Kato's type inequalities for bounded linear operators in Hilbert spaces 9783030174583, 9783030174590

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SPRINGER BRIEFS IN MATHEMATICS

Silvestru Sever Dragomir

Kato’s Type Inequalities for Bounded Linear Operators in Hilbert Spaces

SpringerBriefs in Mathematics Series Editors Nicola Bellomo, Torino, Italy Michele Benzi, Pisa, Italy Palle Jorgensen, Iowa City, IA, USA Tatsien Li, Shanghai, China Roderick Melnik, Waterloo, ON, Canada Otmar Scherzer, Linz, Austria Benjamin Steinberg, New York City, NY, USA Lothar Reichel, Kent, USA Yuri Tschinkel, New York City, NY, USA George Yin, Detroit, MI, USA Ping Zhang, Kalamazoo, MI, USA Editorial Board Luis Gustavo Nonato, São Carlos, Rio Grande do Sul, Brazil Paulo J. S. Silva, Campinas, São Paulo, Brazil

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Silvestru Sever Dragomir

Kato’s Type Inequalities for Bounded Linear Operators in Hilbert Spaces

123

Silvestru Sever Dragomir Department of Mathematics, College of Engineering and Science Victoria University Melbourne, VIC, Australia School of Computer Science and Applied Mathematics, DST-NRF Centre of Excellence in the Mathematical and Statistical Sciences University of the Witwatersrand Johannesburg, South Africa

ISSN 2191-8198 ISSN 2191-8201 (electronic) SpringerBriefs in Mathematics ISBN 978-3-030-17458-3 ISBN 978-3-030-17459-0 (eBook) https://doi.org/10.1007/978-3-030-17459-0 Mathematics Subject Classification (2010): 47A63, 47A50, 47A99 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

This book is dedicated to my granddaughters Sienna and Audrey.

Preface

Linear operator theory in Hilbert spaces plays a central role in contemporary mathematics with numerous applications for partial differential equations, in approximation theory, optimization theory, numerical analysis, probability theory and statistics and other fields. The main aim of this book is to present several results related to Kato’s famous inequality for bounded linear operators on complex Hilbert spaces obtained by the author in a sequence of recent research papers. The book is intended for use by both researchers in various fields of linear operator theory and mathematical inequalities, domains which have grown exponentially in the last decade, as well as by postgraduate students and scientists applying inequalities in their specific areas. The monograph starts with a short introductory chapter where the famous Kato’s inequality is introduced and some simple, however, important particular cases are revealed. In the second chapter, we present several multiplicative and additive generalizations of Kato’s inequality for n-tuple of bounded linear operators on the Hilbert space ðH; h
;
iÞ. Applications for functions of normal operators defined by power series and inequalities for Euclidian norm, s-1-norm and s-1-numerical radius of n-tuples of operators are provided as well. In the third chapter, we present a two-parameter generalization of Kato due to Furuta. Applications for functions of bounded linear operators defined by power series, inequalities for four bounded operators generalizing Furuta’s inequality and some general norm and numerical radius inequalities are given as well. In the fourth chapter, after recalling some fundamental facts on Hilbert–Schmidt operators, trace operators and some properties of traces of such operators, we present a trace version of Kato’s inequality. Some natural functionals associated with this inequality and their superadditivity and monotonicity are established. Several inequalities for sequences of operators and power series of operators are given as well.

vii

viii

Preface

In the fifth chapter, after recalling some fundamental facts on Bochner integral for measurable functions with values in Banach spaces, we provide an integral version of Kato’s inequality. Several norm and numerical radius inequalities with applications for the operator exponential are also given. For the sake of completeness, all the results presented are completely proved and the original references where they have been firstly obtained are mentioned. Melbourne, Australia

Silvestru Sever Dragomir

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2 Inequalities for n-Tuples of Operators . . . . . . . . . . . . . . . 2.1 Multiplicative Inequalities . . . . . . . . . . . . . . . . . . . . . 2.2 Functional Inequalities . . . . . . . . . . . . . . . . . . . . . . . 2.3 Inequalities for the Euclidian Norm . . . . . . . . . . . . . . 2.4 Inequalities for s-1-Norm and s-1-Numerical Radius . . 2.5 Additive Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Inequalities for Functions of Normal Operators . . . . . . 2.7 Applications for the Euclidian Norm . . . . . . . . . . . . . 2.8 Applications for s-1-Norm and s-1-Numerical Radius . 2.9 Other Additive Inequalities . . . . . . . . . . . . . . . . . . . . 2.10 Other Inequalities for Functions of Normal Operators . 2.11 Examples for the Euclidian Norm . . . . . . . . . . . . . . . 2.12 Examples for s-1-Norm and s-1-Numerical Radius . . .

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5 5 9 15 19 24 30 36 37 40 49 55 58

3 Generalizations of Furuta’s Type . . . . . . . . . . . . 3.1 Furuta’s Inequality . . . . . . . . . . . . . . . . . . . 3.2 Functional Generalizations . . . . . . . . . . . . . 3.3 Some Examples . . . . . . . . . . . . . . . . . . . . . 3.4 More Functional Inequalities . . . . . . . . . . . . 3.5 Applications for Some Elementary Functions 3.6 General Vector Inequalities . . . . . . . . . . . . . 3.7 Norm and Numerical Radius Inequalities . . .

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61 61 63 68 70 77 79 84

4 Trace Inequalities . . . . . . . . . . . . . . . . . . . . . . 4.1 Trace of Operators . . . . . . . . . . . . . . . . . 4.2 Trace Inequalities via Kato’s Result . . . . . 4.3 Some Functional Properties . . . . . . . . . . . 4.4 Inequalities for Sequences of Operators . . 4.5 Inequalities for Power Series of Operators

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ix

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5 Integral Inequalities . . . . . . . . . . . . . . . . . . . . 5.1 Some Facts on Bochner Integral . . . . . . . 5.2 Applications of Kato’s Inequality . . . . . . . 5.3 Norm and Numerical Radius Inequalities . 5.4 Applications for the Operator Exponential

Contents

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109 109 111 115 122

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Chapter 1

Introduction

We denote by B (H ) the Banach algebra of all bounded linear operators on a complex Hilbert space (H ; ·, ·) . If P is a positive selfadjoint operator on H, i.e. P x, x ≥ 0 for any x ∈ H, then the following inequality is a generalization of the Schwarz inequality in H |P x, y|2 ≤ P x, x P y, y ,

(1.1)

for any x, y ∈ H. The following inequality is of interest as well, see [21, p. 221]. Let P be a positive selfadjoint operator on H. Then P x2 ≤ P P x, x

(1.2)

for any x ∈ H. The “square root” of a positive bounded selfadjoint operator on H can be defined as follows, see for instance [21, p. 240]: If the operator A ∈ B (H ) is selfadjoint √ and positive, then there exists a unique positive selfadjoint operator B := A ∈ B (H ) such thatB 2 = A.If A is invertible, then so isB. If A ∈ B (H ) , then the operator√A∗ A is selfadjoint and positive. Define the “absolute value” operator by |A| := A∗ A. In 1952, Kato [22] proved the following celebrated generalization of Schwarz inequality for any bounded linear operator T on H :  1−α  α   |T x, y|2 ≤ T ∗ T x, x T T ∗ y, y ,

(1.3)

for any x, y ∈ H, α ∈ [0, 1] . Utilizing the modulus notation introduced before, we can write (1.3) as follows © The Author(s), under exclusive license to Springer Nature Switzerland AG 2019 S. S. Dragomir, Kato’s Type Inequalities for Bounded Linear Operators in Hilbert Spaces, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-030-17459-0_1

1

2

1 Introduction

    2(1−α) |T x, y|2 ≤ |T |2α x, x T ∗  y, y

(1.4)

for any x, y ∈ H, α ∈ [0, 1] . It is useful to observe that, if T = N , a normal operator, i.e., we recall that N N ∗ = N ∗ N , then the inequality (1.4) can be written as    |N x, y|2 ≤ |N |2α x, x |N |2(1−α) y, y ,

(1.5)

and in particular, for selfadjoint operators A we can state it as |Ax, y| ≤ |A|α x |A|1−α y

(1.6)

for any x, y ∈ H, α ∈ [0, 1] . If T = U, a unitary operator, i.e., we recall that UU ∗ = U ∗ U = 1 H , then the inequality (1.4) becomes |U x, y| ≤ x y for any x, y ∈ H, which provides a natural generalization for the Schwarz inequality in H. The symmetric powers in the inequalities above are natural to be considered, so if we choose in (1.4), (1.5) and in (1.6) α = 1/2 then we get for any x, y ∈ H

and

   |T x, y|2 ≤ |T | x, x T ∗  y, y ,

(1.7)

|N x, y|2 ≤ |N | x, x |N | y, y ,

(1.8)

|Ax, y| ≤ |A|1/2 x |A|1/2 y

(1.9)

respectively. It is also worthwhile to observe that, if we take the supremum over y ∈ H, y = 1 in (1.4) then we get   T x2 ≤ T 2(1−α) |T |2α x, x (1.10) for any x ∈ H, or in an equivalent form T x ≤ |T |α x T 1−α

(1.11)

for any x ∈ H. If we take α = 1/2 in (1.10), then we get T x2 ≤ T  |T | x, x

(1.12)

1 Introduction

3

for any x ∈ H, which in the particular case of T = P, a positive operator, provides the result from (1.2). For various interesting generalizations, extension and Kato related results, see the papers [11, 19, 28, 29, 33]. In this monograph we present several recent inequalities related to Kato’s famous result (1.3) obtained by the author in the sequence of research papers [3, 10].

Chapter 2

Inequalities for n-Tuples of Operators

In this chapter we present several multiplicative and additive generalizations of Kato’s inequality for n-tuple of bounded linear operators on the Hilbert space (H ; ·, ·). Applications for functions of normal operators defined by power series and inequalities for Euclidian Norm, s-1-Norm and s-1-Numerical Radius of n-tuples of operators are provided as well.

2.1 Multiplicative Inequalities The following vector inequality holds: Theorem 2.1 (Dragomir [3]) Let (T1 , ..., Tn ) ∈ B (H ) × ... × B (H ) := B (n) (H ) be an n-tuple of bounded linear operators on the Hilbert space (H ; ·, ·) and ( p1 , ..., pn ) ∈ R∗n + an n-tuple of nonnegative weights not all of them equal to zero. Then we have n  j=1

 n α  n 1−α   2   2  2 ∗      p j T j x, y ≤ p j T j x, x p j T j y, y j=1

(2.1)

j=1

for any x, y ∈ H with x = y = 1 and α ∈ [0, 1] . Proof We must prove the inequalities only in the case α ∈ (0, 1), since the case α = 0 or α = 1 follows directly from the corresponding case of Kato’s inequality. Utilizing Kato’s inequality for the operator T j , j ∈ {1, ..., n} we have

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2019 S. S. Dragomir, Kato’s Type Inequalities for Bounded Linear Operators in Hilbert Spaces, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-030-17459-0_2

5

6

2 Inequalities for n-Tuples of Operators n 

n      2  2α 2(1−α) p j  T j x, y  ≤ p j T j  x, x T j∗  y, y

j=1

(2.2)

j=1

≤

n 

  α   1−α 2 2 p j T j  x, x T j∗  y, y

j=1

for any x, y ∈ H with x = y = 1, where for the last inequality we have used the Hölder-McCarthy inequality P r x, x ≤ P x, xr that holds for any positive operator P and any power r ∈ (0, 1) . Now, on making use of the weighted Hölder discrete inequality n 

⎛ pjajbj ≤ ⎝

j=1

n 

⎞1/ p ⎛ p pjaj ⎠

⎝

j=1

n 

⎞1/q q pjbj ⎠

, p, q > 1,

j=1

where (a1 , ..., an ) , (b1 , ..., bn ) ∈ Rn+ , and choose a j =   1−α  ∗ 2 1 , p = α1 and q = 1−α then we get T j  y, y n 

1 1 + = 1, p q

  α T j 2 x, x , b j =

  α   1−α 2 2 p j T j  x, x T j∗  y, y

(2.3)

j=1

⎧ n ⎨

⎫α ⎧ ⎫1−α  n   α 1/α ⎬ ⎨ 1−α 1/(1−α) ⎬  ∗ 2 2 ≤ p j T j  x, x p j T j  y, y ⎩ ⎭ ⎩ ⎭ j=1 j=1 ⎧ ⎫α ⎧ ⎫1−α n n ⎨     ⎬ ⎨ ⎬ 2 2 = p j T j  x, x p j T j∗  y, y ⎩ ⎭ ⎩ ⎭ j=1

=

 n  j=1

 2 p j T j  x, x

α 

j=1

n 

 2 p j T ∗  y, y

1−α

j

j=1

for any x, y ∈ H with x = y = 1. Utilizing (2.2) and (2.3) we deduce the desired inequality (2.1).



Remark 2.2 The inequality (2.1) becomes for y = x the following simpler result that is useful for deriving numerical radius inequalities:

2.1 Multiplicative Inequalities n 

7

 n α  n 1−α   2   2  2 p j  T j x, x  ≤ p j T j  x, x p j T j∗  x, x

j=1

j=1

≤

 n 

j=1

    2  2 p j α T j  + (1 − α) T j∗  x, x

(2.4)



j=1

for any x ∈ H with x = 1. Let (N1 , ..., Nn ) ∈ B (n) (H ) be an n-tuple of normal operators on the Hilbert space (H ; ·, ·) . Then from the above Theorem 2.1 we have the following result that can be utilized in obtaining various inequalities for functions of normal operators defined by power series, namely: n 

 n α  n 1−α   2   2  2       p j N j x, y ≤ p j N j x, x p j N j y, y

j=1

j=1

(2.5)

j=1

for any x, y ∈ H with x = y = 1, α ∈ [0, 1] and any n-tuple weights ( p1 , ..., pn ) ∈ R∗n +. In particular, we get from (2.5) the following inequality for modulus of normal operators  n  n    2  2 p j  N j x, x  ≤ p j  N j  x, x (2.6) j=1

j=1

for any x ∈ H with x = 1. The following result provides upper bounds for the sum

n 

  p j  T j x, y  and has

j=1

important consequences in refining the fundamental triangle inequality for operator norm. Theorem 2.3 (Dragomir [3]) With the assumptions in Theorem 2.1 we have n 

 n 1/2  n 1/2   2α   2(1−α)   ∗      p j T j x, y ≤ p j T j x, x p j Tj y, y

j=1

j=1

(2.7)

j=1

for any x, y ∈ H. Proof From Kato’s inequality for the operator T j , j ∈ {1, ..., n} we have n 

n   1/2   1/2    2α T ∗ 2(1−α) y, y p j  T j x, y  ≤ p j T j  x, x j

j=1

for any x, y ∈ H .

j=1

(2.8)

8

2 Inequalities for n-Tuples of Operators

Now, on making use of the weighted Cauchy–Bunyakovsky–Schwarz discrete inequality ⎞1/2 ⎛ ⎞1/2 ⎛ n n n    pjajbj ≤ ⎝ p j a 2j ⎠ ⎝ p j b2j ⎠ j=1

j=1

j=1

  1/2 2α where (a1 , ..., an ) , (b1 , ..., bn ) ∈ Rn+ , and choose a j = T j  x, x and b j =   1/2  ∗ 2(1−α) y, y , then we get T j  n 

  1/2   1/2 2α T ∗ 2(1−α) y, y p j T j  x, x j

(2.9)

j=1

⎧ n ⎨

⎫1/2 ⎧ ⎫1/2   n 1/2 2 ⎬ ⎨ 1/2 2 ⎬  2α  ∗ 2(1−α) ≤ p j T j  x, x p j T j  y, y ⎩ ⎭ ⎩ ⎭ j=1 j=1 ⎧ ⎫1/2 ⎧ ⎫1/2 n n ⎨     ⎬ ⎨ ⎬ 2α 2(1−α) = p j T j  x, x p j T j∗  y, y ⎩ ⎭ ⎩ ⎭ j=1

=

 n 

 2α p j T j  x, x

1/2  n 

j=1

 2(1−α) p j T ∗  y, y

1/2

j

j=1

j=1

for any x, y ∈ H .



