𝐸({𝜆 0 }). Then (a) ker(𝑇 − 𝜆 0 𝐼) = im(𝐸0 ), (b) 𝜆 0 is an eigenvalue of 𝑇 if and only if 𝐸0 ≠ 0, (c) every isolated point of 𝜎(𝑇) is an eigenvalue of 𝑇, (d) if 𝜎(𝑇) = {𝜆 1 , 𝜆 2 , . . . } is a countable set, then every 𝑥 ∈ 𝐻 has a unique expansion of the form ∞

𝑥 = ∑ 𝑥𝑗 , 𝑗=1

where 𝑇𝑥𝑗 = 𝜆 𝑗 𝑥𝑗 , and 𝑥𝑗 ⊥ 𝑥𝑘 for 𝑗 ≠ 𝑘. (e) If 𝜎(𝑇) has no limit point except possibly 0 and if dim ker(𝑇 − 𝜆𝐼) < ∞, for 𝜆 ≠ 0, then 𝑇 is compact (compare with Section 2.1). Proof. (a) is an immediate consequence of Proposition 9.8 with 𝑓(𝜆) = 𝜆 − 𝜆 0 . (b) follows from (a). (c) If 𝜆 0 is an isolated point of 𝜎(𝑇), then {𝜆 0 } is a nonempty open subset of 𝜎(𝑇) and, by the properties of the spectral decomposition listed above, we get 𝐸0 ≠ 0, therefore (c) follows from (b). (d) Let 𝐸𝑗 = 𝐸({𝜆 𝑗 }) for 𝑗 ∈ ℕ. The projections 𝐸𝑗 have pairwise orthogonal ranges and the mapping 𝜔 󳨃→ 𝐸(𝜔)𝑥 is countably additive (Proposition 9.2), hence for each 𝑥 ∈ 𝐻 we have ∞

∑ 𝐸𝑗 𝑥 = 𝐸(𝜎(𝑇))𝑥 = 𝑥,

𝑗=1

and the series converges in the norm of 𝐻. Now set 𝑥𝑗 = 𝐸𝑗 𝑥 and observe that uniqueness follows from the orthogonality of the vectors 𝑥𝑗 and that 𝑇𝑥𝑗 = 𝜆 𝑗 𝑥𝑗 follows from (a). (e) Let {𝜆 𝑗 } be an enumeration of the nonzero points of 𝜎(𝑇) such that |𝜆 1 | ≥ |𝜆 2 | ≥ . . . and define 𝑓𝑛 (𝜆) = 𝜆 if 𝜆 = 𝜆 𝑗 and 𝑗 ≤ 𝑛, and put 𝑓𝑛 (𝜆) = 0 at the other points of 𝜎(𝑇). We set again 𝐸𝑗 = 𝐸({𝜆 𝑗 }) and obtain 𝑓𝑛 (𝑇) = 𝜆 1 𝐸1 + ⋅ ⋅ ⋅ + 𝜆 𝑛 𝐸𝑛 .

134 | 9 Spectral analysis Since dim im(𝐸𝑗 ) = dim ker(𝑇 − 𝜆 𝑗 𝐼) < ∞, each 𝑓𝑛 (𝑇) is a compact operator. We have |𝜆 − 𝑓𝑛 (𝜆)| ≤ |𝜆 𝑛 | for all 𝜆 ∈ 𝜎(𝑇) and, by the symbolic calculus, ‖𝑇 − 𝑓𝑛 (𝑇)‖ ≤ |𝜆 𝑛 | → 0 as 𝑛 → ∞. By Proposition 2.2, 𝑇 is compact. The symbolic calculus is a powerful tool in operator theory. Finally we mention important applications to positive operators: Definition 9.10. An operator 𝑇 ∈ L(𝐻) is called positive if (𝑇𝑥, 𝑥) ≥ 0 for every 𝑥 ∈ 𝐻. We write 𝑇 ≥ 0. Proposition 9.11. (a) 𝑇 ∈ L(𝐻) is positive if and only if 𝑇 = 𝑇∗ and 𝜎(𝑇) ⊂ [0, ∞). (b) Every positive 𝑇 ∈ L(𝐻) has a unique positive square root 𝑆 ∈ L(𝐻), i.e. 𝑆2 = 𝑇. (c) If 𝑇 ∈ L(𝐻), then 𝑇∗ 𝑇 is positive and the positive square root 𝑃 of 𝑇∗ 𝑇 is the only positive operator in L(𝐻) which satisfies ‖𝑃𝑥‖ = ‖𝑇𝑥‖ for every 𝑥 ∈ 𝐻. (d) If 𝑇 ∈ L(𝐻) is normal, then 𝑇 has a polar decomposition 𝑇 = 𝑈𝑃, where 𝑈 is unitary and 𝑃 is positive. Proof. (a) (𝑇𝑥, 𝑥) and (𝑥, 𝑇𝑥) are complex conjugates of each other. If 𝑇 is positive, (𝑇𝑥, 𝑥) is real, so that (𝑥, 𝑇∗ 𝑥) = (𝑇𝑥, 𝑥) = (𝑥, 𝑇𝑥), for every 𝑥 ∈ 𝐻. Hence 𝑇 = 𝑇∗ (by the proof of Theorem 2.7). Let 𝜆 = 𝛼 + 𝑖𝛽 ∈ 𝜎(𝑇) and put 𝑇𝜆 = 𝑇 − 𝜆𝐼. Then ‖𝑇𝜆 𝑥‖2 = ‖𝑇𝑥 − 𝛼𝑥‖2 + 𝛽2 ‖𝑥‖2 , so that ‖𝑇𝜆 𝑥‖ ≥ |𝛽| ‖𝑥‖. If 𝛽 ≠ 0, it follows that 𝑇𝜆 is invertible, which means 𝜆 ∉ 𝜎(𝑇). So we get that 𝜎(𝑇) lies in the real axis. If 𝜆 > 0, we obtain 𝜆‖𝑥‖2 = (𝜆𝑥, 𝑥) ≤ ((𝑇 + 𝜆𝐼)𝑥, 𝑥) ≤ ‖(𝑇 + 𝜆𝐼)𝑥‖ ‖𝑥‖, so that ‖(𝑇+𝜆𝐼)𝑥‖ ≥ 𝜆‖𝑥‖, which implies that 𝑇+𝜆𝐼 is invertible in L(𝐻), and −𝜆 ∉ 𝜎(𝑇), hence 𝜎(𝑇) ⊂ [0, ∞). Now assume that 𝑇 = 𝑇∗ and 𝜎(𝑇) ⊂ [0, ∞). Let 𝐸 be the spectral decomposition of 𝑇. We have (𝑇𝑥, 𝑥) = ∫ 𝜆 𝑑𝐸𝑥,𝑥 (𝜆). 𝜎(𝑇)

Since 𝐸𝑥,𝑥 is a positive measure and 𝜆 ≥ 0 on 𝜎(𝑇), we obtain that 𝑇 is positive. (b) By (a), Proposition 9.5 and the symbolic calculus, 𝜎(𝑇) is a compact subset of + ℝ and there exists a uniquely determined spectral decomposition 𝐸 such that 𝑇 = ∫ 𝜆 𝑑𝐸(𝜆). 𝜎(𝑇)

9.2 Spectral decomposition of bounded normal operators

|

135

Define 𝑆 = ∫ 𝜆1/2 𝑑𝐸(𝜆). 𝜎(𝑇)

Then 𝑆 is a positive self-adjoint operator with 𝑆2 = 𝑇. In addition there is a sequence of polynomials 𝑝𝑛 such that 𝑝𝑛 (𝜆) → 𝜆1/2 uniformly on 𝜎(𝑇) (see Corollary 12.5) and lim ‖𝑝𝑛 (𝑇) − 𝑆‖ = 0.

𝑛→∞

Let 𝑆 ̃ be an arbitrary positive self-adjoint operator with 𝑆2̃ = 𝑇. Since 𝑇𝑆 ̃ = 𝑆3̃ and ̃ 𝑆𝑇 = 𝑆3̃ , the operator 𝑆 ̃ commutes with 𝑇 and so with polynomials of 𝑇. Hence also ̃ Using that 𝑆𝑆̃ = 𝑆𝑆̃ and 𝑆2 = 𝑆2̃ , with 𝑆 = lim𝑛→∞ 𝑝𝑛 (𝑇). Let 𝑥 ∈ 𝐻 and put 𝑦 = (𝑆 − 𝑆)𝑥. we obtain ̃ 𝑦) = ((𝑆 + 𝑆)(𝑆 ̃ − 𝑆)𝑥, ̃ 𝑦) = ((𝑆2 − 𝑆2̃ )𝑥, 𝑦) = 0. (𝑆𝑦, 𝑦) + (𝑆𝑦, ̃ 𝑦) = 0. Hence 𝑆𝑦 = 𝑆𝑦 ̃ = 0, because (𝑆., .) is Since 𝑆 and 𝑆 ̃ are positive, (𝑆𝑦, 𝑦) = (𝑆𝑦, a positive semidefinite sesquilinear form, for which the Cauchy–Schwarz inequality applies |(𝑆𝑦, 𝑧)|2 ≤ (𝑆𝑦, 𝑦)(𝑆𝑧, 𝑧), for all 𝑧 ∈ 𝐻. Now we get ̃ 𝑥) = 0, ̃ 2 = ((𝑆 − 𝑆)̃ 2 𝑥, 𝑥) = ((𝑆 − 𝑆)𝑦, ‖(𝑆 − 𝑆)𝑥‖ ̃ and 𝑆 = 𝑆.̃ which yields 𝑆𝑥 = 𝑆𝑥 (c) Note first that (𝑇∗ 𝑇𝑥, 𝑥) = (𝑇𝑥, 𝑇𝑥) = ‖𝑇𝑥‖2 ≥ 0, for 𝑥 ∈ 𝐻, so that 𝑇∗ 𝑇 is positive. If 𝑃 ∈ L(𝐻) and 𝑃∗ = 𝑃, then (𝑃2 𝑥, 𝑥) = (𝑃𝑥, 𝑃𝑥) = ‖𝑃𝑥‖2 . Then, by the proof of Theorem 2.7, it follows that ‖𝑃𝑥‖ = ‖𝑇𝑥‖ for every 𝑥 ∈ 𝐻 if and only if 𝑃2 = 𝑇∗ 𝑇. (d) Put 𝑝(𝜆) = |𝜆| and 𝑢(𝜆) = 𝜆/|𝜆| for 𝜆 ≠ 0 and 𝑢(0) = 1. Then 𝑝 and 𝑢 are bounded Borel functions on 𝜎(𝑇). Put 𝑃 = 𝑝(𝑇) and 𝑈 = 𝑢(𝑇). As 𝑝 ≥ 0 we get from (a) that 𝑃 ≥ 0. Since 𝑢𝑢 = 1, we get 𝑈𝑈∗ = 𝑈∗ 𝑈 = 𝐼, and since 𝜆 = 𝑢(𝜆)𝑝(𝜆), the relation 𝑇 = 𝑈𝑃 follows from the symbolic calculus. Remark 9.12. We apply the last results to the 𝜕-Neumann operator 𝑁, to prove the following: Suppose that ◻ : dom(◻) 󳨀→ 𝐿2(0,1) (Ω) is bijective and has a bounded inverse 𝑁, then the basic estimate ∗

‖𝑢‖2 ≤ 𝐶 (‖𝜕𝑢‖2 + ‖𝜕 𝑢‖2 ), must hold.

𝑢 ∈ dom(◻) (4.37)

136 | 9 Spectral analysis 𝑁 is self-adjoint and bounded and, by Proposition 9.11, therefore has a bounded self-adjoint root 𝑁1/2 which is again injective. By Lemma 4.28, 𝑁1/2 has a self-adjoint inverse which will be denoted by 𝑁−1/2 . Let 𝑢 ∈ dom(◻). Then there exists 𝑤 ∈ 𝐿2(0,1) (Ω) such that 𝑁𝑤 = 𝑢. Hence we have 𝑁1/2 𝑣 = 𝑢, where 𝑣 = 𝑁1/2 𝑤 and 𝑁−1/2 𝑣 = 𝑤 = 𝑁−1/2 𝑁−1/2 𝑢 is well defined. Now we get ‖𝑢‖2 = ‖𝑁1/2 𝑣‖2 ≤ 𝐶‖𝑣‖2 = 𝐶 (𝑁−1/2 𝑢, 𝑁−1/2 𝑢) = 𝐶 (𝑁−1/2 𝑁−1/2 𝑢, 𝑢) = 𝐶 (𝑁−1/2 𝑁−1/2 𝑁𝑤, 𝑁𝑤) = 𝐶 (𝑤, 𝑁𝑤) = 𝐶 (◻𝑢, 𝑢) ∗

= 𝐶 (‖𝜕𝑢‖2 + ‖𝜕 𝑢‖2 ), which is the basic estimate (4.37).

9.3 Spectral decomposition of unbounded self-adjoint operators Let Ω be a subset of ℂ and M be a 𝜎-algebra in Ω and let 𝐻 be a Hilbert space. Let 𝐸 : M 󳨀→ L(𝐻) be a resolution of the identity. The symbolic calculus associates to every 𝑓 ∈ 𝐿∞ (𝐸) an operator Ψ(𝑓) ∈ L(𝐻) by the formula (Ψ(𝑓)𝑥, 𝑦) = ∫ 𝑓 𝑑𝐸𝑥,𝑦 , 𝑥, 𝑦 ∈ 𝐻. Ω

Now we will extend this for unbounded measurable functions 𝑓. Lemma 9.13. Let 𝑓 : Ω 󳨀→ ℂ be a measurable function. Put 2

D𝑓 = {𝑥 ∈ 𝐻 : ∫ |𝑓| 𝑑𝐸𝑥,𝑥 < ∞}. Ω

Then D𝑓 is a dense subspace of 𝐻. If 𝑥, 𝑦 ∈ 𝐻, then 1/2

∫ |𝑓| 𝑑|𝐸𝑥,𝑦 | ≤ ‖𝑦‖ [ ∫ |𝑓|2 𝑑𝐸𝑥,𝑥 ] Ω

.

(9.10)

Ω

If 𝑓 is bounded and 𝑢 = Ψ(𝑓)𝑣, for 𝑣 ∈ 𝐻, then 𝑑𝐸𝑥,𝑢 = 𝑓 𝑑𝐸𝑥,𝑣 ,

𝑥 ∈ 𝐻.

Proof. Let 𝑧 = 𝑥 + 𝑦 and 𝜔 ∈ M. Then ‖𝐸(𝜔)𝑧‖2 ≤ (‖𝐸(𝜔)𝑥‖ + ‖𝐸(𝜔)𝑦‖)2 ≤ 2(‖𝐸(𝜔)𝑥‖2 + ‖𝐸(𝜔)𝑦‖2 ).

(9.11)

9.3 Spectral decomposition of unbounded self-adjoint operators

| 137

Recall that 𝐸𝑥,𝑥 (𝜔) = (𝐸(𝜔)𝑥, 𝑥) = (𝐸(𝜔)2 𝑥, 𝑥) = ‖𝐸(𝜔)𝑥‖2 , so we get from above 𝐸𝑧,𝑧 (𝜔) ≤ 2(𝐸𝑥,𝑥 (𝜔) + 𝐸𝑦,𝑦 (𝜔)), which implies that D𝑓 is closed under addition. It is clear that D𝑓 is also closed under scalar multiplication. Therefore D𝑓 is a subspace of 𝐻. For 𝑛 ∈ ℕ, let 𝜔𝑛 be the subset of Ω where |𝑓| < 𝑛. If 𝑥 ∈ im(𝐸(𝜔𝑛 )), then 𝐸(𝜔)𝑥 = 𝐸(𝜔)𝐸(𝜔𝑛 )𝑥 = 𝐸(𝜔 ∩ 𝜔𝑛 )𝑥 , 𝜔 ∈ M. Hence 𝐸𝑥,𝑥 (𝜔) = 𝐸𝑥,𝑥 (𝜔 ∩ 𝜔𝑛 ), therefore ∫ |𝑓|2 𝑑𝐸𝑥,𝑥 = ∫ |𝑓|2 𝑑𝐸𝑥,𝑥 ≤ 𝑛2 ‖𝑥‖2 < ∞. 𝜔𝑛

Ω

Thus im(𝐸(𝜔𝑛 )) ⊂ D𝑓 . Since Ω = ⋃∞ 𝑛=1 𝜔𝑛 , the countable additivity of 𝜔 󳨃→ 𝐸(𝜔)𝑦 implies that 𝑦 = lim𝑛→∞ 𝐸(𝜔𝑛 )𝑦 for every 𝑦 ∈ 𝐻. Hence 𝑦 lies in the closure of D𝑓 and D𝑓 is dense in 𝐻. If 𝑥, 𝑦 ∈ 𝐻 and 𝑓 is bounded and measurable, the Radon–Nikodym Theorem (see for instance [63]) implies that there is a measurable function 𝑔 on Ω such that |𝑔| = 1 on Ω and 𝑔𝑓 𝑑𝐸𝑥,𝑦 = |𝑓| 𝑑|𝐸𝑥,𝑦 |. Hence ∫ |𝑓| 𝑑|𝐸𝑥,𝑦 | = (Ψ(𝑔𝑓)𝑥, 𝑦) ≤ ‖Ψ(𝑔𝑓)𝑥‖ ‖𝑦‖.

(9.12)

Ω

As in Section 9.1 we get ‖Ψ(𝑔𝑓)𝑥‖2 = ∫ |𝑔𝑓|2 𝑑𝐸𝑥,𝑥 = ∫ |𝑓|2 𝑑𝐸𝑥,𝑥 , Ω

Ω

which implies (9.10) for a bounded function 𝑓. The general case is done as in the first assertion of this proposition. To show (9.11) we consider an arbitrary bounded measurable function ℎ and have ∫ ℎ 𝑑𝐸𝑥,𝑢 = (Ψ(ℎ)𝑥, 𝑢) = (Ψ(ℎ)𝑥, Ψ(𝑓)𝑣) Ω

= (Ψ(𝑓)Ψ(ℎ)𝑥, 𝑣) = (Ψ(𝑓ℎ)𝑥, 𝑣) = ∫ ℎ𝑓 𝑑𝐸𝑥,𝑣 . Ω

In the next step we carry over the results of Section 9.1 (symbolic calculus) for unbounded measurable functions.

138 | 9 Spectral analysis Proposition 9.14. Let 𝐸 be a resolution of identity on Ω. (a) To every measurable 𝑓 : Ω 󳨀→ ℂ corresponds a densely defined closed operator Ψ(𝑓) on 𝐻, with domain dom(Ψ(𝑓)) = D𝑓 , which is characterized by (Ψ(𝑓)𝑥, 𝑦) = ∫ 𝑓 𝑑𝐸𝑥,𝑦 , 𝑥 ∈ D𝑓 ,

𝑦∈𝐻

(9.13)

Ω

and which satisfies ‖Ψ(𝑓)𝑥‖2 = ∫ |𝑓|2 𝑑𝐸𝑥,𝑥 ,

𝑥 ∈ D𝑓 .

(9.14)

Ω

(b) If 𝑓 and 𝑔 are measurable, then Ψ(𝑓)Ψ(𝑔) ⊂ Ψ(𝑓𝑔), which means that dom(Ψ(𝑓)Ψ(𝑔)) ⊂ dom(Ψ(𝑓𝑔)) and Ψ(𝑓)Ψ(𝑔) = Ψ(𝑓𝑔) on dom(Ψ(𝑓)Ψ(𝑔)), and dom(Ψ(𝑓)Ψ(𝑔)) = D𝑔 ∩ D𝑓𝑔 . Hence Ψ(𝑓)Ψ(𝑔) = Ψ(𝑓𝑔) if and only if D𝑓𝑔 ⊆ D𝑔 . (c) For every measurable 𝑓 : Ω 󳨀→ ℂ, Ψ(𝑓)∗ = Ψ(𝑓)

and Ψ(𝑓)Ψ(𝑓)∗ = Ψ(|𝑓|2 ) = Ψ(𝑓)∗ Ψ(𝑓).

Proof. Fix 𝑥 ∈ D𝑓 , then the conjugate-linear functional 𝑦 󳨃→ ∫Ω 𝑓 𝑑𝐸𝑥,𝑦 is bounded on 𝐻 (Lemma 9.13). Hence there is a unique element Ψ(𝑓)𝑥 ∈ 𝐻 satisfying (9.13) and ‖Ψ(𝑓)𝑥‖2 ≤ ∫ |𝑓|2 𝑑𝐸𝑥,𝑥 ,

𝑥 ∈ D𝑓 .

(9.15)

Ω

The linearity of Ψ(𝑓) on D𝑓 follows from (9.13) and the fact that 𝐸𝑥,𝑦 is linear in 𝑥. Now we associate with each 𝑓 its truncations 𝑓𝑛 = 𝑓𝜙𝑛 , where 𝜙𝑛 (𝑝) = 1 if |𝑓(𝑝)| ≤ 𝑛, and 𝜙𝑛 (𝑝) = 0, if |𝑓(𝑝)| > 𝑛. Then D𝑓−𝑓𝑛 = D𝑓 , since each 𝑓𝑛 is bounded, and therefore (9.15) shows, using the dominated convergence theorem, that ‖Ψ(𝑓)𝑥 − Ψ(𝑓𝑛 )𝑥‖2 ≤ ∫ |𝑓 − 𝑓𝑛 |2 𝑑𝐸𝑥,𝑥 → 0,

as 𝑛 → ∞,

(9.16)

Ω

for every 𝑥 ∈ D𝑓 . Since 𝑓𝑛 is bounded, (9.14) holds for 𝑓𝑛 . Hence (9.16) implies (9.14) for 𝑓. This proves (a), except for the assertion that Ψ(𝑓) is closed. This will follow from (c) (to be proved) and Lemma 4.5 with 𝑓 instead of 𝑓. (b) First assume that 𝑓 is bounded. Then D𝑓𝑔 ⊂ D𝑔 . If 𝑣 ∈ 𝐻 and 𝑢 = Ψ(𝑓)𝑣, we get from Section 9.1 and (9.11) that (Ψ(𝑓)Ψ(𝑔)𝑥, 𝑣) = (Ψ(𝑔)𝑥, Ψ(𝑓)𝑣) = (Ψ(𝑔)𝑥, 𝑢) = ∫ 𝑔 𝑑𝐸𝑥,𝑢 = ∫ 𝑓𝑔 𝑑𝐸𝑥,𝑣 Ω

= (Ψ(𝑓𝑔)𝑥, 𝑣).

Ω

9.3 Spectral decomposition of unbounded self-adjoint operators

| 139

So we have shown that Ψ(𝑓)Ψ(𝑔)𝑥 = Ψ(𝑓𝑔)𝑥 , 𝑥 ∈ D𝑔 ,

𝑓 ∈ 𝐿∞ .

The last line implies that for 𝑦 = Ψ(𝑔)𝑥 ∫ |𝑓|2 𝑑𝐸𝑦,𝑦 = ∫ |𝑓𝑔|2 𝑑𝐸𝑥,𝑥 , Ω

𝑥 ∈ D𝑔 ,

𝑓 ∈ 𝐿∞ .

(9.17)

Ω

Using truncation we see that (9.17) also holds for arbitrary 𝑓 (possibly unbounded). Since dom(Ψ(𝑓)Ψ(𝑔)) consists of all 𝑥 ∈ D𝑔 such that 𝑦 = Ψ(𝑔)𝑥 ∈ D𝑓 and since (9.17) shows that 𝑦 ∈ D𝑓 if and only if 𝑥 ∈ D𝑓𝑔 , we see that dom(Ψ(𝑓)Ψ(𝑔)) = D𝑔 ∩ D𝑓𝑔 . If 𝑥 ∈ D𝑔 ∩ D𝑓𝑔 , and 𝑦 = Ψ(𝑔)𝑥, and if the truncations 𝑓𝑛 are defined as above, we obtain 𝑓𝑛 → 𝑓 in 𝐿2 (𝐸𝑥,𝑥 ) and 𝑓𝑛 𝑔 → 𝑓𝑔 in 𝐿2 (𝐸𝑦,𝑦 ) and finally Ψ(𝑓)Ψ(𝑔)𝑥 = Ψ(𝑓)𝑦 = lim Ψ(𝑓𝑛 )𝑦 = lim Ψ(𝑓𝑛 𝑔)𝑥 = Ψ(𝑓𝑔)𝑥. 𝑛→∞

𝑛→∞

This proves (b). (c) Suppose that 𝑥 ∈ D𝑓 and 𝑦 ∈ D𝑓 = D𝑓 . It follows from (9.16) and Section 9.1 that (Ψ(𝑓)𝑥, 𝑦) = lim (Ψ(𝑓𝑛 )𝑥, 𝑦) = lim (𝑥, Ψ(𝑓𝑛 )𝑦) = (𝑥, Ψ(𝑓)𝑦). 𝑛→∞

𝑛→∞

∗

Hence 𝑦 ∈ dom(Ψ(𝑓) ), and dom(Ψ(𝑓)) ⊆ dom(Ψ(𝑓)∗ ). If we can show that each 𝑢 ∈ dom(Ψ(𝑓)∗ ) lies in D𝑓 , we obtain Ψ(𝑓)∗ = Ψ(𝑓). Fix 𝑢 for this purpose and put 𝑣 = Ψ(𝑓)∗ 𝑢. Since 𝑓𝑛 = 𝑓𝜙𝑛 , the multiplication theorem yields Ψ(𝑓𝑛 ) = Ψ(𝑓)Ψ(𝜙𝑛 ). Since Ψ(𝜙𝑛 ) is self-adjoint and bounded, we have by Lemma 4.3 and Section 9.1 that Ψ(𝜙𝑛 )Ψ(𝑓)∗ = [Ψ(𝑓)Ψ(𝜙𝑛 )]∗ = Ψ(𝑓𝑛 )∗ = Ψ(𝑓𝑛 ). Hence Ψ(𝜙𝑛 )𝑣 = Ψ(𝑓𝑛 )𝑢,

𝑛 ∈ ℕ.

Since |𝜙𝑛 | ≤ 1 we have now ∫ |𝑓𝑛 |2 𝑑𝐸𝑢,𝑢 = ∫ |𝜙𝑛 |2 𝑑𝐸𝑣,𝑣 ≤ 𝐸𝑣,𝑣 (Ω), Ω

𝑛 ∈ ℕ.

Ω

Hence 𝑢 ∈ D𝑓 . Finally, since D𝑓𝑓 ⊂ D𝑓 , another application of the multiplication theorem gives the last assertion of (c). Definition 9.15. The resolvent set of a linear operator 𝑇 : dom(𝑇) 󳨀→ 𝐻 is the set of all 𝜆 ∈ ℂ such that 𝜆𝐼 − 𝑇 is an injective mapping of dom(𝑇) onto 𝐻 whose inverse belongs to L(𝐻). The spectrum 𝜎(𝑇) of 𝑇 is the complement of the resolvent set of 𝑇. First we collect some information about the spectrum of an unbounded operator.

140 | 9 Spectral analysis Lemma 9.16. If the spectrum 𝜎(𝑇) of an operator 𝑇 does not coincide with the whole of the complex plane ℂ then 𝑇 must be a closed operator. The spectrum of a linear operator is always closed. Moreover, if 𝜁 ∉ 𝜎(𝑇) and 𝑐 := ‖𝑅𝑇 (𝜁)‖ = ‖(𝜁𝐼 − 𝑇)−1 ‖, then the spectrum 𝜎(𝑇) does not intersect the ball {𝑤 ∈ ℂ : |𝜁 − 𝑤| < 𝑐−1 }. The resolvent operator 𝑅𝑇 is a holomorphic operator valued function. Proof. For 𝜁 ∉ 𝜎(𝑇) let 𝑆 = (𝜁𝐼 − 𝑇)−1 which is a bounded operator. Let 𝑥𝑛 ∈ dom(𝑇) with 𝑥 = lim𝑛→∞ 𝑥𝑛 = 𝑥 and lim𝑛→∞ 𝑇𝑥𝑛 = 𝑦 and set 𝑢𝑛 = (𝜁𝐼 − 𝑇)𝑥𝑛 . Then lim 𝑢𝑛 = lim (𝜁𝑥𝑛 − 𝑇𝑥𝑛 ) = 𝜁𝑥 − 𝑦,

𝑛→∞

𝑛→∞

therefore 𝑆(𝜁𝑥 − 𝑦) = lim 𝑆𝑢𝑛 = lim 𝑥𝑛 = 𝑥. 𝑛→∞

𝑛→∞

This implies 𝑥 ∈ dom(𝑇) and (𝜁𝐼 − 𝑇)𝑥 = 𝜁𝑥 − 𝑦, or 𝑇𝑥 = 𝑦. Hence 𝑇 is closed. The remainder of the proof is similar to the case when 𝑇 is bounded, see Lemma 9.6. In the next proposition we refer to the concept of the essential range of a function with respect to a given resolution of the identity (Definition 9.3). Proposition 9.17. Let 𝐸 be a resolution of the identity on Ω and 𝑓 : Ω 󳨀→ ℂ a measurable function. For 𝛼 ∈ ℂ put 𝜔𝛼 = {𝑝 ∈ Ω : 𝑓(𝑝) = 𝛼}. (a) If 𝛼 is in the essential range of 𝑓 and 𝐸(𝜔𝛼 ) ≠ 0, then 𝛼𝐼 − Ψ(𝑓) is not injective. (b) If 𝛼 is in the essential range of 𝑓 but 𝐸(𝜔𝛼 ) = 0, then 𝛼𝐼−Ψ(𝑓) is an injective mapping of D𝑓 onto a proper dense subspace of 𝐻, and there exist vectors 𝑥𝑛 ∈ 𝐻, with ‖𝑥𝑛 ‖ = 1, such that lim [𝛼𝑥𝑛 − Ψ(𝑓)𝑥𝑛 ] = 0. 𝑛→∞

(c) 𝜎(Ψ(𝑓)) is the essential range of 𝑓. One says that 𝛼 lies in the point spectrum of Ψ(𝑓) in case (a) and in the continuous spectrum of Ψ(𝑓) in case (b). Proof. Without loss of generality we can assume that 𝛼 = 0. (a) If 𝐸(𝜔0 ) ≠ 0, there exists 𝑥0 ∈ im(𝐸(𝜔0 )) with ‖𝑥0 ‖ = 1. Let 𝜙0 be the characteristic function of 𝜔0 . Then 𝑓𝜙0 = 0, and Ψ(𝑓)Ψ(𝜙0 ) = 0. Since Ψ(𝜙0 ) = 𝐸(𝜔0 ), it follows that Ψ(𝑓)𝑥0 = Ψ(𝑓)𝐸(𝜔0 )𝑥0 = Ψ(𝑓)Ψ(𝜙0 )𝑥0 = 0. (b) Now we have 𝐸(𝜔0 ) = 0 but 𝐸(𝜔𝑛 ) ≠ 0 for 𝑛 ∈ ℕ where 𝜔𝑛 = {𝑝 ∈ Ω : |𝑓(𝑝)| < 1/𝑛}.

9.3 Spectral decomposition of unbounded self-adjoint operators

| 141

Let 𝑥𝑛 ∈ im(𝐸(𝜔𝑛 )) with ‖𝑥𝑛 ‖ = 1 and let 𝜙𝑛 be the characteristic functions of 𝜔𝑛 . As in (a) we obtain ‖Ψ(𝑓)𝑥𝑛 ‖ = ‖Ψ(𝑓𝜙𝑛 )𝑥𝑛 ‖ ≤ ‖Ψ(𝑓𝜙𝑛 )‖ = ‖𝑓𝜙𝑛 ‖∞ ≤ 1/𝑛. Thus Ψ(𝑓)𝑥𝑛 → 0 although ‖𝑥𝑛 ‖ = 1. If Ψ(𝑓)𝑥 = 0 for some 𝑥 ∈ D𝑓 , then ∫ |𝑓|2 𝑑𝐸𝑥,𝑥 = ‖Ψ(𝑓)𝑥‖2 = 0. Ω

Since |𝑓| > 0 almost everywhere (𝐸𝑥,𝑥 ), we must have 𝐸𝑥,𝑥 (Ω) = 0. But 𝐸𝑥,𝑥 (Ω) = ‖𝑥‖2 . Hence Ψ(𝑓) is injective. Similarly Ψ(𝑓)∗ = Ψ(𝑓) is injective. If 𝑦 ⊥ im(Ψ(𝑓)), then 𝑥 󳨃→ (Ψ(𝑓)𝑥, 𝑦) = 0 is continuous in D𝑓 , hence 𝑦 ∈ dom(Ψ(𝑓)∗ ), and (𝑥, Ψ(𝑓)𝑦) = (Ψ(𝑓)𝑥, 𝑦) = 0 , 𝑥 ∈ D𝑓 . Hence Ψ(𝑓)𝑦 = 0 and 𝑦 = 0. Therefore im(Ψ(𝑓)) is dense in 𝐻. Since Ψ(𝑓) is closed, so is Ψ(𝑓)−1 . If im(Ψ(𝑓)) = 𝐻, the closed graph theorem would imply that Ψ(𝑓)−1 ∈ L(𝐻). This is impossible in view of the sequence (𝑥𝑛 )𝑛 constructed above. Hence (b) is proved. (c) It follows from (a) and (b) that the essential range of 𝑓 is a subset of 𝜎(Ψ(𝑓)). Now assume that 0 is not in the essential range of 𝑓. Then 𝑔 = 1/𝑓 ∈ 𝐿∞ (𝐸), and 𝑓𝑔 = 1, hence Ψ(𝑓)Ψ(𝑔) = Ψ(1) = 𝐼, which proves that im(Ψ(𝑓)) = 𝐻. Since |𝑓| > 0, we have that Ψ(𝑓) is injective, as in the proof of (b). By the closed graph theorem, Ψ(𝑓)−1 ∈ L(𝐻). Therefore 0 ∉ 𝜎(Ψ(𝑓)) and (c) is proved. In the following proposition we describe the change of measure principle. Proposition 9.18. Let M and M󸀠 be 𝜎-algebras in the sets Ω, Ω󸀠 ⊆ ℂ and let 𝐸 : M 󳨀→ L(𝐻) be a resolution of identity, and suppose that Φ : Ω 󳨀→ Ω󸀠 has the property that Φ−1 (𝜔󸀠 ) ∈ M for every 𝜔󸀠 ∈ M󸀠 . If 𝐸󸀠 (𝜔󸀠 ) = 𝐸(Φ−1 (𝜔󸀠 )), then 𝐸󸀠 : M󸀠 󳨀→ L(𝐻) is a resolution of the identity, and 󸀠 ∫ 𝑓 𝑑𝐸𝑥,𝑦 = ∫(𝑓 ∘ Φ) 𝑑𝐸𝑥,𝑦 Ω󸀠

(9.18)

Ω

for every M󸀠 -measurable 𝑓 : Ω󸀠 󳨀→ ℂ for which either of these integrals exists. Proof. A straightforward verification gives that 𝐸󸀠 is again a resolution of the identity. For characteristic functions, (9.18) is just the definition of 𝐸󸀠 . So (9.18) follows for simple functions and also in the general case. In order to derive the general spectral theorem for unbounded self-adjoint operators we will use the Cayley transform.

142 | 9 Spectral analysis The mapping 𝑡−𝑖 𝑡+𝑖 sets up a bijection between the real line and the unit circle minus the point 1. The symbolic calculus developed in Section 9.2 therefore shows that every self-adjoint operator 𝑇 ∈ L(𝐻) gives rise to a unitary operator 𝑡 󳨃→

𝑈 = (𝑇 − 𝑖𝐼)(𝑇 + 𝑖𝐼)−1 , and that every unitary 𝑈 whose spectrum does not contain the point 1 is obtained in this way. This relation will now be extended to unbounded symmetric operators. If 𝑇 is a symmetric operator, we have ‖𝑇𝑥 + 𝑖𝑥‖2 = (𝑇𝑥 + 𝑖𝑥, 𝑇𝑥 + 𝑖𝑥) = ‖𝑥‖2 + ‖𝑇𝑥‖2 = ‖𝑇𝑥 − 𝑖𝑥‖2 , for 𝑥 ∈ dom(𝑇). This implies that (𝑇 + 𝑖𝐼) is injective, and that there is an isometry 𝑈 with dom(𝑈) = im(𝑇 + 𝑖𝐼), and im(𝑈) = im(𝑇 − 𝑖𝐼), defined by 𝑈(𝑇𝑥 + 𝑖𝑥) = 𝑇 − 𝑖𝑥,

𝑥 ∈ dom(𝑇).

Since (𝑇 + 𝑖𝐼)−1 maps dom(𝑈) onto dom(𝑇), we can write 𝑈 = (𝑇 − 𝑖𝐼)(𝑇 + 𝑖𝐼)−1 . This operator 𝑈 is called the Cayley transform of 𝑇. Lemma 9.19. Let 𝑈 be an isometry, i.e. ‖𝑈𝑥‖ = ‖𝑥‖ for all 𝑥 ∈ dom(𝑈). (a) For 𝑥, 𝑦 ∈ dom(𝑈), we have (𝑈𝑥, 𝑈𝑦) = (𝑥, 𝑦). (b) If im(𝐼 − 𝑈) is dense in 𝐻, then 𝐼 − 𝑈 is injective. (c) If any one of the three spaces dom(𝑈), im(𝑈) and G(𝑈) is closed, so are the other two. Proof. (a) Follows from the polarization identity: (𝑥, 𝑦) =

1 (‖𝑥 + 𝑦‖2 − ‖𝑥 − 𝑦‖2 + 𝑖‖𝑥 + 𝑖𝑦‖2 − 𝑖‖𝑥 − 𝑖𝑦‖2 ). 4

(9.19)

(b) Let 𝑥 ∈ dom(𝑈) and (𝐼 − 𝑈)𝑥 = 0. Then 𝑥 = 𝑈𝑥 and (𝑥, (𝐼 − 𝑈)𝑦) = (𝑥, 𝑦) − (𝑥, 𝑈𝑦) = (𝑈𝑥, 𝑈𝑦) − (𝑥, 𝑈𝑦) = 0 for every 𝑦 ∈ dom(𝑈). This implies 𝑥 ⊥ im(𝐼 − 𝑈), so that 𝑥 = 0 if im(𝐼 − 𝑈) is dense in 𝐻. (c) follows from ‖𝑈𝑥 − 𝑈𝑦‖ = ‖𝑥 − 𝑦‖ =

1 ‖(𝑥, 𝑈𝑥) − (𝑦, 𝑈𝑦)‖, √2

𝑥, 𝑦 ∈ dom(𝑈),

where in the last term (𝑥, 𝑈𝑥), (𝑦, 𝑈𝑦) ∈ G(𝑈) are elements of the graph of 𝑈 and ‖(𝑥, 𝑈𝑥) − (𝑦, 𝑈𝑦)‖ = (‖𝑥 − 𝑦‖2 + ‖𝑈𝑥 − 𝑈𝑦‖2 )1/2 .

9.3 Spectral decomposition of unbounded self-adjoint operators

| 143

Proposition 9.20. Let 𝑇 be a symmetric operator on 𝐻 (not necessarily densely defined) and let 𝑈 be its Cayley transform. The following statements are true: (a) 𝑈 is closed if and only if 𝑇 is closed. (b) im(𝐼 − 𝑈) = dom(𝑇), and 𝐼 − 𝑈 is injective, and 𝑇 can be reconstructed from 𝑈 by 𝑇 = 𝑖(𝐼 + 𝑈)(𝐼 − 𝑈)−1 . The Cayley transforms of distinct symmetric operators are distinct. (c) 𝑈 is unitary if and only if 𝑇 is self-adjoint. Conversely, if 𝑉 is an operator in 𝐻 which is an isometry, and if 𝐼 − 𝑉 is injective, then 𝑉 is the Cayley transform of a symmetric operator in 𝐻. Proof. (a) The identity ‖𝑇𝑥 + 𝑖𝑥‖2 = ‖𝑥‖2 + ‖𝑇𝑥‖2 implies that (𝑇 + 𝑖𝐼)𝑥 ↔ (𝑥, 𝑇𝑥) is an isometric one-to-one correspondence between im(𝑇 + 𝑖𝐼) and the graph G(𝑇) of 𝑇. Hence 𝑇 is closed if and only if im(𝑇 + 𝑖𝐼) is closed. By Lemma 9.19, 𝑈 is closed if and only if dom(𝑈) is closed. But, by the definition of the Cayley transform, dom(𝑈) = im(𝑇 + 𝑖𝐼), which proves (a). (b) The one-to-one correspondence 𝑥 ↔ 𝑧 between dom(𝑇) and dom(𝑈) = im(𝑇 + 𝑖𝐼), given by 𝑧 = 𝑇𝑥 + 𝑖𝑥, 𝑈𝑧 = 𝑇𝑥 − 𝑖𝑥 can be written in the form (𝐼 − 𝑈)𝑧 = 2𝑖𝑥,

(𝐼 + 𝑈)𝑧 = 2𝑇𝑥.

Hence (𝐼 − 𝑈) is injective and im(𝐼 − 𝑈) = dom(𝑇), therefore (𝐼 − 𝑈)−1 maps dom(𝑇) onto dom(𝑈), and 2𝑇𝑥 = (𝐼 + 𝑈)𝑧 = (𝐼 + 𝑈)(𝐼 − 𝑈)−1 (2𝑖𝑥),

𝑥 ∈ dom(𝑇).

This proves (b). (c) Assume that 𝑇 is self-adjoint. Then, by Lemma 4.29, im(𝐼 + 𝑇∗ 𝑇) = im(𝐼 + 𝑇2 ) = 𝐻.

(9.20)

We have (𝑇 + 𝑖𝐼)(𝑇 − 𝑖𝐼) = 𝑇2 + 𝐼 = (𝑇 − 𝑖𝐼)(𝑇 + 𝑖𝐼), where all operators have domain dom(𝑇2 ). Hence, (9.20) implies that dom(𝑈) = im(𝑇 + 𝑖𝐼) = 𝐻

(9.21)

im(𝑈) = im(𝑇 − 𝑖𝐼) = 𝐻.

(9.22)

and Now (𝑈∗ 𝑈𝑥, 𝑥) = (𝑈𝑥, 𝑈𝑥) = (𝑥, 𝑥) for every 𝑥 ∈ 𝐻, which implies 𝑈∗ 𝑈 = 𝐼 and 𝑈 is unitary.

144 | 9 Spectral analysis Now assume that 𝑈 is unitary. Then (im(𝐼 − 𝑈))⊥ = ker(𝐼 − 𝑈)∗ = {0}, and dom(𝑇) = im(𝐼−𝑈) is dense in 𝐻. Thus 𝑇∗ is defined and 𝑇 ⊂ 𝑇∗ . Fix 𝑦 ∈ dom(𝑇∗ ). Since im(𝑇 + 𝑖𝐼) = dom(𝑈) = 𝐻, there exists 𝑦0 ∈ dom(𝑇) such that (𝑇∗ + 𝑖𝐼)𝑦 = (𝑇 + 𝑖𝐼)𝑦0 = (𝑇∗ + 𝑖𝐼)𝑦0 . Set 𝑦1 = 𝑦 − 𝑦0 . Then 𝑦1 ∈ dom(𝑇∗ ) and, for every 𝑥 ∈ dom(𝑇) we have ((𝑇 − 𝑖𝐼)𝑥, 𝑦1 ) = (𝑥, (𝑇∗ + 𝑖𝐼)𝑦1 ) = (𝑥, 0) = 0. Thus 𝑦1 ⊥ im(𝑇 − 𝑖𝐼) = im(𝑈) = 𝐻, so 𝑦1 = 0 and 𝑦 = 𝑦0 ∈ dom(𝑇). Hence dom(𝑇) = dom(𝑇∗ ) and (c) is proved. Finally, let 𝑉 be as in the statement of the converse. Then there is a one-to-one correspondence 𝑧 ↔ 𝑥 between dom(𝑉) and im(𝐼 − 𝑉), given by 𝑥 = 𝑧 − 𝑉𝑧. Define 𝑆 on dom(𝑆) = im(𝐼 − 𝑉) by 𝑆𝑥 = 𝑖(𝑧 + 𝑉𝑧) if 𝑥 = 𝑧 − 𝑉𝑧.

(9.23)

If 𝑥, 𝑦 ∈ dom(𝑆), then 𝑥 = 𝑧 − 𝑉𝑧 and 𝑦 = 𝑢 − 𝑉𝑢 for some 𝑧, 𝑢 ∈ dom(𝑉). Since 𝑉 is an isometry, it follows from Lemma 9.19 that (𝑆𝑥, 𝑦) = 𝑖(𝑧 + 𝑉𝑧, 𝑢 − 𝑉𝑢) = 𝑖(𝑉𝑧, 𝑢) − 𝑖(𝑧, 𝑉𝑢) = (𝑧 − 𝑉𝑧, 𝑖𝑢 + 𝑖𝑉𝑢) = (𝑥, 𝑆𝑦). Hence 𝑆 is symmetric. For 𝑧 ∈ dom(𝑉), (9.23) can be written in the form 2𝑖𝑉𝑧 = 𝑆𝑥 − 𝑖𝑥 , 2𝑖𝑧 = 𝑆𝑥 + 𝑖𝑥, hence, if 𝑥 ∈ dom(𝑆), we obtain 𝑉(𝑆𝑥 + 𝑖𝑥) = 𝑆𝑥 − 𝑖𝑥 and that dom(𝑉) = im(𝑆 + 𝑖𝐼). Therefore 𝑉 is the Cayley transform of 𝑆. At this point we use the methods developed above to prove a key result about the spectrum of an unbounded self-adjoint operator. It transfers properties of unbounded selfadjoint operators to the bounded resolvent operators. Proposition 9.21. The spectrum 𝜎(𝑇) of any self-adjoint operator 𝑇 is real and nonempty. If 𝜁 ∉ 𝜎(𝑇) then ‖(𝜁𝐼 − 𝑇)−1 ‖ ≤ |ℑ𝜁|−1 . (9.24) Moreover, (𝜁𝐼 − 𝑇)−1 = ((𝜁𝐼 − 𝑇)−1 )∗ .