Remark 2.4 One of possible vector-valued extensions of (2.7) is as follows: n 

 n 1/2  n 1/2   2α   2(1−α)   ∗ p j  T j x, x  ≤ p j T j  x, x p j T j  x, x

j=1

j=1

 ≤

n  j=1

(2.10)

j=1

⎡   2(1−α) ⎤  T j 2α + T ∗  j ⎥ ⎢ pj ⎣ ⎦ x, x 2

for any x ∈ H. Remark 2.5 The symmetric case for powers, namely the case α = interest since will produce the simpler result n 

1 2

in (2.7) is of

 n 1/2  n 1/2         ∗ p j  T j x, y  ≤ p j T j  x, x p j T j  y, y

(2.11)

j=1

for any x, y ∈ H.

j=1

j=1

2.1 Multiplicative Inequalities

9

In particular, from (2.10) we derive n 

 n 1/2  n 1/2         ∗      p j T j x, x ≤ p j T j x, x p j T j x, x

j=1

j=1

≤

 n  j=1

(2.12)

j=1

⎡     ⎤  T j  + T ∗  j ⎣ ⎦ pj x, x 2

for any x ∈ H. Let (N1 , ..., Nn ) ∈ B (n) (H ) be an n-tuple of normal operators on the Hilbert space (H ; ·, ·) . Then from the above Theorem 2.3 we have n 

 n 1/2  n 1/2   2α   2(1−α)   p j  N j x, y  ≤ p j  N j  x, x p j N j  y, y

j=1

j=1

(2.13)

j=1

for any x, y ∈ H. In particular, we have n 

 n 1/2  n 1/2   2α   2(1−α)   p j  N j x, x  ≤ p j  N j  x, x p j N j  x, x

j=1

j=1

≤

 n 

j=1

pj

!  2α  2(1−α) " N j  + N j  2

j=1

(2.14)

 x, x

for any x ∈ H.

2.2 Functional Inequalities  n Now, by the help of power series f (z) = ∞ n=0 an z we can naturally construct another power series which will have as coefficients values of the coef the absolute n |a | z . It is obvious that this ficient of the original series, namely, f A (z) := ∞ n=0 n new power series will have the same radius of convergence as the original series. We also notice that if all coefficients an ≥ 0, then f A = f , see also [32, p. 246]. As some natural examples that are useful for applications, we can point out that, if f (z) =

∞  (−1)n n=1

n

z n = ln

1 , z ∈ D (0, 1) ; 1+z

(2.15)

10

2 Inequalities for n-Tuples of Operators

g (z) =

∞  (−1)n n=0

(2n)!

z 2n = cos z, z ∈ C;

∞  (−1)n 2n+1 h (z) = = sin z, z ∈ C; z (2n + 1)! n=0

l (z) =

∞ 

(−1)n z n =

n=0

1 , z ∈ D (0, 1) ; 1+z

then the corresponding functions constructed by the use of the absolute values of the coefficients are ∞  1 n 1 f A (z) = z = ln , z ∈ D (0, 1) ; n 1 − z n=1

g A (z) =

∞  n=0

h A (z) =

∞  n=0

l A (z) =

∞ 

(2.16)

1 2n z = cosh z, z ∈ C; (2n)! 1 z 2n+1 = sinh z, z ∈ C; (2n + 1)! zn =

n=0

1 , z ∈ D (0, 1) . 1−z

The following result is a functional generalization of Kato’s inequality for normal operators from (1.5). ∞ n Theorem 2.6 (Dragomir [3]) Let f (z) = n=0 an z be a function defined by power series with real coefficients and convergent on the open disk D (0, R) := {z ∈ C, |z| < R}, R > 0. If N is a normal operator on the Hilbert space H and for α ∈ (0, 1) we have that N 2α , N 2(1−α) < R, then we have the inequalities  # $ 1/2  # 2(1−α) $ 1/2 | f (N ) x, y| ≤ f A |N |2α x, x f A |N | y, y

(2.17)

for any x, y ∈ H. In particular, if N  < R, then | f (N ) x, y| ≤  f A (|N |) x, x1/2  f A (|N |) y, y1/2 for any x, y ∈ H. Proof If N is a normal operator, then for any j ∈ N we have that  j 2 # ∗ $ j  N  = N N = |N |2 j . Now, utilising the inequality (2.13) we can write that

(2.18)

2.2 Functional Inequalities

11

   n     a j N j x, y     j=0 ≤

(2.19)

n     j  a j   N x, y  j=0

 n 1/2  n 1/2     2α     2(1−α) j j a j   N  x, x a j   N  ≤ y, y 

j=0

n    # 2α $ j a j  |N | = x, x

j=0

1/2 

j=0

n    # 2(1−α) $ j a j  |N | y, y

1/2

j=0

for any x, y ∈ H and n ∈ N. Since N 2α , N 2(1−α) < R, then it follows that the series

∞  # $  a j  |N |2α j and j=0

∞  # $  a j  |N |2(1−α) j are absolute convergent in B (H ), and by taking the limit over j=0

n → ∞ in (2.19) we deduce the desired result (2.17).



Remark 2.7 Assume that f, R, N and α are as in Theorem 2.6. If we take the supremum in (2.17) over y ∈ H, y = 1, then we get $ 1/2 % # 2(1−α) $%1/2  # % % f A |N |  f (N ) x ≤ f A |N |2α x, x

(2.20)

for any x ∈ H , which produces the operator norm inequality % # $%1/2 % # $%1/2  f (N ) ≤ % f A |N |2α % % f A |N |2(1−α) % .

(2.21)

If we take y = x in (2.17), then we get $ 1/2  # 2(1−α) $ 1/2  # | f (N ) x, x| ≤ f A |N |2α x, x f A |N | x, x ! #  $ # $" f A |N |2α + f A |N |2(1−α) ≤ x, x 2

(2.22)

for any x ∈ H. This produces the following inequalities for the numerical radius ⎧% # $% % # 2(1−α) $%1/2 2α %1/2 % % ; % ⎪ f A |N | ⎨ f A |N | w ( f (N )) ≤ % % ⎪ f A (|N |2α )+ f A (|N |2(1−α) ) % ⎩% % %. 2

(2.23)

Making use of the examples in (2.15) and (2.16) we can state the vector inequalities:

12

2 Inequalities for n-Tuples of Operators

   ln (1 H + N )−1 x, y   # 1/2  # 1/2 $−1 $−1 ≤ ln 1 H − |N |2α ln 1 H − |N |2α x, x y, y , N  < 1;    (1 H + N )−1 x, y  # 1/2 # 1/2 $−1 $−1 ≤ 1 H − |N |2α 1 H − |N |2α x, x y, y , N  < 1; |sin (N ) x, y|  # $ 1/2  # $ 1/2 ≤ sinh |N |2α x, x , for any N ; sinh |N |2(1−α) y, y |cos (N ) x, y|  # $ 1/2  # $ 1/2 ≤ cosh |N |2α x, x cosh |N |2(1−α) y, y , for any N ; for any x, y ∈ H. We have, for instance, the following norm inequalities as well: % # # $%1/2 % $%1/2 sin (N ) ≤ %sinh |N |2α % %sinh |N |2(1−α) % ; % # # $%1/2 % $%1/2 cos (N ) ≤ %cosh |N |2α % %cosh |N |2(1−α) % for any normal operator N and % % % % # $−1 % $−1 % %1/2 % # %1/2 %ln (1 H + N )−1 % ≤ % %ln 1 H − |N |2α % %ln 1 H − |N |2α % for N with N  < 1. If we utilize the following function as power series representations with nonnegative coefficients: ' (  ∞ 1+z 1 1 ln = z 2n−1 , z ∈ D (0, 1) ; 2 1−z 2n − 1 n=1 $ # ∞   n + 21 −1 sin (z) = z ∈ D (0, 1) ; z 2n+1 , √ π + 1) n! (2n n=0 tanh−1 (z) = 2 F1

(α, β, γ, z) =

∞  n=1 ∞  n=0

1 z 2n−1 , 2n − 1

z ∈ D (0, 1)

 (n + α)  (n + β)  (γ) n z , α, β, γ > 0, n! (α)  (β)  (n + γ)

z ∈ D (0, 1) ;

(2.24)

2.2 Functional Inequalities

13

where  is the Gamma function, then we can state the following vector inequalities: |exp (N ) x, y|  # $ 1/2  # $ 1/2 ≤ exp |N |2α x, x exp |N |2(1−α) y, y ;

(2.25)

 ' (     ln 1 H + N x, y    1H − N * 1/2  ' ( 1/2  ) 1 H + |N |2α 1 H + |N |2(1−α) ≤ ln x, x ; ln y, y 1 H − |N |2α 1 H − |N |2(1−α)  −1   sin (N ) x, y   # $ 1/2  −1 # 2(1−α) $ 1/2 |N | ≤ sin−1 |N |2α x, x ; sin y, y    tanh−1 (N ) x, y   # $ 1/2  # $ 1/2 ≤ tanh−1 |N |2α x, x tanh−1 |N |2(1−α) y, y ; |2 F1 (α, β, γ, N ) x, y| # $ 1/2  # $ 1/2  2(1−α) y, y ; ≤ 2 F1 α, β, γ, |N |2α x, x 2 F1 α, β, γ, |N | for any x, y ∈ H. The first inequality in (2.25) holds for any normal operator N while the other ones request the assumption N  < 1. We also have the norm inequalities % # # $%1/2 % $%1/2 exp (N ) ≤ %exp |N |2α % %exp |N |2(1−α) % ; % # # $%1/2 % $%1/2 cosh (N ) ≤ %cosh |N |2α % %cosh |N |2(1−α) % ; % # # $%1/2 % $%1/2 sinh (N ) ≤ %sinh |N |2α % %sinh |N |2(1−α) % ; for any normal operator N and % ) *%1/2 % ' (% % ' 2α (% 2(1−α) % %1/2 % % % % |N | |N | 1 1 1 + N + + H H H % % % %ln % ≤ %ln %ln % % % % % 2α 2(1−α) % % 1H − N 1 H − |N | 1 H − |N | for N with N  < 1. A similar result is the following one:  n Theorem 2.8 (Dragomir [3]) Let f (z) = ∞ n=0 an z be a function defined by power series with real coefficients and convergent on the open disk D (0, R) ⊂ C, R > 0. If N is a normal operator on the Hilbert space H , z ∈ C such that |z|2 , |z| N ,

14

2 Inequalities for n-Tuples of Operators

N 2 < R, then we have the inequalities # $ # $ α  # 2 $ 1−α | f (z N ) x, y|2 ≤ f A |z|2 f A |N |2 x, x f A |N | y, y

(2.26)

for any x, y ∈ H and α ∈ [0, 1] . In particular, we have # $ # $ 1/2  # 2 $ 1/2 | f (z N ) x, y|2 ≤ f A |z|2 f A |N |2 x, x . f A |N | y, y

(2.27)

Proof By the Cauchy–Bunyakowsky–Schwarz inequality we have  2  n n n       2j     j  j j     a j   N x, y 2 |z| a a z N x, y ≤ j j     j=0 j=0 j=0

(2.28)

for any n ∈ N and x, y ∈ H. Utilising (2.5) we also have  n α  n 1−α n      2     2    j  a j   N x, y 2 ≤ a j   N j  x, x a j   N j  y, y j=0

j=0

j=0

j=0

j=0

(2.29)

α  n 1−α  n     2 j 2 j a j  |N | x, x a j  |N | y, y =

for any n ∈ N and x, y ∈ H. By making use of (2.28) and (2.29) we get  2  n     a j z j N j x, y     j=0 α  n 1−α  n n      2j    2j 2j       a j |z| a j |N | x, x a j |N | y, y ≤ j=0

j=0

(2.30)

j=0

for any n ∈ N and x, y ∈ H. ∞    a j  |N |2 j is absolutely convergent, taking the limit over n → Since the series j=0

∞ in (2.30) produces the desired result (2.26).



Remark 2.9 Assume that f, R, z, N and α are as in Theorem 2.8. If we take the supremum in (2.26) over y ∈ H, y = 1, then we get # $ # $ α % # $%1−α  f (z N ) x2 ≤ f A |z|2 f A |N |2 x, x % f A |N |2 %

(2.31)

2.2 Functional Inequalities

15

for any x ∈ H , which produces the operator norm inequality # $% # $%  f (z N )2 ≤ f A |z|2 % f A |N |2 % .

(2.32)

If we take y = x in (2.26), then we get # $ # $  | f (z N ) x, x|2 ≤ f A |z|2 f A |N |2 x, x

(2.33)

for any x ∈ H. From (2.26) we get the vector inequalities # $ α  # $ 1−α # $ |exp (z N ) x, y|2 ≤ exp |z|2 exp |N |2 x, x exp |N |2 y, y , # $ α  # $ 1−α # $ |sin (z N ) x, y|2 ≤ sinh |z|2 sinh |N |2 x, x sinh |N |2 y, y , and |cos (z N ) x, y|2 # $ α  # $ 1−α # $ ≤ cosh |z|2 cosh |N |2 x, x cosh |N |2 y, y , for any normal operator N , any complex number z and any x, y ∈ H. We have, for instance, from (2.32) the following norm inequalities as well: # $% # $% exp (z N )2 ≤ exp |z|2 %exp |N |2 % and

# $% # $% sin (z N )2 ≤ sinh |z|2 %sinh |N |2 %

for any normal operator N and any complex number z. Similar results can be stated for other functions, however the details are omitted.

2.3 Inequalities for the Euclidian Norm In [30], the author has introduced the following norm on the Cartesian product B (n) (H ) := B (H ) × · · · × B (H ), where B (H ) denotes the Banach algebra of all bounded linear operators defined on the complex Hilbert space H : (T1 , . . . , Tn )e :=

sup

(λ1 ,...,λn )∈Bn

λ1 T1 + · · · + λn Tn  ,

(2.34)

16

2 Inequalities for n-Tuples of Operators

   + , 2  where (T1 , . . . , Tn ) ∈ B (n) (H ) and Bn := (λ1 , . . . , λn ) ∈ Cn  nj=1 λ j  ≤ 1 is the Euclidean closed ball in Cn . It is clear that ·e is a norm on B (n) (H ) and for any (T1 , . . . , Tn ) ∈ B (n) (H ) we have %# $% (T1 , . . . , Tn )e = % T1∗ , . . . , Tn∗ %e , where T j∗ is the adjoint operator of T j , j ∈ {1, . . . , n} . We call this the Euclidian norm of an n-tuple of operators (T1 , . . . , Tn ) ∈ B (n) (H ) . It has been shown in [30] that the following basic inequality for the Euclidian norm holds true: 1 √ n

% % %1 %1 % % %2 %2 % n  ∗ 2 % % n  ∗ 2 % % T  % ≤ (T1 , . . . , Tn )e ≤ % T  % j % j % % % % j=1 % j=1 % %

(2.35)

for any n-tuple (T1 , . . . , Tn ) ∈ B (n) (H ) and the constants √1n and 1 are best possible. In the same paper [30] the author has introduced the Euclidean operator radius of an n-tuple of operators (T1 , . . . , Tn ) by ⎛ ⎞ 21 n   2  T j x, x  ⎠ we (T1 , . . . , Tn ) := sup ⎝ x=1

(2.36)

j=1

and proved that we (·) is a norm on B (n) (H ) and satisfies the double inequality: 1 (T1 , . . . , Tn )e ≤ we (T1 , . . . , Tn ) ≤ (T1 , . . . , Tn )e 2

(2.37)

for each n-tuple (T1 , . . . , Tn ) ∈ B (n) (H ) . As pointed out in [30], the Euclidean numerical radius also satisfies the double inequality: % % %1 %1 % n % n %2 %2   2 %   2 % % 1 % ∗ ∗ T  % ≤ we (T1 , . . . , Tn ) ≤ % T  % √ % j % j % % 2 n% % j=1 % j=1 % %

(2.38)

for any (T1 , . . . , Tn ) ∈ B (n) (H ) and the constants 2√1 n and 1 are best possible. In [2], by utilizing the concept of hypo-Euclidean norm on H n we obtained the following representation for the Euclidian norm: Proposition 2.10 (Dragomir [2]) For any (T1 , . . . , Tn ) ∈ B (n) (H ) we have

2.3 Inequalities for the Euclidian Norm

17

⎛ (T1 , . . . , Tn )e =

sup

y=1,x=1

⎞ 21 n   2  T j y, x  ⎠ . ⎝

(2.39)

j=1

The following different lower bound for the Euclidean operator norm ·e was also obtained in [2]: Proposition 2.11 (Dragomir [2]) For any (T1 , . . . , Tn ) ∈ B (n) (H ), we have 1 (T1 , . . . , Tn )e ≥ √ T1 + · · · + Tn  . n

(2.40)

Utilizing some techniques based on the Boas-Bellman and Bombieri type inequalities we obtained in [2] the following upper bounds: Proposition 2.12 (Dragomir [2]) For any (T1 , . . . , Tn ) ∈ B (n) (H ), we have the inequalities: ⎧ ! "1 +% % , ⎪ # ∗ $ 2  ⎪ 2 ⎪ 2 ⎪ max %T j % + w Tk T j ; ⎪ ⎪ 1≤ j≤n ⎪ 1≤ j =k≤n ⎪ +% % , ⎪ - # $. ⎪ 2 ⎪ ⎪ max %T j % + (n − 1) max w Tk∗ T j ; ⎨ 1≤ j≤n 1≤ j =k≤n +% % , %   %2 (T1 , . . . , Tn )2e ≤  2 % n 2% ⎪ % %  ⎪ max T j % j=1 T j  % ⎪ 1≤ ⎪ j≤n ⎪ ⎪ "1 ⎪ ⎪ 0 2 / ⎪ % % % . -%  ⎪ ⎪ ⎪ + max %T j % %Tk % w Tk T j∗ ⎩ 1≤ j =k≤n

and

(2.41)

1≤ j =k≤n

⎧ 1 n 2  # ∗ $ ⎪ ⎪ max w Tk T j ; ⎪ ⎪ ⎪ 1≤ j≤n k=1 ⎪ ⎪ "1 ! ⎪ ⎪ ⎪ n # ∗ $ 2  ⎪ ⎪ 2 ⎪ w Tk T j ; ⎪ ⎨ j,k=1  n 1 (T1 , . . . , Tn )2e ≤  2# ∗ $ 2 ⎪ ⎪ ⎪ n max w Tk T j ; ⎪ ⎪ 1≤ j≤n k=1 ⎪ ⎪ 1 ⎪ ! " ⎪ ⎪ n ⎪ - 2 # ∗ $. 2  ⎪ ⎪ max w Tk T j . ⎪ ⎩n 1≤k≤n

(2.42)

j=1

Now we can provide now a different upper bound for the Euclidian norm: Proposition 2.13 (Dragomir [3]) Let (T1 , ..., Tn ) ∈ B (n) (H ) be an n-tuple of bounded linear operators on the Hilbert space (H ; ·, ·). Then we have

18

2 Inequalities for n-Tuples of Operators

% %α % %1−α % n % % n %   2 % % % %   T j  % %  T ∗ 2 % (T1 , . . . , Tn )2e ≤ % j % % % % % j=1 % % j=1 %

(2.43)

and ⎡

n   2 T j  x, x we2 (T1 , . . . , Tn ) ≤ sup ⎣ x=1

≤

α  n 1−α ⎤   2 T ∗  x, x ⎦

j=1

j

(2.44)

j=1

⎧ % % % % %n  2 % α %n  ∗ 2 % 1−α ⎪ ⎪ % % % % ⎪ , T  ⎪ % j=1 T j  % ⎨ % j=1 j % %  ⎪  2 % ⎪ %n  2 % ⎪ ⎪ T j  + (1 − α) T ∗  % , ⎩% α j % j=1 %

for any α ∈ [0, 1]. Proof Utilizing the vector inequality (2.1) and taking the supremum over y = 1, x = 1 we have ⎡

⎤α ⎡ ⎤1−α  n  n   2   2 T j  x, x ⎦ ⎣ sup T ∗  y, y ⎦ (T1 , . . . , Tn )2e ≤ ⎣ sup j x=1

j=1

y=1

j=1

(2.45) for any α ∈ [0, 1] and since  sup

x=1

and

%  % % % n  % n  2 %  2 T j  x, x = % T j  % % % % j=1 % j=1

%  %  n % % n   2 % %   2 ∗ ∗ % %   sup T j y, y = % Tj % y=1 j=1 % j=1 %

we get from (2.45) the desired result (2.43). Now from the first inequality in (2.4) we have ⎡ α  n 1−α ⎤ n    2   2 T j  x, x T ∗  x, x ⎦ we2 (T1 , . . . , Tn ) ≤ sup ⎣ j x=1

j=1

j=1

(2.46)

⎤α ⎡ ⎤1−α  n  n   2   2 T j  x, x ⎦ ⎣ sup T ∗  x, x ⎦ ≤ ⎣ sup j ⎡

x=1

j=1

x=1

j=1

2.3 Inequalities for the Euclidian Norm

19

%⎤α ⎡% %⎤1−α ⎡% % n % n % % %  2 % %  ∗  2 % T j  %⎦ ⎣% T  %⎦ = ⎣% j % % % % % j=1 % j=1 % % and from the second inequality in (2.4) we also have we2

  n    2  ∗ 2  α T j  + (1 − α) T j  x, x (T1 , . . . , Tn ) ≤ sup x=1

(2.47)

j=1

% % % % % n  2  ∗ 2 % %     =% α T j + (1 − α) T j % % % j=1 % for any α ∈ [0, 1]. Utilizing (2.46) and (2.47) we get (2.44).



Remark 2.14 The case when α = 1/2 provides the inequalities % %1/2 % %1/2 % % % % n n % % % %      T j 2 % %  T ∗ 2 % (T1 , . . . , Tn )2e ≤ % j % % % % % j=1 % % j=1 %

(2.48)

and ⎡ 1/2  n 1/2 ⎤ n       T j 2 x, x T ∗ 2 x, x ⎦ we2 (T1 , . . . , Tn ) ≤ sup ⎣ j x=1

j=1

(2.49)

j=1

⎧ % % % % %n  2 % 1/2 %n  ∗ 2 % 1/2 ⎪ ⎪ % % % % ⎪ T T ,     ⎪ % j=1 j % % j=1 j % ⎪ ⎨ ≤ % !  2 "% ⎪ % ⎪ |T j |2 +T j∗  % ⎪ % n % ⎪ ⎪ %. ⎩% 2 % j=1 %

2.4 Inequalities for s-1-Norm and s-1-Numerical Radius We can introduce the s-p-norm of the n-tuple of operators (T1 , . . . , Tn ) ∈ B (n) (H ) by ⎡⎛ ⎞ 1p ⎤ n     ⎢⎝  T j y, x  p ⎠ ⎥ (T1 , . . . , Tn )s, p := (2.50) sup ⎣ ⎦. y=1,x=1

j=1

Indeed this is a norm, since by the Minkowski inequality we have

20

2 Inequalities for n-Tuples of Operators

(T1 , . . . , Tn ) + (V1 , . . . , Vn )s, p ⎡⎛ ⎞ 1p ⎤ n    p ⎥ ⎢⎝  T j y, x + V j y, x  ⎠ ⎦ = sup ⎣ y=1,x=1

≤

≤

(2.51)

j=1

sup

⎡⎛ ⎞ 1p ⎛ ⎞ 1p ⎤ n n         ⎢⎝  T j y, x  p ⎠ + ⎝  V j y, x  p ⎠ ⎥ ⎣ ⎦

sup

⎛ ⎞ 1p n    p  T j y, x  ⎠ + ⎝

y=1,x=1

y=1,x=1

j=1

j=1

j=1

⎛ sup

y=1,x=1

⎞ 1p n    p  V j y, x  ⎠ ⎝ j=1

= (T1 , . . . , Tn )s, p + (V1 , . . . , Vn )s, p , which proves the triangle inequality. The other properties of the norm are obvious. For p = 2 we get (T1 , . . . , Tn )s,2 = (T1 , . . . , Tn )e . We are interested in this section in the case p = 1, namely on the s-1-norm defined by n     T j y, x  . (T1 , . . . , Tn )s,1 := sup y=1,x=1 j=1

       Since for any x, y ∈ H we have nj=1  T j y, x  ≥  nj=1 T j y, x , then by the properties of the supremum we get the basic inequality % % % n % n  % % % % % % %T j % . (T ≤ T , . . . , T ≤ ) j% 1 n s,1 % % j=1 % j=1

(2.52)

Similarly, we can also introduce the s-p-numerical radius of the n-tuple of operators (T1 , . . . , Tn ) ∈ B (n) (H ) by ⎡⎛ ⎞ 1p ⎤ n    ⎢  T j x, x  p ⎠ ⎥ ws, p (T1 , . . . , Tn ) := sup ⎣⎝ ⎦, x=1

(2.53)

j=1

which for p = 2 reduces to the Euclidean operator radius introduced previously. We observe that the s- p-numerical radius is also a norm on B (n) (H ) for p ≥ 1 and for p = 1 it satisfies the basic inequality

2.4 Inequalities for s-1-Norm and s-1-Numerical Radius

21

⎞ ⎛ n n   # $ w⎝ T j ⎠ ≤ ws,1 (T1 , . . . , Tn ) ≤ w Tj . j=1

(2.54)

j=1

Proposition 2.15 (Dragomir [3]) Let (T1 , ..., Tn ) ∈ B (n) (H ) be an n-tuple of bounded linear operators on the Hilbert space (H ; ·, ·) . Then we have (T1 , . . . , Tn )s,1

% %1/2 % %1/2 % % % n % % n  2α % %  ∗ 2(1−α) % % T j  % % T  ≤% j % % % % % j=1 % % j=1 %

(2.55)

for any α ∈ [0, 1], and in particular, the following refinement of the triangle inequality for operator norm: % % % n % % % % Tj % % % ≤ (T1 , . . . , Tn )s,1 % j=1 % % %1/2 % %1/2 % % % n % % n  % %  ∗ % T j % %  T % ≤% j % % % % % j=1 % % j=1 % % % %⎤ ⎡% % % % % n n n  % % % %     % % 1  T j % + % T ∗ %⎦ ≤ %T j % . ≤ ⎣% j % % % % 2 % j=1 % % j=1 % j=1

(2.56)

Proof Utilizing the vector inequality (2.7) and taking the supremum over y = 1, x = 1 we have (T1 , . . . , Tn )s,1 (2.57) ⎧ ⎫1/2 ⎧ ⎫1/2  n  n ⎨ ⎬ ⎨ ⎬   2α   2(1−α) T j  x, x T ∗  ≤ sup y, y sup j ⎩x=1 ⎭ ⎩y=1 ⎭ j=1

and since

j=1

%  %  n % %   2α % n  2α % %     T j x, x = % Tj % sup % x=1 j=1 % j=1 %

and

 sup

y=1

%  % % n % n  %  ∗ 2(1−α) %  ∗ 2(1−α) % % T    Tj y, y = % j % % % j=1 j=1

then we get from (2.57) the desired inequality (2.55). The inequality (2.56) follows from (2.55).



22

2 Inequalities for n-Tuples of Operators

The case of normal operators provides a simpler bound: Corollary 2.16 (Dragomir [3]) Let (N1 , ..., Nn ) ∈ B (n) (H ) be an n-tuple of normal operators on the Hilbert space (H ; ·, ·). Then we have (N1 , . . . , Nn )s,1

% %1/2 % %1/2 % n % % n % %  2α % %  2(1−α) % % % % %     Nj % % Nj ≤% % % j=1 % % j=1 %

(2.58)

for any α ∈ [0, 1], and in particular, % % % % % n % n % % n % % %  %  % % % % % %   %N j % . (N ≤ ≤ N N , . . . , N ≤ ) j% 1 n s,1 j % % % % j=1 % % j=1 % j=1

(2.59)

The above results provide an interesting criterion of convergence in the Banach ∞  algebra B (H ) for the series of operators Tj . j=0

- . Criterion 2.17 (Dragomir [3]) Let T j j∈N be a sequence of operators in B (H ). 2(1−α) ∞   ∞   T j 2α and  T ∗  are If there exists an α ∈ (0, 1) such that the series j convergent in the Banach algebra B (H ), then

∞ 

j=0

j=0

T j is convergent in B (H ) and

j=0

% % %1/2 % %1/2 % % %∞ % %∞ % % % ∞ % %  2α % %  ∗ 2(1−α) % % % % % % . %     ≤ T T T j% j j % % % % % % % j=0 % % j=0 % % j=0 In particular, the convergence of the series gence of

∞ 

 ∞   ∞     ∗ T j  and T j  imply the conver-

j=0

j=0

T j in B (H ) with the estimate for the sums as follows:

j=0

%1/2 % %1/2 % % % % %∞ % % %∞ % % ∞ % %  % %  ∗ % % % % % %     Tj % % Tj % Tj % ≤ % % . % % % j=0 % % j=0 % % j=0 The following result for the s-1-numerical radius may be stated as well: Proposition 2.18 (Dragomir [3]) Let (T1 , ..., Tn ) ∈ B (n) (H ) be an n-tuple of bounded linear operators on the Hilbert space (H ; ·, ·) . Then we have

2.4 Inequalities for s-1-Norm and s-1-Numerical Radius

23

ws,1 (T1 , . . . , Tn ) ⎧ 1/2  n 1/2 ⎫ n ⎨  ⎬   2(1−α)  2α p j T j  x, x p j T j∗  x, x ≤ sup ⎩ ⎭ j=1 j=1 ⎧% %1/2 % % % % % n  2(1−α) %1/2 ⎪ ⎪ % n  2α % %  %  ∗ ⎪ ⎪ T j % % T j  % % ; ⎪ ⎪ % % j=1 % ⎨ % j=1 ≤ %  2(1−α) % ⎪ ⎪ 2α   % % ⎪ ⎪ % n |T j | +T j∗  % ⎪ ⎪ % % ⎩% 2 % j=1