(9.25)

9.3 Spectral decomposition of unbounded self-adjoint operators

|

145

Proof. Let 𝜁 = 𝜉 + 𝑖𝜂 and 𝜂 ≠ 0 and set 𝐾 = 𝜂1 (𝑇 − 𝜉𝐼). Using Lemma 4.4, it follows that 𝐾∗ = 𝐾. Let 𝑓 ∈ dom(𝐾) such that 𝐾𝑓 = 𝐾∗ 𝑓 = 𝑖𝑓, then 𝑖(𝑓, 𝑓) = (𝐾𝑓, 𝑓) = (𝑓, 𝐾𝑓) = −𝑖(𝑓, 𝑓), which implies 𝑓 = 0 and that 𝐾 − 𝑖𝐼 is injective. In a similar way one shows that 𝐾 + 𝑖𝐼 is injective. The proof of Proposition 9.20 part (a) implies that im(𝐾 ± 𝑖𝐼) is closed. Now we obtain from Lemma 4.8 that im(𝐾 ± 𝑖𝐼)⊥ = ker(𝐾 ± 𝑖𝐼) = {0}. Therefore (𝐾 ± 𝑖𝐼)−1 is defined on the whole of 𝐻. Since we have ‖𝐾𝑥 ± 𝑖𝑥‖2 = ‖𝐾𝑥‖2 + ‖𝑥‖2 ,

𝑥 ∈ dom(𝐾),

we get ‖(𝐾 ± 𝑖𝐼)−1 𝑦‖ = ‖(𝐾 ± 𝑖𝐼)−1 (𝐾 ± 𝑖𝐼)𝑥‖ = ‖𝑥‖ ≤ ‖(𝐾 ± 𝑖𝐼)𝑥‖ = ‖𝑦‖, for each 𝑦 ∈ 𝐻, which implies that ‖(𝐾 ± 𝑖𝐼)−1 ‖ ≤ 1.

(9.26)

Thus ±𝑖 ∉ 𝜎(𝐾) and hence 𝜁 ∉ 𝜎(𝑇). In addition (9.26) implies (9.24). Now let 𝑥1 , 𝑥2 ∈ dom(𝑇). Then ((𝑇 − 𝜁𝐼)𝑥1 , 𝑥2 ) = (𝑥1 , (𝑇 − 𝜁𝐼)𝑥2 ). Putting 𝑦1 = (𝑇 − 𝜁𝐼)𝑥1 and 𝑦2 = (𝑇 − 𝜁𝐼)𝑥2 and rewriting the last equation in terms of 𝑦1 and 𝑦2 yields (9.25). Finally suppose that 𝜎(𝑇) = 0. Then for any 𝑥, 𝑦 ∈ 𝐻 the complex-valued function 𝑓(𝜁) := ((𝜁𝐼 − 𝑇)−1 𝑥, 𝑦) is holomorphic on ℂ and, by (9.24), vanishes as |ℑ𝜁| → ∞. Liouville’s Theorem now implies that 𝑓 = 0 identically. Since 𝑥, 𝑦 ∈ 𝐻 are arbitrary, we obtain (𝜁𝐼 − 𝑇)−1 is identically zero. This is false, hence 𝜎(𝑇) ≠ 0. The Cayley transform is now used to reduce the construction of the spectral decomposition of an unbounded self-adjoint operator to the spectral decomposition of a unitary operator. Proposition 9.22. Let 𝑇 be an unbounded self-adjoint operator (𝑇 = 𝑇∗ and dom(𝑇) = dom(𝑇∗ )). Then there exists a uniquely determined resolution of the identity 𝐸 on the Borel subsets of ℝ, such that ∞

(𝑇𝑥, 𝑦) = ∫ 𝑡 𝑑𝐸𝑥,𝑦 (𝑡),

𝑥 ∈ dom(𝑇),

𝑦 ∈ 𝐻.

(9.27)

−∞

Moreover, 𝐸 is concentrated on the spectrum 𝜎(𝑇) ⊂ ℝ of 𝑇, in the sense that 𝐸(𝜎(𝑇)) = 𝐼. Proof. Let 𝑈 be the Cayley transform of 𝑇 and let Ω be the unit circle with the point 1 removed. Let 𝐸󸀠 be the spectral decomposition of 𝑈 (Proposition 9.7). By Proposition 9.20 𝐼 − 𝑈 is injective and, by Proposition 9.9 𝐸󸀠 ({1}) = 0. Hence 󸀠 (𝜆), (𝑈𝑥, 𝑦) = ∫ 𝜆 𝑑𝐸𝑥,𝑦 Ω

𝑥, 𝑦 ∈ 𝐻.

(9.28)

146 | 9 Spectral analysis Define 𝑓(𝜆) =

𝑖(1 + 𝜆) , (1 − 𝜆)

𝜆 ∈ Ω.

We define Ψ(𝑓) as in Proposition 9.14 with 𝐸󸀠 in place of 𝐸. 󸀠 (Ψ(𝑓)𝑥, 𝑦) = ∫ 𝑓 𝑑𝐸𝑥,𝑦 ,

𝑥 ∈ D𝑓 ,

𝑦 ∈ 𝐻.

(9.29)

Ω

Since 𝑓 is real-valued, Ψ(𝑓) is self-adjoint (Proposition 9.14). Since 𝑓(𝜆)(1 − 𝜆) = 𝑖(1 + 𝜆), the symbolic calculus gives Ψ(𝑓)(𝐼 − 𝑈) = 𝑖(𝐼 + 𝑈).

(9.30)

im(𝐼 − 𝑈) ⊂ dom(Ψ(𝑓)).

(9.31)

𝑇(𝐼 − 𝑈) = 𝑖(𝐼 + 𝑈)

(9.32)

This implies in particular By Proposition 9.20 and dom(𝑇) = im(𝐼 − 𝑈) ⊂ dom(Ψ(𝑓)). (9.31) and (9.32) imply that Ψ(𝑓) is a self-adjoint extension of the self-adjoint operator 𝑇. Thus we have 𝑇 ⊂ Ψ(𝑓) and Ψ(𝑓) = Ψ(𝑓)∗ ⊂ 𝑇∗ = 𝑇, hence Ψ(𝑓) = 𝑇. In addition we get 󸀠 , 𝑥 ∈ dom(𝑇), (𝑇𝑥, 𝑦) = ∫ 𝑓 𝑑𝐸𝑥,𝑦

𝑦 ∈ 𝐻.

(9.33)

Ω

By Proposition 9.17 (c), 𝜎(𝑇) is the essential range of 𝑓. Thus 𝜎(𝑇) ⊂ ℝ. Since 𝑓 is injective in Ω, we can define 𝐸(𝑓(𝜔)) = 𝐸󸀠 (𝜔) for every Borel set 𝜔 ⊂ Ω, to obtain the desired resolution 𝐸 which converts (9.33) into (9.27). The uniqueness of 𝐸 follows from the uniqueness of the representation (9.28). The symbolic calculus is now used to prove the following assertions. Proposition 9.23. Let 𝑇 be a self-adjoint operator on 𝐻. (a) (𝑇𝑥, 𝑥) ≥ 0 for every 𝑥 ∈ dom(𝑇) (briefly 𝑇 ≥ 0) if and only if 𝜎(𝑇) ⊂ [0, ∞). (b) If 𝑇 ≥ 0, there exists a unique self-adjoint operator 𝑆 ≥ 0 such that dom(𝑆) ⊇ dom(𝑇) and 𝑆2 = 𝑇 on dom(𝑇). The symbolic calculus implies that 𝑥 ∈ dom(𝑇) if and only if 𝑥 ∈ dom(𝑆) and also 𝑆𝑥 ∈ dom(𝑆). In addition dom(𝑇) is a core of 𝑆. Proof. (a) see Proposition 9.11 (a). (b) Assume 𝑇 ≥ 0, so that 𝜎(𝑇) ⊂ [0, ∞), and ∞

(𝑇𝑥, 𝑦) = ∫ 𝑡 𝑑𝐸𝑥,𝑦 (𝑡), 0

𝑥 ∈ dom(𝑇),

𝑦 ∈ 𝐻,

(9.34)

9.3 Spectral decomposition of unbounded self-adjoint operators

147

|

∞

where dom(𝑇) = {𝑥 ∈ 𝐻 : ∫0 𝑡2 𝑑𝐸𝑥,𝑥 (𝑡) < ∞}. Let 𝑠(𝑡) be the nonnegative square root of 𝑡 ≥ 0 and put Ψ(𝑠) = 𝑆, explicitly ∞

(𝑆𝑥, 𝑦) = ∫ 𝑠(𝑡) 𝑑𝐸𝑥,𝑦 (𝑡),

𝑥 ∈ D𝑠 ,

𝑦 ∈ 𝐻.

(9.35)

0

By Proposition 9.14 we obtain that 𝑆2 = 𝑇 on dom(𝑇) and 𝑆 ≥ 0. To prove uniqueness, suppose 𝑅 is self-adjoint, 𝑅 ≥ 0 and 𝑅2 = 𝑇, and 𝐸𝑅 is its spectral decomposition: ∞ 𝑅 (𝑅𝑥, 𝑦) = ∫ 𝑡 𝑑𝐸𝑥,𝑦 (𝑡),

𝑥 ∈ dom(𝑅),

𝑦 ∈ 𝐻.

(9.36)

0

We apply Proposition 9.18 with Ω = [0, ∞) , 𝜙(𝑡) = 𝑡2 , 𝑓(𝑡) = 𝑡, and 𝐸󸀠 (𝜙(𝜔)) = 𝐸𝑅 (𝜔) for 𝜔 ⊂ [0, ∞), to obtain

∞

∞

𝑅 󸀠 = ∫ 𝑡 𝑑𝐸𝑥,𝑦 (𝑡). (𝑇𝑥, 𝑦) = (𝑅2 𝑥, 𝑦) = ∫ 𝑡2 𝑑𝐸𝑥,𝑦(𝑡) 0

(9.37)

(9.38)

0

(9.34) and (9.38) and the uniqueness statement in Proposition 9.22 show that 𝐸󸀠 = 𝐸. By (9.37), 𝐸 determines 𝐸𝑅 , and hence 𝑅. The statement about the domains of the operators 𝑆 and 𝑇 follows from the symbolic calculus and the definition of the domains there (see Proposition 9.14). As 𝑆∗ = 𝑆, Lemma 4.32 immediately implies that dom(𝑇) is a core of 𝑆. As applications of these results we prove a useful characterization of self-adjoint and essentially self-adjoint operators. Proposition 9.24. Let 𝑇 be a closed symmetric operator. Then the following statements are equivalent: (i) 𝑇 is self-adjoint; (ii) ker(𝑇∗ + 𝑖𝐼) = {0} and ker(𝑇∗ − 𝑖𝐼) = {0}; (iii) im(𝑇 + 𝑖𝐼) = 𝐻 and im(𝑇 − 𝑖𝐼) = 𝐻. Proof. (i) implies (ii): By Proposition 9.21 ±𝑖 ∉ 𝜎(𝑇). (ii) implies (iii): Notice that ker(𝑇∗ ± 𝑖𝐼) = {0} if and only if im(𝑇 ∓ 𝑖𝐼) is dense in 𝐻. This follows easily from (𝑇𝑢 ± 𝑖𝑢, 𝑣) = (𝑢, 𝑇∗ ∓ 𝑖𝑣), for 𝑢, 𝑣 ∈ dom(𝑇). So it remains to show that im(𝑇 ∓ 𝑖𝐼) is closed. The symmetry of 𝑇 implies that ‖(𝑇 ∓ 𝑖𝐼)𝑢‖2 = ‖𝑇𝑢‖2 + ‖𝑢‖2 , (9.39) for 𝑢 ∈ dom(𝑇). Now, since 𝑇 is closed, we easily obtain that im(𝑇 ∓ 𝑖𝐼) is closed.

148 | 9 Spectral analysis (iii) implies (i): Let 𝑢 ∈ dom(𝑇∗ ). By (iii) there exists 𝑣 ∈ dom(𝑇) such that (𝑇 − 𝑖𝐼)𝑣 = (𝑇∗ − 𝑖𝐼)𝑢. Since 𝑇 is symmetric, we have also (𝑇∗ − 𝑖𝐼)(𝑣 − 𝑢) = 0. But, if (𝑇 + 𝑖𝐼) is surjective, then (𝑇∗ − 𝑖𝐼) is injective (Lemma 4.8) and we obtain 𝑢 = 𝑣. This proves that 𝑢 ∈ dom(𝑇) and that 𝑇 is self-adjoint. We proved during the assertion that (ii) implies (iii) that Lemma 9.25. If 𝑇 is closed and symmetric, then im(𝑇 ± 𝑖𝐼) is closed. In a similar way we obtain a characterization for essentially self-adjoint operators. Proposition 9.26. Let 𝐴 be a symmetric operator. Then the following statements are equivalent: (i) 𝐴 is essentially self-adjoint; (ii) ker(𝐴∗ + 𝑖𝐼) = {0} and ker(𝐴∗ − 𝑖𝐼) = {0}; (iii) im(𝐴 + 𝑖𝐼) and im(𝐴 − 𝑖𝐼) are dense in 𝐻. Proof. We apply Proposition 9.24 to 𝐴 and notice that 𝐴 is symmetric and that Lemma 4.5 implies that 𝐴∗ = (𝐴)∗ . In addition we use Lemma 9.25. If 𝐴 is also a positive operator, we get Proposition 9.27. Let 𝐴 be a positive, symmetric operator. Then the following statements are equivalent: (i) 𝐴 is essentially self-adjoint; (ii) ker(𝐴∗ + 𝑏𝐼) = {0} for some 𝑏 > 0; (iii) im(𝐴 + 𝑏𝐼) is dense in 𝐻. Proof. We proceed in a similar way as before and notice that for a positive, symmetric operator 𝐴 we have ((𝐴 + 𝑏𝐼)𝑢, 𝑢) ≥ 𝑏‖𝑢‖2 , (9.40) for 𝑢 ∈ dom(𝐴), which is a good substitute for (9.39). By Lemma 4.8, (ii) and (iii) are equivalent. Since the closure of a positive, symmetric operator is again positive and symmetric, it remains to show that a closed, positive symmetric operator 𝑇 is self-adjoint if and only if ker(𝑇∗ + 𝑏𝐼) = {0} for some 𝑏 > 0. We can suppose that 𝑏 = 1. If 𝑇 is self-adjoint, then the spectrum 𝜎(𝑇) ⊆ ℝ+ , hence ker(𝑇 + 𝐼) = ker(𝑇∗ + 𝐼) = {0}. For the converse, we first show that im(𝑇 + 𝐼) is closed: let (𝑦𝑘 )𝑘 ⊂ im(𝑇 + 𝐼) be a convergent sequence. There exists a sequence (𝑥𝑘 )𝑘 ⊂ dom(𝑇) such that 𝑦𝑘 = (𝑇 + 𝐼)𝑥𝑘 . Then (𝑥𝑘 , 𝑦𝑘 ) = (𝑥𝑘 , 𝑇𝑥𝑘 ) + ‖𝑥𝑘 ‖2 ≥ ‖𝑥𝑘 ‖2 , and, by Cauchy–Schwarz, ‖𝑥𝑘 ‖ ≤ ‖𝑦𝑘 ‖.

(9.41)

9.3 Spectral decomposition of unbounded self-adjoint operators

| 149

Since (𝑦𝑘 )𝑘 is convergent, sup𝑘 ‖𝑦𝑘 ‖ < ∞, and, by (9.41), sup𝑘 ‖𝑥𝑘 ‖ < ∞. Now, positivity implies ‖𝑥𝑘 − 𝑥ℓ ‖2 ≤ (𝑥𝑘 − 𝑥ℓ , (𝑇 + 𝐼)(𝑥𝑘 − 𝑥ℓ )) ≤ (‖𝑥𝑘 ‖ + ‖𝑥ℓ ‖)‖𝑦𝑘 − 𝑦ℓ ‖ ≤ 𝐶‖𝑦𝑘 − 𝑦ℓ ‖. Hence (𝑥𝑘 )𝑘 is a Cauchy sequence. Since we supposed that 𝑇 is closed, there exists 𝑥 ∈ dom(𝑇) such that 𝑥 = lim𝑘→∞ 𝑥𝑘 and (𝑇 + 𝐼)𝑥 = 𝑦 = lim𝑘→∞ 𝑦𝑘 . Hence im(𝑇 + 𝐼) is closed. The assumption ker(𝑇∗ + 𝐼) = {0} now gives im(𝑇 + 𝐼) = 𝐻. In order to show that 𝑇 is self-adjoint it suffices to show that dom(𝑇∗ ) ⊆ dom(𝑇). Let 𝑥 ∈ dom(𝑇∗ ). There exists 𝑦 ∈ dom(𝑇) such that (𝑇 + 𝐼)𝑦 = (𝑇∗ + 𝐼)𝑦 = (𝑇∗ + 𝐼)𝑥, since dom(𝑇) ⊆ dom(𝑇∗ ). This implies (𝑇∗ + 𝐼)(𝑥 − 𝑦) = 0, and hence 𝑥 = 𝑦 ∈ dom(𝑇). Finally we mention that every self-adjoint operator is unitarily equivalent to a multiplication operator, which is important for applications to the solution of other spectral problems. We omit the proof, as we will not use this version of the spectral theorem in the sequel. Using the Riesz representation theorem in measure theory (see for instance [63]), this version follows easily from Proposition 9.22. Theorem 9.28. Let 𝑇 be a self-adjoint operator on 𝐻 with spectrum 𝜎(𝑇). Then there exists a finite measure 𝜇 on 𝜎(𝑇) × ℕ and a unitary operator 𝑈 : 𝐻 󳨀→ 𝐿2 (𝜎(𝑇) × ℕ, 𝑑𝜇) with the following properties: if 𝑔 : 𝜎(𝑇) × ℕ 󳨀→ ℝ is the function 𝑔(𝑡, 𝑛) = 𝑡, then 𝑥 ∈ 𝐻 lies in dom(𝑇) if and only if 𝑔⋅𝑈𝑥 ∈ 𝐿2 (𝜎(𝑇)×ℕ, 𝑑𝜇). In addition we have 𝑈𝑇𝑈−1 ℎ = 𝑔ℎ for all ℎ ∈ 𝑈(dom(𝑇)), and

𝑈𝑓(𝑇)𝑈−1 ℎ = 𝑓(𝑔)ℎ

for all bounded Borel functions 𝑓 on 𝜎(𝑇). In particular, 𝑓(𝑇) is a bounded operator and ‖𝑓(𝑇)‖ = ‖𝑓‖∞ .

(9.42)

In the proof of this version of the spectral theorem one starts with a function 𝑓 ∈ C0 (𝜎(𝑇)), a continuous function which vanishes at infinity. Then one considers the

150 | 9 Spectral analysis linear functional Λ(𝑓) := (𝑓(𝑇)𝑥, 𝑥), where 𝑥 ∈ 𝐻 is an appropriate vector (a cyclic vector). The Riesz representation theorem (see [63]) implies that there exists a finite countably additive measure 𝜇 on ℝ such that (𝑓(𝑇)𝑥, 𝑥) = ∫ 𝑓(𝑡) 𝑑𝜇(𝑡). 𝜎(𝑇)

This is the main step. The rest of the proof consists of an application of the symbolic calculus (see [18] for all details).

9.4 Determination of the spectrum In this part we prove some results of the spectral theory of unbounded self-adjoint operators, which are used later on for the applications to the ◻-operator, to Schrödinger operators with magnetic field, and to Pauli and Dirac operators. These results enable us to determine the spectrum of the ◻-operator in some special cases and they yield methods to decide whether the corresponding differential operators are with compact resolvent. First we prove some general results about the spectrum of an unbounded selfadjoint operator. Lemma 9.29. Let 𝑇 be an unbounded self-adjoint operator. Then 𝜆 ∈ 𝜎(𝑇) if and only if there exists a sequence (𝑥𝑘 )𝑘 in dom(𝑇) such that ‖𝑥𝑘 ‖ = 1 for each 𝑘 ∈ ℕ and lim ‖(𝑇 − 𝜆𝐼)𝑥𝑘 ‖ = 0.

𝑘→∞

If 𝑇 and 𝑆 are self-adjoint operators such that 𝑇 = 𝑈−1 𝑆𝑈 for some unitary operator 𝑈, where dom(𝑇) = 𝑈−1 (dom(𝑆)), then 𝜎(𝑇) = 𝜎(𝑆). Proof. If 𝜆 ∈ 𝜎(𝑇), then 𝑇 − 𝜆𝐼 is not injective and one can find 𝑥 ∈ dom(𝑇) such that ‖𝑥‖ = 1 and (𝑇 − 𝜆𝐼)𝑥 = 0. So the constant sequence 𝑥𝑘 = 𝑥 has the desired property. Conversely, suppose that 𝜆 ∉ 𝜎(𝑇). Then (𝑇 − 𝜆𝐼)−1 is a bounded operator. If there exists a sequence (𝑥𝑘 )𝑘 in dom(𝑇) such that ‖𝑥𝑘 ‖ = 1 for each 𝑘 ∈ ℕ and lim𝑘→∞ ‖(𝑇 − 𝜆𝐼)𝑥𝑘 ‖ = 0, then lim (𝑇 − 𝜆𝐼)−1 (𝑇 − 𝜆𝐼)𝑥𝑘 = lim 𝑥𝑘 = 0 𝑘→∞

𝑘→∞

yields a contradiction to ‖𝑥𝑘 ‖ = 1 for each 𝑘 ∈ ℕ. To prove the second assertion take 𝜆 ∈ 𝜎(𝑆) and a sequence (𝑦𝑘 )𝑘 in dom(𝑆) such that ‖𝑦𝑘 ‖ = 1 for each 𝑘 ∈ ℕ and lim ‖(𝑆 − 𝜆𝐼)𝑦𝑘 ‖ = 0.

𝑘→∞

9.4 Determination of the spectrum

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151

Then for 𝑈−1 𝑦𝑘 = 𝑧𝑘 ∈ dom(𝑇) we have ‖𝑧𝑘 ‖ = 1, since 𝑈 is unitary and ‖(𝑇 − 𝜆𝐼)𝑧𝑘 ‖ = ‖𝑈−1 (𝑆𝑦𝑘 − 𝜆𝑦𝑘 )‖ = ‖(𝑆 − 𝜆𝐼)𝑦𝑘 ‖. This shows that 𝜆 ∈ 𝜎(𝑇). Lemma 9.30. Let 𝑇 be an unbounded self-adjoint operator and 𝐸 the uniquely determined resolution of the identity 𝐸 on the Borel subsets of ℝ, such that ∞

(𝑇𝑥, 𝑦) = ∫ 𝑡 𝑑𝐸𝑥,𝑦 (𝑡),

𝑥 ∈ dom(𝑇),

𝑦∈𝐻

−∞

(see Proposition 9.22). Then 𝜎(𝑇) = {𝜆 ∈ ℝ : 𝐸((𝜆 − 𝜖, 𝜆 + 𝜖)) ≠ 0, ∀𝜖 > 0}.

(9.43)

Proof. Let 𝜆 ∈ 𝜎(𝑇). Suppose that there exists 𝜖0 > 0 such that 𝐸((𝜆 − 𝜖0 , 𝜆 + 𝜖0 )) = 0. Then there exists a continuous function 𝑓 on ℝ such that 𝑓(𝑡) = (𝑡−𝜆)−1 on the support of the measure 𝑑𝐸(𝑡). By the symbolic calculus, we obtain a bounded operator (𝑇−𝜆𝐼)−1 and hence a contradiction. Conversely, let 𝜆 belong to the right-hand side of (9.43). For each 𝑘 ∈ ℕ we can now find 𝑥𝑘 ∈ dom(𝑇) such that ‖𝑥𝑘 ‖ = 1 and 𝐸((𝜆 − 1/𝑘, 𝜆 + 1/𝑘))𝑥𝑘 = 𝑥𝑘 . Consider the bounded Borel functions 𝑓𝑘 (𝑡) = (𝑡 − 𝜆) 𝜒(𝜆−1/𝑘,𝜆+1/𝑘) (𝑡), 𝑡 ∈ ℝ,

𝑘 ∈ ℕ,

where 𝜒(𝜆−1/𝑘,𝜆+1/𝑘) is the characteristic function of the interval (𝜆 − 1/𝑘, 𝜆 + 1/𝑘). Then, by (9.42), ‖(𝑇 − 𝜆𝐼)𝑥𝑘 ‖ = ‖(𝑇 − 𝜆𝐼) 𝐸((𝜆 − 1/𝑘, 𝜆 + 1/𝑘))𝑥𝑘 ‖ ≤ 1/𝑘 ‖𝑥𝑘 ‖ = 1/𝑘. Using Lemma 9.29, we get 𝜆 ∈ 𝜎(𝑇). Lemma 9.31. Let 𝑇 be a symmetric operator on 𝐻 with domain dom(𝑇), and suppose that {𝑥𝑘 }𝑘 is a complete orthonormal system in 𝐻. If each 𝑥𝑘 lies in dom(𝑇), and there exist 𝜆 𝑘 ∈ ℝ such that 𝑇𝑥𝑘 = 𝜆 𝑘 𝑥𝑘 for every 𝑘 ∈ ℕ, then 𝑇 is essentially self-adjoint. Moreover the spectrum of 𝑇 is the closure in ℝ of the set of all 𝜆 𝑘 . Proof. If 𝑥 = ∑∞ 𝑘=1 𝛼𝑘 𝑥𝑘 belongs to dom(𝑇), and ∞

𝑦 = 𝑇𝑥 = ∑ 𝛽𝑘 𝑥𝑘 , 𝑘=1

152 | 9 Spectral analysis then 𝛽𝑘 = (𝑦, 𝑥𝑘 ) = (𝑇𝑥, 𝑥𝑘 ) = (𝑥, 𝑇𝑥𝑘 ) = 𝜆 𝑘 (𝑥, 𝑥𝑘 ) = 𝜆 𝑘 𝛼𝑘 . Since 𝑥, 𝑦 ∈ 𝐻 we have

∞

∑ |𝛼𝑘 |2 < ∞,

𝑘=1

and hence

∞

∑ |𝛽𝑘 |2 < ∞

𝑘=1

∞

∑ (1 + 𝜆2𝑘 )|𝛼𝑘 |2 < ∞.

𝑘=1

We define an operator 𝑇̃ as follows: let ∞

∞

𝑘=1

𝑘=1

dom(𝑇)̃ = {𝑥 ∈ 𝐻 : 𝑥 = ∑ 𝛼𝑘 𝑥𝑘 with ∑ (1 + 𝜆2𝑘 )|𝛼𝑘 |2 < ∞} and define

∞

̃ = ∑ 𝛼𝑘 𝜆 𝑘 𝑥𝑘 𝑇𝑥 𝑘=1

̃ It follows that 𝑇̃ is an extension of 𝑇. for 𝑥 ∈ dom(𝑇). Let Σ be the closure of the set {𝜆 𝑘 : 𝑘 ∈ ℕ}. Each 𝜆 𝑘 is an eigenvalue of 𝑇̃ and, by ̃ For 𝑧 ∉ Σ and 𝑥 = ∑∞ 𝛼𝑘 𝑥𝑘 we define the Lemma 9.16, 𝜎(𝑇)̃ is closed, so Σ ⊆ 𝜎(𝑇). 𝑘=1 operator 𝑆 on 𝐻 by ∞

∞

𝑘=1

𝑘=1

𝑆𝑥 = 𝑆( ∑ 𝛼𝑘 𝑥𝑘 ) := ∑ 𝛼𝑘 (𝑧 − 𝜆 𝑘 )−1 𝑥𝑘 . It follows that 𝑆 is injective and that ∞

∞

‖𝑆( ∑ 𝛼𝑘 𝑥𝑘 )‖ ≤ sup |𝑧 − 𝜆 𝑘 |−1 ( ∑ |𝛼𝑘 |2 ) 𝑘=1

𝑘

1/2

= 𝐶 ‖𝑥‖,

𝑘=1

which implies that the operator 𝑆 is bounded. Its range is precisely dom(𝑇)̃ and (𝑧𝐼 − ̃ ̃ 𝑇)𝑆𝑥 = 𝑥 for all 𝑥 ∈ 𝐻. Thus 𝑧 ∉ 𝜎(𝑇)̃ and 𝑆 = (𝑧𝐼 − 𝑇)̃ −1 . This implies Σ = 𝜎(𝑇). Now we claim that 𝑇̃ is the closure of 𝑇. Since 𝜎(𝑇)̃ is not equal to ℂ, Lemma 9.16 implies that 𝑇̃ is a closed operator. Let ∞

𝑢 = ∑ 𝛼𝑘 𝑥𝑘 ∈ dom(𝑇)̃ 𝑘=1

and put 𝑢𝑚 = ∑𝑚 𝑘=1 𝛼𝑘 𝑥𝑘 . Then lim𝑚→∞ 𝑢𝑚 = 𝑢 and 𝑚

∞

𝑘=1

𝑘=1

̃ lim 𝑇𝑢𝑚 = lim ∑ 𝛼𝑘 𝜆 𝑘 𝑥𝑘 = ∑ 𝛼𝑘 𝜆 𝑘 𝑥𝑘 = 𝑇𝑢.

𝑚→∞

Hence 𝑇̃ = 𝑇.

𝑚→∞

9.4 Determination of the spectrum

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153

Finally we prove that 𝑇̃ is self-adjoint. For 𝑥 ∈ dom(𝑇̃ ∗ ) and 𝑇̃ ∗ 𝑥 = 𝑦 we have ̃ 𝑘 ) = 𝜆 𝑘 (𝑥, 𝑥𝑘 ). (𝑦, 𝑥𝑘 ) = (𝑇̃ ∗ 𝑥, 𝑥𝑘 ) = (𝑥, 𝑇𝑥 ∞ ̃ If 𝑥 = ∑∞ 𝑘=1 𝛼𝑘 𝑥𝑘 the above implies that 𝑦 = ∑𝑘=1 𝛼𝑘 𝜆 𝑘 𝑥𝑘 . Hence 𝑥 ∈ dom(𝑇), thus ∗ ∗ ̃ and 𝑇̃ = 𝑇̃ . dom(𝑇̃ ) = dom(𝑇),

Definition 9.32. Let 𝑇 be a self-adjoint operator on 𝐻. The discrete spectrum 𝜎𝑑 (𝑇) of 𝑇 is the set of all eigenvalues 𝜆 of finite multiplicity which are isolated in the sense that the intervals (𝜆 − 𝜖, 𝜆) and (𝜆, 𝜆 + 𝜖) are disjoint from the spectrum for some 𝜖 > 0. The non-discrete part of the spectrum of 𝑇 is called the essential spectrum of 𝑇, and is denoted by 𝜎𝑒 (𝑇). A closed linear subspace 𝐿 of 𝐻 is called invariant if (𝜁𝐼 − 𝑇)−1 (𝐿) ⊆ 𝐿, for all 𝜁 ∉ ℝ. Lemma 9.33. Let 𝑇 be a self-adjoint operator. Then 𝜎𝑑 (𝑇) = {𝜆 ∈ 𝜎(𝑇) : ∃𝜖 > 0, dim im(𝐸((𝜆 − 𝜖, 𝜆 + 𝜖))) < ∞}.

(9.44)

Proof. Let 𝜆 belong to the right-hand side of (9.44). Then there exists 𝜖0 > 0 such that for each 𝜖 ∈ (0, 𝜖0 ) the projection 𝐸((𝜆 − 𝜖, 𝜆 + 𝜖)) becomes a projection with finite range independent of 𝜖. This is actually the projection 𝐸({𝜆}) and we observe that 𝐸((𝜆 − 𝜖0 , 𝜆)) = 0 and 𝐸((𝜆, 𝜆 + 𝜖0 )) = 0. This shows that 𝜆 ∈ 𝜎𝑑 (𝑇). If 𝜆 ∈ 𝜎𝑑 (𝑇), then 𝜆 is an eigenvalue of finite multiplicity and there exists 𝜖 > 0 such that the intervals (𝜆 − 𝜖, 𝜆) and (𝜆, 𝜆 + 𝜖) are disjoint from the spectrum, which means that dim im(𝐸((𝜆 − 𝜖, 𝜆 + 𝜖))) < ∞. As the whole spectrum of a self-adjoint operator is closed, it now follows that the essential spectrum is closed. The following characterization of the essential spectrum of a self-adjoint operator is similar to the characterization of the whole spectrum, compare with Lemma 9.29. Lemma 9.34. Let 𝑇 be a self-adjoint operator. 𝜆 belongs to the essential spectrum 𝜎𝑒 (𝑇) if and only if there exists a sequence (𝑥𝑘 )𝑘 in dom(𝑇) such that ‖𝑥𝑘 ‖ = 1 for each 𝑘 ∈ ℕ, such that 𝑥𝑘 converges weakly to 0 and lim ‖(𝑇 − 𝜆𝐼)𝑥𝑘 ‖ = 0.

𝑘→∞

Proof. The sequence appearing in the lemma is called a Weyl sequence. We say that 𝜆 belongs to the Weyl spectrum 𝑊(𝑇) if there exists an associated Weyl sequence. By Lemma 9.29, we already know that 𝑊(𝑇) ⊆ 𝜎(𝑇).

154 | 9 Spectral analysis Let 𝜆 ∈ 𝑊(𝑇) and suppose that 𝜆 ∈ 𝜎𝑑 (𝑇). Then the spectral projection 𝐸({𝜆}) has finite-dimensional range and hence is compact. So, by Proposition 4.17, lim 𝐸({𝜆})𝑥𝑘 = 0

𝑘→∞

in 𝐻. Now let 𝑦𝑘 := (𝐼 − 𝐸({𝜆}))𝑥𝑘 . Then, as 𝐸({𝜆})𝑇 = 𝑇𝐸({𝜆}) (Proposition 9.22), we get lim𝑘→∞ ‖𝑦𝑘 ‖ = 1 and lim (𝑇 − 𝜆𝐼)𝑦𝑘 = lim (𝐼 − 𝐸({𝜆}))(𝑇 − 𝜆𝐼)𝑥𝑘 = 0.

𝑘→∞

𝑘→∞

But (𝑇 − 𝜆𝐼) is invertible on im(𝐼 − 𝐸({𝜆})), so we obtain lim𝑘→∞ 𝑦𝑘 = 0, which is a contradiction. Hence 𝜆 ∈ 𝜎𝑒 (𝑇). Conversely, let 𝜆 ∈ 𝜎𝑒 (𝑇). Then, by Lemma 9.33, dim im(𝐸((𝜆 − 𝜖, 𝜆 + 𝜖))) = ∞ for any 𝜖 > 0. Let 𝜖𝑘 be a decreasing sequence of positive numbers such that lim𝑘→∞ 𝜖𝑘 = 0. We can now choose an orthonormal system (𝑥𝑘 )𝑘 such that 𝑥𝑘 ∈ im(𝐸((𝜆 − 𝜖𝑘 , 𝜆 + 𝜖𝑘 ))). Then, by Bessel’s inequality (Proposition 1.9), 𝑥𝑘 converges weakly to 0 and the same reasoning as in Lemma 9.30 yields lim ‖(𝑇 − 𝜆𝐼)𝑥𝑘 ‖ = 0.

𝑘→∞

Next, we will characterize the situation when 𝜎𝑒 (𝑇) = 0. For this purpose we need some preparations. Lemma 9.35. Let 𝑇 be a self-adjoint operator on 𝐻 and let 𝜆 ∈ ℝ be an eigenvalue of 𝑇. Then 𝐿 𝜆 = {𝑥 ∈ dom(𝑇) : 𝑇𝑥 = 𝜆𝑥} is a closed invariant subspace of 𝐻. If 𝐿 is an invariant subspace of 𝑇, then 𝐿⊥ is also invariant. Proof. 𝐿 𝜆 = ker(𝑇 − 𝜆𝐼), and 𝑇 − 𝜆𝐼 is a closed operator, hence, by Lemma 4.4, 𝐿 𝜆 is a closed subspace. Now let 𝑥 ∈ 𝐿 𝜆 and 𝜁 ∉ ℝ. Then (𝜁𝐼 − 𝑇)−1 𝑥 = (𝜆 − 𝜁)−1 (𝜁𝐼 − 𝑇)−1 (𝜆 − 𝜁)𝑥 = (𝜆 − 𝜁)−1 (𝜁𝐼 − 𝑇)−1 (𝑇 − 𝜁𝐼)𝑥 = −(𝜆 − 𝜁)−1 𝑥, hence (𝜁𝐼 − 𝑇)−1 (𝐿 𝜆 ) ⊆ 𝐿 𝜆 . Let 𝐿 be an invariant subspace and 𝑦 ∈ 𝐿⊥ . Then, by (9.25) ((𝜁𝐼 − 𝑇)−1 𝑦, 𝑥) = (𝑦, (𝜁𝐼 − 𝑇)−1 𝑥) = 0 for all 𝑥 ∈ 𝐿 and 𝜁 ∉ ℝ. Therefore (𝜁𝐼 − 𝑇)−1 𝑦 ∈ 𝐿⊥ and 𝐿⊥ is invariant.

9.4 Determination of the spectrum

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155

For the following results we always suppose that the underlying Hilbert space 𝐻 is separable and infinite-dimensional, i.e. each complete orthonormal system is countably infinite. Proposition 9.36. Let 𝑇 be a self-adjoint operator on 𝐻. The essential spectrum 𝜎𝑒 (𝑇) of 𝑇 is empty if and only if there exists a complete orthonormal system of eigenvectors {𝑥𝑛 }𝑛 of 𝑇 such that the corresponding eigenvalues 𝜆 𝑛 converge in absolute value to ∞ as 𝑛 → ∞. Proof. If 𝜎𝑒 (𝑇) = 0, then the spectrum 𝜎(𝑇) consists of a set {𝑟𝑛 }𝑛 of isolated eigenvalues of finite multiplicity, which can only converge to ±∞. We enumerate the eigenvalues in order of increasing absolute values and repeat each eigenvalue according to its multiplicity. In this way we get an associated orthonormal system of eigenvectors {𝑥𝑛 }𝑛 of 𝑇. Suppose that this system is not complete. Then, by Lemma 9.35, the subspace 𝐿 = {𝑥 ∈ 𝐻 : (𝑥, 𝑥𝑛 ) = 0 , 𝑛 ∈ ℕ} is invariant with respect to 𝑇 in the sense of Definition 9.31. If 𝑥 ∈ dom(𝑇) ∩ 𝐿, then (𝑇𝑥, 𝑥𝑛 ) = (𝑥, 𝑇𝑥𝑛 ) = 𝑟𝑛 (𝑥, 𝑥𝑛 ) = 0, hence 𝑇(dom(𝑇) ∩ 𝐿) ⊆ 𝐿. The essential spectrum of the restriction of 𝑇 to 𝐿 is also empty. But the spectrum of this restriction is nonempty (Proposition 9.21), therefore 𝑇 has further eigenvalues and eigenvectors not accounted for in the above list. Conversely suppose that a sequence of eigenvalues and eigenvectors with the stated properties exists, and let {𝑠𝑛 }𝑛 be the set of distinct eigenvalues. By the assumption that 𝜆 𝑛 converge in absolute value to ∞ as 𝑛 → ∞, we deduce that 𝑠𝑛 are isolated eigenvalues of finite multiplicity. It follows from Lemma 9.31 that ∞

𝜎(𝑇) = ⋃ {𝑠𝑛 }. 𝑛=1

Thus 𝜎𝑒 (𝑇) = 0. Proposition 9.37. Let 𝑇 be an unbounded self-adjoint operator on 𝐻 which is nonnegative in the sense that 𝜎(𝑇) ⊆ [0, ∞). Then the following conditions are equivalent: (i) The resolvent operator (𝐼 + 𝑇)−1 is compact. (ii) 𝜎𝑒 (𝑇) = 0. (iii) There exists a complete orthonormal system of eigenvectors {𝑥𝑛 }𝑛 of 𝑇 with corresponding eigenvalues 𝜇𝑛 ≥ 0 which converge to +∞ as 𝑛 → ∞. Proof. (i) ⇒ (iii): The operator (𝐼 + 𝑇)−1 : 𝐻 󳨀→ dom(𝑇) is compact and self-adjoint and has dense image. By Proposition 2.7 there exists a complete orthonormal system of eigenvectors 𝑥𝑛 of (𝐼 + 𝑇)−1 and eigenvalues 𝜆 𝑛 (of finite multiplicity) tending to 0 such that ∞

(𝐼 + 𝑇)−1 𝑥 = ∑ 𝜆 𝑛 (𝑥, 𝑥𝑛 )𝑥𝑛 . 𝑛=1

156 | 9 Spectral analysis Since all 𝑥𝑛 ∈ dom(𝑇), we have 𝑇(𝐼 + 𝑇)−1 𝑥𝑛 = 𝜆 𝑛 𝑇𝑥𝑛 . If we add (𝐼 + 𝑇)−1 𝑥𝑛 to this equality we get 𝜆 𝑛 𝑇𝑥𝑛 + (𝐼 + 𝑇)−1 𝑥𝑛 = 𝑇(𝐼 + 𝑇)−1 𝑥𝑛 + (𝐼 + 𝑇)−1 𝑥𝑛 = (𝐼 + 𝑇)(𝐼 + 𝑇)−1 𝑥𝑛 = 𝑥𝑛 , which implies that 𝜆 𝑛 𝑇𝑥𝑛 + 𝜆 𝑛 𝑥𝑛 = 𝑥𝑛 , and therefore 𝑇𝑥𝑛 =

1 − 𝜆𝑛 𝑥 . 𝜆𝑛 𝑛

1−𝜆

Setting 𝜇𝑛 = 𝜆 𝑛 we get (iii). 𝑛 Now suppose that (iii) holds. We rearrange the eigenvectors 𝑥𝑛 so that the sequence {𝜇𝑛 }𝑛 is non-decreasing. Each 𝑥 ∈ 𝐻 can be written in the form 𝑥 = ∑∞ 𝑛=1 (𝑥, 𝑥𝑛 )𝑥𝑛 and we obtain ∞

∞

1 (𝑥, 𝑥𝑛 )𝑥𝑛 . 1 + 𝜇𝑛 𝑛=1

(𝐼 + 𝑇)−1 𝑥 = ∑ (𝑥, 𝑥𝑛 )(𝐼 + 𝑇)−1 𝑥𝑛 = ∑ 𝑛=1

Let

𝑁

1 (𝑥, 𝑥𝑛 )𝑥𝑛 . 𝑛=1 1 + 𝜇𝑛

𝐴𝑁 = ∑

Then the operators 𝐴 𝑁 are of finite rank and, by Bessel’s inequality (1.6), we obtain 󵄩󵄩 󵄩󵄩 ∞ 1 1 ‖((𝐼 + 𝑇)−1 − 𝐴 𝑁 )𝑥‖ = 󵄩󵄩󵄩 ∑ (𝑥, 𝑥𝑛 )𝑥𝑛 󵄩󵄩󵄩 ≤ ‖𝑥‖, 󵄩 1 + 𝜇𝑁 󵄩 𝑛=𝑁+1 1 + 𝜇𝑛 from which we see that 𝐴 𝑁 converges in operator norm to (𝐼 + 𝑇)−1 as 𝑁 → ∞. Now, by Proposition 2.6, (𝐼 + 𝑇)−1 is a compact operator. The equivalence of (ii) and (iii) is a consequence of Proposition 9.36. The following general result explains the approach to the 𝜕-Neumann operator (4.41) by means of the embedding ∗

𝑗 : dom(𝜕) ∩ dom(𝜕 ) 󳨅→ 𝐿2(0,𝑞) (Ω), ∗

where dom(𝜕) ∩ dom(𝜕 ) is endowed with the graph norm ∗

𝑢 󳨃→ (‖𝜕𝑢‖2 + ‖𝜕 𝑢‖2 )1/2 . It will also be crucial for the question of compactness of the 𝜕-Neumann operator, which will be discussed in the following chapters. Proposition 9.38. Let 𝐴 be a nonnegative self-adjoint operator (i.e. 𝜎(𝐴) is contained in [0, ∞)). There exists a unique self-adjoint square root 𝐴1/2 of 𝐴 and dom(𝐴1/2 ) ⊇ dom(𝐴). In addition dom(𝐴) endowed with the norm ‖𝑓‖D := (‖𝐴1/2 𝑓‖2 + ‖𝑓‖2 )1/2

9.4 Determination of the spectrum

|

157

becomes a Hilbert space, the norm ‖.‖D stems from the inner product (𝑓, 𝑔)D = (𝐴1/2 𝑓, 𝐴1/2 𝑔) + (𝑓, 𝑔). Let dom(𝐴) be endowed with the norm ‖.‖D . Then, 𝐴 has compact resolvent if and only if the canonical embedding 𝑗 : dom(𝐴) 󳨅→ 𝐻 is a compact linear operator. Furthermore, 𝐴 has compact resolvent if and only if 𝐴1/2 has compact resolvent. Proof. In Proposition 9.23 we proved existence and uniqueness of the square root of 𝐴. For 𝑛 ∈ ℕ we define the functions 𝑞𝑛 (𝑡) =

𝑛𝑡 , 𝑛+𝑡

𝑡 ∈ [0, ∞).

These are continuous functions with 𝑞𝑛 (𝑡) ≤ 𝑞𝑛+1 (𝑡) and lim𝑛→∞ 𝑞𝑛 (𝑡) = 𝑡. Moreover 𝑞𝑛 (𝑡) ≤ 𝑛 for each 𝑡 ∈ [0, ∞). By Theorem 9.28, the operator 𝑛𝐴(𝑛𝐼 + 𝐴)−1 is bounded on 𝐻 and the function 𝑄𝑛 (𝑥) := (𝑛𝐴(𝑛𝐼 + 𝐴)−1 𝑥, 𝑥),

𝑥∈𝐻

is bounded on the unit ball of 𝐻 and continuous. The functional calculus implies that 𝑄𝑛 (𝑥) increases monotonically to 𝑄(𝑥), where {(𝐴1/2 𝑥, 𝐴1/2 𝑥) for 𝑥 ∈ dom(𝐴), 𝑄(𝑥) = { +∞ otherwise. { A function Θ : 𝐻 󳨀→ (−∞, +∞] is said to be lower semicontinuous if for every convergent sequence 𝑥𝑛 → 𝑥 in 𝐻 we have Θ(𝑥) ≤ lim inf Θ(𝑥𝑛 ). 𝑛→∞

It is easily seen that a function Θ is lower semicontinuous if and only if {𝑥 : Θ(𝑥) > 𝛼} is open for every real 𝛼, and that the pointwise limit of an increasing sequence of continuous functions is a lower semicontinuous function. Therefore the function 𝑄 is lower semicontinuous. Now let {𝑥𝑛 }𝑛 be a Cauchy sequence with respect to ‖.‖D . Then {𝑥𝑛 }𝑛 is also a Cauchy sequence with respect to ‖.‖ and therefore converges to 𝑥 ∈ 𝐻. Given 𝜖 > 0 there exists 𝑁 ∈ ℕ such that 𝑄(𝑥𝑚 − 𝑥𝑛 ) + ‖𝑥𝑚 − 𝑥𝑛 ‖2 < 𝜖2

158 | 9 Spectral analysis for all 𝑚, 𝑛 ≥ 𝑁. Letting 𝑚 → ∞ and using the lower semicontinuity of 𝑄, we deduce that 𝑥 ∈ dom(𝐴) and 𝑄(𝑥 − 𝑥𝑛 ) + ‖𝑥 − 𝑥𝑛 ‖2 ≤ 𝜖2 for all 𝑛 > 𝑁. Hence ‖𝑥 − 𝑥𝑛 ‖D ≤ 𝜖 and dom(𝐴) endowed with the norm ‖.‖D is complete. Since −1 ∉ 𝜎(𝐴), we know that (𝐼 + 𝐴)−1 is a bounded operator on 𝐻. From (9.7) we get that 𝑅𝐴 (−1) = (𝐼 + 𝐴)−1 is compact if and only if 𝑅𝐴 (𝑧) is compact for any 𝑧 ∉ 𝜎(𝐴). Let 𝑢 ∈ 𝐻 and 𝑣 ∈ dom(𝐴). Then (𝑗∗ 𝑢, 𝑣)D = (𝑢, 𝑗𝑣) = (𝑢, 𝑣) = ((𝐼 + 𝐴)(𝐼 + 𝐴)−1 𝑢, 𝑣) = ((𝐼 + 𝐴)−1 𝑢, (𝐼 + 𝐴)𝑣) = ((𝐼 + 𝐴)−1 𝑢, 𝐴𝑣) + ((𝐼 + 𝐴)−1 𝑢, 𝑣) = (𝐴1/2 (𝐼 + 𝐴)−1 𝑢, 𝐴1/2 𝑣) + ((𝐼 + 𝐴)−1 𝑢, 𝑣) = ((𝐼 + 𝐴)−1 𝑢, 𝑣)D . This implies that 𝑗∗ = (𝐼 + 𝐴)−1 as operator on dom(𝐴) and 𝑗 ∘ 𝑗∗ = (𝐼 + 𝐴)−1 as operator on 𝐻. So we deduce the desired conclusion by the fact that 𝑗 is compact if and only if 𝑗 ∘ 𝑗∗ is compact (Theorem 2.5). The last statement follows from (𝑖𝐼 + 𝐴1/2 )∗ = −𝑖𝐼 + 𝐴1/2 and (𝐼 + 𝐴) = (𝑖𝐼 + 𝐴1/2 )(−𝑖𝐼 + 𝐴1/2 ). Our next aim is to compare two self-adjoint, strictly positive operators and to prove Ruelle’s Lemma. Remark 9.39. Let 𝑆 be a self-adjoint operator. Suppose that (𝑆𝑥, 𝑥) ≥ 0 for each 𝑥 ∈ dom(𝑆). We say that 𝑆 is strictly positive, if (𝑆𝑥, 𝑥) > 0 for each 𝑥 ∈ dom(𝑆) \ {0}. Using the fact that dom(𝑆) is a core of 𝑆1/2 (Proposition 9.23), one can easily show that 𝑆 is strictly positive if and only if ‖𝑆1/2 𝑥‖ > 0 for each 𝑥 ∈ dom(𝑆1/2 ) \ {0}. We also indicate that a strictly positive self-adjoint operator 𝑆 is injective and im(𝑆) is dense in 𝐻 and 𝑆−1 is self-adjoint (Lemma 4.28). Lemma 9.40. Let 𝑆 and 𝑇 be strictly positive self-adjoint operators. Then the following assertions are equivalent. (a) dom(𝑇1/2 ) ⊂ dom(𝑆1/2 ) and ‖𝑆1/2 𝑥‖ ≤ ‖𝑇1/2 𝑥‖ for 𝑥 ∈ dom(𝑇1/2 ); (b) dom(𝑇1/2 ) ⊂ dom(𝑆1/2 ) and ‖𝑆1/2 𝑇−1/2 𝑥‖ ≤ ‖𝑥‖ for 𝑥 ∈ dom(𝑇−1/2 ); (c) dom(𝑆−1/2 ) ⊂ dom(𝑇−1/2 ) and ‖𝑇−1/2 𝑆1/2 𝑥‖ ≤ ‖𝑥‖ for 𝑥 ∈ dom(𝑆1/2 ); (d) dom(𝑆−1/2 ) ⊂ dom(𝑇−1/2 ) and ‖𝑇−1/2 𝑥‖ ≤ ‖𝑆−1/2 𝑥‖ for 𝑥 ∈ dom(𝑆−1/2 ). Proof. We show that (a) and (b) are equivalent and that (b) implies (c). The rest of the implications follows by symmetry and by replacing 𝑆 and 𝑇 by 𝑆−1 and 𝑇−1 . Suppose that (a) holds. If 𝑥 ∈ dom(𝑇−1/2 ), then 𝑇−1/2 𝑥 ∈ dom(𝑇1/2 ), so we get 1/2 −1/2 ‖𝑆 𝑇 𝑥‖ ≤ ‖𝑥‖.

9.4 Determination of the spectrum

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159

If (b) holds and 𝑥 ∈ dom(𝑇1/2 ), we have 𝑇1/2 𝑥 ∈ dom(𝑇−1/2 ) and hence ‖𝑆1/2 𝑥‖ = ‖𝑆1/2 𝑇−1/2 𝑇1/2 𝑥‖ ≤ ‖𝑇1/2 𝑥‖. Suppose that (b) holds. We observe that dom(𝑆−1/2 ) = im(𝑆1/2 ) and that we have to show im(𝑆1/2 ) ⊂ dom(𝑇−1/2 ) and ‖𝑇−1/2 𝑆1/2 𝑥‖ ≤ ‖𝑥‖

for 𝑥 ∈ dom(𝑆1/2 ).

If 𝑥 ∈ dom(𝑆1/2 ), then 𝑆1/2 𝑥 ∈ im(𝑆1/2 ). We consider the linear functional 𝜓𝑥 (𝑦) := (𝑆1/2 𝑥, 𝑇−1/2 𝑦)

for 𝑦 ∈ dom(𝑇−1/2 ).

Then |𝜓𝑥 (𝑦)| = |(𝑆1/2 𝑥, 𝑇−1/2 𝑦)| = |(𝑥, 𝑆1/2 𝑇−1/2 𝑦)| ≤ ‖𝑥‖ ‖𝑦‖. This implies that 𝑆1/2 𝑥 ∈ dom((𝑇−1/2 )∗ ) = dom(𝑇−1/2 ). Finally we get |(𝑇−1/2 𝑆1/2 𝑥, 𝑦)| ≤ ‖𝑥‖ ‖𝑦‖,

for 𝑦 ∈ dom(𝑇−1/2 ).

And as dom(𝑇−1/2 ) is dense in 𝐻, we obtain ‖𝑇−1/2 𝑆1/2 𝑥‖ ≤ ‖𝑥‖. For Ruelle’s Lemma we consider strictly positive, self-adjoint operators 𝑆 and 𝑇 and we write 𝑆 ≤ 𝑇, if and only if dom(𝑇) ⊆ dom(𝑆) and (𝑆𝑥, 𝑥) ≤ (𝑇𝑥, 𝑥) for each 𝑥 ∈ dom(𝑇). By Proposition 9.23 the square roots of 𝑆 and 𝑇 exist and are themselves positive, selfadjoint operators. Lemma 9.41 (Ruelle’s Lemma). Let 𝑆 and 𝑇 be strictly positive self-adjoint operators. Suppose that 𝑆 ≤ 𝑇 and that 0 ∈ 𝜌(𝑆). Then 𝑇−1 ≤ 𝑆−1 . Proof. We have dom(𝑇) ⊆ dom(𝑆) and ‖𝑆1/2 𝑥‖2 = (𝑆𝑥, 𝑥) ≤ (𝑇𝑥, 𝑥) = ‖𝑇1/2 𝑥‖2 ,

(9.45)

for all 𝑥 ∈ dom(𝑇). Next we show that dom(𝑇1/2 ) ⊂ dom(𝑆1/2 ). Let 𝑥 ∈ dom(𝑇1/2 ). By Proposition 9.23, dom(𝑇) is a core of dom(𝑇1/2 ). Hence there exists a sequence (𝑥𝑘 )𝑘 in dom(𝑇) such that 𝑥𝑘 → 𝑥 and 𝑇1/2 𝑥𝑘 → 𝑇1/2 𝑥. By (9.45) we have ‖𝑆1/2 (𝑥𝑚 − 𝑥𝑘 )‖ ≤ ‖𝑇1/2 (𝑥𝑚 − 𝑥𝑘 )‖, which implies that (𝑆1/2 𝑥𝑘 )𝑘 is a Cauchy sequence. But 𝑆1/2 is also a closed operator, and so 𝑥 ∈ dom(𝑆1/2 ) and 𝑆1/2 𝑥𝑘 → 𝑆1/2 𝑥. In addition, by (9.45), we have ‖𝑆1/2 𝑥‖ = lim ‖𝑆1/2 𝑥𝑘 ‖ ≤ lim ‖𝑇1/2 𝑥𝑘 ‖ = ‖𝑇1/2 𝑥‖, 𝑘→∞

1/2

𝑘→∞

for each 𝑥 ∈ dom(𝑇 ). Now we can apply Lemma 9.40 (d) and get dom(𝑆−1/2 ) ⊂ dom(𝑇−1/2 ) and ‖𝑇−1/2 𝑥‖ ≤ ‖𝑆−1/2 𝑥‖ for 𝑥 ∈ dom(𝑆−1/2 ). Our assumption 0 ∈ 𝜌(𝑆) implies that 𝐻 = dom(𝑆−1 ) = dom(𝑆−1/2 ), which proves the lemma.

160 | 9 Spectral analysis Example 9.42. We return to the 𝜕-Neumann operator on 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑 ) and remark that the Kohn–Morrey formula from Proposition 6.4 can be written in the form (𝑀𝜑 𝑢, 𝑢)𝜑 ≤ (◻𝜑 𝑢, 𝑢)𝜑 ∗

for a (0, 1)-form 𝑢 ∈ dom (𝜕) ∩ dom (𝜕𝜑 ). So, under the assumptions of Theorem 8.4, we obtain using Ruelle’s Lemma 9.41 that (𝑁𝜑 𝑢, 𝑢)𝜑 ≤ (𝑀𝜑−1 𝑢, 𝑢)𝜑 , setting 𝜕𝑣 = 𝑢 we get ∗

‖𝑣‖2𝜑 = (𝑣, 𝑣)𝜑 = (𝑣, 𝜕𝜑 𝑁𝜑 𝑢)𝜑 = (𝜕𝑣, 𝑁𝜑 𝑢)𝜑 = (𝑢, 𝑁𝜑 𝑢)𝜑 ≤ (𝑀𝜑−1 𝜕𝑣, 𝜕𝑣)𝜑 for each 𝑣 ∈ dom (𝜕) orthogonal to ker (𝜕). This gives a different proof of Hörmander’s 𝐿2 -estimates similar to the Brascamp– Lieb inequality (see [34] and [46]): ∫ |𝑣(𝑧)|2 𝑒−𝜑(𝑧) 𝑑𝜆(𝑧) ≤ ∫ |𝜕𝑣(𝑧)|2𝑖𝜕𝜕𝜑 𝑒−𝜑(𝑧) 𝑑𝜆(𝑧), ℂ𝑛

(9.46)

ℂ𝑛

for each 𝑣 ∈ dom (𝜕) orthogonal to ker (𝜕), where we used the notation in formula (8.7). ∗ Let 1 ≤ 𝑞 ≤ 𝑛. If 𝑢 is a (0, 𝑞)-form in dom (𝜕) ∩ dom (𝜕𝜑 ), we get by (6.11) ∑

󸀠

𝑛

∑ ∫

|𝐾|=𝑞−1 𝑗,𝑘=1 ℂ𝑛

𝜕2 𝜑 𝑢 𝑢 𝑒−𝜑 𝑑𝜆 ≤ (◻𝜑 𝑢, 𝑢)𝜑 . 𝜕𝑧𝑗 𝜕𝑧𝑘 𝑗𝐾 𝑘𝐾

The left-hand side can be written in the form (𝑀̃ 𝜑 𝑢, 𝑢)𝜑 . We suppose that 𝑀̃ 𝜑 is invertible and get as above ‖𝑣‖2𝜑 ≤ (𝑀̃ 𝜑−1 𝜕𝑣, 𝜕𝑣)𝜑 (9.47) for each (0, 𝑞 − 1)-form 𝑣 ∈ dom (𝜕) orthogonal to ker (𝜕).

9.5 Variational characterization of the discrete spectrum Here we explain the max-min principle to describe the lowest part of the spectrum of a self-adjoint operator when it is discrete. This is done in the setting of semibounded operators. Definition 9.43. Let 𝑇 be a symmetric unbounded operator with dom(𝑇). We say that 𝑇 is semibounded (from below) if there exists a constant 𝐶 > 0 such that (𝑇𝑢, 𝑢) ≥ −𝐶‖𝑢‖2 , ∀𝑢 ∈ dom(𝑇). See the next chapter for examples of semibounded operators.

9.5 Variational characterization of the discrete spectrum

|

161

Using Proposition 4.36 one can show that a symmetric semibounded operator 𝑇 admits a self-adjoint extension, for this purpose set for the sesquilinear form from there 𝑎(𝑢, 𝑣) = (𝑇𝑢, 𝑣). The resulting self-adjoint extension is called the Friedrichs extension. Proposition 9.44. Let 𝐴 be a self-adjoint semibounded operator. Let Σ = inf 𝜎𝑒 (𝐴). The set 𝜎(𝐴) ∩ (−∞, Σ) can be described as a sequence (finite or infinite) of eigenvalues 𝜆 𝑗 ordered increasingly. Then one has

and for 𝑘 ≥ 2

𝜆 1 = inf{(𝐴𝜙, 𝜙) ‖𝜙‖−2 : 𝜙 ∈ dom(𝐴), 𝜙 ≠ 0},

(9.48)

𝜆 𝑘 = inf{(𝐴𝜙, 𝜙) ‖𝜙‖−2 : 𝜙 ∈ dom(𝐴) ∩ K⊥𝑘−1 , 𝜙 ≠ 0},

(9.49)

where

K𝑗 = ⨁ ker(𝐴 − 𝜆 𝑚 𝐼). 𝑚≤𝑗

Proof. Let 𝜇1 denote the right-hand side of (9.48). If 𝜙1 is an eigenfunction for the eigenvalue 𝜆 1 , we get 𝜇1 ≤ 𝜆 1 . On the other side we have 𝜎(𝐴 − 𝜆 1 𝐼) ⊆ [0, ∞), hence, by Proposition 9.23, (𝐴𝜙 − 𝜆 1 𝜙, 𝜙) ≥ 0, for each 𝜙 ∈ dom(𝐴), which implies 𝜆 1 ≤ 𝜇1 . Actually, we have shown that if 𝜇1 < Σ, then the spectrum below Σ is not empty. In particular the bottom of the spectrum is an eigenvalue equal to 𝜇1 . For 𝑘 = 2, 3, . . . , we apply the first step of the proof to 𝐴 |dom(𝐴)∩K⊥ and use the 𝑘−1 spectral theorem. Our next aim is to generalize the following min-max lemma for positive, compact operators: Lemma 9.45. Let 𝐴 : 𝐻 󳨀→ 𝐻 be a positive, compact operator (i.e. (𝐴𝑥, 𝑥) ≥ 0 , ∀𝑥 ∈ 𝐻) with spectral decomposition ∞ 𝐴𝑥 = ∑ 𝜆 𝑛 (𝑥, 𝑥𝑛 )𝑥𝑛 , 𝑛=0

where 𝜆 0 ≥ 𝜆 1 ≥ . . . . Then (i)

(𝐴𝑥, 𝑥) , (𝑥, 𝑥) where the maximum is attained by an eigenvector 𝑥0 with eigenvalue 𝜆 0 .

(ii)

𝜆 0 = max 𝑥∈𝐻

𝜆 𝑗 = min max⊥ 𝐿∈𝑁𝑗 𝑥∈𝐿

(𝐴𝑥, 𝑥) , (𝑥, 𝑥)

𝑗 ≥ 1,

where 𝑁𝑗 denotes the set of all 𝑗-dimensional subspaces of 𝐻. The minimum is attained by the subspace 𝐿 = 𝐿 𝑗 = ⟨𝑥0 , . . . , 𝑥𝑗−1 ⟩, i.e. 𝜆 𝑗 = max⊥ 𝑥∈𝐿 𝑗

(𝐴𝑥, 𝑥) . (𝑥, 𝑥)

162 | 9 Spectral analysis Proof. (i) follows directly from the proof of the spectral theorem 2.7. For 𝑗 ≥ 1 we have 𝜆 𝑗 = (𝐴𝑥𝑗 , 𝑥𝑗 )/(𝑥𝑗 , 𝑥𝑗 ). The assertion follows, if we can show that for each 𝑗-dimensional subspace 𝐿 there exists 𝑧0 ⊥ 𝐿 with 𝑧0 ≠ 0 and 𝑧0 = 𝑗 ∑𝑘=0 (𝑧0 , 𝑥𝑘 )𝑥𝑘 . In that case we then have 𝑗

2 (𝐴𝑧0 , 𝑧0 ) ∑𝑘=0 𝜆 𝑘 |(𝑧0 , 𝑥𝑘 )| ≥ 𝜆𝑗, = 𝑗 (𝑧0 , 𝑧0 ) ∑𝑘=0 |(𝑧0 , 𝑥𝑘 )|2

as 𝜆 𝑗 ≤ 𝜆 𝑖 for 0 ≤ 𝑖 ≤ 𝑗 and max⊥ 𝑥∈𝐿

(𝐴𝑥, 𝑥) ≥ 𝜆𝑗. (𝑥, 𝑥)

The existence of 𝑧0 follows from the fact that for a basis {𝑦𝑘 : 𝑘 = 0, . . . , 𝑗 − 1} of 𝐿 the system of linear equations 𝑗

∑ 𝑎𝑖 (𝑥𝑖 , 𝑦𝑘 ) = 0 , 𝑘 = 0, . . . , 𝑗 − 1, 𝑖=0

𝑗

has a non-trivial solution. Set 𝑧0 = ∑𝑖=0 𝑎𝑖 𝑥𝑖 , then 𝑧0 ⊥ 𝐿. So if one has positive compact operators 𝐴 and 𝐵 such that 𝐴 ≤ 𝐵, which means (𝐴𝑥, 𝑥) ≤ (𝐵𝑥, 𝑥), then the eigenvalues of 𝐴 and 𝐵 satisfy 𝜆 𝑗 (𝐴) ≤ 𝜆 𝑗 (𝐵),

𝑗 = 0, 1, 2, . . . .

The corresponding result for unbounded operators yields information about the bottom of the spectrum and of the essential spectrum. Proposition 9.46. Let 𝐻 be a Hilbert space of infinite dimension. Let 𝐴 be a self-adjoint semibounded operator with domain dom(𝐴). Let 𝜇1 (𝐴) = inf{(𝐴𝜙, 𝜙) : 𝜙 ∈ dom(𝐴), ‖𝜙‖ = 1} and for 𝑛 ≥ 2 𝜇𝑛 (𝐴) = sup inf{(𝐴𝜙, 𝜙) : 𝜙 ∈ 𝐿⊥ ∩ dom(𝐴), ‖𝜙‖ = 1} 𝐿∈𝑁𝑛−1

(9.50)

where 𝑁𝑛−1 denotes the set of all subspaces of 𝐻 of dimension ≤ 𝑛 − 1. Then either (a) 𝜇𝑛 (𝐴) is the n-th eigenvalue when the eigenvalues are increasingly ordered (counting the multiplicities) and 𝐴 has a discrete spectrum in (−∞, 𝜇𝑛 (𝐴)], or (b) 𝜇𝑛 (𝐴) corresponds to the bottom of the essential spectrum. In this case we have 𝜇𝑗 (𝐴) = 𝜇𝑛 (𝐴) for all 𝑗 ≥ 𝑛. Proof. Let 𝐸 be the uniquely determined resolution of identity on the Borel subsets of ℝ. First we show that dim im𝐸((−∞, 𝑎)) < 𝑛,

if 𝑎 < 𝜇𝑛 (𝐴),

(9.51)

dim im𝐸((−∞, 𝑎)) ≥ 𝑛,

if 𝑎 > 𝜇𝑛 (𝐴).

(9.52)

9.5 Variational characterization of the discrete spectrum

|

163

To prove (9.51) let 𝑎 < 𝜇𝑛 (𝐴) and suppose that dim im𝐸((−∞, 𝑎)) ≥ 𝑛. If 𝑦 ∈ im𝐸((−∞, 𝑎)), we get 𝑦 = 𝐸((−∞, 𝑎))𝑥, for some 𝑥 ∈ 𝐻 and therefore 𝐸𝑦,𝑦 (𝜔) = (𝐸(𝜔)𝑦, 𝑦) = (𝐸(𝜔)𝐸((−∞, 𝑎))𝑥, 𝐸((−∞, 𝑎))𝑥) = 0,

(9.53)

for each Borel subset 𝜔 of ℝ such that (−∞, 𝑎) ∩ 𝜔 = 0. Since 𝐴 is bounded from below we have (𝐴𝑥, 𝑥) ≥ −𝐶‖𝑥‖2 for all 𝑥 ∈ dom(𝐴) and Proposition 9.23 and Lemma 9.13 imply that ∞

dom(𝐴) = {𝑢 ∈ 𝐻 : ∫ 𝑡2 𝑑𝐸𝑢,𝑢 (𝑡) < ∞}, −𝐶

so we obtain for 𝑦 ∈ im𝐸((−∞, 𝑎)) by (9.53) that ∞

∫ 𝑡2 𝑑𝐸𝑦,𝑦 (𝑡) ≤ 𝐶󸀠 max(𝐶2 , 𝑎2 ) < ∞, −𝐶

which implies that im𝐸((−∞, 𝑎)) ⊆ dom(𝐴). So we can find an 𝑛-dimensional subspace 𝐿 ⊂ dom(𝐴), such that 𝑎

(𝐴𝑢, 𝑢) = ∫ 𝑡 𝑑𝐸𝑢,𝑢 ≤ 𝑎(𝑢, 𝑢), ∀𝑢 ∈ 𝐿.

(9.54)

−𝐶

But then, given any 𝜓1 , . . . , 𝜓𝑛−1 ∈ 𝐻, we can find 𝜙 ∈ 𝐿 ∩ ⟨𝜓1 , . . . , 𝜓𝑛−1 ⟩⊥ such that ‖𝜙‖ = 1 and (𝐴𝜙, 𝜙) ≤ 𝑎. Returning to the definition of 𝜇𝑛 (𝐴) we would have 𝜇𝑛 (𝐴) ≤ 𝑎, which is a contradiction. Hence we have shown (9.51). To prove (9.52), let 𝑎 > 𝜇𝑛 (𝐴) and suppose that dim im𝐸((−∞, 𝑎)) ≤ 𝑛 − 1. Then we can find (𝑛 − 1) generators 𝜓1 , . . . , 𝜓𝑛−1 of this space and any 𝜙 ∈ dom(𝐴) ∩ ⟨𝜓1 , . . . , 𝜓𝑛−1 ⟩⊥ is in im𝐸([𝑎, +∞)), so

(𝐴𝜙, 𝜙) ≥ 𝑎‖𝜙‖2 ,

which is again a contradiction, and we get (9.52). In the next step we will show that 𝜇𝑛 (𝐴) < +∞ for each 𝑛 ∈ ℕ. Since 𝐴 is semibounded from below, 𝜇𝑛 (𝐴) has a uniform lower bound. Suppose that 𝜇𝑛 (𝐴) = +∞. By (9.51), this means that dim im𝐸((−∞, 𝑎)) < 𝑛, for all 𝑎 ∈ ℝ. Hence 𝐻 must be of finite dimension and we arrive at a contradiction.(If 𝐻 is finite dimensional, we have 𝜇𝑛 (𝐴) ≥ ‖𝐴‖.)

164 | 9 Spectral analysis For the rest of the proof we distinguish between the following two cases: dim im𝐸((−∞, 𝜇𝑛 (𝐴) + 𝜖)) = ∞,

∀𝜖 > 0;

(9.55)

for some 𝜖0 > 0.

(9.56)

and dim im𝐸((−∞, 𝜇𝑛 (𝐴) + 𝜖0 )) < ∞,

Assuming (9.55) we claim that the assertion (b) of the proposition holds: using (9.51) in this case we obtain dim im𝐸((𝜇𝑛 (𝐴) − 𝜖, 𝜇𝑛 (𝐴) + 𝜖)) = ∞,

∀𝜖 > 0.

By Lemma 9.33, this shows that 𝜇𝑛 (𝐴) ∈ 𝜎𝑒 (𝐴). Using (9.51) once more, we see that the interval (−∞, 𝜇𝑛 (𝐴)) does not contain any point of the essential spectrum. Hence 𝜇𝑛 (𝐴) = inf{𝜆 : 𝜆 ∈ 𝜎𝑒 (𝐴)}. From (9.50) we obtain that 𝜇𝑛+1 (𝐴) ≥ 𝜇𝑛 (𝐴). But if 𝜇𝑛+1 (𝐴) > 𝜇𝑛 (𝐴), (9.51) would also be satisfied for 𝜇𝑛+1 (𝐴). This is a contradiction to (9.55). Hence assertion (b) is proved. Finally suppose that (9.56) holds. By Lemma 9.33 it is clear that the spectrum of 𝐴 is discrete in (−∞, 𝜇𝑛 (𝐴) + 𝜖0 ). Then, for 𝜖1 > 0 small enough im𝐸((−∞, 𝜇𝑛 (𝐴)]) = im𝐸((−∞, 𝜇𝑛 (𝐴) + 𝜖1 )), and by (9.52) dim im𝐸((−∞, 𝜇𝑛 (𝐴)]) ≥ 𝑛. So, there are at least 𝑛 eigenvalues 𝜆 1 ≤ 𝜆 2 ≤ ⋅ ⋅ ⋅ ≤ 𝜆 𝑛 ≤ 𝜇𝑛 (𝐴) for 𝐴. If 𝜆 𝑛 were strictly less than 𝜇𝑛 (𝐴), then dim im𝐸((−∞, 𝜆 𝑛 ]) = 𝑛, which yields a contradiction to (9.51). So 𝜆 𝑛 = 𝜇𝑛 (𝐴), and 𝜇𝑛 (𝐴) is an eigenvalue. This proves assertion (a). We note that the proof of (9.51) gives Proposition 9.47. Suppose that there exists an 𝑎 and an 𝑛-dimensional subspace 𝐿 ⊂ dom(𝐴) such that (9.54) is satisfied. Then 𝜇𝑛 (𝐴) ≤ 𝑎. Using Proposition 9.46 we now get Corollary 9.48. Under the same assumptions as in Proposition 9.47, if 𝑎 is below the bottom of the essential spectrum of 𝐴, then 𝐴 has at least 𝑛 eigenvalues (counted with multiplicities). The last results permit to compare the spectra of two operators. If (𝐴𝑢, 𝑢) ≤ (𝐵𝑢, 𝑢) for all 𝑢 ∈ dom(𝐵) ⊆ dom(𝐴), then 𝜆 𝑛 (𝐴) ≤ 𝜆 𝑛 (𝐵).

9.6 Notes

|

165

9.6 Notes Most of the material on spectral theory is taken from [65] and [18]. It would be beyond the scope of this book to provide a complete proof of the spectral theorem. The concept of the spectral decomposition is explained in detail for both the bounded and the unbounded self-adjoint operators. For the complete proof some aspects of Banach algebras and the Gelfand transform are needed, which appear to be not important enough for a thorough understanding of the spectral properties of the 𝜕-Neumann operator. See [65] for a complete proof of the spectral theorem and [18] for a different approach using the Helffer–Sjöstrand formula. The use of the square root of the 𝜕Neumann operator and of the ◻-operator, as done in Remark 9.12 and Example 9.42, gives a better understanding of the basic estimate (4.37) and other fundamental results such as Theorem 8.6. Lemma 9.31 is taken from [18]. It is very useful for operators whose eigenvectors can be determined explicitly, it will be of importance in Chapter 14. In the next chapters on operators of mathematical physics certain aspects of spectral theory developed in this chapter will be crucial. Variational methods in spectral analysis (see [37, 62]) are important to explain special properties of Schrödinger operators like diamagnetism and paramagnetism.

10 Schrödinger operators and Witten–Laplacians In this chapter we collect basic informations about Schrödinger operators with magnetic field in real dimension 2 and the Witten–Laplacian for complex dimension ≥ 2. It turns out there is a close relationship between these operators and the weighted ◻-operator of the 𝜕-complex. In addition we include the definition of Dirac and Pauli operators, for which, later on, spectral properties are investigated as applications of corresponding results of the weighted ◻-operator of the 𝜕-complex. First we will study the regularity of the solutions for elliptic differential equations of order two.

10.1 Difference quotients In order to apply Sobolev space theory, we are forced to study difference quotient approximations to weak derivatives. Definition 10.1. Let Ω ⊂ ℝ𝑛 be a bounded domain and 𝑢 ∈ 𝐿2𝑙𝑜𝑐 (Ω) and suppose that 𝑉 ⊂⊂ Ω. The 𝑗-th difference quotient of size ℎ is 𝐷𝑗ℎ 𝑢(𝑥) =

𝑢(𝑥 + ℎ𝑒𝑗 ) − 𝑢(𝑥) ℎ

,

for 𝑗 = 1, . . . , 𝑛, where 𝑥 ∈ 𝑉 and ℎ ∈ ℝ , 0 < |ℎ| < dist(𝑉, 𝑏Ω). Further we define 𝐷ℎ 𝑢 := (𝐷1ℎ 𝑢, . . . , 𝐷𝑛ℎ 𝑢). Proposition 10.2. (i) Let 𝑢 ∈ 𝑊1 (Ω). Then for each 𝑉 ⊂⊂ Ω we have ‖𝐷ℎ 𝑢‖𝐿2 (𝑉) ≤ 𝐶‖∇𝑢‖𝐿2 (Ω)

(10.1)

for some constant 𝐶 > 0 and all ℎ with 0 < |ℎ| < 12 dist(𝑉, 𝑏Ω). (ii) Assume that 𝑢 ∈ 𝐿2 (𝑉), and that there exists a constant 𝐶 > 0 such that ‖𝐷ℎ 𝑢‖𝐿2 (𝑉) ≤ 𝐶 for all ℎ with 0 < |ℎ|
0 such that 𝑛

∑ 𝑎𝑗𝑘 (𝑥)𝑡𝑗 𝑡𝑘 ≥ 𝐶|𝑡|2

(10.6)

𝑗,𝑘=1

for almost every 𝑥 ∈ Ω and all 𝑡 ∈ ℂ𝑛 . Ellipticity means that the symmetric (𝑛 × 𝑛) matrix 𝐴(𝑥) = (𝑎𝑗𝑘 (𝑥))𝑛𝑗,𝑘=1 is positive definite, with smallest eigenvalue greater than or equal to 𝐶. If 𝑎𝑗𝑘 = 𝛿𝑗𝑘 , 𝑏𝑗 = 0, 𝑐 = 0, then 𝐿 = −󳵻. Definition 10.4. Let 𝑓 ∈ 𝐿2 (Ω). We say that a function 𝑢 ∈ 𝐻1 (Ω) is a weak solution to the elliptic partial differential equation 𝐿𝑢 = 𝑓 in Ω, if for the bilinear form 𝑛

𝑛

𝑎(𝑢, 𝑣) = ∫ ( ∑ 𝑎𝑗𝑘 𝑢𝑥𝑗 𝑣𝑥𝑘 + ∑ 𝑏𝑗 𝑢𝑥𝑗 𝑣 + 𝑐𝑢𝑣) 𝑑𝜆 Ω

𝑗,𝑘=1

𝑗=1

we have 𝑎(𝑢, 𝑣) = (𝑓, 𝑣) for all 𝑣 ∈ 𝐻01 (Ω), where (., .) denotes the inner product in 𝐿2 (Ω). In the next proposition we show what is called the interior 𝐻2 -regularity of the operator 𝐿. Proposition 10.5. Let 𝐿 be as in the above definition and 𝑓 ∈ 𝐿2 (Ω). Suppose that 𝑢 ∈ 2 𝐻1 (Ω) is a weak solution of 𝐿𝑢 = 𝑓. Then 𝑢 ∈ 𝐻loc (Ω); and for each open 𝑉 ⊂⊂ Ω we have ̃ ‖𝑢‖𝐻2 (𝑉) ≤ 𝐶(‖𝑓‖ (10.7) 𝐿2 (Ω) + ‖𝑢‖𝐻1 (Ω) ), where 𝐶̃ > 0 is a constant only depending on 𝑉, Ω, and the coefficients of 𝐿.

10.2 Interior regularity | 169

Proof. Choose an open set 𝑊 such that 𝑉 ⊂⊂ 𝑊 ⊂⊂ Ω. Next select a smooth cut-off function 𝜓 with 0 ≤ 𝜓 ≤ 1, 𝜓 = 1 on 𝑉, and 𝜓 = 0 on ℝ𝑛 \ 𝑊. Since 𝑢 is a weak solution of 𝐿𝑢 = 𝑓, we have 𝑎(𝑢, 𝑣) = (𝑓, 𝑣) for all 𝑣 ∈ 𝐻01 (Ω) and hence 𝑛 ∑ ∫ 𝑎 𝑢 𝑣 𝑑𝜆 = ∫ 𝑓𝑣̃ 𝑑𝜆, (10.8) 𝑗𝑘 𝑥𝑗 𝑥𝑘

𝑗,𝑘=1 Ω

Ω

where

𝑛

𝑓 ̃ := 𝑓 − ∑ 𝑏𝑗 𝑢𝑥𝑗 − 𝑐𝑢.

(10.9)

𝑗=1

Let ℓ ∈ {1, . . . , 𝑛} and ℎ ∈ ℝ such that |ℎ| > 0 is small. We substitute 𝑣 = −𝐷ℓ−ℎ (𝜓2 𝐷ℓℎ 𝑢)

(10.10)

into (10.8), where 𝐷ℓℎ 𝑢 denotes the difference quotient (Definition 10.1). For the lefthand side of (10.8) we get 𝑛

𝑛

− ∑ ∫ 𝑎𝑗𝑘 𝑢𝑥𝑗 [𝐷ℓ−ℎ (𝜓2 𝐷ℓℎ 𝑢)]−𝑥𝑘 𝑑𝜆 = ∑ ∫ 𝐷ℓℎ (𝑎𝑗𝑘 𝑢𝑥𝑗 )(𝜓2 𝐷ℓℎ 𝑢)−𝑥𝑘 𝑑𝜆 𝑗,𝑘=1 Ω

𝑗,𝑘=1 Ω

𝑛

ℎ (𝐷ℓℎ 𝑢𝑥𝑗 )(𝜓2 𝐷ℓℎ 𝑢)−𝑥𝑘 + (𝐷ℓℎ 𝑎𝑗𝑘 )𝑢𝑥𝑗 (𝜓2 𝐷ℓℎ 𝑢)−𝑥𝑘 ] 𝑑𝜆, = ∑ ∫[𝑎𝑗𝑘 𝑗,𝑘=1 Ω

where we used the formulas ∫ 𝑣𝐷ℓ−ℎ 𝑤 𝑑𝜆 = − ∫ 𝑤𝐷ℓℎ 𝑣 𝑑𝜆 Ω

Ω

and 𝐷ℓℎ (𝑣𝑤) = 𝑣ℎ 𝐷ℓℎ 𝑤 + 𝑤𝐷ℓℎ 𝑣, for 𝑣ℎ (𝑥) := 𝑣(𝑥 + ℎ𝑒ℓ ). We continue the computation of the left-hand side of (10.8) and obtain 𝑛

𝑛

ℎ ℎ ∑ ∫ 𝑎𝑗𝑘 (𝐷ℓℎ 𝑢𝑥𝑗 )(𝐷ℓℎ 𝑢𝑥𝑘 )− 𝜓2 𝑑𝜆 + ∑ ∫[𝑎𝑗𝑘 (𝐷ℓℎ 𝑢𝑥𝑗 )(𝐷ℓℎ 𝑢)− 2𝜓𝜓𝑥𝑘

𝑗,𝑘=1 Ω

+

𝑗,𝑘=1 Ω ℎ ℎ (𝐷ℓ 𝑎𝑗𝑘 )𝑢𝑥𝑗 (𝐷ℓ 𝑢𝑥𝑘 )− 𝜓2

+ (𝐷ℓℎ 𝑎𝑗𝑘 )𝑢𝑥𝑗 (𝐷ℓℎ 𝑢)− 2𝜓𝜓𝑥𝑘 ] 𝑑𝜆 = 𝑇1 + 𝑇2 .

The first term can be estimated from below using ellipticity (10.6): |𝑇1 | ≥ 𝐶 ∫ 𝜓2 |𝐷ℓℎ ∇𝑢|2 𝑑𝜆.

(10.11)

Ω

For the second term 𝑇2 we have by the assumptions on 𝑎𝑗𝑘 , 𝑏𝑗 and 𝑐 that there exists a constant 𝐶󸀠 > 0 such that |𝑇2 | ≤ 𝐶󸀠 ∫[𝜓|𝐷ℓℎ ∇𝑢| |𝐷ℓℎ 𝑢| + 𝜓|𝐷ℓℎ ∇𝑢| |∇𝑢| + 𝜓|𝐷ℓℎ 𝑢| |∇𝑢|] 𝑑𝜆. Ω

170 | 10 Schrödinger operators and Witten–Laplacians Take into account that 𝜓 = 0 on ℝ𝑛 \ 𝑊 and use the small constant-large constant trick to get 𝐶󸀠 |𝑇2 | ≤ 𝜖 ∫ 𝜓2 |𝐷ℓℎ ∇𝑢|2 𝑑𝜆 + ∫[|𝐷ℓℎ 𝑢|2 + |∇𝑢|2 ] 𝑑𝜆, 𝜖 𝑊

Ω

now choose 𝜖 = 𝐶/2 and the estimate (10.1) to obtain |𝑇2 | ≤

𝐶 ∫ 𝜓2 |𝐷ℓℎ ∇𝑢|2 𝑑𝜆 + 𝐶󸀠󸀠 ∫ |∇𝑢|2 𝑑𝜆. 2 Ω

Ω

Hence, by (10.10), we see that the left-hand side of (10.8) can be estimated from below by 𝐶 (10.12) ∫ 𝜓2 |𝐷ℓℎ ∇𝑢|2 𝑑𝜆 − 𝐶󸀠󸀠 ∫ |∇𝑢|2 𝑑𝜆. 2 Ω

Ω

The absolute value of the right-hand side of (10.8) is certainly less than 𝐶󸀠󸀠 ∫(|𝑓| + |∇𝑢| + |𝑢|)|𝑣| 𝑑𝜆.

(10.13)

Ω

Using Proposition 10.2 (i) we derive that ∫ |𝑣|2 𝑑𝜆 ≤ 𝐶󸀠󸀠 ∫ |∇(𝜓2 𝐷ℓℎ 𝑢)|2 𝑑𝜆 Ω

Ω 󸀠󸀠

≤ 𝐶 ∫(|𝐷ℓℎ 𝑢|2 + 𝜓4 |𝐷ℓℎ ∇𝑢|2 ) 𝑑𝜆 𝑊 󸀠󸀠

≤ 𝐶 ∫(|∇𝑢|2 + 𝜓4 |𝐷ℓℎ ∇𝑢|2 ) 𝑑𝜆. Ω

Again by the small constant-large constant trick and by (10.13) we obtain now that the absolute value of the right-hand side of (10.8) can be estimated from above by 𝜖 ∫ 𝜓2 |𝐷ℓℎ ∇𝑢|2 𝑑𝜆 + Ω

𝐶󸀠 ∫(|𝑓|2 + |𝑢|2 + |∇𝑢|2 ) 𝑑𝜆. 𝜖 Ω

Let 𝜖 = 𝐶/4. Then the absolute value of the right-hand side of (10.8) can be estimated from above by 𝐶 ∫ 𝜓2 |𝐷ℓℎ ∇𝑢|2 𝑑𝜆 + 𝐶󸀠󸀠󸀠 ∫(|𝑓|2 + |𝑢|2 + |∇𝑢|2 ) 𝑑𝜆. 4 Ω

Ω

Finally, combine (10.8), (10.12) and (10.14) to see that ∫ |𝐷ℓℎ ∇𝑢|2 𝑑𝜆 ≤ ∫ 𝜓2 |𝐷ℓℎ ∇𝑢|2 𝑑𝜆 ≤ 𝐶̃ ∫(|𝑓|2 + |𝑢|2 + |∇𝑢|2 ) 𝑑𝜆, 𝑉

Ω

Ω

for some constant 𝐶̃ > 0, for ℓ = 1, . . . , 𝑛 and all sufficiently small |ℎ| ≠ 0.

(10.14)

10.3 Schrödinger operators with magnetic field

| 171

1 2 (Ω), and hence that 𝑢 ∈ 𝐻loc (Ω), Using Proposition 10.2 (ii) we derive that ∇𝑢 ∈ 𝐻loc with the estimate

̃ ‖𝑢‖𝐻2 (𝑉) ≤ 𝐶(‖𝑓‖ 𝐿2 (Ω) + ‖𝑢‖𝐻1 (Ω) ). Remark 10.6. (a) It is not difficult to show that even ̃ ‖𝑢‖𝐻2 (𝑉) ≤ 𝐶(‖𝑓‖ 𝐿2 (Ω) + ‖𝑢‖𝐿2 (Ω) ) holds in Proposition 10.5. 2 (b) The result that 𝑢 ∈ 𝐻loc (Ω) implies that 𝐿𝑢 = 𝑓 almost everywhere in Ω. To see this, note that for each 𝑣 ∈ C∞ 0 (Ω), we have 𝑎(𝑢, 𝑣) = (𝑓, 𝑣), 2 (Ω), we can integrate by parts and obtain and since 𝑢 ∈ 𝐻loc

𝑎(𝑢, 𝑣) = (𝐿𝑢, 𝑣). Thus (𝐿𝑢 − 𝑓, 𝑣) = 0 for all 𝑣 ∈ C∞ 0 (Ω), and so 𝐿𝑢 = 𝑓 almost everywhere in Ω.