(2.60)

for any α ∈ [0, 1], and, in particular, ⎛ w⎝

n 

⎞ T j ⎠ ≤ ws,1 (T1 , . . . , Tn )

j=1

≤

(2.61)

⎧% % %1/2 % % % % n  %1/2 ⎪ ⎪ % n  % %   ∗ % ⎪ ⎪ T j % %  T j % ; % ⎪ ⎪ % % % j=1 ⎨ % j=1 %  % ⎪ ⎪ % n |T j |+T ∗  % ⎪ ⎪ j % % ⎪% ⎪ %. ⎩% 2 % j=1

Remark 2.19 We observe that due to the inequality % % % ⎛ ⎞ % % n  2α  ∗ 2(1−α) % % % n n % % T + T j   %  1% % j % % % ⎝ ⎠ ≤ w T T ≤ % %, j% j % % 2% 2 % j=1 % j=1 % j=1 % the convergence of the series

(2.62)

∞   2(1−α)   |Tk |2α + Tk∗  in the Banach algebra B (H )

k=0

for some α ∈ (0, 1) suffices for the convergence of

∞ 

Tk , which is a slight improve-

k=0

ment of the result from Criterion 2.17. The case α = 21 produces the simpler inequality of interest for the numerical radius of a sum: % % % % ⎛ ⎞ % % % n % n n  %  % 3   ∗ 4% 1% 1  % % %    ⎝ ⎠ (2.63) Tj + Tj % Tj % ≤ w Tj ≤ % %. 2% 2 % % j=1 % % j=1 j=1

24

2 Inequalities for n-Tuples of Operators

2.5 Additive Inequalities Employing the original Kato’s inequality we can state the following new result: Theorem 2.20 (Dragomir et al. [9]) Let (T1 , ..., Tn ) ∈ B (n) (H ) be an n-tuple of bounded linear operators on the Hilbert space (H ; ·, ·) and ( p1 , ..., pn ) ∈ R∗n + an n-tuple of nonnegative weights not all of them equal to zero. Then we have n  j=1

 n )  2α  2(1−α) * 1/2 T j  + T j     p j  T j x, y  ≤ pj x, x 2 j=1 ⎛  2α  2(1−α) ⎞ 1/2  n      ⎜ T j∗  + T j∗  ⎟ pj ⎝ × ⎠ y, y 2 j=1

for any x, y ∈ H, α ∈ [0, 1] and, in particular, for α = n 

1 2

 n 1/2  n 1/2         ∗ p j  T j x, y  ≤ p j T j  x, x p j T j  y, y

j=1

j=1

(2.64)

(2.65)

j=1

for any x, y ∈ H . Proof Utilising Kato’s inequality we have 1/2   1/2      T j x, y  ≤ T j 2α x, x T ∗ 2(1−α) y, y j

and, by replacing α with 1 − α, 1/2   1/2     T ∗ 2α y, y  T j x, y  ≤ T j 2(1−α) x, x , j

which, by summation gives  1/2   1/2      T j x, y  ≤ 1 T j 2α x, x T ∗ 2(1−α) y, y j 2 1/2   1/2    2(1−α) ∗ 2α    Tj x, x y, y + Tj

(2.66)

for any j ∈ {1, ..., n} and x, y ∈ H . By the elementary inequality $1/2 # 2 $1/2 # b + d2 , a, b, c, d ≥ 0 ab + cd ≤ a 2 + c2

(2.67)

2.5 Additive Inequalities

25

we have  1/2   1/2   1/2   1/2   2α T j  x, x T ∗ 2(1−α) y, y T ∗ 2α y, y T j 2(1−α) x, x + j j ≤

0 1/2 /  0 1/2 /      T ∗ 2α + T ∗ 2(1−α) y, y T j 2α + T j 2(1−α) x, x , j

j

which, by (2.66), produces    T j x, y  ≤

)  2α  2(1−α) * T j  + T j 

1/2

x, x 2 ⎞ ⎛    1/2   ∗ 2α  ∗ 2(1−α) + T   T j ⎟ ⎜ j × ⎝ ⎠ y, y 2

(2.68)

for any j ∈ {1, ..., n} and x, y ∈ H . Multiplying the inequalities (2.68) with the positive weights p j , summing over j from 1 to n and utilizing the weighted Cauchy–Buniakowski–Schwarz inequality n 

⎞1/2 ⎛ ⎞1/2 ⎛ n n   pjajbj ≤ ⎝ p j a 2j ⎠ ⎝ p j b2j ⎠

j=1

j=1

j=1

where (a1 , ..., an ) , (b1 , ..., bn ) ∈ Rn+ , we have n  j=1

⎛)   * 1/2   n T j 2α + T j 2(1−α)    p j  T j x, y  ≤ pj ⎝ x, x 2 j=1 ⎞ ⎞ ⎛    1/2   ∗ 2α  ∗ 2(1−α) + T j  ⎟ ⎟ ⎜ T j  ⎟ × ⎝ ⎠ y, y ⎠ 2

≤

 n  j=1

×

 n  j=1

pj

)  2α  2(1−α) * T j  + T j  2

(2.69)

1/2 x, x

⎛  2α  2(1−α) ⎞ 1/2    ∗ + T j∗   T j ⎟ ⎜ pj ⎝ ⎠ y, y 2

for any j ∈ {1, ..., n} and x, y ∈ H , and the inequality in (2.64) is proved.



26

2 Inequalities for n-Tuples of Operators

For vectors of norm one, the second inequality from (2.64) and (2.65) can be refined as follows: Remark 2.21 With the assumptions in Theorem 2.20 we have n  j=1

 n )  2α  2(1−α) * 1/2 T j  + T j     p j  T j x, y  ≤ pj x, x 2 j=1 ⎛  2α  2(1−α) ⎞ 1/2  n      ⎜ T j∗  + T j∗  ⎟ pj ⎝ × ⎠ y, y 2 j=1

(2.70)

⎡ n  )  2α  2(1−α) *⎤1/2 T j  + T j   ⎦ x, x ≤ ⎣ pj 2 j=1

⎡ ⎛  2α  2(1−α) ⎞⎤1/2  n     ∗ + T j∗   ⎢ ⎟⎥ ⎜ T j  × ⎣ pj ⎝ ⎠⎦ y, y 2 j=1 ⎡ ⎡  n 2 )  2α  2(1−α) *⎤1/2 T j  + T j  1 ⎢ ⎣ ⎦ x, x ≤ ⎣ pj 2 2 j=1 2⎤ ⎛  2α  2(1−α) ⎞⎤1/2 ⎡  n   ∗  ∗ + T j  ⎥ ⎟⎥ ⎢ ⎜ T j  + ⎣ pj ⎝ ⎠⎦ y, y ⎥ ⎦ 2 j=1

⎡ )  2α  2(1−α) *  n T j  + T j  1⎣  ≤ pj x, x 2 2 j=1 ⎤ ⎛  2α  2(1−α) ⎞   n      ⎜ T j∗  + T j∗  ⎟ ⎥ pj ⎝ + ⎠ y, y ⎦ 2 j=1 for any x, y ∈ H with x = y = 1. In particular, we have n 

  p j  T j x, y 

j=1

≤

 n  j=1

  p j T j  x, x

(2.71) 1/2  n 

  p j T ∗  y, y j

j=1

1/2

2.5 Additive Inequalities

⎛ ≤ ⎝

n 

27

⎞1/2 ⎞1/2  ⎛ n       ∗ p j T j ⎠ x, x ⎝ p j T ⎠ y, y j

j=1

j=1

⎡ ⎛ ⎤ ⎞1/2 ⎞1/2  n 2 ⎛ n 2    1 ⎢   ⎠ ⎥ ≤ ⎣ ⎝ p j Tj x, x + ⎝ p j T j∗ ⎠ y, y ⎦ 2 j=1 j=1 ⎡   n ⎤ n    1 ⎣    ≤ p j T j x, x + p j T j∗  y, y ⎦ 2 j=1 j=1 for any x, y ∈ H with x = y = 1. The proof follow by utilizing the Hölder-McCarthy inequalities (see for instance [29]) P r x, x ≤ P x, xr and P x, xs ≤ P s x, x that hold for the positive operator P, for r ∈ (0, 1), s ∈ [1, ∞) and x ∈ H with x = 1. The details are omitted. Remark 2.22 We observe also that the choice y = x in the inequality (2.71) produces the result  n 1/2  n 1/2 n         T j x, x  ≤ T j  x, x T ∗  x, x j j=1

j=1

(2.72)

j=1

⎞1/2 ⎞1/2 ⎛ n  ⎛ n      ∗ T j ⎠ x, x ⎝ T ⎠ x, x ≤ ⎝ j

j=1

j=1

⎡ ⎛ ⎤ ⎞1/2 ⎞1/2 2 ⎛ n 2  n   1 ⎢   ⎠ T ∗ ⎠ x, x ⎥ Tj ≤ ⎣ ⎝ x, x + ⎝ ⎦ j 2 j=1 j=1

   n ⎡    ∗  ⎤   T j + T j  ⎣ ⎦ x, x ≤ 2 j=1 for any ∈ H with x = 1.

Remark 2.23 In order to provide some applications for functions of normal operators defined by power series, we need to state the inequality (2.64) for normal operators N j , j ∈ {1, ..., n}, namely

28

2 Inequalities for n-Tuples of Operators n  j=1

 n )  2α  2(1−α) * 1/2 N j  + N j     p j  N j x, y  ≤ pj x, x 2 j=1  n )  2α  2(1−α) * 1/2 N j  + N j   × pj y, y 2 j=1

(2.73)

for any α ∈ [0, 1] and for any x, y ∈ H . From a different perspective that involves quadratics, we can state the following result as well: Theorem 2.24 (Dragomir et al. [9]) Let (T1 , ..., Tn ) ∈ B (n) (H ) be an n-tuple of bounded linear operators on the Hilbert space (H ; ·, ·) and ( p1 , ..., pn ) ∈ R∗n + an n-tuple of nonnegative weights not all of them equal to zero. Then we have n 

 2 p j  T j x, y 

(2.74)

j=1 n %2α % %2(1−α) % ∗ %2α % %2(1−α) 0 1  /% p j %T j x % %T j∗ y % + %T j y % %T j x % 2 j=1 ⎞α ⎛ ⎞1−α ⎡⎛ n n   % % % % 1 2 2 ≤ ⎣⎝ p j %T j x % ⎠ ⎝ p j %T j∗ y % ⎠ 2 j=1 j=1 ⎛ ⎞1−α ⎛ ⎞α ⎤ n n   %2 % % ∗ %2 ⎝ +⎝ p j %T j x % ⎠ p j %T j y % ⎠ ⎦

≤

j=1

≤

1 2

j=1

n 

/% %2 % %2 0 p j %T j x % + %T ∗ y % j

j=1

for any x, y ∈ H with x = y = 1 and α ∈ [0, 1]. Proof We must prove the inequalities only in the case α ∈ (0, 1), since the case α = 0 or α = 1 follows directly from the corresponding case of Kato’s inequality. Utilizing Kato’s inequality for the operator T j , j ∈ {1, ..., n} we have        T j x, y 2 ≤ T j 2α x, x T ∗ 2(1−α) y, y j

(2.75)

and, by replacing α with 1 − α,        T j x, y 2 ≤ T j 2(1−α) x, x T ∗ 2α y, y , j for any x, y ∈ H .

(2.76)

2.5 Additive Inequalities

29

By Hölder-McCarthy inequalities P r x, x ≤ P x, xr , that holds for the positive operator P, for r ∈ (0, 1) and x ∈ H with x = 1 we also have     α   1−α   T j 2α x, x T ∗ 2(1−α) y, y ≤ T j 2 x, x T ∗ 2 y, y

(2.77)

    1−α   α   T ∗ 2 y, y T j 2(1−α) x, x T ∗ 2α y, y ≤ T j 2 x, x

(2.78)

j

and

j

j

j

for any x, y ∈ H with x = y = 1, j ∈ {1, ..., n} and α ∈ (0, 1). If we add (2.75) with (2.76) and make use of (2.77) and (2.78), we deduce α   1−α   α   1−α  2  2 2 2 2 2  T j x, y  ≤ T j  x, x T j∗  y, y + T j∗  y, y T j  x, x (2.79) for any x, y ∈ H with x = y = 1, j ∈ {1, ..., n} and α ∈ (0, 1). Now, if we multiply (2.79) with p j ≥ 0, sum over j from 1 to n we get 2

n 

n   α   1−α  2  2 2 p j  T j x, y  ≤ p j T j  x, x T j∗  y, y

j=1

(2.80)

j=1

+

  α   1−α 2 2 p j T j∗  y, y T j  x, x

n  j=1

for any x, y ∈ H with x = y =1 and α ∈(0, 1).  2 % %2   % %2 2   % % Since T j  x, x = %T j x % and T j∗  y, y = %T j∗ y % , j ∈ {1, ..., n}, then we get from (2.80) the first inequality in (2.74). Now, on making use of the weighted Hölder discrete inequality n  j=1

⎛ pjajbj ≤ ⎝

n  j=1

⎞1/ p ⎛ pjaj ⎠ p

⎝

n 

⎞1/q pjbj ⎠ q

, p, q > 1,

j=1

1 1 + = 1, p q

where (a1 , ..., an ) , (b1 , ..., bn ) ∈ Rn+ , we also have n  j=1

and

⎞α ⎛ ⎞1−α ⎛ n n   %2α % ∗ %2(1−α) %2 % % % ∗ %2 p j %T j x % %T j y % ≤⎝ p j %T j x % ⎠ ⎝ p j %T j y % ⎠ j=1

j=1

30

2 Inequalities for n-Tuples of Operators n 

⎛ ⎞α ⎛ ⎞1−α n n   %2(1−α) %2 % ∗ %2α % % ∗ %2 % p j %T j y % %T j x % ≤⎝ p j %T j y % ⎠ ⎝ p j %T j x % ⎠ .

j=1

j=1

j=1

Summing these two inequalities we deduce the second inequality in (2.74). Finally, on utilizing the Hölder’s inequality $1/ p # q $1/q # b + dq ab + cd ≤ a p + c p , a, b, c, d ≥ 0 where p > 1 and ⎛ ⎝

n 

1 p

+

= 1, we have

⎞α ⎛ ⎞1−α ⎛ ⎞α ⎛ ⎞1−α n n n % % % %   % %2 %2 % ∗ %2 ⎠ % ∗ %2 ⎠ ⎝ % % % % % ⎠ ⎝ ⎝ p j Tj x p j %T y % + p j %T y % p j Tj x ⎠ j

j=1

⎛

≤⎝

j=1

n 

% %2 p j %T j x % +

j=1

=

1 q

n 

j

⎞α ⎛

% %2 % % p j %T j∗ y % ⎠ ⎝

j=1

j=1 n 

% %2 p j %T j x % +

j=1

n 

n % %2 % %2  % % p j %T j x % + p j %T ∗ y % .

j=1

j=1

j=1

n 

⎞1−α % % % ∗ %2 ⎠ p j %T j y %

j=1

j



and the proof is concluded. Remark 2.25 For α = n 

we get from (2.74) that

 2 p j  T j x, y 

j=1

≤

1 2

n 

(2.81)

⎛ ⎞1/2 ⎛ ⎞1/2 n n   %% ∗ % %2 % % % ∗ %2 p j %T j x % %T y % ≤ ⎝ p j %T j x % ⎠ ⎝ p j %T y % ⎠ j

j=1

j

j=1

j=1

%2 % %2 0 1  /% p j %T j x % + %T j∗ y % 2 j=1 n

≤

for any x, y ∈ H with x = y = 1.