10.3 Schrödinger operators with magnetic field We consider differential operators 𝐻(𝐴, 𝑉) of the form (10.15)

𝐻(𝐴, 𝑉) = −󳵻𝐴 + 𝑉, where 𝑉 : ℝ𝑛 󳨀→ ℝ is the electric potential and 𝑛

𝐴 = ∑ 𝐴 𝑗 𝑑𝑥𝑗 , 𝐴 𝑗 : ℝ𝑛 󳨀→ ℝ, 𝑗 = 1, . . . , 𝑛 𝑗=1

is a 1-form, and

2

𝑛

󳵻𝐴 = ∑ ( 𝑗=1

The 2-form 𝐵 = 𝑑𝐴 = ∑ ( 𝑗 0 is a positive constant. Let dom(𝐻(𝐴, 𝑉)) = C∞ 0 (ℝ ). Then 𝐻(𝐴, 𝑉) is a symmetric, semibounded operator on 𝐿2 (ℝ𝑛 ). 𝑛 Proof. For 𝑢 ∈ C∞ 0 (ℝ ) we have

(𝐻(𝐴, 𝑉)𝑢, 𝑢) = ∫ (−󳵻𝐴 𝑢 + 𝑉𝑢) 𝑢 𝑑𝜆 ℝ𝑛 𝑛

= ∫ ∑ |𝑋𝑗 𝑢|2 𝑑𝜆 + ∫ 𝑉|𝑢|2 𝑑𝜆 ℝ𝑛 𝑗=1

ℝ𝑛 2

≥ −𝐶 ‖𝑢‖ . Using the Friedrichs extension 4.36, we obtain Proposition 10.8. Let 𝐻(𝐴, 𝑉) be as in Proposition 10.7. Then 𝐻(𝐴, 𝑉) admits a selfadjoint extension. Proof. Define 𝑎(𝑢, 𝑣) := (𝐻(𝐴, 𝑉)𝑢, 𝑣) + (𝐶 + 1)(𝑢, 𝑣) 𝑛 and define V to be the completion of C∞ 0 (ℝ ) with respect to the inner product 𝑎(𝑢, 𝑣). Then one can apply Proposition 4.36 to get the desired result.

Let 𝑓, 𝑔 ∈ 𝐿1loc (ℝ𝑛 ). We say that 𝑓 ≥ 𝑔 in the distributional sense, if (𝑓, 𝜙) ≥ (𝑔, 𝜙), 𝑛 C∞ 0 (ℝ ).

for all positive 𝜙 ∈ A useful tool for spectral analysis of Schrödinger operators is Kato’s inequality sometimes also called the diamagnetic inequality:

10.3 Schrödinger operators with magnetic field | 173

Proposition 10.9. Let 𝐴 ∈ C2 (ℝ𝑛 , ℝ𝑛 ). Then, for all 𝑓 ∈ 𝐿1loc (ℝ𝑛 ) with (−𝑖∇ − 𝐴)2 𝑓 ∈ 𝐿2loc (ℝ𝑛 ), we have 󳵻|𝑓| ≥ −ℜ(sgn(𝑓)(−𝑖∇ − 𝐴)2 𝑓) = ℜ(sgn(𝑓)󳵻𝐴 𝑓),

(10.18)

in the distributional sense, where sgn is defined as in Chapter 5. Proof. Let 𝐴 1 , . . . , 𝐴 𝑛 be the components of 𝐴. Notice that 𝑛

−󳵻𝐴 𝑓 = (−𝑖∇ − 𝐴)2 𝑓 = ∑ ( − 𝑖 𝑗=1

2 𝜕 − 𝐴 𝑗 ) 𝑓. 𝜕𝑥𝑗

The assumption (−𝑖∇ − 𝐴)2 𝑓 ∈ 𝐿2loc (ℝ𝑛 ), and the regularity property of second-order 2 elliptic operators (see Proposition 10.5) imply that 𝑓 ∈ 𝑊loc (ℝ𝑛 ), in particular 󳵻𝑓, ∇𝑓 ∈ 1 𝑛 𝐿 loc (ℝ ). First suppose that 𝑢 is smooth. Then, with |𝑢|𝜖 = √|𝑢|2 + 𝜖2 − 𝜖, we get ∇|𝑢|𝜖 =

ℜ(𝑢∇𝑢) ℜ(𝑢(∇ − 𝑖𝐴)𝑢) = . √|𝑢|2 + 𝜖2 √|𝑢|2 + 𝜖2

(10.19)

A straightforward calculation shows that for a smooth function 𝑔 we have 𝑔󳵻𝑔 = div(𝑔∇𝑔) − |∇𝑔|2 . Hence we obtain √|𝑢|2 + 𝜖2 󳵻|𝑢|𝜖 = div(√|𝑢|2 + 𝜖2 ∇|𝑢|𝜖 ) − 󵄨󵄨󵄨󵄨∇|𝑢|𝜖 󵄨󵄨󵄨󵄨2 󵄨 󵄨2 = ℜ [∇𝑢 ⋅ (∇ − 𝑖𝐴)𝑢 + 𝑢 div((∇ − 𝑖𝐴)𝑢)] − 󵄨󵄨󵄨∇|𝑢|𝜖 󵄨󵄨󵄨 = ℜ[(∇𝑢 − 𝑖𝐴𝑢) ⋅ (∇ − 𝑖𝐴)𝑢 󵄨 󵄨2 + (−𝑖𝐴𝑢) ⋅ (∇ − 𝑖𝐴)𝑢 + 𝑢 div((∇ − 𝑖𝐴)𝑢)] − 󵄨󵄨󵄨∇|𝑢|𝜖 󵄨󵄨󵄨 󵄨 󵄨2 = |(∇ − 𝑖𝐴)𝑢|2 − 󵄨󵄨󵄨∇|𝑢|𝜖 󵄨󵄨󵄨 + ℜ [(−𝑖𝐴𝑢) ⋅ (∇ − 𝑖𝐴)𝑢 + 𝑢 div((∇ − 𝑖𝐴)𝑢)] . An easy calculation shows that (−𝑖𝐴𝑢) ⋅ (∇ − 𝑖𝐴)𝑢 + 𝑢 div((∇ − 𝑖𝐴)𝑢) = 𝑢 (∇ − 𝑖𝐴)2 𝑢. From (10.19) we get 2 |𝑢|2 |(∇ − 𝑖𝐴)𝑢|2 󵄨2 |𝑢(∇ − 𝑖𝐴)𝑢| 󵄨󵄨 = ≤ |(∇ − 𝑖𝐴)𝑢|2 . 󵄨󵄨∇|𝑢|𝜖 󵄨󵄨󵄨 ≤ 2 2 |𝑢| + 𝜖 |𝑢|2 + 𝜖2

So we finally see that 󳵻|𝑢|𝜖 ≥ ℜ

𝑢 (∇ − 𝑖𝐴)2 𝑢 . √|𝑢|2 + 𝜖2

(10.20)

The rest of the proof uses approximative units and follows the same lines as the proof of the Proposition 5.7.

174 | 10 Schrödinger operators and Witten–Laplacians Using Kato’s inequality and a criterion for essential self-adjointness we obtain Proposition 10.10. Let 𝐴 ∈ C2 (ℝ𝑛 , ℝ𝑛 ) and 𝑉 ∈ 𝐿2loc (ℝ𝑛 ) and 𝑉 ≥ 0. Then the 𝑛 Schrödinger operator 𝐻(𝐴, 𝑉) = −󳵻𝐴 + 𝑉 is essentially self-adjoint on C∞ 0 (ℝ ). In this case the Friedrichs extension is the uniquely determined self-adjoint extension (see Remark 4.27 (b) and Proposition 10.8). Proof. By Proposition 9.27, it is sufficient to show that ker(𝐻(𝐴, 𝑉)∗ + 𝐼) = {0}. Since dom(𝐻(𝐴, 𝑉)∗ ) ⊆ 𝐿2 (ℝ𝑛 ), the triviality of the kernel follows from the statement: if −󳵻𝐴 𝑢 + 𝑉𝑢 + 𝑢 = 0, (10.21) for 𝑢 ∈ 𝐿2 (ℝ𝑛 ), then 𝑢 = 0. If 𝑢 ∈ 𝐿2 (ℝ𝑛 ) and 𝑉 ∈ 𝐿2loc (ℝ𝑛 ), one has 𝑢𝑉 ∈ 𝐿1loc (ℝ𝑛 ). In addition we have the inclusion 𝐿2 (ℝ𝑛 ) ⊂ 𝐿2loc (ℝ𝑛 ) ⊂ 𝐿1loc (ℝ𝑛 ), which follows from the estimate ∫ |𝑢| 𝑑𝜆 ≤ |𝐾|1/2 (∫ |𝑢|2 𝑑𝜆)1/2 . 𝐾

𝐾

Hence we have 𝑢 ∈ 𝐿1loc (ℝ𝑛 ), and, by (10.21), that 󳵻𝑢 ∈ 𝐿1loc (ℝ𝑛 ), where the derivative is taken in the sense of distributions. From (10.18) and (10.21) we obtain 󳵻|𝑢| ≥ ℜ(sgn(𝑢) 󳵻𝐴 𝑢) = ℜ(sgn(𝑢) (𝑉 + 1)𝑢) = |𝑢| (𝑉 + 1) ≥ 0. If (𝜒𝜖 )𝜖 is an approximate unit, we get 󳵻(𝜒𝜖 ∗ |𝑢|) = 𝜒𝜖 ∗ 󳵻|𝑢| ≥ 0.

(10.22)

Since 𝜒𝜖 ∗ |𝑢| ∈ dom(󳵻), we have (󳵻(𝜒𝜖 ∗ |𝑢|), 𝜒𝜖 ∗ |𝑢|) = −‖∇(𝜒𝜖 ∗ |𝑢|)‖2 ≤ 0.

(10.23)

By (10.22), the left side of (10.23) is nonnegative, so ∇(𝜒𝜖 ∗ |𝑢|) = 0 and hence 𝜒𝜖 ∗ |𝑢| = 𝑐 ≥ 0. But |𝑢| ∈ 𝐿2 (ℝ𝑛 ) and 𝜒𝜖 ∗ |𝑢| → |𝑢| in 𝐿2 (ℝ𝑛 ), and so 𝑐 = 0. Hence 𝜒𝜖 ∗ |𝑢| = 0, so |𝑢| = 0 and 𝑢 = 0. Before we return to applications of methods from complex analysis we prove two results on the spectrum of Schrödinger operators:

10.3 Schrödinger operators with magnetic field |

175

Proposition 10.11. Let 𝐴 ∈ C2 (ℝ𝑛 , ℝ𝑛 ) and 𝑉 ∈ 𝐿2loc (ℝ𝑛 ) with 𝑉 ≥ 0. Then inf 𝜎(𝐻(𝐴, 𝑉)) ≥ inf 𝜎(𝐻(0, 𝑉)).

(10.24)

Proof. By Kato’s inequality (10.18), we have −󳵻|𝑓| ≤ ℜ(sgn(𝑓)(−󳵻𝐴 𝑓)), so we get (|𝑓|, 𝐻(0, 𝑉)|𝑓|) ≤ ∫ |𝑓| [ℜ(sgn(𝑓)(−󳵻𝐴 𝑓)) + 𝑉] 𝑑𝜆 ℝ𝑛

≤ ℜ ∫ 𝑓𝐻(𝐴, 𝑉)𝑓 𝑑𝜆 ℝ𝑛

= (𝐻(𝐴, 𝑉)𝑓, 𝑓). Now we can apply Proposition 9.46 to obtain the desired result. Finally we still mention the gauge invariance of the spectrum of 𝐻(𝐴, 𝑉): Proposition 10.12. Let 𝐴, 𝐴󸀠 ∈ C2 (ℝ𝑛 , ℝ𝑛 ) be such that 𝑑𝐴 = 𝑑𝐴󸀠 . Suppose that 𝑉 ∈ 𝐿2loc (ℝ𝑛 ) and 𝑉 ≥ 0. Then 𝜎(𝐻(𝐴, 𝑉)) = 𝜎(𝐻(𝐴󸀠 , 𝑉)). Proof. By the Poincaré Lemma, we have 𝐴󸀠 = 𝐴 + 𝑑𝑔, where 𝑔 ∈ C1 (ℝ𝑛 ). Let 𝑋𝑗 = (−𝑖 𝜕𝑥𝜕 − 𝐴 𝑗 ) and 𝑋𝑗󸀠 = (−𝑖 𝜕𝑥𝜕 − 𝐴󸀠𝑗 ) for 𝑗 = 1, . . . , 𝑛. Then 𝑗

𝑗

𝑋𝑗󸀠 = 𝑒𝑖𝑔 𝑋𝑗 𝑒−𝑖𝑔 . Hence 𝐻(𝐴󸀠 , 𝑉) = 𝑒𝑖𝑔 𝐻(𝐴, 𝑉) 𝑒−𝑖𝑔 . Therefore the operators 𝐻(𝐴󸀠 , 𝑉) and 𝐻(𝐴, 𝑉) are unitarily equivalent and hence, by Lemma 9.29 have the same spectrum. In our first application of methods from complex analysis we will concentrate on Schrödinger operators with magnetic field on ℝ2 of the form −󳵻𝐴 + 𝐵, where the 2-form 𝑑𝐴 is given by 𝑑𝐴 = 𝐵𝑑𝑥 ∧ 𝑑𝑦. So, in this case, the Schrödinger operator −󳵻𝐴 + 𝐵 = 𝐻(𝐴, 𝐵) is completely determined by the vector field 𝐴. Let 𝜑 be a subharmonic C2 -function. We consider the inhomogeneous 𝜕-equation 𝜕𝑢 = 𝑓 for 𝑓 ∈ 𝐿2 (ℂ, 𝑒−𝜑 ). The canonical solution operator to 𝜕 gives a solution with minimal 𝐿2 (ℂ, 𝑒−𝜑 )-norm. We substitute 𝑣 = 𝑢 𝑒−𝜑/2 and 𝑔 = 𝑓 𝑒−𝜑/2 and the equation

176 | 10 Schrödinger operators and Witten–Laplacians becomes 𝐷𝑣 = 𝑔 , where 𝐷 = 𝑒−𝜑/2

𝜕 𝜑/2 𝑒 . 𝜕𝑧

(10.25)

𝑢 is the minimal solution to the 𝜕-equation in 𝐿2 (ℂ, 𝑒−𝜑 ) if and only if 𝑣 is the solution to 𝐷𝑣 = 𝑔 which is minimal in 𝐿2 (ℂ) . ∗ 𝜕 −𝜑/2 The formal adjoint of 𝐷 is 𝐷 = −𝑒𝜑/2 𝜕𝑧 𝑒 . We define dom(𝐷) = {𝑓 ∈ 𝐿2 (ℂ) : 𝐷𝑓 ∈ 𝐿2 (ℂ)} ∗

∗

and likewise for 𝐷 . Then 𝐷 and 𝐷 are closed unbounded linear operators from 𝐿2 (ℂ) to itself. Further we define ∗

∗

∗

dom(𝐷 𝐷 ) = {𝑢 ∈ dom(𝐷 ) : 𝐷 𝑢 ∈ dom(𝐷)} ∗

∗

and we define 𝐷 𝐷 as 𝐷 ∘ 𝐷 on this domain. Any function of the form 𝑒𝜑/2 𝑔, with ∗ ∗ 𝑔 ∈ C20 (ℂ) belongs to dom(𝐷 𝐷 ) and hence dom(𝐷 𝐷 ) is dense in 𝐿2 (ℂ). Since 𝐷=

𝜕 1 𝜕𝜑 + 𝜕𝑧 2 𝜕𝑧

∗

and 𝐷 = −

𝜕 1 𝜕𝜑 + 𝜕𝑧 2 𝜕𝑧

we see that ∗

S = 𝐷𝐷 = −

𝜕2 1 𝜕𝜑 𝜕 1 𝜕𝜑 𝜕 1 − + + 𝜕𝑧𝜕𝑧 2 𝜕𝑧 𝜕𝑧 2 𝜕𝑧 𝜕𝑧 4

󵄨󵄨 𝜕𝜑 󵄨󵄨2 1 𝜕2 𝜑 󵄨󵄨 󵄨󵄨 . 󵄨󵄨 󵄨󵄨 + 2 𝜕𝑧𝜕𝑧 󵄨󵄨 𝜕𝑧 󵄨󵄨

(10.26)

It follows that 4S = − (

𝜕𝜑 2 𝜕𝜑 2 𝑖 𝜕𝜑 𝜕 1 𝜕2 𝜕2 𝑖 𝜕𝜑 𝜕 1 ) ( (( ) ) ) + 󳵻𝜑, + + + − + 𝜕𝑥2 𝜕𝑦2 2 𝜕𝑥 𝜕𝑦 2 𝜕𝑦 𝜕𝑥 4 𝜕𝑥 𝜕𝑦 2

which implies that S is a Schrödinger operator with magnetic field, namely S=

1 (−󳵻𝐴 + 𝐵), 4

(10.27)

where the 1-form 𝐴 = 𝐴 1 𝑑𝑥 + 𝐴 2 𝑑𝑦 is related to the weight 𝜑 by (10.28)

𝐴 1 = −𝜕𝑦 𝜑/2 , 𝐴 2 = 𝜕𝑥 𝜑/2 , 2

󳵻𝐴 = (

2

𝜕 𝜕 − 𝑖𝐴 1 ) + ( − 𝑖𝐴 2 ) , 𝜕𝑥 𝜕𝑦

(10.29)

and the magnetic field 𝐵𝑑𝑥 ∧ 𝑑𝑦 satisfies 𝐵(𝑥, 𝑦) = ∗

∗

1 󳵻𝜑(𝑥, 𝑦) . 2

Both operators 𝐷 𝐷 and 𝐷 𝐷 are nonnegative, self-adjoint operators, see Lemma 4.28 and Lemma 4.29. By Proposition 10.10, we now know that the operator S with

10.3 Schrödinger operators with magnetic field | 177

its domain described above is the uniquely determined self-adjoint extension of the Schrödinger operator with magnetic field 14 (−󳵻𝐴 + 𝐵). ∗ Since 4𝐷 𝐷 = −󳵻𝐴 + 12 󳵻𝜑, it follows that ((−󳵻𝐴 + 12 󳵻𝜑)𝑓, 𝑓) ≥ 0, for 𝑓 ∈ C20 (ℂ). ∗ Similarly one shows that 4𝐷 𝐷 = −󳵻𝐴 − 12 󳵻𝜑, and this implies 1 −2󳵻𝐴 ≥ −󳵻𝐴 + 󳵻𝜑 ≥ −󳵻𝐴 . 2 It follows that

∗

∗

𝐷 𝐷 = 𝑒−𝜑/2 𝜕𝜑 𝜕 𝑒𝜑/2 , and that

∗

∗

𝐷 𝐷 = 𝑒−𝜑/2 𝜕 𝜕𝜑 𝑒𝜑/2 , where

∗ 𝜕𝜑

=

𝜕 − 𝜕𝑧

𝜕𝜑 . 𝜕𝑧

+

(10.30) (10.31)

For 𝑛 = 1 we have ∗

◻𝜑 = 𝜕 𝜕𝜑 , which means that

∗

𝐷 𝐷 = 𝑒−𝜑/2 ◻𝜑 𝑒𝜑/2 .

(10.32)

Theorem 10.13. Let 𝜑 be a subharmonic C2 -function on ℂ such that lim inf 󳵻𝜑(𝑧) > 0. |𝑧|→∞

Then the Schrödinger operator with magnetic field ∗

S = 𝐷𝐷 =

1 1 ( − 󳵻𝐴 + 󳵻𝜑) 4 2

(10.33)

has a bounded inverse on 𝐿2 (ℂ) ∗

(𝐷 𝐷 )−1 = 𝑒−𝜑/2 𝑁𝜑 𝑒𝜑/2 ,

(10.34)

where 𝑁𝜑 = ◻−1 𝜑 . ∗

Proof. We know that 4𝐷 𝐷 = −󳵻𝐴 − 12 󳵻𝜑, which implies that −󳵻𝐴 ≥

1 󳵻𝜑 ≥ 0. 2

Hence, for 𝑢 ∈ C∞ 0 (ℂ), we have ((−󳵻𝐴 + 𝐵)𝑢, 𝑢) = (−󳵻𝐴 𝑢, 𝑢) + (𝐵𝑢, 𝑢) ≥ (𝐵𝑢, 𝑢)

(10.35)

and, by a similar density argument as in Proposition 6.2, we obtain from (10.35) together with the assumption on 𝜑 that ∗

‖𝐷 𝑢‖ ≥ 𝐶‖𝑢‖, ∗

(10.36)

for all 𝑢 ∈ dom(𝐷 ). ∗ Now we take V = dom(𝐷 ) endowed with the topology given by the inner product ∗ ∗ ∗ ∗ (𝐷 𝑢, 𝐷 𝑣), the sesquilinear form 𝑎(𝑢, 𝑣) = (𝐷 𝑢, 𝐷 𝑣), and 𝐻 = 𝐿2 (ℂ), and apply Proposition 4.36 to get the desired result.

178 | 10 Schrödinger operators and Witten–Laplacians

10.4 Witten–Laplacians For several complex variables the situation is more complicated. Let 𝜑 : ℂ𝑛 󳨀→ ℝ be a C2 -weight function. We consider the 𝜕-complex 𝜕

𝜕

𝐿2 (ℂ𝑛 , 𝑒−𝜑 ) 󳨀→ 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑 ) 󳨀→ 𝐿2(0,2) (ℂ𝑛 , 𝑒−𝜑 ) . For 𝑣 ∈ 𝐿2 (ℂ𝑛 ), let

𝑛

𝐷1 𝑣 = ∑ ( 𝑘=1

and for 𝑔 =

∑𝑛𝑗=1

𝑔𝑗 𝑑𝑧𝑗 ∈

𝐿2(0,1) (ℂ𝑛 ),

𝜕𝑣 1 𝜕𝜑 + 𝑣) 𝑑𝑧𝑘 𝜕𝑧𝑘 2 𝜕𝑧𝑘

let 𝑛

∗

𝐷1 𝑔 = ∑ ( 𝑗=1

𝜕𝑔𝑗 1 𝜕𝜑 𝑔𝑗 − ), 2 𝜕𝑧𝑗 𝜕𝑧𝑗

where the derivatives are taken in the sense of distributions. It is easy to see that 𝜕𝑢 = 𝑓 for 𝑢 ∈ 𝐿2 (ℂ𝑛 , 𝑒−𝜑 ) and 𝑓 ∈ 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑 ) if and only if 𝐷1 𝑣 = 𝑔, where 𝑣 = 𝑢 𝑒−𝜑/2 and 𝑔 = 𝑓 𝑒−𝜑/2 . It is also clear that the necessary condition 𝜕𝑓 = 0 for solvability holds if and only if 𝐷2 𝑔 = 0 holds. Here 𝑛

𝐷2 𝑔 = ∑ ( 𝑗,𝑘=1

and

𝜕𝑔𝑗 𝜕𝑧𝑘

𝑛

∗

𝐷2 ℎ = ∑ ( 𝑗,𝑘=1

󸀠

for a suitable (0, 2)-form ℎ = ∑

|𝐽|=2

+

1 𝜕𝜑 𝑔 ) 𝑑𝑧𝑘 ∧ 𝑑𝑧𝑗 2 𝜕𝑧𝑘 𝑗

𝜕ℎ𝑘𝑗 1 𝜕𝜑 ) 𝑑𝑧𝑗 ℎ − 2 𝜕𝑧𝑘 𝑘𝑗 𝜕𝑧𝑘

ℎ𝐽 𝑑𝑧𝐽 .

We consider the corresponding 𝐷-complex : 𝐷1

𝐷2

←󳨀

←󳨀

𝐿2 (ℂ𝑛 ) 󳨀→ 𝐿2(0,1) (ℂ𝑛 ) 󳨀→ 𝐿2(0,2) (ℂ𝑛 ) . ∗ 𝐷1

∗ 𝐷2

and Δ(0,1) are defined by The so-called Witten–Laplacians (see [36]) Δ(0,0) 𝜑 𝜑 ∗

Δ(0,0) = 𝐷1 𝐷1 , 𝜑 ∗

∗

= 𝐷1 𝐷1 + 𝐷2 𝐷2 . Δ(0,1) 𝜑

(10.37)

A computation shows that 𝑛

∗

𝐷1 𝐷1 𝑣 = ∑ ( 𝑗=1

1 𝜕𝜑 𝜕𝑣 1 𝜕𝜑 𝜕𝜑 1 𝜕𝜑 𝜕𝑣 1 𝜕2 𝜑 𝜕2 𝑣 + 𝑣− − 𝑣− ) 2 𝜕𝑧𝑗 𝜕𝑧𝑗 4 𝜕𝑧𝑗 𝜕𝑧𝑗 2 𝜕𝑧𝑗 𝜕𝑧𝑗 2 𝜕𝑧𝑗 𝜕𝑧𝑗 𝜕𝑧𝑗 𝜕𝑧𝑗

and that ∗

𝑛

∗

𝑛

(𝐷1 𝐷1 + 𝐷2 𝐷2 )𝑔 = ∑ [ ∑ ( 𝑘=1

−

𝑗=1

1 𝜕𝜑 𝜕𝑔𝑘 1 𝜕𝜑 𝜕𝜑 + 𝑔 2 𝜕𝑧𝑗 𝜕𝑧𝑗 4 𝜕𝑧𝑗 𝜕𝑧𝑗 𝑘

𝜕2 𝑔𝑘 𝜕2 𝜑 1 𝜕𝜑 𝜕𝑔𝑘 1 𝜕2 𝜑 − 𝑔𝑘 − + 𝑔 )] 𝑑𝑧𝑘 . 2 𝜕𝑧𝑗 𝜕𝑧𝑗 2 𝜕𝑧𝑗 𝜕𝑧𝑗 𝜕𝑧𝑗 𝜕𝑧𝑗 𝜕𝑧𝑗 𝜕𝑧𝑘 𝑗

(10.38)

10.4 Witten–Laplacians 𝜕 𝜕𝑧𝑘

More generally, we set 𝑍𝑘 = ∑󸀠|𝐽|=𝑞

+ 󸀠

1 𝜕𝜑 2 𝜕𝑧𝑘

and 𝑍𝑘∗ = − 𝜕𝑧𝜕 + 𝑘

1 𝜕𝜑 2 𝜕𝑧𝑘

|

179

and we consider (0, 𝑞)-

ℎ𝐽 𝑑𝑧𝐽 , where ∑ means that we sum up only increasing multi-indices forms ℎ = 𝐽 = (𝑗1 , . . . , 𝑗𝑞 ) and where 𝑑𝑧𝐽 = 𝑑𝑧𝑗1 ∧ ⋅ ⋅ ⋅ ∧ 𝑑𝑧𝑗𝑞 . We define 𝑛

󸀠

𝐷𝑞+1 ℎ = ∑ ∑ 𝑍𝑘 (ℎ𝐽 ) 𝑑𝑧𝑘 ∧ 𝑑𝑧𝐽 𝑘=1 |𝐽|=𝑞

and ∗

𝑛

󸀠

𝐷𝑞 ℎ = ∑ ∑ 𝑍𝑘∗ (ℎ𝐽 ) 𝑑𝑧𝑘 ⌋𝑑𝑧𝐽 , 𝑘=1 |𝐽|=𝑞

where 𝑑𝑧𝑘 ⌋𝑑𝑧𝐽 denotes the contraction, or interior multiplication by 𝑑𝑧𝑘 , i.e. we have ⟨𝛼, 𝑑𝑧𝑘 ⌋𝑑𝑧𝐽 ⟩ = ⟨𝑑𝑧𝑘 ∧ 𝛼, 𝑑𝑧𝐽 ⟩ for each (0, 𝑞 − 1)-form 𝛼. The complex Witten–Laplacian on (0, 𝑞)-forms is then given by ∗

∗

= 𝐷𝑞 𝐷𝑞 + 𝐷𝑞+1 𝐷𝑞+1 , Δ(0,𝑞) 𝜑

(10.39)

for 𝑞 = 1, . . . , 𝑛 − 1. The general 𝐷-complex has the form 𝐷𝑞

𝐷𝑞+1

𝐿2(0,𝑞−1) (ℂ𝑛 ) 󳨀→ 𝐿2(0,𝑞) (ℂ𝑛 ) 󳨀→ 𝐿2(0,𝑞+1) (ℂ𝑛 ) . ←󳨀

←󳨀

∗ 𝐷𝑞

∗ 𝐷𝑞+1

It follows that ∗

∗

𝐷𝑞+1 Δ(0,𝑞) = Δ(0,𝑞+1) 𝐷𝑞+1 and 𝐷𝑞+1 Δ(0,𝑞+1) = Δ(0,𝑞) 𝜑 𝜑 𝜑 𝜑 𝐷𝑞+1 . We remark that 𝑛

∗

󸀠

𝐷𝑞 ℎ = ∑ ∑ 𝑍𝑘∗ (ℎ𝐽 ) 𝑑𝑧𝑘 ⌋𝑑𝑧𝐽 = ∑ 𝑘=1 |𝐽|=𝑞

󸀠

𝑛

∑ 𝑍𝑘∗ (ℎ𝑘𝐾 ) 𝑑𝑧𝐾 .

|𝐾|=𝑞−1 𝑘=1

In particular we get for a function 𝑣 ∈ 𝐿2 (ℂ𝑛 ) ∗

𝑛

∗ Δ(0,0) 𝜑 𝑣 = 𝐷1 𝐷1 𝑣 = ∑ 𝑍𝑗 𝑍𝑗 (𝑣), 𝑗=1

and for a (0, 1)-form 𝑔 = ∑𝑛ℓ=1 𝑔ℓ 𝑑𝑧ℓ ∈ 𝐿2(0,1) (ℂ𝑛 ) we obtain ∗

∗

Δ(0,1) 𝜑 𝑔 = (𝐷1 𝐷1 + 𝐷2 𝐷2 )𝑔 𝑛

= ∑ {𝑍𝑗 (𝑍𝑘∗ (𝑔ℓ )) 𝑑𝑧𝑗 ∧ (𝑑𝑧𝑘 ⌋𝑑𝑧ℓ ) + 𝑍𝑘∗ (𝑍𝑗 (𝑔ℓ )) 𝑑𝑧𝑘 ⌋(𝑑𝑧𝑗 ∧ 𝑑𝑧ℓ )} 𝑗,𝑘,ℓ=1

180 | 10 Schrödinger operators and Witten–Laplacians 𝑛

= ∑ {𝑍𝑘∗ (𝑍𝑗 (𝑔ℓ )) (𝑑𝑧𝑗 ∧ (𝑑𝑧𝑘 ⌋𝑑𝑧ℓ ) + 𝑑𝑧𝑘 ⌋(𝑑𝑧𝑗 ∧ 𝑑𝑧ℓ )) 𝑗,𝑘,ℓ=1

+ [𝑍𝑗 , 𝑍𝑘∗ ](𝑔ℓ ) 𝑑𝑧𝑗 ∧ (𝑑𝑧𝑘 ⌋𝑑𝑧ℓ )} 𝑛

𝑛

𝑗,ℓ=1

𝑗,𝑘,ℓ=1

= ∑ 𝑍𝑗∗ 𝑍𝑗 (𝑔ℓ ) 𝑑𝑧ℓ + ∑ = (Δ(0,0) 𝜑

𝜕2 𝜑 𝑔 𝛿 𝑑𝑧 𝜕𝑧𝑗 𝜕𝑧𝑘 ℓ 𝑘ℓ 𝑗

⊗ 𝐼)𝑔 + 𝑀𝜑 𝑔,

where we used that for (0, 1)-forms 𝛼, 𝑎, 𝑏 we have 𝛼⌋(𝑎 ∧ 𝑏) = (𝛼⌋𝑎) ∧ 𝑏 − 𝑎 ∧ (𝛼⌋𝑏), which implies that 𝑑𝑧𝑗 ∧ (𝑑𝑧𝑘 ⌋𝑑𝑧ℓ ) + 𝑑𝑧𝑘 ⌋(𝑑𝑧𝑗 ∧ 𝑑𝑧ℓ ) = 𝑑𝑧𝑗 ∧ (𝑑𝑧𝑘 ⌋𝑑𝑧ℓ ) + (𝑑𝑧𝑘 ⌋𝑑𝑧𝑗 ) ∧ 𝑑𝑧ℓ − 𝑑𝑧𝑗 ∧ (𝑑𝑧𝑘 ⌋𝑑𝑧ℓ ) = (𝑑𝑧𝑘 ⌋𝑑𝑧𝑗 ) ∧ 𝑑𝑧ℓ = 𝛿𝑘ℓ 𝑑𝑧ℓ , and where we set 𝑛

𝜕2 𝜑 𝑔 ) 𝑑𝑧𝑗 𝜕𝑧𝑘 𝜕𝑧𝑗 𝑘 𝑘=1 𝑛

𝑀𝜑 𝑔 = ∑ ( ∑ 𝑗=1

and

𝑛

(Δ(0,0) ⊗ 𝐼) 𝑔 = ∑ Δ(0,0) 𝜑 𝜑 𝑔𝑘 𝑑𝑧𝑘 . 𝑘=1

By Proposition 6.6 we obtain now Theorem 10.14. Let 𝜑 : ℂ𝑛 󳨀→ ℝ be a C2 -plurisubharmonic function and suppose that the lowest eigenvalue 𝜇𝜑 of the Levi matrix 𝑀𝜑 of 𝜑 satisfies lim inf 𝜇𝜑 (𝑧) > 0. |𝑧|→∞

has a bounded inverse on 𝐿2(0,1) (ℂ𝑛 ) Then the operator Δ(0,1) 𝜑 ∗

∗

−1 (𝐷1 𝐷1 + 𝐷2 𝐷2 )−1 = (Δ(0,1) = 𝑒−𝜑/2 𝑁𝜑 𝑒𝜑/2 , 𝜑 )

(10.40)

where 𝑁𝜑 = ◻−1 𝜑 .

10.5 Dirac and Pauli operators There is an interesting connection to Dirac and Pauli operators: recall (10.29) and define the Dirac operator D by

10.5 Dirac and Pauli operators

D = (−𝑖

𝜕 𝜕 − 𝐴 1 ) 𝜎1 + (−𝑖 − 𝐴 2 ) 𝜎2 = A1 𝜎1 + A2 𝜎2 , 𝜕𝑥 𝜕𝑦

|

181

(10.41)

where 0 1

𝜎1 = (

1 ), 0

𝜎2 = (

Hence we can write D=(

0 A1 + 𝑖A2

0 𝑖

A1 − 𝑖A2

0

−𝑖 ) . 0

).

We remark that 𝑖(A2 A1 − A1 A2 ) = 𝐵 and hence it turns out that the square of D is diagonal with the Pauli operators 𝑃± on the diagonal: 2

D =(

=(

A21 − 𝑖(A2 A1 − A1 A2 ) + A22

0 𝑃− 0

0 A21 + 𝑖(A2 A1 − A1 A2 ) + A22

)

0 ), 𝑃+

where 𝑃± = (−𝑖

2 2 𝜕 𝜕 − 𝐴 1 ) + (−𝑖 − 𝐴 2 ) ± 𝐵 = −󳵻𝐴 ± 𝐵. 𝜕𝑥 𝜕𝑦

By Lemma 4.28 and Lemma 4.29 the Pauli operators 𝑃± are nonnegative self-adjoint operators. It follows that 4S = 𝑃+ is the Schrödinger operator with magnetic field and that = 𝑃− . 4 Δ(0,0) 𝜑

(10.42)

In addition we obtain that D2 is self-adjoint and likewise D by the spectral theorem. Finally we consider decoupled weights 𝜑(𝑧1 , . . . , 𝑧𝑛 ) = ∑𝑛𝑗=1 𝜑𝑗 (𝑧𝑗 ). In this case the

operator Δ(0,1) acts diagonally on (0, 1)-forms, each component 𝐸𝑘 of the diagonal be𝜑 ing 1 1 𝐸𝑘 = 𝑃+(𝑘) + ∑ 𝑃−(𝑗) , (10.43) 4 4 𝑗=𝑘̸ where 𝑃±(ℓ) = (−𝑖

2 2 𝜕 𝜕 (ℓ) − 𝐴(ℓ) − 𝐴(ℓ) 1 ) + (−𝑖 2 ) ±𝐵 𝜕𝑥ℓ 𝜕𝑦ℓ 𝜕𝜑

𝜕𝜑

(10.44)

(ℓ) 1 1 (ℓ) ℓ ℓ with 𝑧ℓ = 𝑥ℓ + 𝑖𝑦ℓ , 𝐴(ℓ) = 12 󳵻𝜑ℓ , ℓ = 1, . . . , 𝑛. This 1 = − 2 𝜕𝑦ℓ , 𝐴 2 = 2 𝜕𝑥ℓ , and 𝐵 follows from (10.38) for a decoupled weight and from (10.42).

182 | 10 Schrödinger operators and Witten–Laplacians

10.6 Notes The regularity result on second order elliptic partial differential operators is taken from [20]. Basic facts about Schrödinger operators with magnetic fields can be found in [16, 37, 38, 40]. Kato’s inequality, sometimes also called the diamagnetic inequality, is an important tool for basic questions about Schödinger operators, see for instance [37]. The relationship to the 𝜕-equation appears in [5, 31, 34]. In the following we will describe the spectral analysis of these operators in more details. The weighted 𝜕-complex in several variables leads to the Witten–Laplacian. The Witten–Laplacians were introduced by E. Witten on a compact manifold in connection with a Morse function, which is a function with non-degenerate critical points. The main idea of E. Witten was that the dimension of the kernels of these Laplacians, which are related by the Hodge theory to the Betti numbers, can be estimated from above by the dimension of the eigenspace associated with the small eigenvalues of these operators (see [36, 41, 42, 46, 75, 76]). Dirac and Pauli operators are discussed in [16, 40, 72].

11 Compactness In this chapter we study compactness of the 𝜕-Neumann operator. We give a general characterization using a description of precompact subsets in 𝐿2 -spaces. Therefore we begin this part with a general characterization of precompact subsets in 𝐿2 -spaces, followed by basic facts about Sobolev spaces, such as Gårding’s inequality and the Rellich–Kondrachov Lemma. We point out that our approach includes the case of 𝐿2 (ℂ𝑛 , 𝑒−𝜑 ) where the classical Rellich–Kondrachov Lemma cannot be used. A sufficient condition for compactness of the 𝜕-Neumann operator is given in terms of the behavior at infinity of the eigenvalues of the Levi matrix of the weight function. We also show that property 𝑃 and 𝑃̃ imply compactness of the 𝜕-Neumann operator for smoothly bounded pseudoconvex domains.

11.1 Precompact sets in 𝐿2 -spaces A set 𝐴 is precompact (i.e. 𝐴 is compact) in a Banach space 𝑋 if and only if for every positive number 𝜖 there is a finite subset 𝑁𝜖 of points of 𝑋 such that 𝐴 ⊂ ⋃𝑦∈𝑁𝜖 𝐵𝜖 (𝑦). A set 𝑁𝜖 with this property is called a finite 𝜖-net for 𝐴. We recall the Arzela–Ascoli theorem: Let Ω be a bounded domain in ℝ𝑛 . A subset 𝐾 of C(Ω) is precompact in C(Ω) if the following two conditions hold: (i) There exists a constant 𝑀 such that |𝜙(𝑥)| ≤ 𝑀 holds for every 𝜙 ∈ 𝐾 and 𝑥 ∈ Ω. (Boundedness) (ii) For every 𝜖 > 0 there exists 𝛿 > 0 such that if 𝜙 ∈ 𝐾, 𝑥, 𝑦 ∈ Ω, and |𝑥 − 𝑦| < 𝛿, then |𝜙(𝑥) − 𝜙(𝑦)| < 𝜖. (Equicontinuity) Let (𝜒𝜖 )𝜖 be an approximation to the identity (see Chapter 5.1). Recall that 𝑢 ∗ 𝜒𝜖 ∈ C∞ (ℝ𝑛 ), if 𝑢 ∈ 𝐿1loc (ℝ𝑛 ) (Lemma 5.3). In a similar way one proves the following result: If Ω is a domain in ℝ𝑛 and 𝑢 ∈ 𝐿2 (Ω), then 𝑢 ∗ 𝜒𝜖 ∈ 𝐿2 (Ω) and ‖𝑢 ∗ 𝜒𝜖 ‖2 ≤ ‖𝑢‖2 ,

lim ‖𝑢 ∗ 𝜒𝜖 − 𝑢‖2 = 0.

𝜖→0+

Let Ω ⊆ ℝ𝑛 be a domain and 𝑢 a complex-valued function on Ω. Let {𝑢(𝑥) ̃ 𝑢(𝑥) ={ 0 {

𝑥∈Ω 𝑥 ∈ ℝ𝑛 \ Ω

Theorem 11.1. A bounded subset A of 𝐿2 (Ω) is precompact in 𝐿2 (Ω) if and only if for every 𝜖 > 0 there exists a number 𝛿 > 0 and a subset 𝜔 ⊂⊂ Ω such that for every 𝑢 ∈ A

184 | 11 Compactness and ℎ ∈ ℝ𝑛 with |ℎ| < 𝛿 both of the following inequalities hold: 2 ̃ + ℎ) − 𝑢(𝑥)| ̃ ∫ |𝑢(𝑥 𝑑𝜆(𝑥) < 𝜖2 , Ω

∫ |𝑢(𝑥)|2 𝑑𝜆(𝑥) < 𝜖2 .

(11.1)

Ω\𝜔

Proof. Let 𝜏ℎ 𝑢(𝑥) = 𝑢(𝑥 + ℎ) denote the translate of 𝑢 by ℎ. First assume that A is precompact. Since A has a finite 𝜖/6-net, and since C0 (Ω), the space of continuous functions with compact support in Ω, is dense in 𝐿2 (Ω), there exists a finite set 𝑆 ⊂ C0 (Ω), such that for each 𝑢 ∈ A there exists 𝜙 ∈ 𝑆 satisfying ‖𝑢 − 𝜙‖2 < 𝜖/3. Let 𝜔 be the union of the supports of the finitely many functions in 𝑆. Then 𝜔 ⊂⊂ Ω and the second inequality follows immediately. To prove the first inequality choose a closed ball 𝐵𝑟 of radius 𝑟 centered at the origin and containing 𝜔. Note that (𝜏ℎ 𝜙 − 𝜙)(𝑥) = 𝜙(𝑥 + ℎ) − 𝜙(𝑥) is uniformly continuous and vanishes outside 𝐵𝑟+1 provided |ℎ| < 1. Hence lim ∫ |𝜏ℎ 𝜙(𝑥) − 𝜙(𝑥)|2 𝑑𝜆(𝑥) = 0,

|ℎ|→0

ℝ𝑛

the convergence being uniform for 𝜙 ∈ 𝑆. For |ℎ| sufficiently small, we have ‖𝜏ℎ 𝜙−𝜙‖2 < 𝜖/3. If 𝜙 ∈ 𝑆 satisfies ‖𝑢 − 𝜙‖2 < 𝜖/3, then also ‖𝜏ℎ 𝑢̃ − 𝜏ℎ 𝜙‖2 < 𝜖/3. Hence we have for |ℎ| sufficiently small (independent of 𝑢 ∈ A ), ‖𝜏ℎ 𝑢̃ − 𝑢‖̃ 2 ≤ ‖𝜏ℎ 𝑢̃ − 𝜏ℎ 𝜙‖2 + ‖𝜏ℎ 𝜙 − 𝜙‖2 + ‖𝜙 − 𝑢‖̃ 2 < 𝜖 and the first inequality follows. It is sufficient to prove the converse for the special case Ω = ℝ𝑛 , as it follows for general Ω from its application in this special case to the set à = {𝑢̃ : 𝑢 ∈ A}. Let 𝜖 > 0 be given and choose 𝜔 ⊂⊂ ℝ𝑛 such that for all 𝑢 ∈ A ∫ |𝑢(𝑥)|2 𝑑𝜆(𝑥) < ℝ𝑛 \𝜔

𝜖 . 3

For any 𝜂 > 0 the function 𝑢 ∗ 𝜒𝜂 ∈ C∞ (ℝ𝑛 ) and in particular it belongs to C(𝜔). If 𝜙 ∈ C0 (ℝ𝑛 ), then by Hölder’s inequality 󵄨󵄨 󵄨󵄨2 󵄨󵄨 󵄨 󵄨 |𝜒𝜂 ∗ 𝜙(𝑥) − 𝜙(𝑥)| = 󵄨󵄨 ∫ 𝜒𝜂 (𝑦)(𝜙(𝑥 − 𝑦) − 𝜙(𝑥)) 𝑑𝜆(𝑦)󵄨󵄨󵄨󵄨 󵄨󵄨 𝑛 󵄨󵄨 ℝ 2

≤ ∫ 𝜒𝜂 (𝑦)|𝜏−𝑦 𝜙(𝑥) − 𝜙(𝑥)|2 𝑑𝜆(𝑦). 𝐵𝜂

Hence ‖𝜒𝜂 ∗ 𝜙 − 𝜙‖2 ≤ sup ‖𝜏ℎ 𝜙 − 𝜙‖2 . ℎ∈𝐵𝜂

2

𝑛

If 𝑢 ∈ 𝐿 (ℝ ), let (𝜙𝑗 )𝑗 be a sequence in C0 (ℝ𝑛 ) converging to 𝑢 in 𝐿2 norm. Then (𝜒𝜂 ∗ 𝜙𝑗 )𝑗 is a sequence converging to 𝜒𝜂 ∗ 𝑢 in 𝐿2 (ℝ𝑛 ). Since also 𝜏ℎ 𝜙𝑗 → 𝜏ℎ 𝑢 in 𝐿2 (ℝ𝑛 ), we have ‖𝜒𝜂 ∗ 𝑢 − 𝑢‖2 ≤ sup ‖𝜏ℎ 𝑢 − 𝑢‖2 . ℎ∈𝐵𝜂

11.1 Precompact sets in 𝐿2 -spaces

| 185

From the first inequality in our assumption we derive that lim|ℎ|→0 ‖𝜏ℎ 𝑢 − 𝑢‖2 = 0 uniformly for 𝑢 ∈ A. Hence lim𝜂→0 ‖𝜒𝜂 ∗ 𝑢 − 𝑢‖2 = 0 uniformly for 𝑢 ∈ A. Fix 𝜂 > 0 so that 𝜖 ∫ |𝜒𝜂 ∗ 𝑢(𝑥) − 𝑢(𝑥)|2 𝑑𝜆(𝑥) < 6 𝜔

for all 𝑢 ∈ A. We show that {𝜒𝜂 ∗𝑢 : 𝑢 ∈ A} satisfies the conditions of the Arzela–Ascoli theorem on 𝜔 and hence is precompact in C(𝜔). We have |𝜒𝜂 ∗ 𝑢(𝑥)| ≤ ( sup 𝜒𝜂2 (𝑦))

1/2

‖𝑢‖2 ,

𝑦∈ℝ𝑛

which is bounded uniformly for 𝑥 ∈ ℝ𝑛 and 𝑢 ∈ A since A is bounded in 𝐿2 (ℝ𝑛 ) and 𝜂 is fixed. Similarly |𝜒𝜂 ∗ 𝑢(𝑥 + ℎ) − 𝜒𝜂 ∗ 𝑢(𝑥)| ≤ ( sup 𝜒𝜂2 (𝑦)) 𝑦∈ℝ𝑛

1/2

‖𝜏ℎ 𝑢 − 𝑢‖2

and so lim|ℎ|→0 𝜒𝜂 ∗ 𝑢(𝑥 + ℎ) = 𝜒𝜂 ∗ 𝑢(𝑥) uniformly for 𝑥 ∈ ℝ𝑛 and 𝑢 ∈ A. Thus {𝜒𝜂 ∗ 𝑢 : 𝑢 ∈ A} is precompact in C(𝜔) and there exists a finite set {𝜓1 , . . . , 𝜓𝑚 } of functions in C(𝜔) such that if 𝑢 ∈ A, then for some 𝑗, 1 ≤ 𝑗 ≤ 𝑚, and all 𝑥 ∈ 𝜔 we have |𝜓𝑗 (𝑥) − 𝜒𝜂 ∗ 𝑢(𝑥)| < √

𝜖 . 6|𝜔|

This together with the inequality (|𝑎| + |𝑏|)2 ≤ 2(|𝑎|2 + |𝑏|2 ) implies that ∫ |𝑢(𝑥) − 𝜓̃𝑗 (𝑥)|2 𝑑𝜆(𝑥) = ∫ |𝑢(𝑥)|2 𝑑𝜆(𝑥) + ∫ |𝑢(𝑥) − 𝜓𝑗 (𝑥)|2 𝑑𝑥 ℝ𝑛

ℝ𝑛 \𝜔

𝜔

𝜖 < + 2 ∫(|𝑢(𝑥) − 𝜒𝜂 ∗ 𝑢(𝑥)|2 + |𝜒𝜂 ∗ 𝑢(𝑥) − 𝜓𝑗 (𝑥)|2 ) 𝑑𝜆(𝑥) 3 𝜔

𝜖 𝜖 𝜖 < + 2( + |𝜔|) = 𝜖. 3 6 6|𝜔| Hence A has a finite 𝜖1/2 -net in 𝐿2 (ℝ𝑛 ) and is therefore precompact in 𝐿2 (ℝ𝑛 ). Remark 11.2. (a) With the same proof one gets: A bounded subset A of 𝐿2 (Ω) is precompact in 𝐿2 (Ω) if and only if the following two conditions are satisfied: (i) for every 𝜖 > 0 and for each 𝜔 ⊂⊂ Ω there exists a number 𝛿 > 0 such that for every 𝑢 ∈ A and ℎ ∈ ℝ𝑛 with |ℎ| < 𝛿 the following inequality holds: 2 ̃ + ℎ) − 𝑢(𝑥)| ̃ 𝑑𝜆(𝑥) < 𝜖2 ; ∫ |𝑢(𝑥 𝜔

(11.2)

186 | 11 Compactness (ii) for every 𝜖 > 0 there exists 𝜔 ⊂⊂ Ω such that for every 𝑢 ∈ A ∫ |𝑢(𝑥)|2 𝑑𝜆(𝑥) < 𝜖2 .

(11.3)

Ω\𝜔

(b) An analogous result holds in weighted spaces 𝐿2 (ℂ𝑛 , 𝑒−𝜑 ).

11.2 Sobolev spaces and Gårding’s inequality In the following we will prove a simple version of Gårding’s inequality (coercive estimate), which will be used to investigate compactness of the 𝜕-Neumann operator. For a comprehensive treatment of Gårding’s inequality see for instance [23] or [12]. Definition 11.3. If Ω is a bounded open set in ℝ𝑛 , we define the Sobolev space 𝐻𝑘 (Ω) for 𝑘 a nonnegative integer to be the completion of C∞ (Ω) with respect to the norm 1/2

[ ∑ ∫ |𝜕𝛼 𝑓|2 𝑑𝜆]

(11.4)

,

|𝛼|≤𝑘 Ω

where 𝛼 = (𝛼1 , . . . , 𝛼𝑛 ) is a multi-index, |𝛼| = ∑𝑛𝑗=1 𝛼𝑗 and 𝜕𝛼 𝑓 =

𝜕|𝛼| 𝑓 𝛼 . . . . 𝜕𝑥𝑛𝑛

𝛼 𝜕𝑥1 1

𝐻0𝑘 (Ω) denotes the closure of C∞ 0 (Ω) with respect to the norm (11.4). If Ω is a domain with a C1 -boundary, then 𝐻𝑘 (Ω) coincides with 𝑊𝑘 (Ω) = {𝑓 ∈ 𝐿2 (Ω) : 𝜕𝛼 𝑓 ∈ 𝐿2 (Ω), |𝛼| ≤ 𝑘}, where the derivatives are taken in the sense of distributions (Proposition 5.10). Theorem 11.4. Let Ω be a bounded open set in ℝ𝑛 with a C1 -boundary. Let 𝐷 be a Dirichlet form of order 1 given by 𝑛

𝑛

𝑛

𝑗,𝑘=1

𝑘=1

𝑘=1

𝐷(𝑢, 𝑣) = ∑ (𝜕𝑗 𝑢, 𝑏𝑗𝑘 𝜕𝑘 𝑣) + ∑ (𝜕𝑘 𝑢, 𝑏𝑘 𝑣) + ∑ (𝑢, 𝑏𝑘󸀠 𝜕𝑘 𝑣) + (𝑢, 𝑏𝑣), for 𝑢, 𝑣 ∈ 𝐻1 (Ω), where 𝜕𝑗 =

𝜕 𝜕𝑥𝑗

(11.5)

and 𝑏𝑗𝑘 , 𝑏𝑘 , 𝑏𝑘󸀠 , 𝑏 are C∞ -coefficients and the 𝑏𝑗𝑘 are

real-valued. Suppose that there exists a constant 𝐶0 > 0 such that 𝑛

ℜ ∑ 𝑏𝑗𝑘 (𝑥)𝜉𝑗 𝜉𝑘 ≥ 𝐶0 |𝜉|2 , 𝑗,𝑘=1

we say that 𝐷 is strongly elliptic on Ω.

𝜉 ∈ ℝ𝑛 ,

𝑥 ∈ Ω,

(11.6)

11.2 Sobolev spaces and Gårding’s inequality

| 187

Then there exist constants 𝐶 > 0 and 𝑀 ≥ 0 such that ℜ𝐷(𝑢, 𝑢) ≥ 𝐶‖𝑢‖2𝑊1 (Ω) − 𝑀‖𝑢‖2𝐿2 (Ω) ,

𝑢 ∈ 𝐻1 (Ω),

(11.7)

we say that 𝐷 is coercive over 𝐻1 (Ω) (Gårding’s inequality). Proof. We first set 𝑎𝑗𝑘 = 12 (𝑏𝑗𝑘 + 𝑏𝑘𝑗 ). Since the 𝑏𝑗𝑘 ’s are real, strong ellipticity means that for some constant 𝐶0 > 0, 𝑛

𝑛

𝑗,𝑘=1

𝑗,𝑘=1

∑ 𝑎𝑗𝑘 𝜉𝑗 𝜉𝑘 = ∑ 𝑏𝑗𝑘 𝜉𝑗 𝜉𝑘 ≥ 𝐶0 |𝜉|2

for all 𝜉 ∈ ℝ𝑛 . Thus (𝑎𝑗𝑘 ) is symmetric (𝑎𝑗𝑘 = 𝑎𝑘𝑗 ), so if 𝜉 ∈ ℂ𝑛 is any complex vector, 𝑛

𝑛

𝑗,𝑘=1

𝑗,𝑘=1

ℜ ∑ 𝑏𝑗𝑘 𝜉𝑗 𝜉𝑘 = ∑ 𝑎𝑗𝑘 𝜉𝑗 𝜉𝑘 ≥ 𝐶0 |𝜉|2 . Setting 𝜉 = ∇𝑢, where 𝑢 ∈ 𝐻1 (Ω), we obtain 𝑛

𝑛

𝑗,𝑘=1

𝑘=1

ℜ ∑ 𝑏𝑗𝑘 (𝜕𝑗 𝑢)(𝜕𝑘 𝑢) ≥ 𝐶0 ∑ |𝜕𝑘 𝑢|2 , so integration over Ω yields 𝑛

𝑛

𝑗,𝑘=1

𝑘=1

ℜ ∑ (𝜕𝑗 𝑢, 𝑏𝑗𝑘 𝜕𝑘 𝑣) ≥ 𝐶0 ∑ ‖𝜕𝑘 𝑢‖2𝐿2 (Ω) = 𝐶0 (‖𝑢‖2𝑊1 (Ω) − ‖𝑢‖2𝐿2 (Ω) ). Also, for some 𝐶1 > 0 (independent of 𝑢) we have |(𝜕𝑘 𝑢, 𝑏𝑘 𝑢)| ≤ ‖𝑢‖𝑊1 (Ω) ‖𝑏𝑘 𝑢‖𝐿2 (Ω) ≤ 𝐶1 ‖𝑢‖𝑊1 (Ω) ‖𝑢‖𝐿2 (Ω) , |(𝑢, 𝑏𝑘󸀠 𝜕𝑘 𝑢)| ≤ ‖𝑢‖𝐿2 (Ω) ‖𝑏𝑘󸀠 𝜕𝑘 𝑢‖𝐿2 (Ω) ≤ 𝐶1 ‖𝑢‖𝑊1 (Ω) ‖𝑢‖𝐿2 (Ω) , |(𝑢, 𝑏𝑢)| ≤ 𝐶1 ‖𝑢‖2𝐿2 (Ω) ≤ 𝐶1 ‖𝑢‖𝑊1 (Ω) ‖𝑢‖𝐿2 (Ω) .

If we set 𝐶2 = (2𝑛 + 1)𝐶1 , we have ℜ𝐷(𝑢, 𝑢) ≥ 𝐶0 (‖𝑢‖2𝑊1 (Ω) − ‖𝑢‖2𝐿2 (Ω) ) − 𝐶2 ‖𝑢‖𝑊1 (Ω) ‖𝑢‖𝐿2 (Ω) . But since 𝑐𝑑 ≤ 12 (𝑐2 + 𝑑2 ) for all 𝑐, 𝑑 > 0, 𝐶2 ‖𝑢‖𝑊1 (Ω) ‖𝑢‖𝐿2 (Ω) ≤ so ℜ𝐷(𝑢, 𝑢) ≥

𝐶2 𝐶0 ‖𝑢‖2𝑊1 (Ω) + 2 ‖𝑢‖2𝐿2 (Ω) , 2 2𝐶0

2𝐶02 + 𝐶22 𝐶0 ‖𝑢‖2𝐿2 (Ω) , ‖𝑢‖2𝑊1 (Ω) − 2 2𝐶0

which proves Gårding’s inequality.

(11.8)

188 | 11 Compactness Our next aim is to prove the classical Rellich–Kondrachov Lemma, which states that the embedding of 𝑊1 (Ω) into 𝐿2 (Ω) is compact, provided that Ω is a bounded domain with a C1 -boundary. Lemma 11.5 (Rellich–Kondrachov). Let Ω be a bounded domain with a C1 -boundary. Then the embedding 𝑗 : 𝑊1 (Ω) 󳨀→ 𝐿2 (Ω) is compact. Proof. We have to show that the unit ball in 𝑊1 (Ω) is precompact in 𝐿2 (Ω). For this purpose we apply Proposition 5.11 and consider the extension of elements of the unit ball in 𝑊1 (Ω) to ℝ𝑛 . Let F denote the set of all these extensions. Then, by Proposition 5.11 (iii), F is a bounded set in 𝐿2 (ℝ𝑛 ). By Lemma 5.3 we know that for each 𝜖 > 0 there exists a number 𝑁 > 0 such that (11.9)

‖𝜒1/𝑘 ∗ 𝑓 − 𝑓‖𝐿2 (ℝ𝑛 ) ≤ 𝜖, for each 𝑓 ∈ F and for each 𝑘 > 𝑁. By Hölder’s inequality we have ‖𝜒1/𝑘 ∗ 𝑓‖𝐿∞ (ℝ𝑛 ) ≤ 𝐶𝑘 ‖𝑓‖𝐿2 (ℝ𝑛 ) ,

(11.10)

for all 𝑓 ∈ F, where 𝐶𝑘 = ‖𝜒1/𝑘 ‖𝐿2 (ℝ𝑛 ) . Hence we can now verify the second condition in Theorem 11.1: Given 𝜖 > 0 there exists 𝜔 ⊂⊂ Ω such that ‖𝑓‖𝐿2 (Ω\𝜔) < 𝜖, for each 𝑓 in the unit ball of 𝑊1 (Ω): indeed, we consider the extensions to ℝ𝑛 and write ‖𝑓‖𝐿2 (Ω\𝜔) ≤ ‖𝑓 − 𝜒1/𝑘 ∗ 𝑓‖𝐿2 (ℝ𝑛 ) + ‖𝜒1/𝑘 ∗ 𝑓‖𝐿2 (Ω\𝜔) , we use Proposition 5.11 (iii) and (11.9), and, in view of (11.10) we have to choose 𝜔 such that |Ω \ 𝜔| is small enough. We are left to verify the first condition of Remark 11.2: Let 𝜔 ⊂⊂ Ω and 𝜖 > 0 and consider first a function 𝑢 ∈ C∞ (Ω). Let ℎ ∈ ℝ𝑛 such that |ℎ| < dist(𝜔, 𝑏Ω) and set 𝑣(𝑡) := 𝑢(𝑥 + 𝑡ℎ),

𝑡 ∈ [0, 1].

Then 𝑣󸀠 (𝑡) = ℎ ⋅ ∇𝑢(𝑥 + 𝑡ℎ) and 1

1 󸀠

𝑢(𝑥 + ℎ) − 𝑢(𝑥) = 𝑣(1) − 𝑣(0) = ∫ 𝑣 (𝑡) 𝑑𝑡 = ∫ ℎ ⋅ ∇𝑢(𝑥 + 𝑡ℎ) 𝑑𝑡. 0

0

Hence we obtain

1 2

2

|𝑢(𝑥 + ℎ) − 𝑢(𝑥)| ≤ |ℎ| ∫ |∇𝑢(𝑥 + 𝑡ℎ)|2 𝑑𝑡 0

(11.11)

11.2 Sobolev spaces and Gårding’s inequality

| 189

and 1

∫ |𝑢(𝑥 + ℎ) − 𝑢(𝑥)|2 𝑑𝜆(𝑥) ≤ |ℎ|2 ∫ ∫ |∇𝑢(𝑥 + 𝑡ℎ)|2 𝑑𝑡 𝑑𝜆(𝑥) 𝜔

𝜔 0 1 2

= |ℎ| ∫ ∫ |∇𝑢(𝑥 + 𝑡ℎ)|2 𝑑𝜆(𝑥) 𝑑𝑡 0 𝜔 1

= |ℎ|2 ∫ ∫ |∇𝑢(𝑥)|2 𝑑𝜆(𝑥) 𝑑𝑡. 0 𝜔+𝑡ℎ

If |ℎ| < dist(𝜔, 𝑏Ω), there exists 𝜔󸀠 ⊂⊂ Ω such that 𝜔 + 𝑡ℎ ⊂ 𝜔󸀠 for each 𝑡 ∈ [0, 1]. Therefore we get the estimate ‖𝜏ℎ 𝑢 − 𝑢‖2𝐿2 (𝜔) ≤ |ℎ|2 ∫ |∇𝑢(𝑥)|2 𝑑𝜆(𝑥).

(11.12)

𝜔󸀠

If 𝑢 belongs to the unit ball in 𝑊1 (Ω), we approximate 𝑢 by functions in C∞ (Ω) (Proposition 5.10), apply (11.12) and pass to the limit getting ‖𝜏ℎ 𝑢 − 𝑢‖2𝐿2 (𝜔) ≤ |ℎ|2 ∫ |∇𝑢(𝑥)|2 𝑑𝜆(𝑥) ≤ |ℎ|2 , 𝜔󸀠

which shows that the first condition of Remark 11.2 holds. Remark 11.6. (a) If Ω ⊂ ℝ𝑛 , 𝑛 ≥ 2, is a bounded domain with a C1 -boundary, it even follows that 𝑊1 (Ω) ⊂ 𝐿𝑞 (Ω) , 𝑞 ∈ [1, 2𝑛/(𝑛 − 2)) and that the embedding is also compact (see for instance [8]). (b) It follows immediately from Lemma 11.5 that the embedding of 𝜄 : 𝐻01 (Ω) 󳨀→ 𝐿2 (Ω) is compact. By Theorem 2.5 the adjoint 𝜄∗ : 𝐿2 (Ω) 󳨀→ (𝐻01 (Ω))󸀠 =: 𝐻0−1 (Ω) is also compact. In the following we will still describe 𝐻0−1 (Ω) as a certain space of distributions, which will be helpful later on. Recall that the dual-norm is given by ‖𝑓‖𝐻0−1 (Ω) = sup{|𝑓(𝑢)| : 𝑢 ∈ 𝐻01 (Ω) , ‖𝑢‖𝐻01 (Ω) ≤ 1}.

190 | 11 Compactness Proposition 11.7. Let 𝑓 ∈ 𝐻0−1 (Ω). Then there exist functions 𝑓0 , 𝑓1 , . . . , 𝑓𝑛 in 𝐿2 (Ω) such that 𝑛

(11.13)

𝑓(𝑣) = ∫(𝑓0 𝑣 + ∑ 𝑓𝑗 𝑣𝑥𝑗 ) 𝑑𝜆 , 𝑗=1

Ω

1/2

𝑛

‖𝑓‖𝐻0−1 (Ω) = inf {( ∫ ∑ |𝑓𝑗2 | 𝑑𝜆) Ω

: 𝑓 satisfies (11.13)}.

(11.14)

𝑗=0

We write

𝑛

𝑓 = 𝑓0 − ∑

𝑗=1

𝜕𝑓𝑗 𝜕𝑥𝑗

,

whenever (11.13) holds. Proof. For 𝑢, 𝑣 ∈ 𝐻01 (Ω), the inner product is given by (𝑢, 𝑣) = ∫ ∇𝑢 ⋅ ∇𝑣 𝑑𝜆 + ∫ 𝑢𝑣 𝑑𝜆. Ω

Ω

If 𝑓 ∈ 𝐻0−1 (Ω), the Riesz representation theorem 1.6 implies that there exists a unique function 𝑢 ∈ 𝐻01 (Ω), such that 𝑓(𝑣) = (𝑢, 𝑣) , ∀𝑣 ∈ 𝐻01 (Ω), hence 𝑓(𝑣) = ∫ ∇𝑢 ⋅ ∇𝑣 𝑑𝜆 + ∫ 𝑢𝑣 𝑑𝜆, Ω

(11.15)

Ω

which gives (11.13), where 𝑓0 = 𝑢 and 𝑓𝑗 = 𝑢𝑥𝑗 , 𝑗 = 1, . . . , 𝑛. By Cauchy–Schwarz we obtain 𝑛

1/2

‖𝑓‖𝐻0−1 (Ω) ≤ ( ∫ ∑ |𝑓𝑗 |2 𝑑𝜆) Ω

,

𝑗=0

and setting 𝑣 = 𝑢/‖𝑢‖𝐻01 (Ω) in (11.15) we deduce 𝑛

1/2

‖𝑓‖𝐻0−1 (Ω) = ( ∫ ∑ |𝑓𝑗 |2 𝑑𝜆) Ω

,

𝑗=0

which gives (11.14).

11.3 Compactness in weighted spaces Now we will apply the description of compact subsets in 𝐿2 -spaces to derive a general characterization of compactness of the 𝜕-Neumann operator and a sufficient condition for compactness in terms of the weight function. In a certain sense we adopt the real method using Sobolev spaces to the situation in ℂ𝑛 for the complex ◻-operator.

11.3 Compactness in weighted spaces

| 191

Definition 11.8. Let 𝑄𝜑

W

∗

= {𝑢 ∈ 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑 ) : 𝑢 ∈ dom(𝜕) ∩ dom(𝜕𝜑 )}

with norm

∗

‖𝑢‖𝑄𝜑 = (‖𝜕𝑢‖2𝜑 + ‖𝜕𝜑 𝑢‖2𝜑 )1/2 .

(11.16) ∗

Remark: W𝑄𝜑 coincides with the form domain dom(𝜕) ∩ dom(𝜕𝜑 ) of 𝑄𝜑 (see Proposition 6.6). Proposition 11.9. Suppose that the weight function 𝜑 is plurisubharmonic and that the lowest eigenvalue 𝜇𝜑 of the Levi matrix 𝑀𝜑 satisfies lim 𝜇𝜑 (𝑧) = +∞ .

(11.17)

𝑗𝜑 : W𝑄𝜑 󳨅→ 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑 )

(11.18)

|𝑧|→∞

Then the embedding is compact. Proof. For 𝑢 ∈ W𝑄𝜑 we have by Proposition 6.4 ∗

‖𝜕𝑢‖2𝜑 + ‖𝜕𝜑 𝑢‖2𝜑 ≥ (𝑀𝜑 𝑢, 𝑢)𝜑 . This implies ∗

‖𝜕𝑢‖2𝜑 + ‖𝜕𝜑 𝑢‖2𝜑 ≥ ∫ 𝜇𝜑 (𝑧) |𝑢(𝑧)|2 𝑒−𝜑(𝑧) 𝑑𝜆(𝑧).

(11.19)

ℂ𝑛

We show that the unit ball in W𝑄𝜑 is relatively compact in 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑 ). For this purpose we use the characterization of compact subsets in 𝐿2 -spaces from Theorem 11.1. We start with the first condition in Theorem 11.1. Let 𝑢 = ∑𝑛𝑗=1 𝑢𝑗 𝑑𝑧𝑗 be a (0, 1)-form with coefficients in C∞ 0 . For each 𝑢𝑗 and for 𝑡 ∈ ℝ and ℎ = (ℎ1 , . . . , ℎ𝑛 ) ∈ ℂ𝑛 let 𝑣𝑗 (𝑡) := 𝑢𝑗 (𝑧 + 𝑡ℎ). Note that

󵄨󵄨 𝜕𝑢 󵄨󵄨2 󵄨󵄨 𝜕𝑢 󵄨󵄨2 1/2 𝑛 󵄨 𝑗 󵄨 󵄨 𝑗 󵄨 (𝑧 + 𝑡ℎ)󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨 (𝑧 + 𝑡ℎ)󵄨󵄨󵄨󵄨 )] , |𝑣𝑗󸀠 (𝑡)| ≤ |ℎ|[ ∑ (󵄨󵄨󵄨󵄨 󵄨 𝜕𝑥𝑘 󵄨󵄨 󵄨󵄨 𝜕𝑦𝑘 󵄨󵄨 𝑘=1 󵄨

where 𝑧𝑘 = 𝑥𝑘 + 𝑖𝑦𝑘 , for 𝑘 = 1, . . . , 𝑛. By the fact that 1

𝑢𝑗 (𝑧 + ℎ) − 𝑢𝑗 (𝑧) = 𝑣𝑗 (1) − 𝑣𝑗 (0) = ∫ 𝑣𝑗󸀠 (𝑡) 𝑑𝑡 0

192 | 11 Compactness we can now estimate for |ℎ| < 𝑅 ∫ |𝜏ℎ 𝑢𝑗 (𝑧) − 𝑢𝑗 (𝑧)|2 𝑒−𝜑(𝑧) 𝑑𝜆(𝑧) 𝔹𝑅

= ∫ |𝜏ℎ (𝜒𝑅 𝑢𝑗 )(𝑧) − 𝜒𝑅 𝑢𝑗 (𝑧)|2 𝑒−𝜑(𝑧) 𝑑𝜆(𝑧) 𝔹𝑅 1

󵄨󵄨 𝜕(𝜒 𝑢 ) 󵄨󵄨2 󵄨󵄨 𝜕(𝜒 𝑢 ) 󵄨󵄨2 𝑛 󵄨 󵄨 󵄨 󵄨 𝑅 𝑗 𝑅 𝑗 ≤ |ℎ| ∫ [ ∫ ∑ (󵄨󵄨󵄨󵄨 (𝑧 + 𝑡ℎ)󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨 (𝑧 + 𝑡ℎ)󵄨󵄨󵄨󵄨 ) 𝑑𝑡]𝑒−𝜑(𝑧) 𝑑𝜆(𝑧) 󵄨󵄨 𝜕𝑥𝑘 󵄨󵄨 󵄨󵄨 𝜕𝑦𝑘 󵄨󵄨 0 𝑘=1 𝔹 2

𝑅

󵄨󵄨 𝜕(𝜒 𝑢 ) 󵄨󵄨2 󵄨󵄨 𝜕(𝜒 𝑢 ) 󵄨󵄨2 𝑛 󵄨 󵄨 󵄨 󵄨 𝑅 𝑗 𝑅 𝑗 (𝑧)󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨 (𝑧)󵄨󵄨󵄨󵄨 )𝑒−𝜑(𝑧) 𝑑𝜆(𝑧) ≤ 𝐶𝜑,𝑅 |ℎ|2 ∫ ∑ (󵄨󵄨󵄨󵄨 󵄨󵄨 𝜕𝑥𝑘 󵄨󵄨 󵄨󵄨 𝜕𝑦𝑘 󵄨󵄨 𝔹 𝑘=1 3𝑅

for 𝑗 = 1, . . . , 𝑛, where 𝜒𝑅 is a C∞ cut-off function which is identically 1 on 𝔹2𝑅 and zero outside 𝔹3𝑅 . It is clear that the corresponding Dirichlet form of ◻𝜑 satisfies the assumptions of Theorem 11.4 in 𝔹3𝑅 (by (6.11), the second order terms reduce to the Laplacian), so by Gårding’s inequality for 𝔹3𝑅 , ∗

󸀠 (‖𝜕(𝜒𝑅 𝑢)‖2𝜑 + ‖𝜕𝜑 (𝜒𝑅 𝑢)‖2𝜑 + ‖𝜒𝑅 𝑢‖2𝜑 ) ‖𝜒𝑅 𝑢‖2𝑊1 ,𝜑 ≤ 𝐶𝜑,𝑅 ∗

󸀠󸀠 ≤ 𝐶𝜑,𝑅 (‖𝜕𝑢‖2𝜑 + ‖𝜕𝜑 𝑢‖2𝜑 + ‖𝑢‖2𝜑 )

we can control the last integral by the norm ‖𝑢‖2𝑄𝜑 . Since we started from the unit ball in W𝑄𝜑 we get that the first condition in Theorem 11.1 is satisfied. The second condition is satisfied for the unit ball of W𝑄𝜑 since we have ∫ |𝑢(𝑧)|2 𝑒−𝜑(𝑧) 𝑑𝜆(𝑧) ≤ ∫ ℂ𝑛 \𝔹𝑅

ℂ𝑛 \𝔹𝑅

𝜇𝜑 (𝑧)|𝑢(𝑧)|2 inf{𝜇𝜑 (𝑧) : |𝑧| ≥ 𝑅}

𝑒−𝜑(𝑧) 𝑑𝜆(𝑧).

So formula (11.19) together with assumption (11.17) shows that 2 −𝜑(𝑧)

∫ |𝑢(𝑧)| 𝑒 ℂ𝑛 \𝔹𝑅

𝑑𝜆(𝑧) ≤

‖𝑢‖2𝑄𝜑 inf{𝜇𝜑 (𝑧) : |𝑧| ≥ 𝑅}

< 𝜖,

(11.20)

if 𝑅 is big enough. We remark that 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑 ) ≅ 𝐿2 (ℂ𝑛 , 𝑒−𝜑 ) ⊕ ⋅ ⋅ ⋅ ⊕ 𝐿2 (ℂ𝑛 , 𝑒−𝜑 ), with 𝑛 factors, and that precompactness of a subset in 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑 ) is equivalent to being contained in a product of precompact subsets of 𝐿2 (ℂ𝑛 , 𝑒−𝜑 ). We are now able to give a concise proof of the main result in [34] or [28], see [33], the original proof there uses a number of more involved methods from real analysis. Proposition 11.10. Let 𝜑 be a plurisubharmonic C2 -weight function. If the lowest eigenvalue 𝜇𝜑 (𝑧) of the Levi matrix 𝑀𝜑 satisfies (11.17), then 𝑁𝜑 is compact.

11.3 Compactness in weighted spaces

|

193

Proof. By Proposition 11.9, the embedding W𝑄𝜑 󳨅→ 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑 ) is compact. The inverse 𝑁𝜑 of ◻𝜑 is continuous as an operator from 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑 ) into W𝑄𝜑 , this follows from Proposition 6.6. Therefore we have that 𝑁𝜑 is compact as an operator from 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑 ) into itself (see Remark 6.7). Proposition 11.11. Let 𝜑 be a plurisubharmonic C2 -weight function. Let 1 ≤ 𝑞 ≤ 𝑛 and suppose that the sum 𝑠𝑞 of any 𝑞 (equivalently: the smallest 𝑞) eigenvalues of 𝑀𝜑 satisfies lim 𝑠𝑞 (𝑧) = +∞.

|𝑧|→∞

(11.21)

Then 𝑁𝜑,𝑞 : 𝐿2(0,𝑞) (ℂ𝑛 , 𝑒−𝜑 ) 󳨀→ 𝐿2(0,𝑞) (ℂ𝑛 , 𝑒−𝜑 ) is compact. Proof. For (0, 𝑞)-forms use (6.11) to show that Gårding’s inequality can be applied and Proposition 6.8 to show that ∗

‖𝜕𝑢‖2𝜑 + ‖𝜕𝜑 𝑢‖2𝜑 ≥ ∫ 𝑠𝑞 (𝑧) |𝑢(𝑧)|2 𝑒−𝜑(𝑧) 𝑑𝜆(𝑧). ℂ𝑛

Now one can continue as in the proof of Proposition 11.9. Example: We consider the plurisubharmonic weight function 𝜑(𝑧, 𝑤) = |𝑧|2 |𝑤|2 + |𝑤|4 on ℂ2 . The Levi matrix of 𝜑 has the form (

|𝑤|2 𝑤𝑧

𝑧𝑤 ) |𝑧| + 4|𝑤|2 2

and the eigenvalues are 𝜇𝜑,1 (𝑧, 𝑤) =

1 (5|𝑤|2 + |𝑧|2 − √9|𝑤|4 + 10|𝑧|2 |𝑤|2 + |𝑧|4 ) 2

and

1 (5|𝑤|2 + |𝑧|2 + √9|𝑤|4 + 10|𝑧|2 |𝑤|2 + |𝑧|4 ) . 2 It follows that (11.17) fails, but 1 𝑠2 (𝑧, 𝑤) = 󳵻𝜑(𝑧, 𝑤) = |𝑧|2 + 5|𝑤|2 , 4 hence (11.21) is satisfied. We point out that since 𝜇𝜑,1 (𝑧, 0) = 0 for all 𝑧 ∈ ℂ, we have 𝜇𝜑,2 (𝑧, 𝑤) =

lim inf 𝜇𝜑,1 (𝑧, 𝑤) = 0,

|(𝑧,𝑤)|→∞

so condition (6.5) is not satisfied, and it is not even clear that 𝑁𝜑 is bounded. Notice that

𝑁𝜑 : 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑 ) 󳨀→ 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑 )

can be written in the form where is the adjoint operator to 𝑗𝜑 .

𝑁𝜑 = 𝑗𝜑 ∘ 𝑗𝜑∗ , 𝑗𝜑∗ : 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑 ) 󳨀→ W𝑄𝜑

(11.22)

194 | 11 Compactness This means that 𝑁𝜑 is compact if and only if 𝑗𝜑 is compact and summarizing the above results we get the following Theorem 11.12. Let 1 ≤ 𝑞 ≤ 𝑛 and let 𝜑 : ℂ𝑛 󳨀→ ℝ+ be a plurisubharmonic C2 -weight function and suppose that lim inf 𝑠𝑞 (𝑧) > 0. |𝑧|→∞

The 𝜕-Neumann operator 𝑁𝜑,𝑞 : 𝐿2(0,𝑞) (ℂ𝑛 , 𝑒−𝜑 ) 󳨀→ 𝐿2(0,𝑞) (ℂ𝑛 , 𝑒−𝜑 ) is compact if and only if for each 𝜖 > 0 there exists 𝑅 > 0 such that ∗

∫ |𝑢(𝑧)|2 𝑒−𝜑(𝑧) 𝑑𝜆(𝑧) ≤ 𝜖(‖𝜕𝑢‖2𝜑 + ‖𝜕𝜑 𝑢‖2𝜑 )

(11.23)

ℂ𝑛 \𝔹𝑅 ∗

for each 𝑢 ∈ 𝑑𝑜𝑚 (𝜕) ∩ 𝑑𝑜𝑚 (𝜕𝜑 ). Next we adapt a method from ([71, Proposition 4.5]) and use the last result to show that compactness percolates up the weighted 𝜕-complex. Proposition 11.13. Let 𝜑 be a plurisubharmonic C2 -function. Suppose that 1 ≤ 𝑞 ≤ 𝑛 − 1 and that (11.21) holds. If 𝑁𝜑,𝑞 is compact, then 𝑁𝜑,𝑞+1 is also compact. ∗

Proof. For 𝑢 = ∑󸀠|𝐽|=𝑞+1 𝑢𝐽 𝑑𝑧𝐽 ∈ 𝑑𝑜𝑚 (𝜕) ∩ 𝑑𝑜𝑚 (𝜕𝜑 ) we define (0, 𝑞)-forms 󸀠

𝑣𝑘 := ∑ 𝑢𝑘𝐾 𝑑𝑧𝐾 ,

𝑘 = 1, . . . , 𝑛.

|𝐾|=𝑞

∗

We claim that 𝑣𝑘 ∈ 𝑑𝑜𝑚 (𝜕) ∩ 𝑑𝑜𝑚 (𝜕𝜑 ). The components of 𝜕𝑣𝑘 are linear combinations ∗

of terms 𝜕𝑢𝐽 /𝜕𝑧𝑗 , and their weighted 𝐿2 -norms are controlled by ‖𝜕𝑢‖2𝜑 + ‖𝜕𝜑 𝑢‖2𝜑 , since by (6.11) we have 𝑛 󵄩 󵄩󵄩 𝜕𝑢𝐽 󵄩󵄩󵄩2 ∗ 󸀠 󵄩󵄩󵄩 . ‖𝜕𝑢‖2𝜑 + ‖𝜕𝜑 𝑢‖2𝜑 ≥ ∑ ∑ 󵄩󵄩󵄩󵄩 󵄩 𝜕𝑧𝑗 󵄩󵄩󵄩𝜑 |𝐽|=𝑞+1 𝑗=1 󵄩 ∗

This shows that 𝑣𝑘 ∈ 𝑑𝑜𝑚 (𝜕). To prove that 𝑣𝑘 ∈ 𝑑𝑜𝑚 (𝜕𝜑 ), we first observe that for 𝛼 = ∑󸀠|𝐾|=𝑞 𝛼𝐾 𝑑𝑧𝐾 ∈ 𝐿2(0,𝑞) (ℂ𝑛 , 𝑒−𝜑 ) and for 𝑘 fixed we have 󸀠

󸀠

(𝑑𝑧𝑘 ∧ 𝛼, 𝑢)𝜑 = ( ∑ 𝛼𝐾 (𝑑𝑧𝑘 ∧ 𝑑𝑧𝐾 ), ∑ 𝑢𝐽 𝑑𝑧𝐽 ) |𝐾|=𝑞

|𝐽|=𝑞+1

𝜑

= (𝛼, 𝑣𝑘 )𝜑 , where we used the inner products in 𝐿2(0,𝑞+1) (ℂ𝑛 , 𝑒−𝜑 ) and 𝐿2(0,𝑞) (ℂ𝑛 , 𝑒−𝜑 ), respectively. Now we get for 𝛽 ∈ 𝑑𝑜𝑚 (𝜕) (𝜕𝛽, 𝑣𝑘 )𝜑 = (𝑑𝑧𝑘 ∧ 𝜕𝛽, 𝑢)𝜑 = −(𝜕(𝑑𝑧𝑘 ∧ 𝛽), 𝑢)𝜑 ∗

= −(𝑑𝑧𝑘 ∧ 𝛽, 𝜕𝜑 𝑢)𝜑 = −(𝛽, 𝛾𝑘 )𝜑 ,

11.3 Compactness in weighted spaces

|

195

∗

where 𝛾𝑘 = ∑󸀠|𝐿|=𝑞−1 (𝜕𝜑 𝑢)𝑘𝐿 𝑑𝑧𝐿 . The last equality follows as before and we get that ∗

𝑣𝑘 ∈ 𝑑𝑜𝑚 (𝜕𝜑 ), and that

∗

(11.24)

𝜕𝜑 𝑣𝑘 = −𝛾𝑘 .

Since 𝑁𝜑,𝑞 is compact, we can apply Theorem 11.12: for each 𝜖 > 0 there exists 𝑅 > 0 such that ∗

∗

∫ |𝑣𝑘 (𝑧)|2 𝑒−𝜑(𝑧) 𝑑𝜆(𝑧) ≤ 𝜖(‖𝜕𝑣𝑘 ‖2𝜑 + ‖𝜕𝜑 𝑣𝑘 ‖2𝜑 ) ≤ 𝐶𝜖(‖𝜕𝑢‖2𝜑 + ‖𝜕𝜑 𝑢‖2𝜑 ) ℂ𝑛 \𝔹𝑅

for 𝑘 = 1, . . . , 𝑛, where the last inequality follows from (6.11) and 𝑛 󵄩 󵄩󵄩 𝜕𝑢 󵄩󵄩󵄩2 ∗ 󸀠 ‖𝜕𝑣𝑘 ‖2𝜑 = ∑ ∑ 󵄩󵄩󵄩󵄩 𝑘𝐾 󵄩󵄩󵄩󵄩 ≤ 𝐶󸀠 (‖𝜕𝑢‖2𝜑 + ‖𝜕𝜑 𝑢‖2𝜑 ) 󵄩 𝜕𝑧𝑗 󵄩󵄩𝜑 |𝐾|=𝑞 𝑗=1 󵄩

and from

∗

󸀠

∗

∗

‖𝜕𝜑 𝑣𝑘 ‖2𝜑 = ‖𝛾𝑘 ‖2𝜑 = ∑ ‖(𝜕𝜑 𝑢)𝑘𝐿 ‖2𝜑 ≤ ‖𝜕𝜑 𝑢‖2𝜑 . |𝐿|=𝑞−1

From the definition of 𝑣𝑘 it follows that ∫ |𝑢(𝑧)|2 𝑒−𝜑(𝑧) 𝑑𝜆(𝑧) = ℂ𝑛 \𝔹𝑅

1 𝑛 ∑ ∫ |𝑣𝑘 (𝑧)|2 𝑒−𝜑(𝑧) 𝑑𝜆(𝑧) 𝑞 + 1 𝑘=1 ℂ𝑛 \𝔹𝑅

≤

∗ 𝐶𝜖̃ (‖𝜕𝑢‖2𝜑 + ‖𝜕𝜑 𝑢‖2𝜑 ). 𝑞+1 ∗

Since 𝑢 was an arbitrary (0, 𝑞)-form in dom (𝜕) ∩ dom (𝜕𝜑 ) we can again apply Theorem 11.12 to get the desired conclusion. For a further study of compactness we define weighted Sobolev spaces and prove, under suitable conditions, a Rellich–Kondrachov Lemma for these weighted Sobolev spaces. We will also have to consider their dual spaces, which already appeared in [7] and [47]. Definition 11.14. For 𝑘 ∈ ℕ let 𝑊𝑘 (ℂ𝑛 , 𝑒−𝜑 ) := {𝑓 ∈ 𝐿2 (ℂ𝑛 , 𝑒−𝜑 ) : 𝐷𝛼 𝑓 ∈ 𝐿2 (ℂ𝑛 , 𝑒−𝜑 ) for |𝛼| ≤ 𝑘}, where 𝐷𝛼 =

𝜕|𝛼| 𝜕𝛼1 𝑥1 ...𝜕𝛼2𝑛 𝑦𝑛

for (𝑧1 , . . . , 𝑧𝑛 ) = (𝑥1 , 𝑦1 , . . . , 𝑥𝑛 , 𝑦𝑛 ) with norm ‖𝑓‖2𝑘,𝜑 = ∑ ‖𝐷𝛼 𝑓‖2𝜑 . |𝛼|≤𝑘

We will also need weighted Sobolev spaces with negative exponent. But it turns out that for our purposes it is more reasonable to consider the dual spaces of the following spaces.

196 | 11 Compactness Definition 11.15. Let 𝑋𝑗 =

𝜕𝜑 𝜕 − 𝜕𝑥𝑗 𝜕𝑥𝑗

and 𝑌𝑗 =

𝜕𝜑 𝜕 − , 𝜕𝑦𝑗 𝜕𝑦𝑗

for 𝑗 = 1, . . . , 𝑛 and define 𝑊1 (ℂ𝑛 , 𝑒−𝜑 , ∇𝜑) = {𝑓 ∈ 𝐿2 (ℂ𝑛 , 𝑒−𝜑 ) : 𝑋𝑗 𝑓, 𝑌𝑗 𝑓 ∈ 𝐿2 (ℂ𝑛 , 𝑒−𝜑 ), 𝑗 = 1, . . . , 𝑛}, with norm

𝑛

‖𝑓‖2𝜑,∇𝜑 = ‖𝑓‖2𝜑 + ∑(‖𝑋𝑗 𝑓‖2𝜑 + ‖𝑌𝑗 𝑓‖2𝜑 ). 𝑗=1

In the next step we will analyze the dual space of 𝑊1(ℂ𝑛 , 𝑒−𝜑 , ∇𝜑), compare with Proposition 11.7. By the mapping 𝑓 󳨃→ (𝑓, 𝑋𝑗 𝑓, 𝑌𝑗 𝑓), the space 𝑊1 (ℂ𝑛 , 𝑒−𝜑 , ∇𝜑) can be identified with a product of 𝐿2 (ℂ𝑛 , 𝑒−𝜑 ), hence each continuous linear functional 𝐿 on 𝑊1 (ℂ𝑛 , 𝑒−𝜑 , ∇𝜑) is represented (in a non-unique way) by 𝑛

𝐿(𝑓) = ∫ 𝑓𝑔0 𝑒−𝜑 𝑑𝜆 + ∑ ∫ ((𝑋𝑗 𝑓)𝑔𝑗 + (𝑌𝑗 𝑓)ℎ𝑗 )𝑒−𝜑 𝑑𝜆, ℂ𝑛

𝑗=1

ℂ𝑛

𝑛 for some 𝑔𝑗 , ℎ𝑗 ∈ 𝐿2 (ℂ𝑛 , 𝑒−𝜑 ). For 𝑓 ∈ C∞ 0 (ℂ ) it follows that 𝑛

𝐿(𝑓) = ∫ 𝑓𝑔0 𝑒−𝜑 𝑑𝜆 − ∑ ∫ 𝑓 ( ℂ𝑛

𝑗=1

ℂ𝑛

𝜕𝑔𝑗 𝜕𝑥𝑗

+

𝜕ℎ𝑗 𝜕𝑦𝑗

) 𝑒−𝜑 𝑑𝜆.

𝑛 1 𝑛 −𝜑 Since C∞ 0 (ℂ ) is dense in 𝑊 (ℂ , 𝑒 , ∇𝜑) we have shown

Lemma 11.16. Each element 𝑢 ∈ (𝑊1 (ℂ𝑛 , 𝑒−𝜑 , ∇𝜑))󸀠 = 𝑊−1 (ℂ𝑛 , 𝑒−𝜑 , ∇𝜑) can be represented in a non-unique way by 𝑛

𝑢 = 𝑔0 + ∑ ( 𝑗=1

𝜕𝑔𝑗 𝜕𝑥𝑗

+

𝜕ℎ𝑗 𝜕𝑦𝑗

),

where 𝑔𝑗 , ℎ𝑗 ∈ 𝐿2 (ℂ𝑛 , 𝑒−𝜑 ). The dual norm ‖𝑢‖−1,𝜑,∇𝜑 := sup{|𝑢(𝑓)| : ‖𝑓‖𝜑,∇𝜑 ≤ 1} can be expressed in the form 𝑛

‖𝑢‖2−1,𝜑,∇𝜑 = inf{‖𝑔0 ‖2 + ∑ (‖𝑔𝑗 ‖2 + ‖ℎ𝑗 ‖2 )}, 𝑗=1

where the infimum is taken over all families (𝑔𝑗 , ℎ𝑗 ) in 𝐿2 (ℂ𝑛 , 𝑒−𝜑 ) representing the functional 𝑢. In particular each function in 𝐿2 (ℂ𝑛 , 𝑒−𝜑 ) can be identified with an element of 𝑊−1 (ℂ𝑛 , 𝑒−𝜑 , ∇𝜑).