2.6 Inequalities for Functions of Normal Operators  n Now, by the help of power series f (z) = ∞ n=0 an z we can naturally construct another power series which will have as coefficients values of the coef the absolute n |a | z . It is obvious that this ficient of the original series, namely, f A (z) := ∞ n=0 n

2.6 Inequalities for Functions of Normal Operators

31

new power series will have the same radius of convergence as the original series. We also notice that if all coefficients an ≥ 0, then f A = f . The following result is a functional inequality for normal operators that can be obtained from (2.64).  n Theorem 2.26 (Dragomir et al. [9]) Let f (z) = ∞ n=0 an z be a function defined by power series with complex coefficients and convergent on the open disk D (0, R) ⊂ C, R > 0. If N is a normal operator on the Hilbert space H and for α ∈ (0, 1) we have that N 2α , N 2(1−α) < R, then we have the inequalities # $4 1/2 1 3 # 2α $ f A |N | + f A |N |2(1−α) x, x 2 # $4 1/2 3 # 2α $ × f A |N | + f A |N |2(1−α) y, y

| f (N ) x, y| ≤

(2.82)

for any x, y ∈ H . In particular, if N  < R, then | f (N ) x, y| ≤  f A (|N |) x, x1/2  f A (|N |) y, y1/2

(2.83)

for any x, y ∈ H . Proof If N is a normal operator, then for any j ∈ N we have that  j 2 # ∗ $ j  N  = N N = |N |2 j . Now, utilizing the inequality (2.75) we can write that    n     a j N j x, y     j=0 ≤

n     j  a j   N x, y  j=0



) * 1/2 n    |N |2 jα + |N |2 j(1−α)   ≤ aj x, x 2 j=0  n ) * 1/2    |N |2 jα + |N |2 j(1−α) a j  × y, y 2 j=0 for any x, y ∈ H and n ∈ N.

(2.84)

32

2 Inequalities for n-Tuples of Operators

Since N 2α , N 2(1−α) < R, then it follows that the series

∞  # $  a j  |N |2α j and j=0

∞  # $  a j  |N |2(1−α) j are absolute convergent in B (H ), and by taking the limit over j=0

n → ∞ in (2.84) we deduce the desired result (2.82).



Remark 2.27 With the assumptions in Theorem 2.26, if we take the supremum over y ∈ H, y = 1, then we get the vector inequality # $4 1/2 1 3 # 2α $ f A |N | + f A |N |2(1−α) x, x %2 # $ # $% × % f A |N |2α + f A |N |2(1−α) %

 f (N ) x ≤

(2.85)

for any x ∈ H , which in its turn produces the norm inequality  f (N ) ≤

# $ # $% 1% % f A |N |2α + f A |N |2(1−α) % 2

(2.86)

for any α ∈ [0, 1]. Moreover, if we take y = x in (2.82), then we have | f (N ) x, x| ≤

# $4  1 3 # 2α $ f A |N | + f A |N |2(1−α) x, x 2

(2.87)

for any x ∈ H , which, by taking the supremum over x ∈ H, x = 1 generates the numerical radius inequality w ( f (N )) ≤

# $4 1 3 # 2α $ w f A |N | + f A |N |2(1−α) 2

(2.88)

for any α ∈ [0, 1]. We can state the following particular vector inequalities:    ln (1 H + N )−1 x, y  1/2 $−1 $−1  # 1  # ≤ ln 1 H − |N |2α x, x + ln 1 H − |N |2(1−α) 2 1/2  # $−1 $−1  # y, y × ln 1 H − |N |2α + ln 1 H − |N |2(1−α) , and

(2.89)

2.6 Inequalities for Functions of Normal Operators

   (1 H + N )−1 x, y  1/2 $−1 # $−1  1 # ≤ 1 H − |N |2α x, x + 1 H − |N |2(1−α) 2 1/2  # $−1 $−1  # y, y × ln 1 H − |N |2α + ln 1 H − |N |2(1−α) ,

33

(2.90)

for any x, y ∈ H and N  < 1. We also have the inequalities # $ # $4 1/2 1 3 sinh |N |2α + sinh |N |2(1−α) x, x 2 $ # $4 1/2 3 # × sinh |N |2α + sinh |N |2(1−α) y, y

|sin (N ) x, y| ≤

(2.91)

and # $ # $4 1/2 1 3 cosh |N |2α + cosh |N |2(1−α) x, x 2 3 # $ # $4 1/2 × cosh |N |2α + cosh |N |2(1−α) y, y

|cos (N ) x, y| ≤

(2.92)

for any x, y ∈ H and N a normal operator. If we utilize functions as power series representations with nonnegative coefficients, then we can state the following vector inequalities: # $ # $4 1/2 1 3 |exp (N ) x, y| ≤ exp |N |2α + exp |N |2(1−α) x, x 2 $ # $4 1/2 3 # × exp |N |2α + exp |N |2(1−α) y, y

(2.93)

for any x, y ∈ H and N a normal operator. If N  < 1, then we also have the inequalities  ' (     ln 1 H + N x, y    1H − N ) *" 1/2 ! ' ( 1 1 H + |N |2(1−α) 1 H + |N |2α ≤ + ln x, x ln 2 1 H − |N |2α 1 H − |N |2(1−α) ) ! ' *" 1/2 ( 1 H + |N |2(1−α) 1 H + |N |2α + ln × ln y, y 1 H − |N |2α 1 H − |N |2(1−α)

(2.94)

34

2 Inequalities for n-Tuples of Operators

   tanh−1 (N ) x, y 

(2.95)

# $ # $4 1/2 1 3 ≤ tanh−1 |N |2α + tanh−1 |N |2(1−α) x, x 2 # $ # $4 1/2 3 × tanh−1 |N |2α + tanh−1 |N |2(1−α) y, y and |2 F1 (α, β, γ, N ) x, y| # $ # $4 1/2 1 3 2α ≤ +2 F1 α, β, γ, |N |2(1−α) x, x 2 F1 α, β, γ, |N | 2 # $ # $4 1/2 3 × 2 F1 α, β, γ, |N |2α +2 F1 α, β, γ, |N |2(1−α) y, y

(2.96)

for any x, y ∈ H . From a different perspective, we also have: Theorem 2.28 (Dragomir et al. [9]) With the assumption of Theorem 2.26 and if N is a normal operator on the Hilbert space H and z ∈ C such that N 2 , |z|2 < R, then we have the inequalities α  # 2 $ 1−α 1 # 2 $  # 2 $ f A |z| f A |N | x, x f A |N | y, y 2  # 2$ 1−α  # 2 $ α  + f A |N | x, x f A |N | y, y

| f (z N ) x, y|2 ≤

≤

(2.97)

  # $ $ 1 # 2 $ # # 2 $ f A |z| f A |N | x, x + f A |N |2 y, y 2

for any x, y ∈ H with x = y = 1 and α ∈ [0, 1]. In particular, for α = 21 we have # $ # $ 1/2  # 2 $ 1/2 | f (z N ) x, y|2 ≤ f A |z|2 f A |N |2 x, x f A |N | y, y   # $ $ 1 # $ # # 2 $ ≤ f A |z|2 f A |N | x, x + f A |N |2 y, y 2

(2.98)

for any x, y ∈ H with x = y = 1. Proof If we use the second and third inequality from (2.74) for powers of operators we have n     j  a j   N x, y 2 j=0

⎞α ⎛ ⎞1−α ⎡⎛ n n %# $ %2  %   1 ⎣⎝   % 2 j % a j  % a j %N j x % ⎠ ⎝ ≤ % N ∗ y% ⎠ 2 j=0 j=0

(2.99)

2.6 Inequalities for Functions of Normal Operators

35

⎛

⎞1−α ⎛ ⎞α ⎤ n n %2     % j %2  % # $ j % a j  % N x % ⎠ a j  % ⎝ +⎝ % N ∗ y% ⎠ ⎦ j=0

≤

1 2

n 

j=0

' (   % j %2 % # $j % %2 a j  % N x % + % % N ∗ y%

j=0

for any x, y ∈ H with x = y = 1 and α ∈ [0, 1]. Since N is a normal operator on the Hilbert space H , then  % j %2  j 2  % N x % =  N  x, x = |N |2 j x, x and

  %# $ %2 # $ 2   2 j  % ∗ j %  ∗ j y % =  N  y, y =  N ∗  y, y = |N |2 j y, y % N

for any j ∈ {0, ..., n} and for any x, y ∈ H with x = y = 1. Then from (2.99) we have n     j  a j   N x, y 2

(2.100)

j=0

⎡⎛ ⎞α ⎛ n ⎞1−α n       1 a j  |N |2 j x, x ⎠ ⎝ a j  |N |2 j y, y ⎠ ≤ ⎣⎝ 2 j=0 j=0 ⎛ ⎞1−α ⎛  ⎞α ⎤ n n       a j  |N |2 j x, x ⎠ a j  |N |2 j y, y ⎠ ⎦ ⎝ +⎝ j=0

j=0

⎛   n ⎞ n   1 ⎝    a j  |N |2 j y, y ⎠ a j |N |2 j x, x + ≤ 2 j=0 j=0 for any x, y ∈ H with x = y = 1 and α ∈ [0, 1]. By the weighted Cauchy–Buniakowski–Schwarz inequality we also have  2   n n   n    2j     j  j j   ≤ a j  |z| a j   N x, y 2 a z N x, y j    j=0  j=0 j=0

(2.101)

for any x, y ∈ H with x = y = 1. ∞ ∞   ∞      a j  |z|2 j , a j  |N |2 j are convergent, Now, since the series ajz j N j, j=0

j=0

j=0

then by (2.100) and (2.101), on letting n → ∞, we deduce the desired result (2.97). 

36

2 Inequalities for n-Tuples of Operators

Similar inequalities for some particular functions of interest can be stated. However, the details are left to the interested reader.

2.7 Applications for the Euclidian Norm We can state now the following result: Theorem 2.29 (Dragomir et al. [9]) For any (T1 , . . . , Tn ) ∈ B (n) (H ) we have %⎞α ⎛% %⎞1−α ⎡⎛% % % % % n n % % % %     1 2 2 2 ∗  %⎠ % % %    ⎠ ⎝ ⎣ ⎝ (T1 , . . . , Tn )e ≤ Tj % Tj % % % 2 % j=1 % % % j=1 %⎞1−α ⎛% %⎞α ⎤ ⎛% % n % n % % %  ∗ 2 % %  2 % T j  %⎠ T  %⎠ ⎦ ⎝% + ⎝% j % % % % % j=1 % j=1 % % % % %⎤ ⎡% % % n % % n %  ∗  2 % 1 %  2 % %+% T  %⎦ T ≤ ⎣% j j % % % 2 % % % j=1 % % j=1

(2.102)

and we2 (T1 , . . . , Tn ) ⎧⎛ ⎡ ⎞α ⎛ n ⎞1−α ⎫ n ⎨ ⎬       1 T j 2 x, x ⎠ ⎝ T ∗ 2 x, x ⎠ ≤ ⎣ sup ⎝ j ⎭ 2 x=1 ⎩ j=1 j=1 ⎧⎛ ⎞1−α ⎛ n ⎞α ⎫⎤ n ⎨  ⎬   2  2 T j  x, x ⎠ T ∗  x, x ⎠ ⎦ ⎝ + sup ⎝ j ⎭ x=1 ⎩ j=1 j=1 %⎞α ⎛% %⎞1−α ⎡⎛% % % % % n n % % % %     1 T j 2 %⎠ ⎝% T ∗ 2 %⎠ ≤ ⎣⎝% j % % % % 2 % j=1 % % % j=1 %⎞1−α ⎛% %⎞α ⎤ ⎛% % n % n % % %   2 % %  ∗ 2 % % % %     ⎠ ⎠ ⎦ ⎝ ⎝ Tj % Tj % + % % % % j=1 % j=1 % % for any α ∈ [0, 1]. Proof We have from the second inequality in (2.74)

(2.103)

2.7 Applications for the Euclidian Norm

37

⎡⎛ ⎞α ⎛ n ⎞1−α n n     2   2 2 1  T j x, y  ≤ ⎣⎝ T j  x, x ⎠ ⎝ T ∗  y, y ⎠ j 2 j=1 j=1 j=1 ⎛ ⎞1−α ⎛ n  ⎞α ⎤ n    2  2 T j  x, x ⎠ T ∗  y, y ⎠ ⎦ ⎝ +⎝ j j=1

(2.104)

j=1

for any x, y ∈ H with x = y = 1 and α ∈ [0, 1]. Taking the supremum over x = y = 1 we have (T1 , . . . , Tn )2e ⎡⎛  ⎞α ⎛ ⎞1−α  n  n   2   2 1 ⎣⎝ T j  x, x ⎠ ⎝ sup T ∗  y, y ⎠ ≤ sup j 2 x=1 j=1 y=1 j=1 ⎛ ⎞1−α ⎛ ⎞α ⎤  n  n   2   2 T j  x, x ⎠ T ∗  y, y ⎠ ⎦ ⎝ sup + ⎝ sup x=1

y=1

j=1

j

j=1

%⎞α ⎛% %⎞1−α ⎡⎛% % % % % n % n  ∗ 2 %  2 % 1 ⎣⎝% % % %     ⎠ ⎠ ⎝ Tj % Tj % = % % % 2 % j=1 % % % j=1 %⎞1−α ⎛% %⎞α ⎤ ⎛% % n % n % % %   2 % %  ∗ 2 % T j  %⎠ T  %⎠ ⎦ , ⎝% + ⎝% j % % % % % j=1 % j=1 % %

which proves the first part of (2.102). The second part follows by the elementary inequality a α b1−α ≤ αa + (1 − α) b for a, b ≥ 0 and α ∈ [0, 1]. The inequality (2.103) follows from (2.104) by taking y = x and then the supremum over x = 1. 