11.3 Compactness in weighted spaces

|

197

Proposition 11.17. Suppose that 𝜑 is a C2 -function satisfying lim (𝜃|∇𝜑(𝑧)|2 + 󳵻𝜑(𝑧)) = +∞,

|𝑧|→∞

for some 𝜃 ∈ (0, 1), where 𝑛 󵄨󵄨 𝜕𝜑 󵄨󵄨2 󵄨󵄨 𝜕𝜑 󵄨󵄨2 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 |∇𝜑(𝑧)|2 = ∑ (󵄨󵄨󵄨 󵄨 +󵄨 󵄨 ). 󵄨 𝜕𝑥𝑘 󵄨󵄨󵄨 󵄨󵄨󵄨 𝜕𝑦𝑘 󵄨󵄨󵄨 𝑘=1 󵄨

Then the embedding of 𝑊1 (ℂ𝑛 , 𝑒−𝜑 , ∇𝜑) into 𝐿2 (ℂ𝑛 , 𝑒−𝜑 ) is compact. Proof. We adapt methods from Proposition 11.9 (see also [7, 46] and [47]) and use the general result that an operator between Hilbert spaces is compact if and only if the image of a weakly convergent sequence is strongly convergent (Proposition 4.17). For the vector fields 𝑋𝑗 from Definition 11.15 and their formal adjoints 𝑋𝑗∗ in the

weighted space 𝐿2 (ℂ𝑛 , 𝑒−𝜑 ) we have 𝑋𝑗∗ = − 𝜕𝑥𝜕 and 𝑗

(𝑋𝑗 + 𝑋𝑗∗ )𝑓 = − 𝑛 for 𝑓 ∈ C∞ 0 (ℂ ), and

𝜕𝜑 𝜕2 𝜑 𝑓 and [𝑋𝑗 , 𝑋𝑗∗ ]𝑓 = − 2 𝑓, 𝜕𝑥𝑗 𝜕𝑥𝑗

([𝑋𝑗 , 𝑋𝑗∗ ]𝑓, 𝑓)𝜑 = ‖𝑋𝑗∗ 𝑓‖2𝜑 − ‖𝑋𝑗 𝑓‖2𝜑 ,

(11.25)

‖(𝑋𝑗 + 𝑋𝑗∗ )𝑓‖2𝜑 ≤ (1 + 1/𝜖)‖𝑋𝑗 𝑓‖2𝜑 + (1 + 𝜖)‖𝑋𝑗∗ 𝑓‖2𝜑

(11.26)

for each 𝜖 > 0. Similar relations hold for the vector fields 𝑌𝑗 . Now we set Ψ(𝑧) = |∇𝜑(𝑧)|2 + (1 + 𝜖)󳵻𝜑(𝑧). By (11.25) and (11.26), it follows that 𝑛

(Ψ𝑓, 𝑓)𝜑 ≤ (2 + 𝜖 + 1/𝜖) ∑ (‖𝑋𝑗 𝑓‖2𝜑 + ‖𝑌𝑗 𝑓‖2𝜑 ). 𝑗=1

𝑛 1 𝑛 −𝜑 Since C∞ 0 (ℂ ) is dense in 𝑊 (ℂ , 𝑒 , ∇𝜑) by definition, this inequality holds for all 1 𝑛 −𝜑 𝑓 ∈ 𝑊 (ℂ , 𝑒 , ∇𝜑). If (𝑓𝑘 )𝑘 is a sequence in 𝑊1 (ℂ𝑛 , 𝑒−𝜑 , ∇𝜑) converging weakly to 0, then (𝑓𝑘 )𝑘 is a bounded sequence in 𝑊1 (ℂ𝑛 , 𝑒−𝜑 , ∇𝜑) and our assumption implies that

Ψ(𝑧) = |∇𝜑(𝑧)|2 + (1 + 𝜖)󳵻𝜑(𝑧) is positive in a neighborhood of ∞. So we obtain ∫ |𝑓𝑘 (𝑧)|2 𝑒−𝜑(𝑧) 𝑑𝜆(𝑧) ≤ ∫ |𝑓𝑘 (𝑧)|2 𝑒−𝜑(𝑧) 𝑑𝜆(𝑧) + ∫ ℂ𝑛

|𝑧| 0 such that ∗

∫ |𝑢(𝑧)|2 𝑒−𝜑(𝑧) 𝑑𝜆(𝑧) ≤ 𝜖(‖𝜕𝑢‖2𝜑 + ‖𝜕𝜑 𝑢‖2𝜑 ) ℂ𝑛 \𝔹𝑅 ∗

for all 𝑢 ∈ dom (𝜕) ∩ dom (𝜕𝜑 ). (5) The operators ∗

𝜕𝜑 𝑁1,𝜑 : 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑 ) ∩ ker(𝜕) 󳨀→ 𝐿2 (ℂ𝑛 , 𝑒−𝜑 ) and

∗

𝜕𝜑 𝑁2,𝜑 : 𝐿2(0,2) (ℂ𝑛 , 𝑒−𝜑 ) ∩ ker(𝜕) 󳨀→ 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑 ) are both compact.

11.3 Compactness in weighted spaces

| 199

Proof. (1) and (4) are equivalent by Theorem 11.12. Next we show that (1) and (5) are equivalent: suppose that 𝑁1,𝜑 is compact. For 𝑓 ∈ 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑 ) it follows that ∗

‖𝜕𝜑 𝑁1,𝜑 𝑓‖2𝜑 ≤ ⟨𝑓, 𝑁1,𝜑 𝑓⟩𝜑 ≤ 𝜖‖𝑓‖2𝜑 + 𝐶𝜖 ‖𝑁1,𝜑 𝑓‖2𝜑 . ∗

Hence, by Proposition 2.1, 𝜕𝜑 𝑁1,𝜑 is compact. Applying the formula ∗

∗

∗

∗

𝑁1,𝜑 = (𝜕𝜑 𝑁1,𝜑 )∗ (𝜕𝜑 𝑁1,𝜑 ) + (𝜕𝜑 𝑁2,𝜑 )(𝜕𝜑 𝑁2,𝜑 )∗ , ∗

(see Proposition 4.63), we get that also 𝜕𝜑 𝑁2,𝜑 is compact. The converse follows easily from the same formula. Now we show (5) 󳨐⇒ (3) 󳨐⇒ (2) 󳨐⇒ (1). We follow the lines of [71], where the case of a bounded pseudoconvex domain is handled. Assume (5): if (3) does not hold, then there exists 𝜖0 > 0 and a sequence (𝑢𝑛 )𝑛 in ∗ dom (𝜕) ∩dom (𝜕𝜑 ) with ‖𝑢𝑛 ‖𝜑 = 1 and ∗

‖𝑢𝑛 ‖2𝜑 ≥ 𝜖0 (‖𝜕𝑢𝑛 ‖2𝜑 + ‖𝜕𝜑 𝑢𝑛 ‖2𝜑 ) + 𝑛‖𝑢𝑛 ‖2−1,𝜑,∇𝜑 −1 for each 𝑛 ≥ 1, which implies that 𝑢𝑛 → 0 in 𝑊(0,1) (ℂ𝑛 , 𝑒−𝜑 , ∇𝜑). Since 𝑢𝑛 can be written in the form ∗ ∗ ∗ 𝑢𝑛 = (𝜕𝜑 𝑁1,𝜑 )∗ 𝜕𝜑 𝑢𝑛 + (𝜕𝜑 𝑁2,𝜑 ) 𝜕𝑢𝑛 ,

(5) implies there exists a subsequence of (𝑢𝑛 )𝑛 converging in 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑 ) and the limit must be 0, which contradicts ‖𝑢𝑛 ‖𝜑 = 1. To show that (3) implies (2) we consider a bounded sequence in dom (𝜕) ∩ ∗ dom (𝜕𝜑 ). By Proposition 6.4 this sequence is also bounded in 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑 ). Now Proposition 11.17 implies that it has a subsequence converging in −1 𝑊(0,1) (ℂ𝑛 , 𝑒−𝜑 , ∇𝜑). Finally use (3) to show that this subsequence is a Cauchy sequence in 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑 ), therefore (2) holds. Assume (2): by Proposition 6.4 and the basic facts about 𝑁1,𝜑 , it follows that ∗

𝑁1,𝜑 : 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑 ) 󳨀→ dom (𝜕) ∩ dom (𝜕𝜑 ) is continuous in the graph topology, hence ∗

𝑁1,𝜑 : 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑 ) 󳨀→ dom (𝜕) ∩ dom (𝜕𝜑 ) 󳨅→ 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑 ) is compact. Remark 11.21. If lim 𝜇𝜑 (𝑧) = +∞,

|𝑧|→∞

then the condition of the Rellich–Kondrachov Lemma 11.17 is satisfied. This follows from the fact that we have for the trace tr(𝑀𝜑 ) of the Levi matrix tr(𝑀𝜑 ) =

1 󳵻𝜑, 4

200 | 11 Compactness and since for any invertible (𝑛 × 𝑛) matrix 𝑇 tr(𝑀𝜑 ) = tr(𝑇𝑀𝜑 𝑇−1 ), it follows that tr(𝑀𝜑 ) equals the sum of all eigenvalues of 𝑀𝜑 . Hence our assumption on the lowest eigenvalue 𝜇𝜑 of the Levi matrix implies that the assumption of Proposition 11.17 is satisfied. Remark 11.22. We mention that for the weight 𝜑(𝑧) = |𝑧|2 the 𝜕-Neumann operator fails to be compact (see Chapter 15), but the condition lim (𝜃|∇𝜑(𝑧)|2 + 󳵻𝜑(𝑧)) = +∞

|𝑧|→∞

of the Rellich–Kondrachov Lemma is satisfied. Remark 11.23. Let 𝐴2(0,1) (ℂ𝑛 , 𝑒−𝜑 ) denote the space of (0, 1)-forms with holomorphic coefficients belonging to 𝐿2 (ℂ𝑛 , 𝑒−𝜑 ). We point out that assuming (11.17) implies directly – without use of Sobolev spaces – that the embedding of the space ∗

𝐴2(0,1) (ℂ𝑛 , 𝑒−𝜑 ) ∩ dom (𝜕𝜑 ) ∗

provided with the graph norm 𝑢 󳨃→ (‖𝑢‖2𝜑 + ‖𝜕𝜑 𝑢‖2𝜑 )1/2 into 𝐴2(0,1) (ℂ𝑛 , 𝑒−𝜑 ) is compact. Compare 11.20 (2). ∗

For this purpose let 𝑢 ∈ 𝐴2(0,1) (ℂ𝑛 , 𝑒−𝜑 ) ∩ dom (𝜕𝜑 ). Then we obtain from the proof of 6.4 that 𝑛 ∗ 𝜕2 𝜑 ‖𝜕𝜑 𝑢‖2𝜑 = ∫ ∑ 𝑢𝑗 𝑢𝑘 𝑒−𝜑 𝑑𝜆. 𝜕𝑧 𝜕𝑧 𝑗 𝑘 𝑛 𝑗,𝑘=1 ℂ

∑𝑛𝑗=1

Let us for 𝑢 = 𝑢𝑗 𝑑𝑧𝑗 identify 𝑢(𝑧) with the vector (𝑢1 (𝑧), . . . , 𝑢𝑛 (𝑧)) ∈ ℂ𝑛 . Then, if we denote by ⟨., .⟩ the standard inner product in ℂ𝑛 , we have 𝜕2 𝜑(𝑧) 𝑢 (𝑧)𝑢𝑘 (𝑧). 𝜕𝑧𝑗 𝜕𝑧𝑘 𝑗 𝑗,𝑘=1

𝑛

𝑛

⟨𝑢(𝑧), 𝑢(𝑧)⟩ = ∑ |𝑢𝑗 (𝑧)|2 and ⟨𝑀𝜑 𝑢(𝑧), 𝑢(𝑧)⟩ = ∑ 𝑗=1

Note that the lowest eigenvalue 𝜇𝜑 of the Levi matrix 𝑀𝜑 can be expressed as 𝜇𝜑 (𝑧) = inf

⟨𝑀𝜑 𝑢(𝑧), 𝑢(𝑧)⟩

𝑢(𝑧)=0̸

⟨𝑢(𝑧), 𝑢(𝑧)⟩

.

So we get 𝜇𝜑 (𝑧)]−1 ∫ ⟨𝑢, 𝑢⟩𝑒−𝜑 𝑑𝜆 ≤ ∫ ⟨𝑢, 𝑢⟩𝑒−𝜑 𝑑𝜆 + [ inf 𝑛 ℂ𝑛

𝔹𝑅

ℂ \𝔹𝑅

∫ 𝜇𝜑 (𝑧) ⟨𝑢, 𝑢⟩𝑒−𝜑 𝑑𝜆 ℂ𝑛 \𝔹𝑅

𝜇𝜑 (𝑧)]−1 ∫ ⟨𝑀𝜑 𝑢, 𝑢⟩𝑒−𝜑 𝑑𝜆. ≤ ∫ ⟨𝑢, 𝑢⟩𝑒−𝜑 𝑑𝜆 + [ inf 𝑛 𝔹𝑅

ℂ \𝔹𝑅

ℂ𝑛

11.3 Compactness in weighted spaces

|

201

For a given 𝜖 > 0 choose 𝑅 so large that 𝜇𝜑 (𝑧)]−1 < 𝜖. [ inf 𝑛 ℂ \𝔹𝑅

For the sake of simplicity we can also use polycylinders 𝔻𝑟 of the form 𝔻𝑟 := {(𝑧1 , . . . , 𝑧𝑛 ) : |𝑧𝑗 | < 𝑟 , 𝑗 = 1, . . . , 𝑛} instead of the balls 𝔹𝑅 . Now we use the fact that for Bergman spaces of holomorphic functions the embedding of 𝐴2 (𝔻𝑠 ) into 𝐴2 (𝔻𝑟 ) is compact for 𝑟 < 𝑠 : in order to show this we consider the orthonormal basis 𝜙𝛼𝑠 (𝑧) = (

𝛼

𝛼 (𝛼1 + 1) . . . (𝛼𝑛 + 1) 1/2 𝑧1 1 . . . 𝑧𝑛 𝑛 ) 𝜋𝑛 𝑠|𝛼|+𝑛

in 𝐴2 (𝔻𝑠 ), and observe that the squares of the norms of 𝜙𝛼𝑠 in 𝐴2 (𝔻𝑟 ) are 𝑟 2|𝛼|+2𝑛 ( ) . 𝑠 As

𝑟 2|𝛼|+2𝑛 ∑( ) < ∞, 𝑠 𝛼

we get from Proposition 2.10 that the embedding of 𝐴2 (𝔻𝑠 ) into 𝐴2 (𝔻𝑟 ) is even a Hilbert–Schmidt operator and we can conclude the proof of our assertion as in Proposition 11.17. Inspired by a result on Schrödinger operators with magnetic field of Iwatsuka [44] we prove another characterization of compactness, which will be used later. Proposition 11.24. Let 𝜑 : ℂ𝑛 󳨀→ ℝ+ be a plurisubharmonic C2 -weight function. The 𝜕-Neumann operator 𝑁𝜑 : 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑 ) 󳨀→ 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑 ) is compact if and only if there is a smooth function Λ : ℂ𝑛 󳨀→ ℝ such that Λ(𝑧) → ∞ as |𝑧| → ∞ and ∗

‖𝜕𝑢‖2𝜑 + ‖𝜕𝜑 𝑢‖2𝜑 ≥ ∫ Λ |𝑢|2 𝑒−𝜑 𝑑𝜆

(11.27)

ℂ𝑛

for each 𝑢 ∈ W𝑄𝜑 . Proof. Suppose (11.27) holds. Then for each 𝜖 > 0 there exists a number 𝑅 > 0 such that Λ ≥ 1/𝜖 on ℂ𝑛 \ 𝔹𝑅 . This implies ∗

‖𝜕𝑢‖2𝜑 + ‖𝜕𝜑 𝑢‖2𝜑 ≥ ∫ Λ |𝑢|2 𝑒−𝜑 𝑑𝜆 ≥ ℂ𝑛

which means that (11.23) holds.

1 𝜖

∫ |𝑢|2 𝑒−𝜑 𝑑𝜆, ℂ𝑛 \𝔹𝑅

202 | 11 Compactness We indicate that the condition of Theorem 11.12 can be written in the following form: for each 𝜖 > 0 there exists 𝑅(𝜖) > 0 such that ‖𝑢‖𝐿2

(0,1)

(ℂ𝑛 \𝔹𝑅(𝜖) ,𝜑)

≤ 𝜖‖𝑢‖𝑄𝜑 .

Hence for all 𝑢 ∈ W𝑄𝜑 and for 𝑗 ∈ ℕ we have 2𝑗

∫

|𝑢|2 𝑒−𝜑 𝑑𝜆 ≤

ℂ𝑛 \𝔹𝑅(1/2𝑗 )

1 ‖𝑢‖2𝑄𝜑 2𝑗

and hence ∫ |𝑢|2 𝑒−𝜑 𝑑𝜆 ≤ ∫ 1 ⋅ |𝑢|2 𝑒−𝜑 𝑑𝜆 + ℂ𝑛

𝔹𝑅(1/2)

2 ⋅ |𝑢|2 𝑒−𝜑 𝑑𝜆

𝔹𝑅(1/4) \𝔹𝑅(1/2)

∫

+

∫

4 ⋅ |𝑢|2 𝑒−𝜑 𝑑𝜆 + ⋅ ⋅ ⋅

𝔹𝑅(1/8) \𝔹𝑅(1/4)

≤ (𝐶 + 1)‖𝑢‖2𝑄𝜑 . Now it is easy to define a smooth function Λ tending to ∞ as |𝑧| tends to ∞ such that (11.27) holds.

11.4 Bounded pseudoconvex domains Finally we investigate compactness of the 𝜕-Neumann operator of a bounded pseudoconvex domain. For this purpose we consider the following potential theoretic concepts. Definition 11.25. Let Ω ⊂⊂ ℂ𝑛 be a smoothly bounded pseudoconvex domain. Ω satisfies property (P), if for each 𝑀 > 0 there exists a neighborhood 𝑈 of 𝜕Ω and a plurisubharmonic function 𝜑𝑀 ∈ C2 (𝑈) with 0 ≤ 𝜑𝑀 ≤ 1 on 𝑈 such that 𝑛

𝜕2 𝜑𝑀 (𝑝)𝑡𝑗 𝑡𝑘 ≥ 𝑀‖𝑡‖2 , 𝜕𝑧 𝜕𝑧 𝑗 𝑘 𝑗,𝑘=1 ∑

for all 𝑝 ∈ 𝑈 and for all 𝑡 ∈ ℂ𝑛 . ˜ ) if the following holds: there is a constant 𝐶 Definition 11.26. Ω satisfies property (P 2 such that for all 𝑀 > 0 there exists a C -function 𝜑𝑀 in a neighborhood 𝑈 (depending on 𝑀) of 𝑏Ω with 󵄨󵄨2 󵄨󵄨 𝑛 𝑛 𝜕𝜑 𝜕2 𝜑𝑀 󵄨 󵄨󵄨 󵄨󵄨 ∑ 𝑀 (𝑧)𝑡𝑗 󵄨󵄨󵄨 ≤ 𝐶 ∑ (𝑧)𝑡𝑗 𝑡𝑘 (i) 󵄨󵄨 󵄨󵄨 𝜕𝑧𝑗 𝜕𝑧𝑘 󵄨 󵄨 𝑗=1 𝜕𝑧𝑗 𝑗,𝑘=1 and 𝑛 𝜕2 𝜑𝑀 ∑ (ii) (𝑧)𝑡𝑗 𝑡𝑘 ≥ 𝑀‖𝑡‖2 , 𝑗=1 𝜕𝑧𝑗 𝜕𝑧𝑘 for all 𝑧 ∈ 𝑈 and for all 𝑡 ∈ ℂ𝑛 .

11.4 Bounded pseudoconvex domains

|

203

Remark 11.27. In [9] Catlin showed that property (P) implies compactness of the 𝜕˜ ) also implies comoperator 𝑁 on 𝐿2(0,1) (Ω) and McNeal ([59]) showed that property (P 2 pactness of the 𝜕-operator 𝑁 on 𝐿 (0,1) (Ω). It is not difficult to show that property (P) ˜ ): If (𝜑𝑀 ) is the family of functions from the definition of property implies property (P 𝜑𝑀 ˜ ), see also [71]. (P), then (𝑒 ) will work for (P We can now use a similar approach as for weighted spaces to prove Catlin’s result. We use again Theorem 11.1. ∗ In order to show that the unit ball in dom(𝜕) ∩ dom(𝜕 ) in the graph norm 𝑓 󳨃→ 1 ∗ (‖𝜕𝑓‖2 + ‖𝜕 𝑓‖2 ) 2 satisfies condition (i) of Theorem 11.1 we first remark that com∗ ∗ pactly supported forms are not dense in dom(𝜕) ∩ dom(𝜕 ), but forms in dom(𝜕 ) with coefficients in C∞ (Ω) are dense (Proposition 5.14). So if 𝜔 ⊂⊂ Ω, we choose 𝜔 ⊂⊂ 𝜔1 ⊂⊂ 𝜔2 ⊂⊂ Ω and a cut-off function 𝜓 with 𝜓(𝑧) = 1 for 𝑧 ∈ 𝜔1 and 𝜓(𝑧) = 0 for ∗ 𝑧 ∈ Ω \ 𝜔2 . For 𝑢 ∈ dom(𝜕) ∩ dom(𝜕 ) we define 𝑢̃ = 𝜓𝑢 and remark that the domain ∗ of 𝜕 is preserved under multiplication by a function in C1 (Ω) (Remark 5.15), therefore ∗ 𝑢̃ has compactly supported coefficients and belongs to dom(𝜕) ∩ dom(𝜕 ). The graph norm of 𝑢̃ is bounded by a constant 𝐶 depending only on 𝜔, 𝜔1 , 𝜔2 , Ω, if 𝑢 belongs to the unit ball in the graph norm. By construction we have ‖𝜏ℎ 𝑢 − 𝑢‖𝐿2 (𝜔) = ‖𝜏ℎ 𝑢̃ − 𝑢‖̃ 𝐿2 (𝜔) , if |ℎ| is small enough, hence we can use Gårding’s inequality for 𝜔 ⊂⊂ Ω to show that condition (i) holds. To verify condition (ii) we use property (P) and the following version of the Kohn– Morrey formula 𝑛

∗ 𝜕2 𝜑𝑀 𝑢𝑗 𝑢𝑘 𝑒−𝜑𝑀 𝑑𝜆 ≤ ‖𝜕𝑢‖2𝜑𝑀 + ‖𝜕𝜑𝑀 𝑢‖2𝜑𝑀 , 𝜕𝑧 𝜕𝑧 𝑗 𝑘 𝑗,𝑘=1

∫ ∑ Ω

(11.28)

here we used that Ω is pseudoconvex, which means that the boundary terms in the Kohn–Morrey formula can be neglected, this follows easily from Theorem 7.2. Now we point out that the weighted 𝜕-complex is equivalent to the unweighted one and that ∗ 𝜕𝜑 the expression ∑𝑛𝑗=1 𝜕𝑧𝑀 𝑢𝑗 which appears in 𝜕𝜑𝑀 𝑢, can be controlled by the complex Hessian

𝜕2 𝜑 ∑𝑛𝑗,𝑘=1 𝜕𝑧 𝜕𝑧𝑀 𝑗 𝑘

𝑗

𝑢𝑗 𝑢𝑘 , which follows from the fact that property (P) implies prop-

˜ ). Of course we also use that the weight 𝜑𝑀 is bounded on Ω ⊂⊂ ℂ𝑛 . In this erty (P way the same reasoning as in the weighted case shows that property (P) implies con∗ dition (11.3). Therefore condition (P) gives that the unit ball of dom(𝜕) ∩ dom(𝜕 ) in the 1 ∗ graph norm 𝑓 󳨃→ (‖𝜕𝑓‖2 + ‖𝜕 𝑓‖2 ) 2 is relatively compact in 𝐿2(0,1) (Ω) and hence that the 𝜕-Neumann operator is compact. Now let ∗ 𝑗 : dom(𝜕) ∩ dom(𝜕 ) 󳨅→ 𝐿2(0,1) (Ω) denote the embedding. By (4.41), we know that 𝑁 = 𝑗 ∘ 𝑗∗ .

204 | 11 Compactness ∗

Hence 𝑁 is compact if and only if 𝑗 is compact, where dom(𝜕) ∩ dom(𝜕 ) is endowed 1 ∗ with the graph norm 𝑓 󳨃→ (‖𝜕𝑓‖2 + ‖𝜕 𝑓‖2 ) 2 . Theorem 11.28. Let Ω ⊂⊂ ℂ𝑛 be a smoothly bounded pseudoconvex domain. The 𝜕Neumann operator 𝑁 is compact if and only if for each 𝜖 > 0 there exists 𝜔 ⊂⊂ Ω such that ∗ ∫ |𝑢(𝑧)|2 𝑑𝜆(𝑧) ≤ 𝜖(‖𝜕𝑢‖2 + ‖𝜕 𝑢‖2 ) Ω\𝜔 ∗

for each 𝑢 ∈ dom (𝜕) ∩ dom (𝜕 ). This follows from the above remarks about the embedding 𝑗 and the fact that the two conditions (11.1) are also necessary for a bounded set in 𝐿2 to be relatively compact. Remark 11.29. (a) Let 1 ≤ 𝑞 ≤ 𝑛. We say that Ω satisfies property (𝑃𝑞 ) if the following holds: for every positive number 𝑀, there exists a neighborhood 𝑈 of 𝑏Ω and a C2 -function 𝜑𝑀 on 𝑈, such that 0 ≤ 𝜑𝑀 (𝑧) ≤ 1, 𝑧 ∈ 𝑈, and such that for any 𝑧 ∈ 𝑈, the sum of any 𝑞 𝜕2 𝜑 (equivalently: the smallest 𝑞) eigenvalues of the Hermitian form ( 𝜕𝑧 𝜕𝑧𝑀 (𝑧))𝑗,𝑘 is at 𝑗

𝑘

least 𝑀. Using the methods from above one can show that property (𝑃𝑞 ) implies compactness of the operator 𝑁𝑞 , compare with Proposition 11.11 and [71]. For this purpose we consider the following version of the Kohn–Morrey formula ∫ ∑

∗ 𝜕2 𝜑𝑀 𝑢𝑗𝐾 𝑢𝑘𝐾 𝑒−𝜑𝑀 𝑑𝜆 ≤ ‖𝜕𝑢‖2𝜑𝑀 + ‖𝜕𝜑𝑀 𝑢‖2𝜑𝑀 , 𝜕𝑧 𝜕𝑧 𝑗 𝑘 𝑗,𝑘=1 𝑛

󸀠

∑

Ω |𝐾|=𝑞−1

(11.29)

∗

where 𝑢 ∈ dom (𝜕) ∩ dom (𝜕 ) is a (0, 𝑞)-form and follow the lines of Proposition 11.11 (see also [71, Theorem 4.8]). (b) In view of Remark 11.6 (a) it is interesting to compare the Sobolev embedding 𝑊1 (Ω) ⊂ 𝐿𝑞 (Ω) , 𝑞 ∈ [1, 2𝑛/(𝑛 − 2)) where the derivatives are taken with respect to the real variables 𝑥𝑗 = ℜ𝑧𝑗 and 𝑦𝑗 = ℑ𝑧𝑗 for 𝑗 = 1, . . . , 𝑛, with the embedding of the space ∗

2

W(Ω) := {𝑢 ∈ 𝐿 (0,1) (Ω) : 𝑢 ∈ dom (𝜕) ∩ dom (𝜕 )}

endowed with graph norm, into 𝐿2(0,1) (Ω). We have the following result: If Ω ⊂⊂ ℂ𝑛 is a smoothly bounded pseudoconvex domain and the inequality ∗

‖𝑢‖𝑞 ≤ 𝐶(‖𝜕𝑢‖2 + ‖𝜕 𝑢‖2 )1/2 ∗

(11.30)

for some 𝑞 > 2 and for all 𝑢 ∈ dom (𝜕) ∩ dom (𝜕 ) holds, then the 𝜕-Neumann operator 𝑁 : 𝐿2(0,1) (Ω) 󳨀→ 𝐿2(0,1) (Ω) is compact.

11.5 Notes

| 205

To show this we have to check that the unit ball in W(Ω) is precompact in 𝐿2(0,1) (Ω). By Theorem 11.28, we have to show that for each 𝜖 > 0 there exists 𝜔 ⊂⊂ Ω such that ∫ |𝑢(𝑧)|2 𝑑𝜆(𝑧) < 𝜖2 , Ω\𝜔

for all 𝑢 in the unit ball of W(Ω). By (11.30) and Hölder’s inequality we have 1 2

2

1 𝑞

𝑞

1

( ∫ |𝑢(𝑧)| 𝑑𝜆(𝑧)) ≤ ( ∫ |𝑢(𝑧)| 𝑑𝜆(𝑧)) ⋅ |Ω \ 𝜔| 2 Ω\𝜔

− 𝑞1

Ω\𝜔 1

≤ 𝐶 |Ω \ 𝜔| 2

− 𝑞1

.

Now we can choose 𝜔 ⊂⊂ Ω such that the last term is < 𝜖. In a similar way as for Proposition 11.20 one obtains compactness estimates for the 𝜕Neumann operator on a smoothly bounded domain. Here we use the standard Sobolev spaces 𝑊1 (Ω) and the classical Rellich–Kondrachov Lemma without weights. Proposition 11.30. Let Ω be a smoothly bounded pseudoconvex domain. Then the following statements are equivalent. (1) The 𝜕-Neumann operator 𝑁1 is a compact operator from 𝐿2(0,1) (Ω) into itself. ∗

(2) The embedding of the space dom (𝜕) ∩ dom (𝜕 ), provided with the graph norm 𝑢 󳨃→ ∗ (‖𝑢‖2 + ‖𝜕𝑢‖2 + ‖𝜕 𝑢‖2 )1/2 , into 𝐿2(0,1) (Ω) is compact. (3) For every positive 𝜖 there exists a constant 𝐶𝜖 such that ∗

‖𝑢‖2 ≤ 𝜖(‖𝜕𝑢‖2 + ‖𝜕 𝑢‖2 ) + 𝐶𝜖 ‖𝑢‖2−1 , ∗

for all 𝑢 ∈ dom (𝜕) ∩ dom (𝜕 ). (4) For every positive 𝜖 there exists 𝜔 ⊂⊂ Ω such that ∗

∫ |𝑢(𝑧)|2 𝑑𝜆(𝑧) ≤ 𝜖(‖𝜕𝑢‖2 + ‖𝜕 𝑢‖2 ) Ω\𝜔 ∗

for all 𝑢 ∈ dom (𝜕) ∩ dom (𝜕 ). (5) The operators ∗ 𝜕 𝑁1 : 𝐿2(0,1) (Ω) ∩ ker(𝜕) 󳨀→ 𝐿2 (Ω) and ∗

𝜕 𝑁2 : 𝐿2(0,2) (Ω) ∩ ker(𝜕) 󳨀→ 𝐿2(0,1) (Ω) are both compact.

11.5 Notes In this chapter we tried to point out some parallel aspects in the classical theory of Sobolev spaces and in the properties of the 𝜕-Neumann operator. The treatment of

206 | 11 Compactness compact subsets in 𝐿2 -spaces stems from [1], basic properties of Sobolev spaces and the Rellich–Kondrachov Lemma are taken from [8] and [20]. We refer the reader to these books for more general results on higher derivatives and 𝐿𝑝 -spaces. The sufficient condition for compactness of the 𝜕-Neumann operator in weighted 𝐿2 -spaces was originally discovered from the theory of Schrödinger operators with magnetic field and from the Witten–Laplacian (see [34]), here we presented the complex analytic approach from [33] and [28]. A thorough treatment of compactness of the 𝜕-Neumann operator for bounded pseudoconvex domains can be found in the book of E. Straube [71], with all its important implications concerning regularity of the 𝜕-Neumann operator in Sobolev spaces and far-reaching connections to potential theory. A characterization of compactness of the 𝜕-Neumann operator in terms of geometric properties of pseudoconvex domains or in terms of the weight functions is still open. In [25] Fu and Straube solved the problem for bounded convex domains. It turned out that in this case compactness of the 𝜕-Neumann operator on the level of holomorphic (0, 𝑞)forms already implies compactness on the whole of 𝐿2(0,𝑞) (Ω). In counterexamples to compactness, the obstruction often already occurs on the level of holomorphic (0, 𝑞)forms. We will draw attention to this phenomenon in the next chapters.

12 The 𝜕-Neumann operator and the Bergman projection In this chapter we investigate the connection between the 𝜕-Neumann operator and commutators of the Bergman projection with multiplication operators. In [10] it is shown that compactness of the 𝜕-Neumann operator 𝑁 on 𝐿2(0,1) (Ω) implies compactness of the commutator [𝑃, 𝑀], where 𝑃 is the Bergman projection and 𝑀 is a pseudodifferential operator of order 0. Here we show that compactness of the 𝜕-Neumann operator 𝑁 restricted to (0, 1)-forms with holomorphic coefficients is equivalent to compactness of the commutator [𝑃, 𝑀] defined on the whole 𝐿2 (Ω). In addition we derive a formula for the 𝜕-Neumann operator restricted to (0, 1)-forms with holomorphic coefficients expressed by commutators of the Bergman projection and the multiplication operators by 𝑧 and 𝑧.̄ To show the equivalences to compactness mentioned above we will need a complex version of the Stone–Weierstraß Theorem.

12.1 The Stone–Weierstraß Theorem In the sequel we will need a generalization of the well-known theorem of Weierstraß that any real-valued continuous function on a compact interval [𝑎, 𝑏] is the uniform limit of polynomials on [𝑎, 𝑏]. We will consider a compact subset 𝐾 in ℂ𝑛 and will show that any complex valued continuous function on 𝐾 can be uniformly approximated on 𝐾 by polynomials in the variables 𝑧 = (𝑧1 , . . . , 𝑧𝑛 ) and 𝑧 = (𝑧1 , . . . , 𝑧𝑛 ). We prove a more general version and will need some appropriate concepts and several lemmas. A subset A of the space C(𝐾) of continuous real (complex)-valued functions (endowed with the sup-norm) is said to separate points if for every 𝑧, 𝑤 ∈ 𝐾 with 𝑧 ≠ 𝑤 there exists 𝑓 ∈ A such that 𝑓(𝑧) ≠ 𝑓(𝑤). We call A an algebra if it is a real (complex) vector subspace of C(𝐾) such that 𝑓𝑔 ∈ A (pointwise multiplication) whenever 𝑓 ∈ A and 𝑔 ∈ A. If A is a subset of the space C(𝐾) of real-valued continuous functions, A is called a lattice if max(𝑓, 𝑔) and min(𝑓, 𝑔) are in A for all 𝑓, 𝑔 ∈ A. Since the algebra and lattice operations are continuous in sup-norm of C(𝐾), one easily sees that if A is an algebra or a lattice, so is its closure A in the sup-norm. Lemma 12.1. For each 𝜖 > 0 there is a polynomial 𝑝 on ℝ such that 𝑝(0) = 0 and | |𝑥| − 𝑝(𝑥)| < 𝜖 for 𝑥 ∈ [−1, 1]. Proof. For |𝑡| < 1 the function (1 − 𝑡)1/2 has the Taylor series expansion ∞

(−1) 1 2𝑘 − 3 𝑡𝑘 ... 2 2 2 𝑘! 𝑘=1

(1 − 𝑡)1/2 = 1 + ∑ ∞

= 1 − ∑ 𝑐𝑘 𝑡𝑘 , 𝑘=1

208 | 12 The 𝜕-Neumann operator and the Bergman projection where 𝑐𝑘 > 0. It is also absolutely convergent for 𝑡 = −1. The monotone convergence theorem applied to the counting measure on ℕ implies that ∞

∞

∑ 𝑐𝑘 = lim ∑ 𝑐𝑘 𝑡𝑘 = 1 − lim(1 − 𝑡)1/2 = 1,

𝑘=1

𝑡→1

𝑘=1

𝑡→1

𝑘 from which we get that the series 1 − ∑∞ 𝑘=1 𝑐𝑘 𝑡 converges absolutely and uniformly on 1/2 [−1, 1] to (1 − 𝑡) . Given 𝜖 > 0 we can find a suitable partial sum 𝑞(𝑡) such that

|(1 − 𝑡)1/2 − 𝑞(𝑡)| < 𝜖/2 , 𝑡 ∈ [−1, 1]. We set 𝑡 = 1 − 𝑥2 and 𝑟(𝑥) = 𝑞(1 − 𝑥2 ) and obtain a polynomial 𝑟 such that | |𝑥| − 𝑟(𝑥)| < 𝜖/2 , 𝑥 ∈ [−1, 1]. In particular, |𝑟(0)| < 𝜖/2, so if we set 𝑝(𝑥) = 𝑟(𝑥) − 𝑟(0), we get the desired polynomial. First we consider the space of real-valued continuous functions. Lemma 12.2. Let A be a closed subalgebra of the space C(𝐾) of real-valued continuous functions. Then |𝑓| ∈ A, whenever 𝑓 ∈ A, and A is a lattice. Proof. We write ‖𝑓‖ = sup𝑧∈𝐾 |𝑓(𝑧)|, for 𝑓 ∈ C(𝐾). If 𝑓 ∈ A and 𝑓 ≠ 0, then ℎ := 𝑓/‖𝑓‖ maps 𝐾 into [−1, 1]. If 𝑝 is as in Lemma 12.1 we have ‖ |ℎ| − 𝑝 ∘ ℎ‖ < 𝜖. Since 𝑝(0) = 0, 𝑝 has no constant term, therefore 𝑝 ∘ ℎ ∈ A, as A is an algebra. Since A is closed, it follows that |ℎ| ∈ A and |𝑓| = ‖𝑓‖ |ℎ| ∈ A. This proves the first assertion. The second follows from max(𝑓, 𝑔) =

1 1 (𝑓 + 𝑔 + |𝑓 − 𝑔|) , min(𝑓, 𝑔) = (𝑓 + 𝑔 − |𝑓 − 𝑔|). 2 2

Lemma 12.3. Let A be a closed lattice in the space C(𝐾) of real-valued continuous functions. Let 𝑓 ∈ C(𝐾) and suppose that for every 𝑧, 𝑤 ∈ 𝐾 there exists 𝑔𝑧𝑤 ∈ A such that 𝑓(𝑧) = 𝑔𝑧𝑤 (𝑧) and 𝑓(𝑤) = 𝑔𝑧𝑤 (𝑤). Then 𝑓 ∈ A. Proof. For a given 𝜖 > 0 and 𝑧, 𝑤 ∈ 𝐾 let 𝑈𝑧𝑤 = {𝜁 ∈ 𝐾 : 𝑓(𝜁) < 𝑔𝑧𝑤 (𝜁) + 𝜖} , 𝑉𝑧𝑤 = {𝜁 ∈ 𝐾 : 𝑓(𝜁) > 𝑔𝑧𝑤 (𝜁) − 𝜖}. These sets are open and contain 𝑧 and 𝑤. Fix 𝑤 ∈ 𝐾 : then {𝑈𝑧𝑤 }𝑧∈𝐾 covers 𝐾, so there is a finite subcover {𝑈𝑧𝑗 𝑤 }𝑛𝑗=1 . Let 𝑔𝑤 = max(𝑔𝑧1 𝑤 , . . . , 𝑔𝑧𝑛 𝑤 ). Then 𝑓 < 𝑔𝑤 + 𝜖 on 𝐾 and 𝑓 > 𝑔𝑤 − 𝜖 on 𝑉𝑤 = ⋂𝑛𝑗=1 𝑉𝑧𝑗 𝑤 , which is open and contains 𝑤. So {𝑉𝑤 }𝑤∈𝐾 is an open covering of 𝐾, which has a finite subcover {𝑉𝑤𝑘 }𝑚 𝑘=1 . Let 𝑔 = min(𝑔𝑤1 , . . . , 𝑔𝑤𝑚 ). Then ‖𝑓 − 𝑔‖ < 𝜖, and, since A is a closed lattice, 𝑔 ∈ A and 𝑓 ∈ A.

12.1 The Stone–Weierstraß Theorem

| 209

Before we prove the first version of the Stone–Weierstraß Theorem we remark that ℝ2 as an algebra under coordinatewise addition and multiplication has only the following subalgebras: ℝ2 and {(0, 0)} and the linear spans of (1, 0), (0, 1) and (1, 1), because a nonzero subalgebra A ⊂ ℝ2 has the property that with (0, 0) ≠ (𝑎, 𝑏) ∈ A the pair (𝑎2 , 𝑏2 ) ∈ A, so if 𝑎 ≠ 0, 𝑏 ≠ 0, and 𝑎 ≠ 𝑏, then (𝑎, 𝑏) and (𝑎2 , 𝑏2 ) are linearly independent, so A = ℝ2 . The other cases – 𝑎 ≠ 0 = 𝑏, 𝑎 = 0 ≠ 𝑏, and 𝑎 = 𝑏 ≠ 0 – give the other three subalgebras. With this in mind, we can now prove the first version of the Stone–Weierstraß Theorem. Theorem 12.4. If A is a closed subalgebra of the space C(𝐾) of real-valued continuous functions, which separates points, then either A = C(𝐾) or A = {𝑓 ∈ C(𝐾) : 𝑓(𝑧0 ) = 0} for some 𝑧0 ∈ 𝐾. The first alternative holds if and only if A contains the constant functions. Proof. Given 𝑧, 𝑤 ∈ 𝐾 with 𝑧 ≠ 𝑤, let A𝑧𝑤 = {(𝑓(𝑧), 𝑓(𝑤)) : 𝑓 ∈ A}. Then, since 𝑓 󳨃→ (𝑓(𝑧), 𝑓(𝑤)) is an algebra homomorphism, A𝑧𝑤 is a subalgebra of ℝ2 in the sense of the remark from above. If A𝑧𝑤 = ℝ2 for all 𝑧, 𝑤 ∈ 𝐾, then Lemma 12.1 and Lemma 12.2 imply that A = C(𝐾). Otherwise there exist 𝑧, 𝑤 ∈ 𝐾, for which A𝑧𝑤 is a proper subalgebra of ℝ2 . It cannot be {(0, 0)} or the linear span of (1, 1) because A separates points. So A𝑧𝑤 is the linear span of (1, 0) or (0, 1). In either case there exists 𝑧0 ∈ 𝐾 such that 𝑓(𝑧0 ) = 0 for all 𝑓 ∈ A. There is only one such 𝑧0 since A separates points, so if neither 𝑧 nor 𝑤 is 𝑧0 , we have A𝑧𝑤 = ℝ2 . Then Lemma 12.1 and Lemma 12.2 imply that A = {𝑓 ∈ C(𝐾) : 𝑓(𝑧0 ) = 0}. If A contains constant functions this cannot be the case, so A then equals C(𝐾). The set of polynomials is an algebra which separates points and contains constants. Hence we get Corollary 12.5. The restrictions of the real polynomial functions to 𝐾 are dense in the space C(𝐾) of real-valued continuous functions. Theorem 12.4 is false for the space C(𝐾) of complex-valued continuous functions. Nevertheless, there is a complex version of the Stone–Weierstraß Theorem Theorem 12.6. If A is a closed complex subalgebra of the space C(𝐾) of complex-valued continuous functions which separates points and is closed under complex conjugation, then either A = C(𝐾) or A = {𝑓 ∈ C(𝐾) : 𝑓(𝑧0 ) = 0} for some 𝑧0 ∈ 𝐾. Proof. Since ℜ𝑓 = (𝑓 + 𝑓)/2 and ℑ𝑓 = (𝑓 − 𝑓)/2𝑖, the set Aℝ of real and imaginary parts of functions in A is a subalgebra of the space of real-valued continuous functions to which Theorem 12.4 applies. Since A = {𝑓 + 𝑖𝑔 : 𝑓, 𝑔 ∈ Aℝ }, the desired result follows.

210 | 12 The 𝜕-Neumann operator and the Bergman projection Corollary 12.7. The set of all polynomials 𝑝 of the form 𝑝(𝑧, 𝑧) = ∑ 𝜆 𝛼 𝑧𝛼1 𝑧𝛼2 , |𝛼|≤𝑁

2𝑛

where 𝛼 = (𝛼1 , 𝛼2 ) is a multi-index in ℕ , is dense in the space C(𝐾) of complex-valued continuous functions. Proof. It is clear that the set of polynomials 𝑝 of the above form separates points and is invariant under complex conjugation. Therefore we can apply Theorem 12.6.

12.2 Commutators of the Bergman projection Let Ω be a bounded pseudoconvex domain in ℂ𝑛 and let 𝐴2(0,1) (Ω) denote the space of all (0, 1)-forms with holomorphic coefficients belonging to 𝐿2 (Ω). In Section 3.2 we showed that the canonical solution operator 𝑆 : 𝐴2(0,1) (Ω) 󳨀→ 𝐿2 (Ω) has the form 𝑆(𝑔)(𝑧) = ∫ 𝐾(𝑧, 𝑤) < 𝑔(𝑤), 𝑧 − 𝑤 > 𝑑𝜆(𝑤),

(12.1)

Ω

where 𝐾 denotes the Bergman kernel of Ω and 𝑔 = ∑𝑛𝑗=1 𝑔𝑗 𝑑𝑧𝑗 is a (0, 1)-form belonging to 𝐴2(0,1) (Ω) and 𝑛

< 𝑔(𝑤), 𝑧 − 𝑤 >= ∑ 𝑔𝑗 (𝑤)(𝑧𝑗 − 𝑤𝑗 ), 𝑗=1

for 𝑧 = (𝑧1 , . . . , 𝑧𝑛 ) and 𝑤 = (𝑤1 , . . . , 𝑤𝑛 ). The restriction of the canonical solution operator to forms with holomorphic coefficients has many interesting aspects, which in most cases correspond to certain growth properties of the Bergman kernel. It is also of great interest to clarify to what extent compactness of the restriction already implies compactness of the original solution operator to 𝜕. This is the case for convex domains, see [25]. There are many other examples of noncompactness where the obstruction already occurs for forms with holomorphic coefficients (see Chapter 14 and [52, 54]). We define the following operator 𝑇 : 𝐿2(0,1) (Ω) 󳨀→ 𝐿2 (Ω), by 𝑇𝑓(𝑧) = ∫ 𝐾(𝑧, 𝑤)⟨𝑓(𝑤), 𝑧 − 𝑤⟩ 𝑑𝜆(𝑤),

(12.2)

Ω

where 𝑓 = ∑𝑛𝑘=1 𝑓𝑘 𝑑𝑧𝑘 and ⟨𝑓(𝑤), 𝑧 − 𝑤⟩ = ∑𝑛𝑘=1 𝑓𝑘 (𝑤)(𝑧𝑘 − 𝑤𝑘 ). The operator 𝑇 can be written as a sum of commutators 𝑛

𝑛

𝑇𝑓 = ∑ [𝑀𝑘 , 𝑃]𝑓𝑘 , 𝑓 = ∑ 𝑓𝑘 𝑑𝑧𝑘 𝑘=1

(12.3)

𝑘=1

where 𝑀𝑘 𝑣(𝑧) = 𝑧𝑘 𝑣(𝑧), 𝑣 ∈ 𝐿2 (Ω), 𝑘 = 1, . . . , 𝑛, and 𝑃 is the Bergman projection, see (1.5).

12.2 Commutators of the Bergman projection

| 211

Let P : 𝐿2(0,1) (Ω) 󳨀→ 𝐴2(0,1) (Ω) be the orthogonal projection on the space of (0, 1)forms with holomorphic coefficients. We claim that 𝑇𝑓 = 𝑇P𝑓 , 𝑓 ∈ 𝐿2(0,1) (Ω). It suffices to show that 𝑇𝑔 = 0, for 𝑔⊥𝐴2(0,1) (Ω) : 𝑛

𝑛

𝑇𝑔(𝑧) = − ∑ 𝑃𝑀𝑘 𝑔𝑘 (𝑧) = − ∑ ∫ 𝐾(𝑧, 𝑤)𝑤𝑘 𝑔𝑘 (𝑤) 𝑑𝜆(𝑤) 𝑘=1

𝑘=1 Ω

𝑛

−

= − ∑ ∫ 𝑔𝑘 (𝑤) [𝐾(𝑤, 𝑧)𝑤𝑘 ] 𝑑𝜆(𝑤) = 0, 𝑘=1 Ω

because 𝑤 󳨃→ 𝐾(𝑤, 𝑧)𝑤𝑘 is holomorphic and 𝑔𝑘 ⊥𝐴2 (Ω), for 𝑘 = 1, . . . , 𝑛. Now, let 𝑆 denote the canonical solution operator to 𝜕 restricted to 𝐴2(0,1) (Ω). From (12.1) we have for 𝑓 ∈ 𝐿2(0,1) (Ω) 𝑆(P𝑓) = 𝑇(P𝑓) = 𝑇𝑓.

(12.4)

Hence we have proved the following Theorem 12.8. If 𝑓 ∈ 𝐿2(0,1) (Ω), then 𝑇(P𝑓) = 𝑇𝑓. The operator 𝑆 is compact as an operator from 𝐴2(0,1) (Ω) to 𝐿2 (Ω), if and only if the operator 𝑇 is compact as an operator from 𝐿2(0,1) (Ω) to 𝐿2 (Ω). Remark 12.9. The adjoint operator 𝑇∗ : 𝐿2 (Ω) 󳨀→ 𝐿2(0,1) (Ω) is given by 𝑛

𝑇∗ (𝑔) = ∑ [𝑃, 𝑀𝑘 ] 𝑔 𝑑𝑧𝑘 , 𝑔 ∈ 𝐿2 (Ω),

(12.5)

𝑘=1

where 𝑀𝑘 𝑣(𝑧) = 𝑧𝑘 𝑣(𝑧). Here we have 𝑇∗ (𝐼 − 𝑃)(𝑔) = 𝑇∗ (𝑔), since [𝑃, 𝑀𝑘 ] 𝑃𝑔 = 𝑃𝑀𝑘 𝑃𝑔 − 𝑀𝑘 𝑃𝑔 = 0. In a similar way the following results can be proved Lemma 12.10. (1) 𝑃𝑀𝑗 𝑃 = 𝑀𝑗 𝑃, (2) 𝑃𝑀𝑗 𝑃 = 𝑃𝑀𝑗 . Let 2 𝐵(0,1) (Ω) = {𝑓 ∈ 𝐿2(0,1) (Ω) : 𝑓 ∈ ker𝜕}.

Now suppose that Ω is a bounded pseudoconvex domain in ℂ𝑛 . The 𝜕-Neumann 2 2 operator 𝑁 can be viewed as an operator from 𝐵(0,1) (Ω) to 𝐵(0,1) (Ω). The operator ∗

2 𝜕 𝑁 : 𝐵(0,1) (Ω) 󳨀→ 𝐴2 (Ω)⊥

is the canonical solution operator to 𝜕 (see Proposition 4.63 ).

212 | 12 The 𝜕-Neumann operator and the Bergman projection 2 Theorem 12.11. If 𝑓 = ∑𝑛𝑘=1 𝑓𝑘 𝑑𝑧𝑘 ∈ 𝐵(0,1) (Ω), then 𝑛

𝑛

𝑘=1

𝑗=1

P𝑁P𝑓 = ∑ ( ∑ (𝑃𝑀𝑘 𝑀𝑗 𝑃𝑓𝑗 − 𝑀𝑘 𝑃𝑀𝑗 𝑓𝑗 ))𝑑𝑧𝑘 .

(12.6)

If 𝑓 = ∑𝑛𝑘=1 𝑓𝑘 𝑑𝑧𝑘 ∈ 𝐴2(0,1) (Ω), then 𝑛

𝑛

𝑘=1

𝑗=1

P𝑁𝑓 = ∑ [𝑃, 𝑀𝑘 ]( ∑ 𝑀𝑗 𝑓𝑗 )𝑑𝑧𝑘 .

(12.7)

2 (Ω) we have Proof. First we observe that for 𝑓 ∈ 𝐵(0,1) ∗

∗

𝑁𝜕𝜕 𝑁𝑓 = 𝑁(𝐼 − 𝜕 𝜕𝑁)𝑓 = 𝑁𝑓, where we used the fact that 2 2 𝑁 : 𝐵(0,1) (Ω) 󳨀→ 𝐵(0,1) (Ω).

If 𝑓 ∈ 𝐴2(0,1) (Ω), then by Theorem 12.8 it follows that ∗

𝜕 𝑁𝑓 = 𝑇𝑓. 2 Let 𝑓 ∈ 𝐴2(0,1) (Ω) and 𝑔 ∈ 𝐵(0,1) (Ω) with orthogonal decomposition 𝑔 = ℎ + ℎ,̃ where 2 ℎ ∈ 𝐴 (0,1) (Ω) and ℎ̃ = (𝐼 − P)𝑔, then ∗ ∗ ̃ 𝑇𝑓) = (𝜕∗ 𝑁ℎ, 𝑇𝑓) + (𝜕∗ 𝑁ℎ,̃ 𝑇𝑓) (𝑔, 𝑁𝜕𝜕 𝑁𝑓) = (𝜕 𝑁(ℎ + ℎ), ∗

∗

= (𝑇ℎ, 𝑇𝑓) + (𝜕 𝑁ℎ,̃ 𝑇𝑓) = (𝑇𝑔, 𝑇𝑓) + (𝜕 𝑁ℎ,̃ 𝑇𝑓) ∗ = (𝑔, 𝑇∗ 𝑇𝑓) + (𝜕 𝑁ℎ,̃ 𝑇𝑓).

Since

∗

(𝜕 𝑁ℎ,̃ 𝑇𝑓) = (𝑁ℎ,̃ 𝜕𝑇𝑓) = (𝑁ℎ,̃ 𝑓) = (ℎ,̃ 𝑁𝑓), we obtain ∗ (𝑔, 𝑁𝑓) = (𝑔, 𝑁𝜕𝜕 𝑁𝑓) = (𝑔, 𝑇∗ 𝑇𝑓) + (ℎ,̃ 𝑁𝑓)

= (𝑔, 𝑇∗ 𝑇𝑓) + ((𝐼 − P)𝑔, 𝑁𝑓) = (𝑔, 𝑇∗ 𝑇𝑓) + (𝑔, (𝐼 − P)𝑁𝑓). 2 Now, since 𝑔 ∈ 𝐵(0,1) (Ω) was arbitrary, we get

𝑁𝑓 = 𝑇∗ 𝑇𝑓 + 𝑁𝑓 − P𝑁𝑓, and therefore ∗

P𝑁𝑓 = 𝑇 𝑇𝑓. 2 If we take into account that for 𝑓 ∈ 𝐵(0,1) (Ω) we have 𝑇𝑓 = 𝑇P𝑓, we can now apply the last formula for P𝑓 and get ∗

P𝑁P𝑓 = 𝑇 𝑇𝑓.

12.2 Commutators of the Bergman projection

| 213

2 (Ω), then It remains to compute 𝑇∗ 𝑇. If 𝑓 ∈ 𝐵(0,1) 𝑛

𝑛

𝑘=1

𝑗=1

𝑇∗ 𝑇𝑓 = ∑ [𝑃, 𝑀𝑘 ]( ∑[𝑀𝑗 , 𝑃]𝑓𝑗 )𝑑𝑧𝑘 𝑛

𝑛

= ∑ ( ∑(𝑃𝑀𝑘 𝑀𝑗 𝑃 − 𝑀𝑘 𝑃𝑀𝑗 𝑃 − 𝑃𝑀𝑘 𝑃𝑀𝑗 + 𝑀𝑘 𝑃𝑀𝑗 )𝑓𝑗 )𝑑𝑧𝑘 𝑗=1

𝑘=1

𝑛

𝑛

= ∑ ( ∑ (𝑃𝑀𝑘 𝑀𝑗 𝑃𝑓𝑗 − 𝑀𝑘 𝑃𝑀𝑗 𝑓𝑗 ))𝑑𝑧𝑘 , 𝑘=1

𝑗=1

where we used Lemma 12.10. If 𝑓 ∈ 𝐴2(0,1) (Ω), then 𝑃𝑓𝑗 = 𝑓𝑗 and we obtain the second formula in Theorem 12.11. Using the last results we get the criterion for compactness of the commutators [𝑃, 𝑀𝑘 ] : Theorem 12.12. Let Ω be a bounded pseudoconvex domain in ℂ𝑛 . Then the following conditions are equivalent: (1) 𝑁 |𝐴2 (Ω) is compact; ∗

(0,1)

(2) 𝜕 𝑁 |𝐴2

(0,1)

(Ω)

is compact;

(3) [𝑃, 𝑀𝑘 ] is compact on 𝐿2 (Ω) for 𝑘 = 1, . . . , 𝑛; (4) (𝐼 − 𝑃)𝑀𝑘 𝑃 is compact on 𝐿2 (Ω) for 𝑘 = 1, . . . , 𝑛; (5) [𝑀𝜑 , 𝑃] is compact on 𝐿2 (Ω) for each continuous function 𝜑 on Ω. ∗

2 (Ω) 󳨀→ 𝐴2 (Ω)⊥ be the canonical solution operator to 𝜕 Proof. Let 𝑆1 = 𝜕 𝑁1 : 𝐵(0,1) ∗

2 2 and similarly 𝑆2 = 𝜕 𝑁2 : 𝐵(0,2) (Ω) 󳨀→ 𝐵(0,1) (Ω)⊥ , then

𝑁1 = 𝑆1∗ 𝑆1 + 𝑆2 𝑆2∗ (see Proposition 4.63). Since 𝑆2∗ |𝐴2

(0,1)

𝑁1 |𝐴2

(Ω) =

(0,1)

0, we have

(Ω) =

𝑆1∗ 𝑆1 |𝐴2

(0,1)

(Ω) ,

and (1) is equivalent to (2). ∗ Now suppose that (2) holds. Then, since the restriction of 𝜕 𝑁 to 𝐴2(0,1) (Ω) is of the form ∗

𝑛

𝜕 𝑁𝑓 = ∑ [𝑀𝑘 , 𝑃]𝑓𝑘 , 𝑘=1

∑𝑛𝑘=1

𝐴2(0,1) (Ω), 2

𝑓𝑘 𝑑𝑧𝑘 ∈ then by Theorem 12.8 it follows that the operators where 𝑓 = [𝑀𝑘 , 𝑃] are compact on 𝐿 (Ω). Since [𝑀𝑘 , 𝑃]∗ = [𝑃, 𝑀𝑘 ], we obtain property (3). It is also clear by Theorem 12.8 that (3) implies (2).

214 | 12 The 𝜕-Neumann operator and the Bergman projection Now suppose that (3) holds. It follows that [𝑀𝑘 , 𝑃]𝑃 is also compact, and since [𝑀𝑘 , 𝑃]𝑃 = 𝑀𝑘 𝑃 − 𝑃𝑀𝑘 𝑃 = (𝐼 − 𝑃)𝑀𝑘 𝑃, the Hankel operators (𝐼 − 𝑃)𝑀𝑘 𝑃 are compact. So we have shown that (3) implies (4). Suppose that (4) holds. The Hankel operators 𝐻𝑧𝑗 𝑧𝑘 with symbol 𝑧𝑗 𝑧𝑘 can be written in the form 𝐻𝑧𝑗 𝑧𝑘 = (𝐼 − 𝑃)𝑀𝑗 (𝑃 + (𝐼 − 𝑃))𝑀𝑘 𝑃 = (𝐼 − 𝑃)𝑀𝑗 (𝐼 − 𝑃)𝑀𝑘 𝑃, hence it follows that 𝐻𝑧𝑗 𝑧𝑘 is compact. Similarly one can show that for any polynomial 𝑝(𝑧, 𝑧) = ∑ 𝜆 𝛼 𝑧𝛼1 𝑧𝛼2 , |𝛼|≤𝑁

2𝑛

where 𝛼 = (𝛼1 , 𝛼2 ) is a multi-index in ℕ , the corresponding Hankel operator 𝐻𝑝 = (𝐼− 𝑃)𝑀𝑝 𝑃 is compact. Now let 𝜑 ∈ C(Ω). Then, by Corollary 12.7, there exists a polynomial 𝑝 of the above form such that ‖𝜑 − 𝑝‖∞ < 𝜖. Hence ‖𝐻𝜑 − 𝐻𝑝 ‖ = ‖(𝐼 − 𝑃)𝑀𝜑−𝑝 𝑃‖ ≤ ‖𝜑 − 𝑝‖∞ , where ‖ . ‖ denotes the operator norm and ‖ . ‖∞ the sup-norm on Ω. Since the compact operators form a closed two-sided ideal in the operator norm (Corollary 2.4) and since for 𝑔 = 𝑔1 + 𝑔2 where 𝑔1 ∈ 𝐴2 (Ω) and 𝑔2 ∈ 𝐴2 (Ω)⊥ we have [𝑀𝜑 , 𝑃]𝑔 = −𝐻𝜑∗ 𝑔2 + 𝐻𝜑 𝑔1 , it follows that [𝑀𝜑 , 𝑃] is compact. Remark 12.13. (a) If Ω is a bounded convex domain, then compactness of ∗

𝜕 𝑁 |𝐴2

(0,1)

(Ω)

∗

implies already compactness of 𝜕 𝑁 on all of 𝐿2(0,1) (Ω) (see [25]), hence, in this case property (1) of Theorem 12.12 can be replaced by 𝑁 being compact on 𝐿2(0,1) (Ω) and ∗

property (2) of Theorem 12.12 can be replaced by 𝜕 𝑁 being compact on 𝐿2(0,1) (Ω). (b) For 𝑛 ≥ 2 and 1 ≤ 𝑞 ≤ 𝑛 let 2 𝐵(0,𝑞) (Ω) = {𝑓 ∈ 𝐿2(0,𝑞) (Ω) : 𝑓 ∈ ker𝜕}.

From Proposition 4.64 we know that the orthogonal projection 2 (Ω) 𝑃𝑞 : 𝐿2(0,𝑞) (Ω) 󳨀→ 𝐵(0,𝑞)

can be written in the form

∗

𝑃𝑞 = 𝐼 − 𝜕 𝑁𝑞+1 𝜕. Recently Celik and Sahutoglu [11] showed that for 2 ≤ 𝑞 ≤ 𝑛 the following statements are equivalent:

12.3 Notes

| 215

(1) 𝑁𝑞 is compact on 𝐿2(0,𝑞) (Ω), ∗

2 (Ω), (2) 𝜕 𝑁𝑞 is compact on 𝐵(0,𝑞) 2 (Ω) for each 𝜑 ∈ C(Ω). (3) [𝑀𝜑 , 𝑃𝑞 ] is compact on 𝐵(0,𝑞) 2 This is mainly due to the fact that 𝐵(0,𝑞) (Ω) is a much larger space than the space of (0, 𝑞)-forms with holomorphic coefficients belonging to 𝐿2 (Ω). Whether the same is true for 𝑞 = 1 is still not known, compare with Theorem 12.12, where in the first assertion one has compactness only on 𝐴2(0,1) (Ω).

12.3 Notes The proof of the Stone–Weierstraß Theorem stems from [22]. N. Salinas [66] showed ∗ that compactness of the canonical solution operator 𝜕 𝑁𝑞+1 restricted to forms with holomorphic coefficients is a consequence of compactness of the commutators [𝑀𝑗 , 𝑃𝑞 ], 1 ≤ 𝑗 ≤ 𝑛. Compactness of 𝑁𝑞 for 1 ≤ 𝑞 ≤ 𝑛 can also be characterized in terms of 𝐶∗ algebras (see [66, 67] and [11]). Catlin and D’Angelo [10] used compactness of the commutators [𝑀𝜑 , 𝑃] in conjunction with a complex variables analogue of Hilbert’s 17th problem. They showed that compactness of 𝑁1 implies that the commutators [𝑀, 𝑃] are compact for all tangential pseudodifferential operators 𝑀 of order 0. Theorem 12.11 and Theorem 12.12 are from [32].

13 Compact resolvents In this chapter we return to the differential operators introduced in Chapter 10 and discuss the question whether these operators are with compact resolvents. For this purpose we use the general characterization of compactness of the 𝜕-Neumann operator.

13.1 Schrödinger operators Here we characterize compactness of the 𝜕-Neumann operator 𝑁𝜑 on 𝐿2 (ℂ, 𝑒−𝜑 ) which was originally done in [34] using methods from Schrödinger operators, and later in [58] using estimates of the Bergman kernel in 𝐴2 (ℂ, 𝑒−𝜑 ). Here we give a direct proof using methods of Chapter 11. Theorem 13.1. Let 𝜑 be a subharmonic C2 -function such that lim inf 󳵻𝜑(𝑧) > 0. |𝑧|→∞

The 𝜕-Neumann operator 𝑁𝜑 is compact on 𝐿2 (ℂ, 𝑒−𝜑 ) if and only if 󳵻𝜑(𝑧) → ∞ as |𝑧| → ∞. Proof. Suppose that 󳵻𝜑(𝑧) → ∞ as |𝑧| → ∞. In Chapter 10 we showed that ◻𝜑 = ∗

∗

𝑒𝜑/2 𝐷 𝐷 𝑒−𝜑/2 and that 𝐷 𝐷 = − 14 󳵻𝐴 + 18 󳵻𝜑. We also proved that −󳵻𝐴 ≥ implies that − 14 󳵻𝐴 + 18 󳵻𝜑 ≥ 14 󳵻𝜑 and hence for 𝑓 ∈ C∞ 0 (ℂ) we obtain

1 2

󳵻𝜑, which

∗

(◻𝜑 𝑓, 𝑓)𝜑 = (𝑒𝜑/2 𝐷 𝐷 𝑒−𝜑/2 𝑓, 𝑓)𝜑 ∗

= (𝑒−𝜑/2 𝐷 𝐷 𝑒−𝜑/2 𝑓, 𝑓) ∗

= (𝐷 𝐷 𝑒−𝜑/2 𝑓, 𝑒−𝜑/2 𝑓) and setting 𝑔 = 𝑒−𝜑/2 𝑓 we get 1 1 (󳵻𝜑 𝑔, 𝑔) = (󳵻𝜑 𝑓, 𝑓)𝜑 4 4 and we can apply Proposition 11.24 to see that 𝑁𝜑 is compact. If 𝑁𝜑 is compact, we know by Proposition 11.24 ∗

(◻𝜑 𝑓, 𝑓)𝜑 = (𝐷 𝐷 𝑔, 𝑔) ≥

(◻𝜑 𝑢, 𝑢)𝜑 ≥ ∫ Λ |𝑢|2 𝑒−𝜑 𝑑𝜆,

(13.1)

ℂ

where Λ is a smooth function such that Λ(𝑧) → ∞ as |𝑧| → ∞. Equation (6.9) gives 󵄨󵄨󵄨 𝜕𝑢 󵄨󵄨󵄨2 1 (◻𝜑 𝑢, 𝑢)𝜑 = ∫ 󵄨󵄨󵄨 󵄨󵄨󵄨 𝑒−𝜑 𝑑𝜆 + ∫ 󳵻𝜑 |𝑢|2 𝑒−𝜑 𝑑𝜆, 4 󵄨󵄨 𝜕𝑧 󵄨󵄨 ℂ

for 𝑢 ∈ dom(◻𝜑 ).

ℂ

(13.2)

13.1 Schrödinger operators

| 217

Now let Λ 0 be the supremum over all continuous functions Λ such that (13.1) holds. Then Λ 0 is a lower semicontinuous function. This follows from the fact that the supremum of any collection of lower semicontinuous functions is lower semicontinuous and (13.2) implies that Λ 0 ≥ 14 󳵻𝜑. The function Λ 0 − 14 󳵻𝜑 is lower semicontinuous. Hence the sets 1 𝐸 := {𝑧 : Λ 0 (𝑧) − 󳵻𝜑(𝑧) > 0} 4

1 and 𝐸𝑘 := {𝑧 : Λ 0 (𝑧) − 󳵻𝜑(𝑧) > 1/𝑘}, 4

for 𝑘 ∈ ℕ, are open. Furthermore 𝐸𝑘 ↗ 𝐸 and lim𝑘→∞ |𝐸𝑘 | = |𝐸|. Suppose that |𝐸| > 0. Then there exists 𝑘0 ∈ ℕ such that |𝐸𝑘0 | > 0. By (13.1) and (13.2) it follows that 󵄨󵄨 𝜕𝑢 󵄨󵄨2 1 󵄨 󵄨 ∫ 󵄨󵄨󵄨 󵄨󵄨󵄨 𝑒−𝜑 𝑑𝜆 ≥ ∫ (Λ 0 − 󳵻𝜑)|𝑢|2 𝑒−𝜑 𝑑𝜆 ≥ 1/𝑘0 ∫ |𝑢|2 𝑒−𝜑 𝑑𝜆. 󵄨󵄨 𝜕𝑧 󵄨󵄨 4

ℂ

(13.3)

𝐸𝑘0

ℂ

Let {𝜒𝑁 }𝑁∈ℕ be a sequence of cut-off functions which are identically one on 𝔹𝑁 = {𝑧 : |𝑧| < 𝑁}, which are supported in 𝔹𝑁+1 and which satisfy sup |∇𝜒𝑁 (𝑧)| ≤ 2. Without loss of generality we assume that 𝐸𝑘0 ⊂ 𝔹1 . Now let ℎ ∈ 𝐴2 (ℂ, 𝑒−𝜑 ) be a non-trivial entire function (by Theorem 8.11 we even know that 𝐴2 (ℂ, 𝑒−𝜑 ) is of infinite dimension). Then 𝜒𝑁 ℎ ∈ C∞ 0 (ℂ) and we can apply (13.3) to get ∫ 𝑠𝑢𝑝𝑝∇𝜒𝑁

󵄨󵄨 𝜕𝜒 󵄨󵄨2 󵄨 󵄨 |ℎ|2 󵄨󵄨󵄨 𝑁 󵄨󵄨󵄨 𝑒−𝜑 𝑑𝜆 ≥ 1/𝑘0 ∫ |𝜒𝑁 ℎ|2 𝑒−𝜑 𝑑𝜆 = 1/𝑘0 ∫ |ℎ|2 𝑒−𝜑 𝑑𝜆. 󵄨󵄨 𝜕𝑧 󵄨󵄨 𝐸𝑘0

(13.4)

𝐸𝑘0

Using the assumption on ∇𝜒𝑁 we obtain for arbitrary 𝑁 ∈ ℕ 4

∫

|ℎ|2 𝑒−𝜑 𝑑𝜆 ≥ 1/𝑘0 ∫ |ℎ|2 𝑒−𝜑 𝑑𝜆 > 𝛿 > 0,

(13.5)

𝐸𝑘0

𝔹𝑁+1 \𝔹𝑁

for some 𝛿 > 0, where we used that 𝐸𝑘0 is open and ℎ is a non-trivial entire function. Now we see that ∞

‖ℎ‖2𝜑 = ∫ |ℎ|2 𝑒−𝜑 𝑑𝜆 + ∑

𝑁=1

𝔹1

∫

|ℎ|2 𝑒−𝜑 𝑑𝜆

𝔹𝑁+1 \𝔹𝑁

2

is infinite by (13.5). This contradicts ℎ ∈ 𝐴 (ℂ, 𝑒−𝜑 ). Hence |𝐸| = 0 and Λ 0 = 1/4󳵻𝜑, almost everywhere, which implies that 󳵻𝜑(𝑧) → ∞ as |𝑧| → ∞. Corollary 13.2. Let 𝜑 be a subharmonic C2 -function such that lim inf 󳵻𝜑(𝑧) > 0. |𝑧|→∞

The canonical solution operator 𝑆𝜑 to 𝜕 is compact on 𝐿2 (ℂ, 𝑒−𝜑 ) if and only if 󳵻𝜑(𝑧) → ∞ as |𝑧| → ∞.

218 | 13 Compact resolvents Proof. Since 𝑁𝜑 = 𝑆𝜑∗ 𝑆𝜑 , the result follows from Theorem 13.1 and Theorem 2.5. Remark 13.3. Using the notations of Chapter 8 and the assumptions on 𝜑, we can express the last theorem in the following way: The Schrödinger operator with magnetic field 1 S = (−󳵻𝐴 + 𝐵), 4 where 𝜕 𝜕 1 𝑖 𝜕𝜑 2 𝑖 𝜕𝜑 2 + ) +( − ) and 𝐵 = 󳵻𝜑 Δ𝐴 = ( 𝜕𝑥 2 𝜕𝑦 𝜕𝑦 2 𝜕𝑥 2 has compact resolvent if and only if 󳵻𝜑 → ∞ as |𝑧| → ∞.

13.2 Dirac and Pauli operators We return to the Dirac and Pauli operators related with the weight function 𝜑 : D=(−𝑖 𝜕𝜑

1 𝜕𝜑 2 𝜕𝑥

where 𝐴 1 = − 12 𝜕𝑦 , 𝐴 2 =

𝜕 𝜕 − 𝐴 1 ) 𝜎1 + ( − 𝑖 − 𝐴 2 ) 𝜎2 , 𝜕𝑥 𝜕𝑦

and

𝜎1 = (

0 1

1 ), 0

𝜎2 = (

0 𝑖

−𝑖 ) . 0

The square of D is diagonal with the Pauli operators 𝑃± on the diagonal: 2

D =(

where 𝑃± = (−𝑖

𝑃− 0

0 ), 𝑃+

2 2 𝜕 𝜕 − 𝐴 1 ) + (−𝑖 − 𝐴 2 ) ± 𝐵 = −Δ 𝐴 ± 𝐵, 𝜕𝑥 𝜕𝑦

where 𝐵 = 12 󳵻𝜑. Theorem 13.4. Suppose that |𝑧|2 󳵻𝜑(𝑧) → +∞ as |𝑧| → ∞. Then the corresponding Dirac operator D has noncompact resolvent. Proof. By Proposition 9.38, D2 has compact resolvent, if and only if D has compact resolvent. Suppose that D has compact resolvent. Since 2

D =(

𝑃− 0

0 ), 𝑃+

this would imply that both 𝑃± have compact resolvent.

13.2 Dirac and Pauli operators

219

|

We know from (10.30) that ∗

∗

𝑃− = 4𝐷 𝐷 = 4𝑒−𝜑/2 𝜕𝜑 𝜕 𝑒𝜑/2 and that 𝑃− is a nonnegative self-adjoint operator. It follows from Theorem 8.11 that the space of entire functions 𝐴2 (ℂ, 𝑒−𝜑 ) is of infinite dimension. This means that 0 belongs to the essential spectrum of 𝑃− . Hence, by Proposition 9.37, 𝑃− fails to have compact resolvent and we arrive at a contradiction. A similar conclusion can be drawn in several variables for the Witten–Laplacian ∗

∗

Δ(0,0) = 𝐷1 𝐷1 = 𝑒−𝜑/2 𝜕𝜑 𝜕 𝑒𝜑/2 , 𝜑 if lim|𝑧|→∞ |𝑧|2 𝜇𝜑 (𝑧) = +∞, then Δ(0,0) fails to have compact resolvent. (𝜇𝜑 is the lowest 𝜑 eigenvalue of the Levi matrix 𝑀𝜑 .) Remark 13.5. A non-negative Borel measure 𝜇 on ℂ is doubling, if there exists a constant 𝐶 > 0 such that for any 𝑧 ∈ ℂ and any 𝑟 > 0 𝜇(𝐷(𝑧, 𝑟)) ≤ 𝐶𝜇(𝐷(𝑧, 𝑟/2)).

(13.6)

𝜇(𝐷(𝑧, 2𝑟)) ≥ (1 + 𝐶−3 )𝜇(𝐷(𝑧, 𝑟)),

(13.7)

It can be shown that

for each 𝑧 ∈ ℂ and for each 𝑟 > 0; in particular 𝜇(ℂ) = ∞, unless 𝜇(ℂ) = 0 (see [70]). Example: if 𝑝(𝑧, 𝑧) is a polynomial on ℂ of degree 𝑑, then 𝑑𝜇(𝑧) = |𝑝(𝑧, 𝑧)|𝑎 𝑑𝜆(𝑧),

𝑎>−

1 𝑑

is a doubling measure on ℂ, see [70]. Let 𝜑 : ℂ 󳨀→ ℝ+ be a subharmonic C2 -function. Suppose that 𝑑𝜇 = 󳵻𝜑 𝑑𝜆 is a non-trivial doubling measure. We show that the weighted space of entire functions 𝐴2 (ℂ, 𝑒−𝜑 ) = {𝑓 entire : ‖𝑓‖2𝜑 = ∫ |𝑓|2 𝑒−𝜑 𝑑𝜆 < ∞} ℂ

is of infinite dimension. We know that 𝜇(ℂ) = ∫ 󳵻𝜑 𝑑𝜆 = ∞.

(13.8)

ℂ

Let 𝐾𝜑 (𝑧, 𝑤) denote the Bergman kernel of 𝐴2 (ℂ, 𝑒−𝜑 ). So, if 𝑓 ∈ 𝐴2 (ℂ, 𝑒−𝜑 ), then 𝑓(𝑧) = ∫ 𝐾𝜑 (𝑧, 𝑤)𝑓(𝑤)𝑒−𝜑(𝑤) 𝑑𝜆(𝑤). ℂ

(13.9)

220 | 13 Compact resolvents We claim that

∞

∫ 𝐾𝜑 (𝑧, 𝑧)𝑒−𝜑(𝑧) 𝑑𝜆(𝑧) = ∑ ‖𝑓𝑘 ‖2𝜑 ,

(13.10)

𝑘=1

ℂ

for any complete orthonormal basis (𝑓𝑘 )𝑘 of 𝐴2 (ℂ, 𝑒−𝜑 ), finite or infinite. The partial sums 𝑁

𝐹𝑁 (𝑧) = ∑ |𝑓𝑘 (𝑧)|2 𝑘=1

form a pointwise non-decreasing sequence: 𝐹𝑁 (𝑧) ≤ 𝐹𝑁+1 (𝑧), for each 𝑧 ∈ ℂ and for each 𝑁 ∈ ℕ. In addition we have ∞

lim 𝐹𝑁 (𝑧) = ∑ 𝑓𝑘 (𝑧)𝑓𝑘 (𝑧) = 𝐾𝜑 (𝑧, 𝑧),

𝑁→∞

𝑘=1

see (1.9). The monotone convergence theorem implies that ∞

∞

𝑘=1

ℂ 𝑘=1

∑ ‖𝑓𝑘 (𝑧)‖2𝜑 = ∫ ∑ |𝑓𝑘 (𝑧)|2 𝑒−𝜑(𝑧) 𝑑𝜆(𝑧) = ∫ 𝐾𝜑 (𝑧, 𝑧)𝑒−𝜑(𝑧) 𝑑𝜆(𝑧).

(13.11)

ℂ

Now we use the following pointwise inequality for the Bergman kernel which is due to Marzo and Ortega-Cerd𝑎̀ ([58]): there exists a constant 𝐶 > 0 such that 𝐶−1 󳵻𝜑(𝑧) ≤ 𝐾𝜑 (𝑧, 𝑧)𝑒−𝜑(𝑧) ≤ 𝐶 󳵻𝜑(𝑧),

(13.12)

for each 𝑧 ∈ ℂ, to conclude that 𝐴2 (ℂ, 𝑒−𝜑 ) is of infinite dimension. (13.8) and (13.11) imply that the corresponding Dirac operator D fails to be with compact resolvent, if 𝜑 is a subharmonic function, such that 𝑑𝜇 = 󳵻𝜑 𝑑𝜆 is a nontrivial doubling measure.

13.3 Notes A. Iwatsuka [44] and J. Avron, I. Herbst and B. Simon [2] were the first to obtain results on the problem of compact resolvence of Schrödinger operators with magnetic field. Theorem 13.1 was inspired by the work of A. Iwatsuka. M. Christ ([13]) initiated the study of weighted spaces of entire functions, where the weight 𝜑 has the property that 𝑑𝜇 = 󳵻𝜑 𝑑𝜆 is a non-trivial doubling measure. There are two other approaches to characterize compactness of 𝑁𝜑 , in [34] by means of the Fefferman–Phong inequality and in [58] by means of estimates of the Bergman kernel of 𝐴2 (ℂ, 𝑒−𝜑 ). The results there read as follows: Suppose that 󳵻𝜑 defines a doubling measure, then 𝑁𝜑 is compact if and only if ∫𝔹(𝑧,1) 󳵻𝜑 𝑑𝜆 → ∞, as |𝑧| → ∞, where 𝔹(𝑧, 1) = {𝑤 : |𝑤 − 𝑧| < 1}. Witten–Laplacians, as they appear in the context of semiclassical analysis, are studied in [36, 39] and [46]. A thorough treatment of Dirac and Pauli operators can be found in [16, 40] and [72].

14 Spectrum of ◻ on the Fock space In this chapter we will investigate the spectral properties of the ◻-operator for the Fock space, using solutions of corresponding partial differential equations without boundary conditions. It is important that in this case the ◻-operator acts diagonally, which is related to the fact that the weight function is decoupled – a property which turns out to be an obstruction for compactness (see Chapter 15). We show that the spectrum of the ◻-operator for the Fock space consists of positive integers each of which is of infinite multiplicity and indicate consequences of this result related to the essential spectrum.

14.1 The general setting We consider the weighted 𝜕-complex 𝜕

𝜕

←󳨀

←󳨀

𝐿2(0,𝑞−1) (ℂ𝑛 , 𝑒−𝜑 ) 󳨀→ 𝐿2(0,𝑞) (ℂ𝑛 , 𝑒−𝜑 ) 󳨀→ 𝐿2(0,𝑞+1) (ℂ𝑛 , 𝑒−𝜑 ), ∗ 𝜕𝜑

∗ 𝜕𝜑

𝑛 where for (0, 𝑞)-forms 𝑢 = ∑󸀠|𝐽|=𝑞 𝑢𝐽 𝑑𝑧𝐽 with coefficients in C∞ 0 (ℂ ) we have 󸀠

𝑛

𝜕𝑢 = ∑ ∑ |𝐽|=𝑞 𝑗=1

𝜕𝑢𝐽 𝑑𝑧 ∧ 𝑑𝑧𝐽 , 𝜕𝑧𝑗 𝑗

and ∗

𝜕𝜑 𝑢 = − ∑

󸀠

𝑛

∑ 𝛿𝑘 𝑢𝑘𝐾 𝑑𝑧𝐾 ,

|𝐾|=𝑞−1 𝑘=1

𝜕𝜑 . 𝜕𝑧𝑘

where 𝛿𝑘 = 𝜕𝑧𝜕 − 𝑘 The complex Laplacian on (0, 𝑞)-forms is defined as ∗

∗

◻𝜑,𝑞 := 𝜕 𝜕𝜑 + 𝜕𝜑 𝜕. ◻𝜑,𝑞 is a self-adjoint and positive operator, which means that (◻𝜑,𝑞 𝑓, 𝑓)𝜑 ≥ 0 , for 𝑓 ∈ dom(◻𝜑 ). The associated Dirichlet form is denoted by ∗

∗

𝑄𝜑 (𝑓, 𝑔) = (𝜕𝑓, 𝜕𝑔)𝜑 + (𝜕𝜑 𝑓, 𝜕𝜑 𝑔)𝜑 , ∗

(14.1)

for 𝑓, 𝑔 ∈ dom(𝜕) ∩ dom(𝜕𝜑 ). The weighted 𝜕-Neumann operator 𝑁𝜑,𝑞 is – if it exists – the bounded inverse of ◻𝜑,𝑞 .

222 | 14 Spectrum of ◻ on the Fock space ∗

We indicate that a square integrable (0, 1)-form 𝑓 = ∑𝑛𝑗=1 𝑓𝑗 𝑑𝑧𝑗 belongs to dom(𝜕𝜑 ) if and only if 𝑛 𝜕 (𝑓𝑗 𝑒−𝜑 ) ∈ 𝐿2 (ℂ𝑛 , 𝑒−𝜑 ), 𝑒𝜑 ∑ 𝜕𝑧 𝑗 𝑗=1 where the derivative is to be taken in the sense of distributions, and that forms with ∗ 𝑛 2 coefficients in C∞ 0 (ℂ ) are dense in dom(𝜕) ∩ dom(𝜕𝜑 ) in the graph norm 𝑓 󳨃→ (‖𝜕𝑓‖𝜑 + ∗

1

‖𝜕𝜑 𝑓‖2𝜑 ) 2 (see Chapter 6 and [28]). We consider the Levi matrix

𝑀𝜑 = (

𝜕2 𝜑 ) 𝜕𝑧𝑗 𝜕𝑧𝑘 𝑗𝑘

of 𝜑 and suppose that the sum 𝑠𝑞 of any 𝑞 (equivalently: the smallest 𝑞) eigenvalues of 𝑀𝜑 satisfies lim inf 𝑠𝑞 (𝑧) > 0. (14.2) |𝑧|→∞

In Proposition 6.8 we showed that (14.2) implies that there exists a continuous linear operator 𝑁𝜑,𝑞 : 𝐿2(0,𝑞) (ℂ𝑛 , 𝑒−𝜑 ) 󳨀→ 𝐿2(0,𝑞) (ℂ𝑛 , 𝑒−𝜑 ), such that ◻𝜑,𝑞 ∘ 𝑁𝜑,𝑞 𝑢 = 𝑢, for any 𝑢 ∈ 𝐿2(0,𝑞) (ℂ𝑛 , 𝑒−𝜑 ). If we suppose that the sum 𝑠𝑞 of any 𝑞 (equivalently: the smallest 𝑞) eigenvalues of 𝑀𝜑 satisfies lim 𝑠𝑞 (𝑧) = ∞, (14.3) |𝑧|→∞

then the 𝜕-Neumann operator 𝑁𝜑,𝑞 : 𝐿2(0,𝑞) (ℂ𝑛 , 𝑒−𝜑 ) 󳨀→ 𝐿2(0,𝑞) (ℂ𝑛 , 𝑒−𝜑 ) is compact (see Proposition 11.11). To find the canonical solution to 𝜕𝑓 = 𝑢, where 𝑢 ∈ 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑 ) is a given (0, 1)∗

form such that 𝜕𝑢 = 0, one can take 𝑓 = 𝜕𝜑 𝑁𝜑,1 𝑢 and 𝑓 will also satisfy 𝑓 ⊥ ker𝜕. If the weight function is 𝜑(𝑧) = |𝑧|2 , the corresponding Levi matrix 𝑀𝜑 is the iden2

2

tity. The space 𝐴2 (ℂ𝑛 , 𝑒−|𝑧| ) of entire functions belonging to 𝐿2 (ℂ𝑛 , 𝑒−|𝑧| ) is the Fock space, which plays an important role in quantum mechanics. In this case 𝑛 ∗ 1 ◻𝜑,0 𝑢 = 𝜕𝜑 𝜕𝑢 = − 󳵻𝑢 + ∑ 𝑧𝑗 𝑢𝑧𝑗 , 4 𝑗=1

(14.4)

2

where 𝑢 ∈ dom(◻𝜑,0 ) ⊆ 𝐿2 (ℂ𝑛 , 𝑒−|𝑧| ) and 𝑛 ∗ 1 ◻𝜑,𝑛 𝑢 = 𝜕 𝜕𝜑 𝑢 = − 󳵻𝑢 + ∑ 𝑧𝑗 𝑢𝑧𝑗 + 𝑛 𝑢, 4 𝑗=1 2

where 𝑢 ∈ dom(◻𝜑,𝑛 ) ⊆ 𝐿2(0,𝑛) (ℂ𝑛 , 𝑒−|𝑧| ).

(14.5)

14.2 Determination of the spectrum

223

|

For 𝑛 > 1 and 1 ≤ 𝑞 ≤ 𝑛 − 1 the 𝜕-Neumann Laplacian ◻𝜑,𝑞 acts diagonally (see Chapter 10): for 2 󸀠 𝑢 = ∑ 𝑢𝐽 𝑑𝑧𝐽 ∈ dom(◻𝜑,𝑞 ) ⊆ 𝐿2(0,𝑞) (ℂ𝑛 , 𝑒−|𝑧| ) |𝐽|=𝑞

we have ∗

∗

󸀠

◻𝜑,𝑞 𝑢 = (𝜕 𝜕𝜑 + 𝜕𝜑 𝜕) 𝑢 = ∑ ( − |𝐽|=𝑞

𝑛 1 󳵻𝑢𝐽 + ∑ 𝑧𝑗 𝑢𝐽𝑧𝑗 + 𝑞 𝑢𝐽 ) 𝑑𝑧𝐽 . 4 𝑗=1

(14.6)

14.2 Determination of the spectrum In order to determine the spectrum of ◻𝜑,𝑞 for 𝜑(𝑧) = |𝑧|2 we use Lemma 9.31. For the sake of simplicity and in order to explain the general method, we start with the complex one-dimensional case. Looking for the eigenvalues 𝜇 of ◻𝜑,0 we have to consider: ◻𝜑,0 𝑢 = −𝑢𝑧 𝑧 + 𝑧𝑢𝑧 = 𝜇𝑢. (14.7) 2

It is clear that the space 𝐴2 (ℂ𝑛 , 𝑒−|𝑧| ) is contained in the eigenspace of the eigenvalue 𝜇 = 0. For any positive integer 𝑘 the anti-holomorphic monomial 𝑧𝑘 is an eigenfunction for the eigenvalue 𝜇 = 𝑘. In the following we denote by ℕ0 = ℕ ∪ {0}. Lemma 14.1. Let 𝑛 = 1. For 𝑘 ∈ ℕ0 and 𝑚 ∈ ℕ the functions 𝑚

𝑢𝑘,𝑚 (𝑧, 𝑧) = 𝑧𝑘+𝑚 𝑧𝑚 + ∑

𝑗=1

(−1)𝑗 (𝑘 + 𝑚)! 𝑚! 𝑧𝑘+𝑚−𝑗 𝑧𝑚−𝑗 𝑗! (𝑘 + 𝑚 − 𝑗)! (𝑚 − 𝑗)!

(14.8)

are eigenfunctions for the eigenvalue 𝑘 + 𝑚 of the operator ◻𝜑,0 𝑢 = −𝑢𝑧 𝑧 + 𝑧𝑢𝑧 . For 𝑘 ∈ ℕ and 𝑚 ∈ ℕ0 the functions 𝑘

(−1)𝑗 (𝑘 + 𝑚)! 𝑘! 𝑧𝑘−𝑗 𝑧𝑘+𝑚−𝑗 𝑗=1 𝑗! (𝑘 + 𝑚 − 𝑗)! (𝑘 − 𝑗)!