2.8 Applications for s-1-Norm and s-1-Numerical Radius We can state the following result: Theorem 2.30 (Dragomir et al. [9]) For any (T1 , . . . , Tn ) ∈ B (n) (H ) we have

38

2 Inequalities for n-Tuples of Operators

% % % n )  2α  2(1−α) *%1/2 % T j % + Tj % (T1 , . . . , Tn )s,1 ≤ % % % 2 % j=1 % % ⎛  2α  2(1−α) ⎞%1/2 % n % % T j∗  + T j∗  % % ⎟% ⎜ ×% ⎠% ⎝ % % 2 % j=1 % ⎡% ) *%     % n 2α 2(1−α) %      % % T T + 1 j j % ≤ ⎣% % % 2 % j=1 2 % % ⎛  2α  2(1−α) ⎞%⎤ % n % % T j∗  + T j∗  % % ⎟%⎥ ⎜ +% ⎠%⎦ ⎝ % % 2 % j=1 %

(2.105)

and ws,1 (T1 , . . . , Tn ) % ⎛   2α  2(1−α) ⎞%   % n % % T j 2α + T j 2(1−α) + T j∗  + T j∗  % % ⎟% ⎜ ≤% ⎠% . ⎝ % % 4 % j=1 %

(2.106)

Proof From (2.64) we have  n )  2α  2(1−α) * 1/2 n   T j  + T j     T j x, y  ≤ x, x 2 j=1 j=1 ⎛  2α  2(1−α) ⎞ 1/2  n      ⎜ T j∗  + T j∗  ⎟ × ⎠ y, y ⎝ 2 j=1 for any x, y ∈ H . Taking the supremum over y = 1, x = 1 in (2.107) we have ⎡ (T1 , . . . , Tn )s,1 ≤ ⎣ sup

x=1

 n )  2α  2(1−α) *  T j  + T j  j=1

2

⎤1/2 x, x ⎦

⎤ ⎛  2α  2(1−α) ⎞  1/2  n    ∗ T j  + T j∗    ⎟ ⎥ ⎜ ⎢ × ⎣ sup ⎠ y, y ⎦ ⎝ 2 y=1 j=1 ⎡

(2.107)

2.8 Applications for s-1-Norm and s-1-Numerical Radius

39

% % % n )  2α  2(1−α) *%1/2 % T j % + Tj % =% % % 2 % j=1 % % ⎛  2α  2(1−α) ⎞%1/2 % n % % T j∗  + T j∗  % % ⎟% ⎜ ×% ⎠% ⎝ % % 2 % j=1 % and the first inequality (2.105) is proved. The second part follows by the arithmetic mean-geometric mean inequality. Now, if we take y = x in (2.107), then we get  n )  2α  2(1−α) * 1/2 n   T j  + T j     T j x, x  ≤ x, x 2 j=1 j=1 ⎛  2α  2(1−α) ⎞ 1/2  n      ⎜ T j∗  + T j∗  ⎟ × ⎠ x, x ⎝ 2 j=1 ⎛   2α  2(1−α) ⎞     n T j 2α + T j 2(1−α) + T ∗  + T ∗   j j 1 ⎟ ⎜ ≤ ⎠ x, x . ⎝ 2 j=1 2 Taking the supremum over x = 1 we deduce the desired result (2.106). Remark 2.31 If we take α =

1 2



in the first inequality in (2.105), then we deduce

(T1 , . . . , Tn )s,1

% %1/2 % %1/2 % n % % n % %  % %  ∗ %  T j % %  T % ≤% j % % % % % j=1 % % j=1 %

(2.108)

from where we get the following refinement of the generalized triangle inequality % % %1/2 % %1/2 % % n % n % % n % % %  % % % %  ∗ % % % % % % %     (T ≤ T T T , . . . , T ≤ ) j% 1 n s,1 j % j % % % % % j=1 % j=1 % % % j=1 % % % %⎤ ⎡% % % n % % n n  %  ∗  % % % 1 %  % % % %   %T j % . ⎦ + ≤ T T ≤ ⎣% j % j % % % 2 % j=1 % % j=1 % j=1

40

2 Inequalities for n-Tuples of Operators

From (2.106) we also have for α =

1 2

that

% ⎛     ⎞% % T j  + T ∗  % n % % j % ⎝ ⎠% . ws,1 (T1 , . . . , Tn ) ≤ % % 2 % j=1 %

(2.109)

2.9 Other Additive Inequalities The following result holds: Theorem 2.32 (Dragomir et al. [10]) Let (T1 , ..., Tn ) ∈ B (n) (H ) be an n-tuple of bounded linear operators on the Hilbert space (H ; ·, ·) and ( p1 , ..., pn ) ∈ R∗n + an n-tuple of nonnegative weights not all of them equal to zero, then      ' (  n    n   T j + T j∗ T j + T j∗     ≤ x, y x, y p p j j     2 2  j=1  j=1  ⎤ ⎡     T j x, y  +  T ∗ x, y  n  j ⎦ ≤ pj ⎣ 2 j=1 ≤

 n  j=1

×

 n  j=1

(2.110)

⎡   2(1−α) ⎤ 1/2 T j 2α + T ∗  j ⎥ ⎢ pj ⎣ ⎦ x, x 2

⎡   2(1−α) ⎤ 1/2 T j 2α + T ∗  j ⎥ ⎢ pj ⎣ ⎦ y, y 2

for any α ∈ [0, 1] and, in particular, for α =

1 2

     ' (  n    n   T j + T j∗ T j + T j∗    (2.111) x, y  ≤ x, y  pj pj   2 2  j=1  j=1  ⎤ ⎡    ∗    n T T + x, y x, y    j j ⎣ ⎦ ≤ pj 2 j=1

2.9 Other Additive Inequalities

41

⎡     ⎤ 1/2 T j  + T ∗  j ⎦ x, x ≤ pj ⎣ 2 j=1 ⎡     ⎤  n 1/2 T j  + T ∗   j ⎦ y, y × pj ⎣ 2 j=1 

n 

for any x, y ∈ H . Proof The first two inequalities are obvious by the properties of the modulus. Utilising Kato’s inequality we have 1/2   1/2     T ∗ 2(1−α) y, y  T j x, y  ≤ T j 2α x, x j

(2.112)

and, by replacing x with y we have 1/2   1/2      T j y, x  ≤ T j 2α y, y T ∗ 2(1−α) x, x j

i.e.,

1/2   1/2  ∗    T j 2α y, y  T x, y  ≤ T ∗ 2(1−α) x, x j

j

(2.113)

for any j ∈ {1, ..., n} and x, y ∈ H . Adding the inequalities (2.112) and (2.113) and utilizing the elementary inequality $1/2 # 2 $1/2 # b + d2 , a, b, c, d ≥ 0 ab + cd ≤ a 2 + c2 we get 1/2   1/2       T ∗ 2(1−α) y, y  T j x, y  +  T ∗ x, y  ≤ T j 2α x, x j

j

(2.114)

  1/2   1/2 2(1−α) T j 2α y, y + T ∗  x, x j

1/2    2(1−α)  2α x, x ≤ T j  + T ∗  j

1/2    2(1−α)  2α y, y × T j  + T ∗  j

for any j ∈ {1, ..., n} and x, y ∈ H . Multiplying the inequalities (2.114) by p j ≥ 0 and then summing over j from 1 to n and utilizing the weighted Cauchy–Buniakowski–Schwarz inequality we have n  j=1

3   4 p j  T j x, y  +  T j∗ x, y 

(2.115)

42

2 Inequalities for n-Tuples of Operators

≤

n 

pj

   1/2    1/2     T j 2α + T ∗ 2(1−α) x, x T j 2α + T ∗ 2(1−α) y, y j

j

j=1

≤

 n 

j

j=1

×

1/2

   2(1−α)  2α x, x p j T j  + T ∗ 

 n 

   2(1−α)  2α y, y p j T j  + T ∗ 

1/2

j

j=1

for x, y ∈ H , which is equivalent with the third inequality in (2.110).



Remark 2.33 The particular case y = x is of interest for providing numerical radii inequalities and can be stated as:      ( '  n    n  T j + T j∗  T j + T j∗     ≤ x, x p p x, x j j     2 2  j=1  j=1 ≤

n 

  p j  T j x, x 

j=1

≤

(2.116)

 n  j=1

⎡   2(1−α) ⎤  T j 2α + T ∗  j ⎥ ⎢ pj ⎣ ⎦ x, x 2

for any α ∈ [0, 1] and, for α = 21 ,    n  ' (  n       T j + T j∗ T j + T j∗     ≤ x, x x, x p p j j     2 2  j=1  j=1 ≤

n 

  p j  T j x, x 

j=1

≤

 n  j=1

⎡     ⎤  T j  + T ∗  j ⎦ x, x pj ⎣ 2

for any x ∈ H . The case of unitary vectors provides more refinements as follows: Remark 2.34 With the assumptions in Theorem 2.32, we have

(2.117)

2.9 Other Additive Inequalities n  j=1

 ⎤ ⎡     T j x, y  +  T ∗ x, y  j ⎦ pj ⎣ 2

 ≤

n  j=1

×

 n  j=1

⎡   2(1−α) ⎤ 1/2 T j 2α + T ∗  j ⎥ ⎢ pj ⎣ ⎦ x, x 2 ⎡   2(1−α) ⎤ 1/2 T j 2α + T ∗  j ⎥ ⎢ pj ⎣ ⎦ y, y 2

⎡  ⎛  2(1−α) ⎤⎞1/2  n  T j 2α + T ∗   j ⎥⎟ ⎢ ⎜ ≤ ⎝ pj ⎣ ⎦⎠ x, x 2 j=1 ⎛ ⎡   2(1−α) ⎤⎞1/2  n  2α  ∗   T + T j  ⎜ ⎢ j ⎥⎟ × ⎝ pj ⎣ ⎦⎠ y, y 2 j=1 ⎡ ⎛ 2 ⎡   2(1−α) ⎤⎞1/2   n T j 2α + T ∗   ⎢ j 1 ⎥⎟ ⎢ ⎜ ≤ ⎢ pj ⎣ ⎦⎠ x, x ⎝ 2⎣ 2 j=1 2⎤ ⎡  ⎛  2(1−α) ⎤⎞1/2  n  2α   ∗ T  + T  ⎥ j ⎥⎟ ⎜ ⎢ j + ⎝ pj ⎣ ⎦⎠ y, y ⎥ ⎦ 2 j=1

⎡

⎡   2(1−α) ⎤   n 2α     + T j∗  1 ⎢  ⎢ Tj ⎥ ≤ ⎣ pj ⎣ ⎦ x, x 2 2 j=1  +

n  j=1

⎤ ⎡   2(1−α) ⎤  T j 2α + T ∗  j ⎥ ⎥ ⎢ pj ⎣ ⎦ y, y ⎦ 2

for any α ∈ [0, 1] and, in particular,

43

(2.118)

44

2 Inequalities for n-Tuples of Operators

 ⎤ ⎡     T j x, y  +  T ∗ x, y  j ⎦ pj ⎣ (2.119) 2 j=1 ⎡     ⎤ ⎡     ⎤ 1/2  n 1/2  n T j  + T ∗  T j  + T ∗    j j ⎦ x, x ⎦ y, y pj ⎣ pj ⎣ ≤ 2 2 j=1 j=1 ⎡     ⎤⎞1/2 ⎡     ⎤⎞1/2 ⎛ n  ⎛ n  ∗  T j  + T ∗   T + T    j j j ⎣ ⎣ ⎝ ⎝ ⎦ ⎠ ⎦ ⎠ ≤ pj x, x pj y, y 2 2 j=1 j=1 ⎡ ⎡     ⎤⎞1/2 2 ⎛ n T j  + T ∗   ⎢ j 1 ⎝ ⎦⎠ x, x ≤ ⎢ pj ⎣ 2⎣ 2 j=1

n 

⎤ ⎡     ⎤⎞1/2 ⎛ n 2 ∗   T j + T j   ⎥ ⎦⎠ y, y ⎥ + ⎝ pj ⎣ ⎦ 2 j=1 ⎡     ⎤ ⎡     ⎤ ⎡   n ⎤ ∗   T j  + T ∗  n  j 1 ⎣  ⎣ T j + T j  ⎦ ⎦ y, y ⎦ ≤ pj pj ⎣ x, x + 2 2 2 j=1 j=1 for any x, y ∈ H with x = y = 1. The proofs follow by utilizing the Hölder-McCarthy inequalities P r x, x ≤ P x, xr and P x, xs ≤ P s x, x that hold for the positive operator P, for r ∈ (0, 1), s ∈ [1, ∞) and x ∈ H with x = 1. The details are omitted. In order to employ the above result in obtaining some inequalities for functions of normal operators defined by power series, we need the following version of (2.110). Remark 2.35 If we write the inequality (2.110) for the normal operators N j , j ∈ {1, ..., n} then we get    n  ' (  n       N j + N ∗j N j + N ∗j     ≤ x, y x, y p p j j     2 2  j=1  j=1  ⎤ ⎡     N j x, y  +  N ∗ x, y  n  j ⎦ ≤ pj ⎣ 2 j=1

(2.120)

2.9 Other Additive Inequalities

45

 ≤

n 

pj

j=1

×

 n 

pj

!  2α  2(1−α) " N j  + N j  2 !  2α  2(1−α) " N j  + N j  2

j=1

for any α ∈ [0, 1] and, in particular, for α =

1/2 x, x 1/2 y, y

1 2

     ' (  n    n   N j + N ∗j N j + N ∗j    (2.121) x, y  ≤ x, y  pj pj   2 2  j=1  j=1  ⎤ ⎡    ∗   +  N x, y  n N x, y  j j ⎦ ≤ pj ⎣ 2 j=1 1/2  n 1/2  n       ≤ p j  N j  x, x p j  N j  y, y j=1

j=1

for any x, y ∈ H . The following results involving quadratics also holds: Theorem 2.36 (Dragomir et al. [10]) Let (T1 , ..., Tn ) ∈ B (n) (H ) be an n-tuple of bounded linear operators on the Hilbert space (H ; ·, ·) and ( p1 , ..., pn ) ∈ R∗n + an n-tuple of nonnegative weights not all of them equal to zero, then n 

 2  2  p j  T j x, y  +  T j∗ x, y 

(2.122)

j=1

≤

n 

% %2α % %2(1−α) % %2α % %2(1−α)  p j %T j x % %T j∗ y % + %T j y % %T j∗ x %

j=1

⎛ ≤⎝ ⎛ +⎝

n 

⎞α ⎛ ⎞1−α n  %2 % % ∗ %2 p j %T j x % ⎠ ⎝ p j %T y % ⎠

j=1

j=1

j

n 

⎞α ⎛ ⎞1−α n  %2 % % ∗ %2 p j %T j y % ⎠ ⎝ p j %T x % ⎠

j=1

j=1

⎛ ≤⎝

j

n 

⎞α ⎛ ⎞1−α n % %   %2 % %2  % % % 2 2 p j %T j x % + %T j y % ⎠ ⎝ p j %T ∗ y % + %T ∗ x % ⎠

j=1

j=1

j

for any x, y ∈ H with x = y = 1 and α ∈ [0, 1].

j

46

2 Inequalities for n-Tuples of Operators

Proof We must prove the inequalities only in the case α ∈ (0, 1), since the case α = 0 or α = 1 follows directly from the corresponding case of Kato’s inequality. Utilising Kato’s inequality we have        T j x, y 2 ≤ T j 2α x, x T ∗ 2(1−α) y, y j

(2.123)

and, by replacing x with y we have      ∗   T x, y 2 ≤ T ∗ 2(1−α) x, x T j 2α y, y j j

(2.124)

for any j ∈ {1, ..., n} and x, y ∈ H . By Hölder-McCarthy inequality P r x, x ≤ P x, xr for r ∈ (0, 1) and x ∈ H with x = 1 we also have       α   1−α T j 2α x, x T ∗ 2(1−α) y, y ≤ T j 2 x, x T ∗ 2 y, y

(2.125)

    α   1−α   T ∗ 2(1−α) x, x T j 2α y, y ≤ T j 2 y, y T ∗ 2 x, x

(2.126)

j

and

j

j

j

for any j ∈ {1, ..., n} and x, y ∈ H with x = y = 1. We then obtain by summation      T j x, y 2 +  T ∗ x, y 2 j   α   1−α   α   1−α 2 2 2 2 ≤ T j  x, x T j∗  y, y + T j  y, y T j∗  x, x

(2.127)

for any j ∈ {1, ..., n} and x, y ∈ H with x = y = 1. Now, if we multiply (2.127) with p j ≥ 0, sum over j from 1 to n we get n 

 2  2  p j  T j x, y  +  T j∗ x, y 

j=1

≤

n 

  α   1−α 2 2 p j T j  x, x T j∗  y, y

j=1

+

n 

  α   1−α 2 2 p j T j  y, y T j∗  x, x

j=1

for any x, y ∈ H with x = y = 1 and α ∈ (0, 1).

(2.128)

2.9 Other Additive Inequalities

47

  % %2     % %2  2 % %2 2 2 % % Since T j  x, x = %T j x % , T j∗  y, y = %T j∗ y % , T j  y, y = %T j y %    % %2 % %  ∗ 2 and T j  x, x = %T j∗ x % j ∈ {1, ..., n}, then we get from (2.128) the first part of (2.122). Now, on making use of the weighted Hölder discrete inequality n 

⎛ pjajbj ≤ ⎝

j=1

n 

⎞1/ p ⎛ pjaj ⎠

⎝

p

n 

j=1

⎞1/q pjbj ⎠ q

, p, q > 1,

j=1

1 1 + = 1, p q

where (a1 , ..., an ) , (b1 , ..., bn ) ∈ Rn+ , we also have n 

⎛ ⎞α ⎛ ⎞1−α n n   %2α % ∗ %2(1−α) %2 % % % ∗ %2(1−α) ⎠ p j %T j x % %T j y % ≤⎝ p j %T j x % ⎠ ⎝ p j %T j y %

j=1

j=1

j=1

and n 

⎞α ⎛ ⎞1−α ⎛ n n   %2α % ∗ %2(1−α) %2 % % % ∗ %2 p j %T j y % %T j x % ≤⎝ p j %T j y % ⎠ ⎝ p j %T j x % ⎠ .

j=1

j=1

j=1

Summing these two inequalities we deduce the second inequality in (2.122). Finally, on utilizing the Hölder inequality $1/ p # q $1/q # b + dq , a, b, c, d ≥ 0 ab + cd ≤ a p + c p where p > 1 and ⎛ ⎝

1 p

+

1 q

= 1, we have

n 

⎞α ⎛ ⎞1−α n  %2 % % ∗ %2 p j %T j x % ⎠ ⎝ p j %T y % ⎠

j=1

j=1

j

⎛

+⎝

n 

⎞α ⎛ ⎞1−α n  %2 % % ∗ %2 p j %T j y % ⎠ ⎝ p j %T x % ⎠

j=1

j=1

j

⎞α ⎛ ⎞1−α ⎛ n n n n   %2  %2 % % % ∗ %2  % ∗ %2 ≤⎝ p j %T j x % + p j %T j y % ⎠ ⎝ p j %T j y % + p j %T j x % ⎠ j=1

j=1

j=1

j=1

and the proof is completed. Remark 2.37 Utilizing the elementary inequality for complex numbers



48

2 Inequalities for n-Tuples of Operators

   z + w 2 |z|2 + |w|2   , z, w ∈ C  2  ≤ 2 we have n  j=1

⎡   2 ⎤  !' ( 2 "   T j x, y 2 +  T ∗ x, y  n   T j + T j∗ j ⎥ ⎢ x, y  ≤ p j  pj ⎣ ⎦ (2.129) 2 2 j=1

and by the weighted arithmetic mean-geometric mean inequality a α b1−α ≤ αa + (1 − α) b, a, b ≥ 0, α ∈ [0, 1] we also have ⎛ ⎝

n 

⎞α ⎛ ⎞1−α n % %   %2 % %2  % % % 2 2 p j %T j x % + %T j y % ⎠ ⎝ p j %T ∗ y % + %T ∗ x % ⎠

j=1

j=1

j

≤α

n 

j

1 2

and use (2.113), (2.129) and (2.130) we derive

!' ( 2 "  T j + T j∗   x, y  pj  2 j=1 ⎡   2 ⎤   T j x, y 2 +  T ∗ x, y  n  j ⎥ ⎢ pj ⎣ ≤ ⎦ 2 j=1

n 

n %% % % %% %4 1  3% p j %T j x % %T j∗ y % + %T j y % %T j∗ x % ≤ 2 j=1 ⎞1/2 ⎞1/2 ⎛ ⎛ n n  %2 % ∗ % 1 ⎝ % ≤ p j %T j x % ⎠ ⎝ p j % T j y %⎠ 2 j=1 j=1 ⎞1/2 ⎛ ⎞1/2 ⎛ n n  %2 % ∗ %2 1 ⎝ % + p j %T j y % ⎠ ⎝ p j %T j x % ⎠ 2 j=1 j=1

≤⎝

j=1

j

j=1

If we choose α =

n 

(2.130)

n % %  %2 % %2  %2 % %2  % % % % + (1 − α) p j Tj x + Tj y p j %T ∗ y % + %T ∗ x % .

j=1

⎛

j

(2.131)

⎛ ⎡% % % % ⎤⎞1/2 !% % % % "⎞1/2 % ∗ %2 % ∗ %2 n y + x % %T % T j x %2 + % T j y %2 %T  j % ⎥⎟ ⎢ j ⎠ ⎜ pj pj ⎣ ⎦⎠ ⎝ 2 2 j=1

2.9 Other Additive Inequalities

≤

n  j=1

49

⎡% % %2 % % ⎤ % % % % ∗ %2 ∗ % % T j x %2 + % T j y %2 + % y + x %T % %T j j % ⎥ ⎢ pj ⎣ ⎦ 4

⎡ ⎤         2  ∗ 2   2  ∗ 2 n n T T + + T T     j j j j 1⎢ ⎥ = ⎣ x, x + y, y ⎦ pj pj 2 j=1 2 2 j=1 for any x, y ∈ H with x = y = 1. Remark 2.38 The case of normal operators N j , j ∈ {1, ..., n} is of interest for functions of operators and maybe stated as follows: !' ( 2 "  N j + N ∗j  x, y  p j  2 j=1 ⎡   2 ⎤   N j x, y 2 +  N ∗ x, y  n  j ⎥ ⎢ pj ⎣ ≤ ⎦ 2 j=1

n 

(2.132)

n %2α % %2(1−α) % %2α % %2(1−α)  1  % p j % N j x % % N j y% + % N j y% % N j x % ≤ 2 j=1 ⎞α ⎛ ⎞1−α ⎛ n n  %2 %2 % 1 ⎝ % ≤ p j %N j x % ⎠ ⎝ p j % N j y% ⎠ 2 j=1 j=1 ⎞α ⎛ ⎞1−α ⎛ n n  %2 %2 % 1 ⎝ % + p j % N j y% ⎠ ⎝ p j %N j x % ⎠ 2 j=1 j=1

≤

n %2 % %2  1  % p j % N j x % + % N j y% 2 j=1

for any x, y ∈ H with x = y = 1 and α ∈ [0, 1].

2.10 Other Inequalities for Functions of Normal Operators  n Now, by the help of power series f (z) = ∞ n=0 an z we can naturally construct another power series which will have as coefficients values of the coef the absolute n ficient of the original series, namely, f A (z) := ∞ n=0 |an | z . It is obvious that this new power series will have the same radius of convergence as the original series. We also notice that if all coefficients an ≥ 0, then f A = f .

50

2 Inequalities for n-Tuples of Operators

The following result is a functional inequality for normal operators that can be obtained from (2.110).  n Theorem 2.39 (Dragomir et al. [10]) Let f (z) = ∞ n=0 an z be a function defined by power series with complex coefficients and convergent on the open disk D (0, R) ⊂ C, R > 0. If N is a normal operator on the Hilbert space H and for α ∈ [0, 1] we have that N 2α , N 2(1−α) < R, then we have the inequalities 1/2 # $* ' (  ) # 2α $   f (N ) + f (N ∗ ) f A |N | + f |N |2(1−α)   x, y  ≤ (2.133) x, x  2 2 ) ×

1/2 # $ # $* f A |N |2α + f |N |2(1−α) y, y 2

for any x, y ∈ H . Proof If N is a normal operator, then for any j ∈ N we have that  j 2 # ∗ $ j  N  = N N = |N |2 j . Utilising the inequality (2.120) we have  ) *    j ∗ j   n N + (N )  aj x, y   2   j=0    n   N j + (N ∗ ) j    |a| j  ≤ x, y    2 j=0 !  j    " n  N x, y  +  (N ∗ ) j x, y   |a| j ≤ 2 j=0  n !# 1/2 $j # $j "  |N |2α + |N |2(1−α) |a| j ≤ x, x 2 j=0  n !# 1/2 $j # $j "  |N |2α + |N |2(1−α) |a|0 × y, y 2 j=1 for any α ∈ [0, 1], n ∈ N and any x, y ∈ H. Since N 2α , N 2(1−α) < R, then it follows that the series

(2.134)

∞  # $  a j  |N |2α j and j=0

∞  # $  a j  |N |2(1−α) j are absolute convergent in B (H ), and by taking the limit over j=0

n → ∞ in (2.134) we deduce the desired result (2.133).



2.10 Other Inequalities for Functions of Normal Operators

51

Remark 2.40 With the assumptions in Theorem 2.39, if we take the supremum over y ∈ H, y = 1, then we get the vector inequality %' ( % % 1 # # 2α $ % f (N ) + f (N ∗ ) # 2(1−α) $$ 1/2 % x% x, x % ≤ 2 f A |N | + f |N | % 2 % # $ # $% × % f A |N |2α + f |N |2(1−α) %

(2.135)

for any x ∈ H , which in its turn produces the norm inequality % % % f (N ) + f (N ∗ ) % 1 % # 2α $ # $% % % ≤ % f A |N | + f |N |2(1−α) % % % 2 2

(2.136)

for any α ∈ [0, 1]. Moreover, if we take y = x in (2.133), then we have    1 3 # 2α $  f (N ) + f (N ∗ ) # $4  ≤  x, x f A |N | + f |N |2(1−α) x, x   2 2

(2.137)

for any x ∈ H , which, by taking the supremum over x ∈ H, x = 1 generates the numerical radius inequality ' w

f (N ) + f (N ∗ ) 2

( ≤

# $4 1 3 # 2α $ w f A |N | + f |N |2(1−α) 2

(2.138)

for any α ∈ [0, 1]. We can state the particular vector inequalities: ! "    ln (1 + N )−1 + ln (1 + N ∗ )−1 H H   x, y     2 1/2 $−1 $−1  # 1  # ≤ ln 1 H − |N |2α x, x + ln 1 H − |N |2(1−α) 2 1/2  # $−1 $−1  # y, y × ln 1 H − |N |2α + ln 1 H − |N |2(1−α) ,

(2.139)

and ! "    (1 + N )−1 + (1 + N ∗ )−1 H H   x, y     2  1/2 $−1 # $−1  1 # ≤ 1 H − |N |2α x, x + 1 H − |N |2(1−α) 2 1/2  # $−1 $−1  # y, y × ln 1 H − |N |2α + ln 1 H − |N |2(1−α) , for any x, y ∈ H and N  < 1.

(2.140)

52

2 Inequalities for n-Tuples of Operators

We also have the inequalities      sin (N ) + sin (N ∗ )  x, y   2 3 # $ # $4 1/2 1 ≤ sinh |N |2α + sinh |N |2(1−α) x, x 2 $ # $4 1/2 3 # × sinh |N |2α + sinh |N |2(1−α) y, y

(2.141)

and      cos (N ) + cos (N ∗ )  x, y   2 # $ # $4 1/2 1 3 ≤ cosh |N |2α + cosh |N |2(1−α) x, x 2 $ # $4 1/2 3 # × cosh |N |2α + cosh |N |2(1−α) y, y

(2.142)

for any x, y ∈ H and N a normal operator. If we utilize function as power series representations with nonnegative coefficients, then we can state the following vector inequalities as well:      exp (N ) + exp (N ∗ )   x, y   2 # $ # $4 1/2 1 3 ≤ exp |N |2α + exp |N |2(1−α) x, x 2 $ # $4 1/2 3 # × exp |N |2α + exp |N |2(1−α) y, y

(2.143)

for any x, y ∈ H and N a normal operator. If N  < 1, then we also have the inequalities 0 / 0⎤ ⎡ /   ln 1 H +N + ln 1 H +N ∗ ∗   1 H −N 1 H −N  ⎣ ⎦ x, y    2   ) *" 1/2 ! ' ( 1 1 H + |N |2(1−α) 1 H + |N |2α ≤ + ln x, x ln 2 1 H − |N |2α 1 H − |N |2(1−α) ) ! ' *" 1/2 ( 1 H + |N |2(1−α) 1 H + |N |2α + ln × ln y, y 1 H − |N |2α 1 H − |N |2(1−α)

(2.144)

2.10 Other Inequalities for Functions of Normal Operators

     tanh−1 (N ) + tanh−1 (N ∗ )  x, y   2

53

(2.145)

# $ # $4 1/2 1 3 tanh−1 |N |2α + tanh−1 |N |2(1−α) x, x 2 # $ # $4 1/2 3 × tanh−1 |N |2α + tanh−1 |N |2(1−α) y, y ≤

and      2 F1 (α, β, γ, N ) +2 F1 (α, β, γ, N ∗ )  x, y   2 # $ # $4 1/2 1 3 2α ≤ +2 F1 α, β, γ, |N |2(1−α) x, x 2 F1 α, β, γ, |N | 2 # $ # $4 1/2 3 × 2 F1 α, β, γ, |N |2α +2 F1 α, β, γ, |N |2(1−α) y, y

(2.146)

for any x, y ∈ H . From a different perspective, we also have: Theorem 2.41 (Dragomir et al. [10]) With the assumption of Theorem 2.39 and if N is a normal operator on the Hilbert space H and z ∈ C such that N 2 , |z|2 < R, then we have the inequalities ' ( 2  f (z N ) + f (z N ∗ )   x, y   2 α  # 2 $ 1−α 1 # 2 $  # 2 $ ≤ f A |z| f A |N | x, x f A |N | y, y 2 $ α  # $ 1−α   # + f A |N |2 y, y f A |N |2 x, x ≤

(2.147)

  # $ 4 1 # 2 $ 3 # 2 $ f A |z| f A |N | x, x + f A |N |2 y, y 2

for any x, y ∈ H with x = y = 1 and α ∈ [0, 1]. In particular, for α = 21 we have ' ( 2   f (z N ) + f (z N ∗ )   x, y   2 # 2$  # 2$ 1/2  # 2 $ 1/2 ≤ f A |z| f A |N | x, x f A |N | y, y $   # $ 4 1 # $ 3 # ≤ f A |z|2 f A |N |2 x, x + f A |N |2 y, y 2 for any x, y ∈ H with x = y = 1.