𝑣𝑘,𝑚 (𝑧, 𝑧) = 𝑧𝑘 𝑧𝑘+𝑚 + ∑

(14.9)

are eigenfunctions for the eigenvalue 𝑘 of the operator ◻𝜑,0 𝑢 = −𝑢𝑧 𝑧 + 𝑧𝑢𝑧 . Proof. To prove (14.8) we set 𝑢𝑘,𝑚 (𝑧, 𝑧) = 𝑧𝑘+𝑚 𝑧𝑚 + 𝑎1 𝑧𝑘+𝑚−1 𝑧𝑚−1 + 𝑎2 𝑧𝑘+𝑚−2 𝑧𝑚−2 + ⋅ ⋅ ⋅ + 𝑎𝑚−1 𝑧𝑘+1 𝑧 + 𝑎𝑚 𝑧𝑘 and compute 𝜕2 𝑢 (𝑧, 𝑧) 𝜕𝑧𝜕𝑧 𝑘,𝑚 = (𝑘 + 𝑚)𝑚𝑧𝑘+𝑚−1 𝑧𝑚−1 + 𝑎1 (𝑘 + 𝑚 − 1)(𝑚 − 1)𝑧𝑘+𝑚−2 𝑧𝑚−2 + ⋅ ⋅ ⋅ + 𝑎𝑚−1 (𝑘 + 1)𝑧𝑘

224 | 14 Spectrum of ◻ on the Fock space as well as 𝑧

𝜕 𝑢 (𝑧, 𝑧) 𝜕𝑧 𝑘,𝑚 = (𝑘 + 𝑚)𝑧𝑘+𝑚 𝑧𝑚 + 𝑎1 (𝑘 + 𝑚 − 1)𝑧𝑘+𝑚−1 𝑧𝑚−1 + ⋅ ⋅ ⋅ + 𝑎𝑚−1 (𝑘 + 1)𝑧𝑘+1 𝑧 + 𝑎𝑚 𝑘𝑧𝑘 ,

which implies that the function 𝑢𝑘,𝑚 is an eigenfunction for the eigenvalue 𝜇 = 𝑘 + 𝑚 and from (14.7) we obtain, comparing the highest exponents of 𝑧 and 𝑧, (𝑘 + 𝑚)𝑚 − 𝑎1 (𝑘 + 𝑚 − 1) = −(𝑘 + 𝑚)𝑎1 , hence 𝑎1 = −(𝑘 + 𝑚)𝑚. Comparing the next lower exponents we get 𝑎1 (𝑘 + 𝑚 − 1)(𝑚 − 1) − 𝑎2 (𝑘 + 𝑚 − 2) = −𝑎2 (𝑘 + 𝑚) and 𝑎2 =

1 2

(𝑘 + 𝑚)(𝑘 + 𝑚 − 1)𝑚(𝑚 − 1). In general we find that for 𝑗 = 1, 2, . . . , 𝑚 𝑎𝑗 =

(−1)𝑗 (𝑘 + 𝑚)! 𝑚! , 𝑗! (𝑘 + 𝑚 − 𝑗)! (𝑚 − 𝑗)!

which proves (14.8). In order to show (14.9) we set 𝑣𝑘,𝑚 (𝑧, 𝑧) = 𝑧𝑘 𝑧𝑘+𝑚 + 𝑏1 𝑧𝑘−1 𝑧𝑘+𝑚−1 + 𝑏2 𝑧𝑘−2 𝑧𝑘+𝑚−2 + ⋅ ⋅ ⋅ + 𝑏𝑘−1 𝑧 𝑧𝑚+1 + 𝑏𝑘 𝑧𝑚 and compute 𝜕2 𝑣 (𝑧, 𝑧) 𝜕𝑧𝜕𝑧 𝑘,𝑚 = 𝑘(𝑘 + 𝑚)𝑧𝑘−1 𝑧𝑘+𝑚−1 + 𝑏1 (𝑘 − 1)(𝑘 + 𝑚 − 1)𝑧𝑘−2 𝑧𝑘+𝑚−2 + ⋅ ⋅ ⋅ + 𝑏𝑘−1 𝑧 𝑧𝑚+1 as well as 𝑧

𝜕 𝑣 (𝑧, 𝑧) = 𝑘𝑧𝑘 𝑧𝑘+𝑚 + 𝑏1 (𝑘 − 1)𝑧𝑘−1 𝑧𝑘+𝑚−1 + ⋅ ⋅ ⋅ + 𝑏𝑘−1 𝑧 𝑧𝑚+1 𝜕𝑧 𝑘,𝑚

which implies that the function 𝑣𝑘,𝑚 is an eigenfunction for the eigenvalue 𝜇 = 𝑘, for each 𝑚 ∈ ℕ and from (14.7) we obtain, comparing the highest exponents of 𝑧 and 𝑧, 𝑘(𝑘 + 𝑚) − 𝑏1 (𝑘 − 1) = −𝑘𝑏1 , hence 𝑏1 = −(𝑘 + 𝑚)𝑘. Comparing the next lower exponents we get 𝑏1 (𝑘 − 1)(𝑘 + 𝑚 − 1) − 𝑏2 (𝑘 − 2) = −𝑏2 𝑘 and 𝑏2 =

1 2

(𝑘 + 𝑚)(𝑘 + 𝑚 − 1)𝑘(𝑘 − 1). In general we find that for 𝑗 = 1, 2, . . . , 𝑘 𝑏𝑗 =

which proves (14.9).

(−1)𝑗 (𝑘 + 𝑚)! 𝑘! , 𝑗! (𝑘 + 𝑚 − 𝑗)! (𝑘 − 𝑗)!

14.2 Determination of the spectrum

|

225

Now we are able to prove Theorem 14.2. Let 𝑛 = 1 and 𝜑(𝑧) = |𝑧|2 . The spectrum of ◻𝜑,0 consists of all nonnegative integers {0, 1, 2, . . . } each of which is of infinite multiplicity, so 0 is the bottom of the essential spectrum. The spectrum of ◻𝜑,1 consists of all positive integers {1, 2, 3, . . . } each of which is of infinite multiplicity. 2

Proof. We already know that the whole Bergman space 𝐴2 (ℂ, 𝑒−|𝑧| ) is contained in the eigenspace of the eigenvalue 0 of the operator ◻𝜑,0 and, for any positive integer 𝑘, the anti-holomorphic monomial 𝑧𝑘 is an eigenfunction for the eigenvalue 𝜇 = 𝑘. In addition, all functions of the form 𝑧𝜈 𝑧𝜅 with 𝜈, 𝜅 ∈ ℕ0 can be expressed as a linear combination of functions of the form (14.8) and (14.9). For a fixed 𝑘 ∈ ℕ the functions of the form (14.9) have infinite multiplicity as the parameter 𝑚 ∈ ℕ0 is free. So these eigenvalues are of infinite multiplicity. All the eigenfunctions considered so far yield 2 a complete orthogonal basis of 𝐿2 (ℂ, 𝑒−|𝑧| ), since the Hermite polynomials {𝐻0 (𝑥)𝐻𝑘 (𝑦), 𝐻1 (𝑥)𝐻𝑘−1 (𝑦), . . . , 𝐻𝑘 (𝑥)𝐻0 (𝑦)} 2

2

for 𝑘 = 0, 1, 2, . . . form a complete orthogonal system in 𝐿2 (ℝ2 , 𝑒−𝑥 −𝑦 ) (see for instance [22]) and since 𝑥 = 1/2(𝑧 + 𝑧) , 𝑦 = 𝑖/2(𝑧 − 𝑧) we can apply Lemma 9.31 and obtain that the spectrum of ◻𝜑,0 is ℕ0 . The statement for the spectrum of ◻𝜑,1 follows from (14.5). For several variables we can adopt the method from above to obtain the following result Theorem 14.3. Let 𝑛 > 1 and 𝜑(𝑧) = |𝑧1 |2 + ⋅ ⋅ ⋅ + |𝑧𝑛 |2 and 0 ≤ 𝑞 ≤ 𝑛. The spectrum of ◻𝜑,𝑞 consists of all integers {𝑞, 𝑞 + 1, 𝑞 + 2, . . . } each of which is of infinite multiplicity. Proof. Recall that the 𝜕-Neumann Laplacian ◻𝜑,𝑞 acts diagonally and that 󸀠

◻𝜑,𝑞 𝑢 = ∑ ( − |𝐽|=𝑞

𝑛 1 󳵻𝑢𝐽 + ∑ 𝑧𝑗 𝑢𝐽𝑧𝑗 + 𝑞 𝑢𝐽 ) 𝑑𝑧𝐽 . 4 𝑗=1

The factor 𝑞 in the last formula is responsible for the fact that the eigenvalues start with 𝑞, which can be seen, in each component separately, by 𝑛 1 − 󳵻𝑢𝐽 + ∑ 𝑧𝑗 𝑢𝐽𝑧𝑗 = (𝜇 − 𝑞) 𝑢𝐽 . 4 𝑗=1

Now let 𝑘1 , 𝑘2 , . . . , 𝑘𝑛 ∈ ℕ0 and 𝑚1 , 𝑚2 , . . . , 𝑚𝑛 ∈ ℕ. Then the function 𝑢𝑘1 ,𝑚1 (𝑧1 , 𝑧1 ) 𝑢𝑘2 ,𝑚2 (𝑧2 , 𝑧2 ) . . . 𝑢𝑘𝑛 ,𝑚𝑛 (𝑧𝑛 , 𝑧𝑛 ) is the component of an eigenfunction for the eigenvalue ∑𝑛𝑗=1 (𝑘𝑗 + 𝑚𝑗 ) of the operator ◻𝜑,𝑞 , which follows from (14.6) and (14.8).

226 | 14 Spectrum of ◻ on the Fock space Similarly it follows from (14.6) and (14.9) that for 𝑘1 , 𝑘2 , . . . , 𝑘𝑛 𝑚1 , 𝑚2 , . . . , 𝑚𝑛 ∈ ℕ0 the function

∈

ℕ and

𝑣𝑘1 ,𝑚1 (𝑧1 , 𝑧1 ) 𝑣𝑘2 ,𝑚2 (𝑧2 , 𝑧2 ) . . . 𝑣𝑘𝑛 ,𝑚𝑛 (𝑧𝑛 , 𝑧𝑛 ) is an eigenfunction for the eigenvalue ∑𝑛𝑗=1 𝑘𝑗 . All other possible 𝑛-fold products with factors 𝑢𝑘𝑗 ,𝑚𝑗 or 𝑣𝑘𝑗 ,𝑚𝑗 (also mixed) appear as eigenfunctions of ◻𝜑,𝑞 . 𝛽

𝛼

From this we obtain that all expressions of the form 𝑧1 1 𝑧1 1 . . . 𝑧𝑛𝛼𝑛 𝑧𝛽𝑛𝑛 for arbitrary 𝛼𝑗 , 𝛽𝑗 ∈ ℕ0 , 𝑗 = 1, . . . , 𝑛, can be written as a linear combination of eigenfunctions of ◻𝜑,𝑞 , which proves that all these eigenfunctions constitute a complete basis in 2

𝐿2(0,𝑞) (ℂ𝑛 , 𝑒−|𝑧| ), see the proof of Theorem 14.2. So we can again apply Lemma 9.31.

Remark 14.4. (i) Since in all cases the essential spectrum is nonempty, the corresponding ◻𝜑,𝑞 operator fails to be with compact resolvent (see Proposition 9.37). (ii) If one considers the weight function 𝜑(𝑧) = (|𝑧1 |2 + |𝑧2 |2 + ⋅ ⋅ ⋅ + |𝑧𝑛 |2 )𝛼 for 𝛼 > 1 the situation is completely different: The operators ◻𝜑,𝑞 , 1 ≤ 𝑞 ≤ 𝑛, are with compact resolvent (see Proposition 11.11 ), so the essential spectrum must be empty. We can use the results from above to settle the corresponding questions for the socalled Witten–Laplacian which is defined on 𝐿2 (ℂ𝑛 ). 𝜕𝜑 𝜕𝜑 For this purpose we set 𝑍𝑘 = 𝜕𝑧𝜕 + 12 𝜕𝑧 and 𝑍𝑘∗ = − 𝜕𝑧𝜕 + 12 𝜕𝑧 and we consider (0, 𝑞)𝑘

𝑘

𝑘

𝑘

forms ℎ = ∑󸀠|𝐽|=𝑞 ℎ𝐽 𝑑𝑧𝐽 , where ∑󸀠 means that we sum up only increasing multi-indices 𝐽 = (𝑗1 , . . . , 𝑗𝑞 ) and where 𝑑𝑧𝐽 = 𝑑𝑧𝑗1 ∧ ⋅ ⋅ ⋅ ∧ 𝑑𝑧𝑗𝑞 . We define 𝑛

󸀠

𝐷𝑞+1 ℎ = ∑ ∑ 𝑍𝑘 (ℎ𝐽 ) 𝑑𝑧𝑘 ∧ 𝑑𝑧𝐽 𝑘=1 |𝐽|=𝑞

and ∗

𝑛

󸀠

𝐷𝑞 ℎ = ∑ ∑ 𝑍𝑘∗ (ℎ𝐽 ) 𝑑𝑧𝑘 ⌋𝑑𝑧𝐽 , 𝑘=1 |𝐽|=𝑞

where 𝑑𝑧𝑘 ⌋𝑑𝑧𝐽 denotes the contraction, or interior multiplication by 𝑑𝑧𝑘 , i.e. we have ⟨𝛼, 𝑑𝑧𝑘 ⌋𝑑𝑧𝐽 ⟩ = ⟨𝑑𝑧𝑘 ∧ 𝛼, 𝑑𝑧𝐽 ⟩ for each (0, 𝑞 − 1)-form 𝛼. The complex Witten–Laplacian on (0, 𝑞)-forms is then given by ∗

∗

= 𝐷𝑞 𝐷𝑞 + 𝐷𝑞+1 𝐷𝑞+1 , Δ(0,𝑞) 𝜑 for 𝑞 = 1, . . . , 𝑛 − 1.

14.2 Determination of the spectrum

|

227

The general 𝐷-complex has the form 𝐷𝑞

𝐷𝑞+1

𝐿2(0,𝑞−1) (ℂ𝑛 ) 󳨀→ 𝐿2(0,𝑞) (ℂ𝑛 ) 󳨀→ 𝐿2(0,𝑞+1) (ℂ𝑛 ) . ←󳨀

←󳨀

∗ 𝐷𝑞

∗ 𝐷𝑞+1

It follows that ∗

𝐷𝑞+1 Δ(0,𝑞) = Δ(0,𝑞+1) 𝐷𝑞+1 𝜑 𝜑

∗

and 𝐷𝑞+1 Δ(0,𝑞+1) = Δ(0,𝑞) 𝜑 𝜑 𝐷𝑞+1 .

We remark that ∗

𝑛

󸀠

𝐷𝑞 ℎ = ∑ ∑ 𝑍𝑘∗ (ℎ𝐽 ) 𝑑𝑧𝑘 ⌋𝑑𝑧𝐽 = ∑ 𝑘=1 |𝐽|=𝑞

󸀠

𝑛

∑ 𝑍𝑘∗ (ℎ𝑘𝐾 ) 𝑑𝑧𝐾 .

|𝐾|=𝑞−1 𝑘=1

In particular we get for a function 𝑣 ∈ 𝐿2 (ℂ𝑛 ) 𝑛

∗

∗ Δ(0,0) 𝜑 𝑣 = 𝐷1 𝐷1 𝑣 = ∑ 𝑍𝑗 𝑍𝑗 (𝑣),

(14.10)

𝑗=1

and for a (0, 1)-form 𝑔 = ∑𝑛ℓ=1 𝑔ℓ 𝑑𝑧ℓ ∈ 𝐿2(0,1) (ℂ𝑛 ) we obtain ∗

∗

(0,0) Δ(0,1) ⊗ 𝐼)𝑔 + 𝑀𝜑 𝑔, 𝜑 𝑔 = (𝐷1 𝐷1 + 𝐷2 𝐷2 )𝑔 = (Δ 𝜑

where we set

𝑛

𝑛

𝑀𝜑 𝑔 = ∑ ( ∑ 𝑗=1

𝑘=1

(14.11)

𝜕2 𝜑 𝑔 ) 𝑑𝑧𝑗 𝜕𝑧𝑘 𝜕𝑧𝑗 𝑘

and ⊗ 𝐼) 𝑔 = ∑𝑛𝑘=1 Δ(0,0) (Δ(0,0) 𝜑 𝜑 𝑔𝑘 𝑑𝑧𝑘 . In general we have that = 𝑒−𝜑/2 ◻𝜑,𝑞 𝑒𝜑/2 , Δ(0,𝑞) 𝜑

(14.12)

for 𝑞 = 0, 1, . . . , 𝑛. In our case 𝜑(𝑧) = |𝑧1 |2 + ⋅ ⋅ ⋅ + |𝑧𝑛 |2 we get 󸀠

Δ(0,𝑞) 𝜑 ℎ = ∑ (− |𝐽|=𝑞

for

1 1 𝑛 1 𝑛 󳵻ℎ𝐽 + ∑(𝑧𝑗 ℎ𝐽𝑧𝑗 − 𝑧𝑗 ℎ𝐽𝑧𝑗 ) + |𝑧|2 ℎ𝐽 + (𝑞 − ) ℎ𝐽 ) 𝑑𝑧𝐽 , (14.13) 4 2 𝑗=1 4 2 󸀠

2 𝑛 ℎ = ∑ ℎ𝐽 𝑑𝑧𝐽 ∈ dom (Δ(0,𝑞) 𝜑 ) ⊆ 𝐿 (0,𝑞) (ℂ ). |𝐽|=𝑞

Using (14.12) and Lemma 9.29 we get that Δ(0,𝑞) and ◻𝜑,𝑞 have the same spectrum. 𝜑 Hence by Theorem 14.3 we obtain Theorem 14.5. Let 𝜑(𝑧) = |𝑧1 |2 + ⋅ ⋅ ⋅ + |𝑧𝑛 |2 and 0 ≤ 𝑞 ≤ 𝑛. The spectrum of the Witten– Laplacian Δ(0,𝑞) consists of all integers {𝑞, 𝑞 + 1, 𝑞 + 2, . . . } each of which is of infinite 𝜑 multiplicity.

228 | 14 Spectrum of ◻ on the Fock space

14.3 Notes Only in a few cases it is possible to determine the exact form of the spectrum of a ◻operator. G. Folland [21] described it for the unit ball in ℂ𝑛 using spherical harmonics, in this case the spectrum is discrete consisting of eigenvalues of finite multiplicity. With the help of Bessel functions S. Fu [24] showed that the spectrum of the ◻-operator for a polycylinder contains eigenvalues of finite and infinite multiplicity. The spectrum of Δ(0,0) 𝜑 , even in a more general form, was calculated by X. Ma and G. Marinescu, [56, 57]. They use directly Hermite polynomials and the fact that the Hermite polynomials form a complete orthogonal system of the weighted space 2 𝐿2 (ℝ, 𝑒−𝑥 ) and general properties of the Kodaira–Laplacian.

15 Obstructions to compactness In this chapter we give some examples of domains or weights, for which the corresponding 𝜕-Neumann operator or the canonical solution operator to 𝜕 fails to be compact. For this purpose we will mainly use the characterization of compactness from Chapter 11. The standard examples for noncompactness are on the one side polydiscs and on the other side so-called decoupled weights.

15.1 The bidisc First we consider the the canonical solution operator to 𝜕 for the bidisc 𝔻 × 𝔻 (see [52]): We know from Chapter 1 that the monomials 𝜑𝑛 (𝑧) = √

𝑛+1 𝑛 𝑧 , 𝜋

𝑛 = 0, 1, 2, . . .

constitute a complete orthonormal system in 𝐴2 (𝔻). Consider the following (0, 1)forms 𝛼𝑛 in 𝐿2(0,1) (𝔻 × 𝔻) with holomorphic coefficients: 𝛼𝑛 (𝑧1 , 𝑧2 ) = 𝜑𝑛 (𝑧1 ) 𝑑𝑧2 . They are 𝜕-closed and their norms in 𝐿2(0,1) (𝔻 × 𝔻) are ‖𝛼𝑛 ‖ = √𝜋 , 𝑛 = 0, 1, 2, . . . The canonical solution to 𝜕𝑢 = 𝛼𝑛 is given by 𝑢𝑛 (𝑧1 , 𝑧2 ) = 𝜑𝑛 (𝑧1 ) 𝑧2 , this means that 𝑢𝑛 ∈ 𝐴2 (𝔻 × 𝔻)⊥ , which follows by the fact that for each ℎ ∈ 𝐴2 (𝔻 × 𝔻) we have ∫ 𝑢𝑛 (𝑧1 , 𝑧2 ) ℎ(𝑧1 , 𝑧2 ) 𝑑𝜆(𝑧1 , 𝑧2 ) = ∫ 𝜑𝑛 (𝑧1 ) (∫ 𝑧2 ℎ(𝑧1 , 𝑧2 ) 𝑑𝜆(𝑧2 ))− 𝑑𝜆(𝑧1 ), 𝔻×𝔻

𝔻

𝔻

where the inner integral vanishes by Cauchy’s theorem applied to the holomorphic function 𝑧2 󳨃→ 𝑧2 ℎ(𝑧1 , 𝑧2 ). Finally, ‖𝑢𝑛 ‖ = √

𝜋 2

and 𝑢𝑛 ⊥𝑢𝑚

if 𝑛 ≠ 𝑚,

which follows from (𝑢𝑛 , 𝑢𝑚 ) = ∫ |𝑧2 |2 𝑑𝜆(𝑧2 ) ∫ 𝜑𝑛 (𝑧1 )𝜑𝑚 (𝑧1 ) 𝑑𝜆(𝑧1 ) 𝔻

and (𝜑𝑛 , 𝜑𝑚 ) = 𝛿𝑛,𝑚 .

𝔻

230 | 15 Obstructions to compactness Thus {𝑢𝑛 } has no convergent subsequence in 𝐿2 (𝔻×𝔻). This shows that the canon∗ ical solution operator 𝜕 𝑁 to 𝜕 fails to be compact. Further obstructions to compactness on weakly pseudoconvex domains can be found in [25, 26] and [71].

15.2 Weighted spaces We continue to calculate the integrals in (11.23) for the weight 𝜑(𝑧) = |𝑧|𝛼 in ℂ. We set 𝛽 = 𝛼/2 and 𝑢𝑘 (𝑧) = 𝑧𝑘 for 𝑘 ∈ ℕ. The left-hand side of (11.23) is ∞

𝛼

𝛼

∫ |𝑢𝑘 (𝑧)|2 𝑒−|𝑧| 𝑑𝜆(𝑧) = 2𝜋 ∫ 𝑟2𝑘+1 𝑒−𝑟 𝑑𝑟, 𝑅

ℂ\𝐵𝑅

we indicate that

∞

𝛼

∫ 𝑟2𝑘+1 𝑒−𝑟 𝑑𝑟 = 0

1 𝑘 1 Γ( + ). 2𝛽 𝛽 𝛽

The right-hand side of (11.23) reads ∗

𝛼

𝛼

∫ |𝜕𝜑 𝑢𝑘 (𝑧)|2 𝑒−|𝑧| 𝑑𝜆(𝑧) = ∫ | − 𝑘𝑧𝑘−1 + 𝛽𝑧𝛽+𝑘−1 𝑧𝛽 |2 𝑒−|𝑧| 𝑑𝜆(𝑧) ℂ

ℂ ∞

2𝛽

= 2𝜋 ∫(𝑘2 𝑟2𝑘−1 − 2𝑘𝛽𝑟2𝛽+2𝑘−1 + 𝛽2 𝑟4𝛽+2𝑘−1 ) 𝑒−𝑟 𝑑𝑟 0

= 2𝜋[

𝛽 𝑘2 𝑘 𝑘 𝑘 Γ ( ) − 𝑘 Γ ( + 1) + Γ ( + 2) ] 2𝛽 𝛽 𝛽 2 𝛽

= 𝜋𝛽 Γ (

𝑘 + 1) . 𝛽

If 𝛼 = 2, it follows that condition (11.23) is not satisfied. For this purpose we consider the integral ∞

2

∫ 𝑟2𝑘+1 𝑒−𝑟 𝑑𝑟 𝑅 2

and substitute 𝑟 = 𝑠 obtaining ∞

∞

2

∫ 𝑟2𝑘+1 𝑒−𝑟 𝑑𝑟 = ∫ 𝑠𝑘 𝑒−𝑠 𝑑𝑠. 𝑅

𝑅2

Now we apply 𝑘-times integration by parts and get ∞

2

∞

2

𝑘

𝑅2𝑗 . 𝑗=0 𝑗!

∫ 𝑠𝑘 𝑒−𝑠 𝑑𝑠 = 𝑒−𝑅 𝑅2𝑘 + 𝑘 ∫ 𝑠𝑘−1 𝑒−𝑠 𝑑𝑠 = 𝑒−𝑅 𝑘! ∑ 𝑅2

𝑅2

15.2 Weighted spaces

| 231

Observe that for 𝛽 = 1 we have Γ(

𝑘 1 𝑘 + 1) = Γ ( + ) = 𝑘! 𝛽 𝛽 𝛽

and as there is 𝜖0 > 0 such that for each 𝑅 > 0 there exists 𝑘 ∈ ℕ such that 2

𝑘

𝑒−𝑅 ∑ 𝑗=0

𝑅2𝑗 > 𝜖0 , 𝑗!

condition (11.23) is not satisfied for 𝛼 = 2. This means that the 𝜕-Neumann 𝑁𝜑 operator 2

on 𝐿2 (ℂ, 𝑒−|𝑧| ) fails to be compact, and as 𝑁𝜑 = 𝑆∗ 𝑆, where 𝑆 is the canonical solution operator to 𝜕, the canonical solution operator 𝑆 also fails to be compact (compare Theorem 2.22). Another proof for this fact uses spectral theory: from (10.3) we know that ∗

◻𝜑 𝑢 = 𝜕 𝜕𝜑 𝑢 = −

𝜕2 𝑢 𝜕𝑢 +𝑧 + 𝑢, 𝜕𝑧𝜕𝑧 𝜕𝑧 2

hence it follows immediately that the whole space 𝐴2 (ℂ, 𝑒−|𝑧| ) is a subspace of the eigenspace to the eigenvalue 1 of the operator ◻𝜑 , which means that the essential spectrum of ◻𝜑 is nonempty and 𝑁𝜑 fails to be compact by Proposition 9.37. In the next examples we consider decoupled C2 -weights 𝜑(𝑧1 , 𝑧2 , . . . , 𝑧𝑛 ) = 𝜑1 (𝑧1 ) + 𝜑2 (𝑧2 ) + ⋅ ⋅ ⋅ + 𝜑𝑛 (𝑧𝑛 ) and follow an idea of G. Schneider [68], (see also the results in Chapter 3). Proposition 15.1. Suppose that 𝑛 ≥ 2 and that there exists ℓ such that 𝐴2 (ℂ, 𝑒−𝜑ℓ ) is infinite dimensional. Suppose also that 1 ∈ 𝐿2 (ℂ, 𝑒−𝜑𝑗 ) for all 𝑗. Suppose finally that for some 𝑘 ≠ ℓ, 𝑧𝑘 ∈ 𝐿2 (ℂ, 𝑒−𝜑𝑘 ). Then the canonical solution operator to 𝜕 fails to be compact even on the space 𝐴2(0,1) (ℂ𝑛 , 𝑒−𝜑 ). Proof. Let 𝑃𝑘 denote the Bergman projection 𝑃𝑘 : 𝐿2 (ℂ, 𝑒−𝜑𝑘 ) 󳨀→ 𝐴2 (ℂ, 𝑒−𝜑𝑘 ). It is clear that the function (𝑧𝑘 − 𝑃𝑘 𝑧𝑘 ) is not zero. Let (𝑓𝜈 )𝜈 be an infinite orthonormal system in 𝐴2 (ℂ, 𝑒−𝜑ℓ ) and define ℎ𝜈 (𝑧) := 𝑓𝜈 (𝑧ℓ )(𝑧𝑘 − 𝑃𝑘 𝑧𝑘 ). Then (ℎ𝜈 )𝜈 is an orthogonal family in 𝐴2 (ℂ𝑛 , 𝑒−𝜑 )⊥ . To see this let 𝑔 ∈ 𝐴2 (ℂ𝑛 , 𝑒−𝜑 ) and note that (𝑣, 𝑃𝑘 𝑧𝑘 )𝜑𝑘 = (𝑣, 𝑧𝑘 )𝜑𝑘 for 𝑣 ∈ 𝐴2 (ℂ, 𝑒−𝜑𝑘 ), which implies that (𝑔, ℎ𝜈 )𝜑 = 0. In addition we have 𝜕ℎ𝜈 = 𝑓𝜈 (𝑧ℓ )𝑑𝑧𝑘 . Hence (𝜕ℎ𝜈 )𝜈 constitutes a bounded sequence in 𝐴2(0,1) (ℂ𝑛 , 𝑒−𝜑 ), and for the canonical solution operator 𝑆 we have 𝑆(𝑓𝜈 (𝑧ℓ )𝑑𝑧𝑘 ) = ℎ𝜈 and since (ℎ𝜈 )𝜈 is an orthogonal family, it has no convergent subsequence, which implies the result.

232 | 15 Obstructions to compactness Remark 15.2. If the conditions of Proposition 15.1 are satisfied, then the corresponding 𝜕-Neumann operator 𝑁𝜑,1 also fails to be compact, which follows from Proposition 11.20. In the following example we consider the 𝜕-Neumann operator 𝑁𝜑,1 for a decoupled weight 𝜑: Example. Let 𝜑(𝑧1 , 𝑧2 ) = |𝑧1 |2 + |𝑧2 |2 and consider the corresponding 𝜕-Neumann operator 𝑁𝜑,1 . We will investigate the following sequence of (0, 1)-forms 𝑢𝑘 (𝑧1 , 𝑧2 ) = 𝜓𝑘 (𝑧1 ) 𝑑𝑧2 , where 𝜓𝑘 (𝑧1 ) =

𝑧1𝑘 , √𝜋𝑘!

for 𝑘 ∈ ℕ. It follows that 𝜕𝑢𝑘 = 0 for each 𝑘 ∈ ℕ and ∗

𝜕𝜑 𝑢𝑘 (𝑧1 , 𝑧2 ) = 𝑧2 𝜓𝑘 (𝑧1 ). This implies and 𝑁𝜑,1 𝑢𝑘 = 𝑢𝑘

◻𝜑,1 𝑢𝑘 = 𝑢𝑘

for each 𝑘 ∈ ℕ. The set {𝑢𝑘 : 𝑘 ∈ ℕ} is a bounded set of mutually orthogonal (0, 1)forms in 𝐿2(0,1) (ℂ𝑛 , 𝑒−𝜑 ). As 𝑁𝜑,1 𝑢𝑘 = 𝑢𝑘 , it follows that 𝑁𝜑,1 fails to be compact. The following computation shows that condition (11.23) is not satisfied for the (0, 1)-forms 𝑢𝑘 , where we consider ∫ℂ2 \𝑄 instead of ∫ℂ2 \𝔹 , where 𝑅

𝑅

𝑄𝑅 = {(𝑧1 , 𝑧2 ) : |𝑧1 | < 𝑅 , |𝑧2 | < 𝑅}. We have 2

𝑅

2

∫ |𝑢𝑘 (𝑧1 , 𝑧2 )|2 𝑒−|𝑧1 | −|𝑧2 | 𝑑𝜆(𝑧1 , 𝑧2 ) =

∞

2 2 4𝜋 ∫ ( ∫ 𝑟12𝑘+1 𝑒−𝑟1 𝑑𝑟1 )𝑟2 𝑒−𝑟2 𝑑𝑟2 𝑘!

0

ℂ2 \𝑄𝑅

+

𝑅

∞

∞

𝑅

0

2 2 4𝜋 ∫ ( ∫ 𝑟12𝑘+1 𝑒−𝑟1 𝑑𝑟1 )𝑟2 𝑒−𝑟2 𝑑𝑟2 . 𝑘!

After the substitution 𝑟12 = 𝑠 the first integral is equal to 𝑅

∞

2 2𝜋 ∫ ( ∫ 𝑠𝑘 𝑒−𝑠 𝑑𝑠)𝑟2 𝑒−𝑟2 𝑑𝑟2 . 𝑘!

0

𝑅2

As in the example from above we get ∞

2

𝑘

∫ 𝑠𝑘 𝑒−𝑠 𝑑𝑠 = 𝑒−𝑅 𝑘! ∑ 𝑅2

𝑗=0

𝑅2𝑗 , 𝑗!

15.2 Weighted spaces

|

233

and finally substituting 𝑟22 = 𝑡 𝑅

𝑅2

∞

𝑘 2 2 2𝜋 𝑅2𝑗 ∫ ( ∫ 𝑠𝑘 𝑒−𝑠 𝑑𝑠)𝑟2 𝑒−𝑟2 𝑑𝑟2 = 𝜋𝑒−𝑅 ∑ ∫ 𝑒−𝑡 𝑑𝑡 𝑘! 𝑗! 𝑗=0 0

0

𝑅2

2

𝑘

2𝑗

2 𝑅 (1 − 𝑒−𝑅 ). 𝑗=0 𝑗!

= 𝜋𝑒−𝑅 ∑ On the right-hand side of (11.23) we only have the term

2 2 1 ∫ |𝑧1 |2𝑘 |𝑧2 |2 𝑒−|𝑧1 | −|𝑧2 | 𝑑𝜆(𝑧1 , 𝑧2 ) 𝜋𝑘!

ℂ2

=

∞

∞

0

0

2 2 4𝜋 ∫ 𝑟12𝑘+1 𝑒−𝑟1 𝑑𝑟1 ∫ 𝑟23 𝑒−𝑟2 𝑑𝑟2 𝑘!

= 𝜋. This implies 2

∫ |𝑢𝑘 (𝑧1 , 𝑧2 )|2 𝑒−|𝑧1 |

−|𝑧2 |2

2

𝑘

𝑑𝜆(𝑧1 , 𝑧2 ) ≥ 𝜋𝑒−𝑅 ∑

𝑗=0

ℂ2 \𝑄𝑅

2 𝑅2𝑗 (1 − 𝑒−𝑅 ). 𝑗!

As there is 𝜖0 > 0 such that for each 𝑅 > 0 there exists 𝑘 ∈ ℕ such that 2

𝑘

𝑒−𝑅 ∑ 𝑗=0

2 𝑅2𝑗 (1 − 𝑒−𝑅 ) > 𝜖0 , 𝑗!

condition (11.23) is not satisfied. Finally we discuss compactness of 𝑁𝜑,1 and 𝑁𝜑,2 in ℂ2 for a more general setting: let 𝜑(𝑧1 , 𝑧2 ) = 𝜑1 (𝑧1 ) + 𝜑2 (𝑧2 ). The eigenvalues of the Levi matrix are 𝜕2 𝜑1 𝜕𝑧1 𝜕𝑧1

and

𝜕2 𝜑2 . 𝜕𝑧2 𝜕𝑧2

If the (0, 1)-form 𝑢 = 𝑢1 𝑑𝑧1 + 𝑢2 𝑑𝑧2 belongs to dom(◻𝜑,1 ), then ◻𝜑,1 𝑢 = ( −

𝜕𝜑 𝜕𝑢1 𝜕𝜑2 𝜕𝑢1 𝜕2 𝜑1 𝜕2 𝑢1 𝜕2 𝑢1 − + 1 + + 𝑢 ) 𝑑𝑧1 𝜕𝑧1 𝜕𝑧1 𝜕𝑧2 𝜕𝑧2 𝜕𝑧1 𝜕𝑧1 𝜕𝑧2 𝜕𝑧2 𝜕𝑧1 𝜕𝑧1 1

+(−

𝜕𝜑 𝜕𝑢2 𝜕𝜑2 𝜕𝑢2 𝜕2 𝜑2 𝜕2 𝑢2 𝜕2 𝑢2 − + 1 + + 𝑢 ) 𝑑𝑧2 𝜕𝑧1 𝜕𝑧1 𝜕𝑧2 𝜕𝑧2 𝜕𝑧1 𝜕𝑧1 𝜕𝑧2 𝜕𝑧2 𝜕𝑧2 𝜕𝑧2 2

and for 𝑉 = 𝑣 𝑑𝑧1 ∧ 𝑑𝑧2 ∈ dom(◻𝜑,2 ) we have that ◻𝜑,2 𝑉 equals to (−

𝜕𝜑 𝜕𝑣 𝜕𝜑 𝜕𝑣 𝜕2 𝜑1 𝜕2 𝜑2 𝜕2 𝑣 𝜕2 𝑣 − + 1 + 2 + 𝑣+ 𝑣) 𝑑𝑧1 ∧ 𝑑𝑧2 . 𝜕𝑧1 𝜕𝑧1 𝜕𝑧2 𝜕𝑧2 𝜕𝑧1 𝜕𝑧1 𝜕𝑧2 𝜕𝑧2 𝜕𝑧1 𝜕𝑧1 𝜕𝑧2 𝜕𝑧2

234 | 15 Obstructions to compactness Now suppose that 𝐴2 (ℂ, 𝑒−𝜑1 ) is infinite dimensional, that 1 ∈ 𝐿2 (ℂ, 𝑒−𝜑𝑗 ) for 𝑗 = 1, 2, that 𝑧2 ∈ 𝐿2 (ℂ, 𝑒−𝜑2 ) and finally that 𝜕2 𝜑1 (𝑧1 ) 𝜕2 𝜑2 (𝑧2 ) + → ∞ as |𝑧1 |2 + |𝑧2 |2 → ∞. 𝜕𝑧1 𝜕𝑧1 𝜕𝑧2 𝜕𝑧2 Then 𝑁𝜑,2 is compact, but 𝑁𝜑,1 fails to be compact. Our assumptions imply that 𝑁𝜑,2 is compact by Proposition 11.11. In addition we have that 𝑁𝜑,2 = 𝑆2∗ 𝑆2 , where 𝑆2 is the canonical solution operator for 𝜕 for (0, 2)-forms. Hence 𝑆2 is also compact. Now suppose that 𝑁𝜑,1 is compact. Since 𝑁𝜑,1 = 𝑆1∗ 𝑆1 + 𝑆2 𝑆2∗ this would imply that 𝑆1 is compact, contradicting Proposition 15.1. We get the same conclusion if we apply Proposition 11.20. The above assumptions are all satisfied for instance for the weight functions 𝜑(𝑧1 , 𝑧2 ) = |𝑧1 |2𝑘 + |𝑧2 |2𝑘 , 𝑘 = 2, 3, . . . .

15.3 Notes A general characterization of compactness of the 𝜕-Neumann operator for smoothly bounded domains in terms of geometric properties of the boundary of the domain is not known. Christ and Fu [14] showed: If Ω ⊂ ℂ2 is a smoothly bounded pseudoconvex Hartogs domain (i.e. with (𝑧, 𝑤) ∈ Ω we have (𝑧, 𝑒𝑖𝜃 𝑤) ∈ Ω for every 𝜃 ∈ ℝ ), then the 𝜕-Neumann operator 𝑁 is compact if and only if property (P) holds. Also analytic discs in the boundary are an obstruction to compactness. But there are examples of Hartogs domains which fail property (P) although their boundaries contain no analytic discs. In the case of a weighted space 𝐿2 (ℂ, 𝑒−𝜑 ), compactness of the 𝜕-Neumann operator can be characterized in terms of the weight function (see Theorem 13.1), a different approach to this problem, using estimates of the Bergman kernel, is contained in [58]. An analogous characterization for compactness of the 𝜕-Neumann operator on 𝐿2 (ℂ𝑛 , 𝑒−𝜑 ) , 𝑛 ≥ 2 has also not been found up to now.

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Index 𝐴2 (Ω), 1 𝑇𝑝0,1 (𝑏Ω), 73

𝑇𝑝1,0 (𝑏Ω), 73 Δ(0,0) 𝜑 , 178 (0,𝑞)

Δ 𝜑 , 179, 226 𝜕-Neumann operator, 81, 105, 107, 110, 186, 201–204, 207, 211, 216 𝜖-net, 183, 185 S𝑝 , 40 K(𝐻1 , 𝐻2 ), 16 L(𝐻1 , 𝐻2 ), 16 V-elliptic form, 63 𝜕-complex, 76, 104, 178, 203 𝜎-algebra, 127 ◻, 77, 81 ◻𝜑 , 104 𝑖𝜕𝜕, 73 adjoint operator, 27, 50, 104, 211 algebra, 207 approximation to the identity, 88 Arzela-Ascoli theorem, 183, 185 Baire category theorem, 52 Banach algebra, 129, 165 basic estimate, 81, 119 Bergman kernel, 5, 9, 37, 210, 216, 220, 234 Bergman projection, 6, 207 Bergman space, 1 Bessel’s inequality, 6, 22 biholomorphic map, 12 Borel set, 127 boundedness, 183 Brascamp-Lieb inequality, 160 canonical solution operator, 26, 37, 39, 49, 84, 107, 175, 210, 211, 217 Cauchy-Riemann equation, 16 Cayley transform, 141, 142, 145 closable operator, 50 closed graph, 74, 75 closed graph theorem, 53, 54, 141 closed operator, 50, 76, 140 closed range, 81 closure of an operator, 50 coercive, 186, 187

commutator, 113, 213 compact operator, 16, 19, 27, 107, 133, 191, 192, 194, 201, 203, 204, 213, 214, 216, 217 compact resolvent, 218 complex Laplacian, 77, 104 complex measure, 127 connection, 112 continuous form, 63 continuous spectrum, 140 core of the operator, 62 curvature, 112 cut-off function, 169 decoupled, 231 derivative of a distribution, 66 diamagnetic inequality, 172 difference quotient, 166 Dirac Delta distribution, 66 Dirac operator, 180, 218 Dirichlet form, 105, 186, 192 discrete spectrum, 153 distribution, 65 doubling measure, 219 dual norm, 4 electric potential, 171 elliptic partial differential operator, 168, 171 equicontinuity, 183 essential range, 128, 140 essential spectrum, 153, 219 essential supremum, 128 essentially bounded, 128 essentially self-adjoint operator, 60, 148, 151 exact sequence, 71 exhaustion function, 74 Fefferman-Phong inequality, 220 Fock space, 11, 34, 221 Friedrichs extension, 62, 161 Friedrichs’ Lemma, 90 fundamental form, 112 fundamental solution, 67 Gårding’s inequality, 186, 187, 192, 203 Gauß-Green Theorem, 69, 72 gauge invariance, 175

240 | Index Gelfand transform, 131, 165 Gram-Schmidt process, 6 Hölder’s inequality, 184, 205 Hörmander 𝐿2 -estimate, 119, 160 Hahn-Banach Theorem, 114 Hankel operator, 214 Helffer-Sjöstrand formula, 165 Hermitian form, 63 higher-order reflection, 94 Hilbert-Schmidt operator, 23, 41, 43, 48, 201 holomorphic vector bundle, 112 interior multiplication, 112, 179, 226 invariant subspace, 153 Kähler manifold, 112 Kato’s inequality, 172 lattice, 207 Levi form, 73 Liouville’s Theorem, 145 lower semicontinuous function, 157, 217 magnetic field, 171, 176, 218 meager set, 53 Minkowski’s inequality, 88 Nakano vanishing theorem, 113 non-coercive, 79 normal operator, 130 normal subalgebra, 129 nowhere dense set, 53 Ohsawa-Takegoshi extension theorem, 118 open-mapping theorem, 52, 53, 56 operator norm, 16 orthogonal projection, 3, 127 orthonormal, 6 orthonormal basis, 7 parallelogram rule, 2, 19 Parseval’s equation, 7 Pauli operator, 180, 218 plurisubharmonic, 74, 104, 110, 120–122, 124, 125, 180, 192, 193, 201, 202 plurisubharmonic 108 point spectrum, 140 polar decomposition, 134 polarization identity, 14, 142

positive definite matrix, 168 positive operator, 134, 148 precompact, 183–185 property (P˜ ), 202 property (P), 202, 203, 234 pseudoconvex, 73, 80, 86, 98, 115, 119, 121, 122, 202, 203, 210, 211, 213 pseudodifferential operator, 207 quadratic form, 82 Radon-Nikodym theorem, 137 Rellich-Kondrachov Lemma, 188, 195 reproducing property, 5 resolution of the identity, 127, 136 resolvent of an operator, 130 resolvent set, 130, 139 restriction of a distribution, 67 Riemann mapping, 14 Riesz representation theorem, 4, 150 Ruelle’s Lemma, 159, 160 𝑠-number, 23 Schatten-class, 23, 39, 40, 47, 48 Schrödinger operator, 104, 172, 176, 218 Schrödinger operator with magnetic field, 166, 172, 177 self-adjoint operator, 3, 6, 19, 40, 50, 60, 62, 77, 144–147, 149, 151, 155, 156, 159, 219 semibounded operator, 160 sesquilinear form, 63 set of the second category, 53 sgn-function, 91 simple function, 129 Sobolev space, 68, 186 spectral decomposition, 132 spectral theorem, 127, 131 spectrum, 129, 139, 144 spectrum of a bounded operator, 130 square root, 134 Stirling’s formula, 30 Stone-Weierstraß Theorem, 207 strictly positive operator, 158 strictly pseudoconvex, 74 strongly elliptic, 186 subharmonic function, 175, 177 support of a distribution, 67 symbolic calculus, 132 symmetric operator, 60, 151

Index | 241

tangential Cauchy-Riemann operator, 73 test function, 65 total variation, 127 totally bounded set, 16 trace, 96 trivial line bundle, 112 twist factor, 115 twisted 𝜕-complex, 114

weak null-sequence, 55 weak solution, 168 weighted Sobolev space, 195 Weyl sequence, 153 Weyl spectrum, 153 Witten Laplacian, 104, 166, 178

uniform boundedness principle, 54, 128

Zorn’s lemma, 7

De Gruyter Expositions in Mathematics

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