(2.148)

54

2 Inequalities for n-Tuples of Operators

Proof If we use the third and the fourth inequality in (2.132) we have ⎡) * 2 ⎤ n      N j + (N ∗ ) j  a j  ⎣ x, y  ⎦   2 j=0 ⎞α ⎛ ⎞1−α ⎛ n n  %2   % j %2 1 ⎝   % j a j  % N y % ⎠ a j %N x % ⎠ ⎝ ≤ 2 j=0 j=0 ⎞α ⎛ ⎞1−α ⎛ n n  % %   % 1 ⎝   % 2 2 a j  % N j x % ⎠ a j % N j y% ⎠ ⎝ + 2 j=0 j=0

(2.149)

%2 % %2  1    % a j % N j x % + % N j y% 2 j=0 n

≤

for any x, y ∈ H with x = y = 1 and α ∈ [0, 1]. Since N is a normal operator on the Hilbert space H , then  % j %2  j 2  % N x % =  N  x, x = |N |2 j x, x for any j ∈ {0, ..., n} and for any x ∈ H with x = 1. Then from (2.149) we get ⎡) * 2 ⎤ n      N j + (N ∗ ) j  a j  ⎣ x, y  ⎦   2 j=0 ⎛ ⎞α ⎛ n ⎞1−α n   1 ⎝    a j  |N |2 j y, y ⎠ a j |N |2 j x, x ⎠ ⎝ ≤ 2 j=0 j=0 ⎞α ⎛ ⎛  ⎞1−α n n    1 ⎝    2j 2 j a j  |N | x, x ⎠ a j |N | y, y ⎠ ⎝ + 2 j=0 j=0 ⎡   n ⎤ n       1 a j  |N |2 j x, x + a j  |N |2 j y, y ⎦ ≤ ⎣ 2 j=0 j=0 for any x, y ∈ H with x = y = 1 and α ∈ [0, 1].

(2.150)

2.10 Other Inequalities for Functions of Normal Operators

55

By the weighted Cauchy–Buniakowski–Schwarz inequality we also have  ) * 2  n j ∗ j    N + (N )  ajz j x, y   2  j=0  ⎡) * 2 ⎤ n n     2j     N j + (N ∗ ) j  a j  |z| a j  ⎣ ≤ x, y  ⎦   2 j=0 j=0

(2.151)

for any x, y ∈ H with x = y = 1. ∞ ∞ ∞   ∞      a j  |z|2 j ,  a j  |N |2 j are Now, since the series ajz j N j, a j z j (N ∗ ) j , j=0

j=0

j=0

j=0

convergent, then by (2.150) and (2.151) on letting n → ∞, we deduce the desired result (2.147).  Similar inequalities for some particular functions of interest can be stated. However, the details are left to the interested reader.

2.11 Examples for the Euclidian Norm We have: Theorem 2.42 (Dragomir et al. [10]) For any (T1 , . . . , Tn ) ∈ B (n) (H ) we have % %α % %1−α % n % % n % %' 2 ∗ (%   % % % % % % T1 + T1∗     T + T n n % % %  T j 2 % %  T ∗ 2 % , . . . , ≤ (2.152) j % % % % % % 2 2 % j=1 % % j=1 % e % % % % % n % n % % %  ∗  2 % %  2 % % % %     T j % + (1 − α) % Tj % ≤ α% % % j=1 % j=1 % % and '

( T1 + T1∗ Tn + Tn∗ ,..., 2 2 ⎡ α  n 1−α ⎤ n   2   2 T j  x, x T ∗  x, x ⎦ ≤ sup ⎣ j

we2

x=1

j=1

j=1

(2.153)

56

2 Inequalities for n-Tuples of Operators

≤

⎧% %α % % % 2 %1−α n   % % n  ⎪ ⎪ 2 % % %   ⎪ % T j  % % T ∗  % ⎪ % ⎪ ⎪ % % j=1 j % ⎨ % j=1

% % ⎪ ⎪ %   % n  ⎪  ⎪ % n  2  ∗ 2 % ⎪ ⎪ α T j + (1 − α) T j  % ⎩% % j=1 % j=1 % % % % % n % n % % %  ∗  2 % %  2 % T j  % + (1 − α) % T  % ≤ α% j % % % % % j=1 % j=1 % %

for any α ∈ [0, 1]. Proof Making use of the inequalities (2.122) and (2.129) we have !' ( 2 " n   T j + T j∗   x, y   2 j=1 α  n 1−α  n   2 1   2 ∗  T j x, x T j y, y ≤ 2 j=1 j=1 α  n 1−α  n   2 1   2 ∗ T  x, x T j y, y + j 2 j=1 j=1 for any x, y ∈ H with x = y = 1 and α ∈ [0, 1]. Taking the supremum over x = y = 1 in (2.154) we get %' (%2 % T1 + T1∗ Tn + Tn∗ % % % ,..., % % 2 2 e α 1−α  n  n   2   2 1 ∗    T j x, x T j y, y ≤ sup sup 2 x=1 j=1 y=1 j=1 α 1−α  n  n   2   2 1 ∗ T j  y, y T  x, x + sup sup j 2 y=1 j=1 x=1 j=1 % %α % %1−α % n % % n % %  2 % %  ∗ 2 % % % %     Tj % % Tj % =% % % j=1 % % j=1 % and the inequality (2.152) is proved.

(2.154)

2.11 Examples for the Euclidian Norm

57

Now, if we take y = x in (2.154) we get !' ( 2 " n   T j + T j∗   x, x   2 j=1 α  n 1−α  n   2   2 ∗    T j x, x T x, x ≤

(2.155)

j

j=1

j=1

⎤ ⎡ n  n   2   ∗ 2 T j  + (1 − α) T  ⎦ x, x ≤ ⎣α j j=1

j=1

for any x ∈ H with x = 1 and α ∈ [0, 1]. Taking the supremum over x = 1 in (2.155) we get the desired result. Remark 2.43 In the particular case α =

1 2



we get

% %1/2 % %1/2 % %' % % % n n 2 ∗ (% % % T1 + T1∗ % % % %     T + T n n % % ≤%  T j 2 % %  T ∗ 2 % , . . . , j % % % % % % 2 2 % j=1 % % j=1 % e % % %⎤ ⎡% % % n % % n   2 % %  ∗ 2 % 1 ⎣% % T j  % + % T  %⎦ ≤ j % % % 2 % % % j=1 % % j=1

(2.156)

and ( Tn + Tn∗ T1 + T1∗ ,..., 2 2 ⎡ 1/2  n 1/2 ⎤ n   2   2 T j  x, x T ∗  x, x ⎦ ≤ sup ⎣ j '

we2

x=1

≤

j=1

j=1

⎧% %1/2 % % % % % n  2 %1/2 ⎪ ⎪ % n  2 % %   ∗ % ⎪ ⎪ T j % % T j  % % ⎪ ⎪ % % j=1 % ⎨ % j=1

%  2 % ⎪ ⎪ % n |T j |2 +T ∗  % ⎪ ⎪ j % % ⎪% ⎪ % ⎩% 2 % j=1 % % %⎤ % ⎡ % % % % n n % % % %     1 ⎣% 2% ∗ 2 %⎦ %    ≤ Tj % + % Tj % . % 2 % j=1 % % j=1 %

(2.157)

58

2 Inequalities for n-Tuples of Operators

2.12 Examples for s-1-Norm and s-1-Numerical Radius We have: Theorem 2.44 (Dragomir et al. [10]) For any (T1 , . . . , Tn ) ∈ B (n) (H ) we have % ⎡   2(1−α) ⎤% % n % 2α  ∗ %' (%   % % ∗ ∗ T + T j   % T1 + T1 Tn + Tn % % ⎥% ⎢ j % % (2.158) ≤ , . . . , % % ⎦ ⎣ % % % % 2 2 2 s,1 % j=1 % % % %⎤ ⎡% % % n % % n   % % % %     1 ⎣% 2α % ∗ 2(1−α) %⎦ %    ≤ Tj % + % Tj % % 2 % j=1 % % j=1 % and ' ws,1

T1 + T1∗ Tn + Tn∗ ,..., 2 2

( ≤ ws,1 (T1 , . . . , Tn ) (2.159) % ⎡   2(1−α) ⎤% % n % % T j 2α + T j∗  % % ⎥% ⎢ ≤% ⎦% ⎣ % % 2 % j=1 % % % %⎤ ⎡% % % n % % n   % % % %     1 T j 2α % + % T ∗ 2(1−α) %⎦ ≤ ⎣% j % % % % 2 % j=1 % % j=1 %

for any α ∈ [0, 1]. Proof Utilizing the inequality (2.110) we have  n     T j + T j∗   x, y   2 j=1 ⎡   2(1−α) ⎤  n 1/2 2α  ∗   T + T j  ⎢ j ⎥ ≤ ⎦ x, x ⎣ 2 j=1

(2.160)

⎡   2(1−α) ⎤ 1/2  n 2α     + T j∗   ⎢ Tj ⎥ × ⎦ y, y ⎣ 2 j=1 for any x, y ∈ H and α ∈ [0, 1]. Taking the supremum in (2.160) over x = y = 1 we get the first inequality in (2.158). The second part follows by the triangle inequality.

2.12 Examples for s-1-Norm and s-1-Numerical Radius

59

By the inequality (2.116) we have   n  n   T j + T j∗      T j x, x  ≤ x, x   2 j=1 j=1 ⎡   2(1−α) ⎤   n 2α    ⎢ T j  + T j∗  ⎥ ≤ ⎦ x, x ⎣ 2 j=1 for any x ∈ H . Taking the supremum over x = 1 we deduce the desired result (2.159). Remark 2.45 The case α =

1 2



produces the following chains of inequalities

% % % ' (% %' (% % n T j + T j∗ % % T1 + T1∗ Tn + Tn∗ % % %≤% % ,..., % % % % 2 2 2 % j=1 % s,1 % ⎛     ⎞% % T j  + T ∗  % % n % j ⎝ ⎠% ≤% % % 2 % j=1 % % % %⎤ ⎡% % % n % % n    % %  ∗ % 1 ⎣% % % %     ⎦ ≤ Tj % + % Tj % % 2 % % % j=1 % % j=1

(2.161)

and ⎛ ⎞ ' ( n ' ∗ ∗(  T j + T j∗ ⎠ ≤ ws,1 T1 + T1 , . . . , Tn + Tn w⎝ 2 2 2 j=1 ≤ ws,1 (T1 , . . . , Tn ) % ⎛     ⎞% % T j  + T ∗  % % n % j % ⎝ ⎠% ≤% % 2 % j=1 % % % %⎤ ⎡% % % n % % n   % % % %     1  T j % + % T ∗ %⎦ . ≤ ⎣% j % % % % 2 % j=1 % % j=1 %

(2.162)

Chapter 3

Generalizations of Furuta’s Type

In this chapter we present a two parameter generalization of Kato due to Furuta. Applications for functions of bounded linear operators defined by power series and inequalities for four bounded operators generalizing Furuta’s inequality and provide some general Norm and Numerical Radius inequalities are given as well.

3.1 Furuta’s Inequality In the following we denote by B (H ) the Banach algebra of all bounded linear operators on a complex Hilbert space (H ; ·, ·) . In 1952, Kato [22] proved the following celebrated generalization of Schwarz inequality for any bounded linear operator T on H :  1−α  α   |T x, y|2 ≤ T ∗ T x, x T T ∗ y, y ,

(K)

for any x, y ∈ H and α ∈ [0, 1]. Utilizing the modulus notation, we can write (K) as follows     2(1−α) |T x, y|2 ≤ |T |2α x, x T ∗  y, y (3.1) for any x, y ∈ H and α ∈ [0, 1]. In order to generalize this result, in 1994 Furuta [19] obtained the following result:         T |T |α+β−1 x, y 2 ≤ |T |2α x, x T ∗ 2β y, y

(F)

for any x, y ∈ H and α, β ∈ [0, 1] with α + β ≥ 1. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2019 S. S. Dragomir, Kato’s Type Inequalities for Bounded Linear Operators in Hilbert Spaces, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-030-17459-0_3

61

62

3 Generalizations of Furuta’s Type

From the proof in [19], one realizes that the condition α, β ∈ [0, 1] is taken only to fit with the result from the Heinz-Kato inequality |T x, y| ≤  Aα x B 1−α y ,

(HK)

for any x, y ∈ H and α ∈ [0, 1] , where A and B are positive operators such that T x ≤ Ax and T ∗ y ≤ By for all x, y ∈ H . Therefore, one can state the more general result: Theorem 3.1 (Furuta Inequality [19]) Let T ∈ B (H ) and α, β ≥ 0 with α + β ≥ 1. Then for any x, y ∈ H we have the inequality (F). We observe that this fact allows for some particular instances of interest that were not possible in the case when α, β ∈ [0, 1]. If we take β = α in (F) then we get         T |T |2α−1 x, y 2 ≤ |T |2α x, x T ∗ 2α y, y

(3.2)

for any x, y ∈ H and α ≥ 21 . In particular, for α = 1 we get     2 |T |T | x, y|2 ≤ |T |2 x, x T ∗  y, y

(3.3)

for any x, y ∈ H. If we take T = N a normal operator, i.e., we recall that N N ∗ = N ∗ N , then we get from (F) the following inequality for normal operators       N |N |α+β−1 x, y 2 ≤ |N |2α x, x |N |2β y, y

(3.4)

for any x, y ∈ H and α, β ≥ 0 with α + β ≥ 1. This implies the inequalities       N |N |2α−1 x, y 2 ≤ |N |2α x, x |N |2α y, y for any x, y ∈ H and α ≥

1 2

and, in particular,

   |N |N | x, y|2 ≤ |N |2 x, x |N |2 y, y for any x, y ∈ H. Making y = x in (3.5) produces      N |N |2α−1 x, x  ≤ |N |2α x, x for any x ∈ H and α ≥

1 2

(3.5)

and, in particular,

(3.6)

3.1 Furuta’s Inequality

63

  |N |N | x, x| ≤ |N |2 x, x for any x ∈ H.

3.2 Functional Generalizations

n Now, by the help of power series f (z) = ∞ n=0 an z we can naturally construct another power series which has as coefficients values of the coefficients

the absolute n |a | z . It is obvious that this new of the original series, namely, f A (z) := ∞ n n=0 power series has the same radius of convergence as the original series. We also notice that if all coefficients an ≥ 0, then f A = f.

n Theorem 3.2 (Dragomir [4]) Let f (z) = ∞ n=0 an z be a function defined by power series with real coefficients and convergent on the open disk D (0, R) ⊂ C, R > 0. Let T ∈ B (H ), α, β ≥ 0 with α + β ≥ 1 and such that T 2α , T 2β < R,

(3.7)

  α+β  α+β−1 2  T f |T | |T | x, y         2β  ∗ 2β T  y, y ≤ f A |T |2α |T |2α x, x f A T ∗ 

(3.8)

then we have the inequality

for any x, y ∈ H. Proof Since α, β ≥ 0 with α + β ≥ 1, then nα + nβ ≥ 1 for any n ≥ 1. From Furuta’s inequality (F) we have the power inequality 1/2        T |T |nα+nβ−1 x, y  ≤ |T |2nα x, x 1/2 T ∗ 2nβ y, y ,

(3.9)

for any natural numbers n ≥ 1 and x, y ∈ H. If we multiply this inequality with the positive quantities |an−1 |, use the triangle inequality and the Cauchy–Bunyakowsky–Schwarz discrete inequality we have successively  k       (3.10) an−1 T |T |n(α+β)−1 x, y     n=1

≤

k  n=1

  |an−1 |  T |T |n(α+β)−1 x, y 

64

3 Generalizations of Furuta’s Type

≤

k 

1/2  1/2  ∗ 2nβ T  y, y |an−1 | |T |2nα x, x

n=1

≤

k 

|an−1 | |T |2nα x, x

1/2 k 

n=1

 2nβ |an−1 | T ∗  y, y

1/2

n=1

for any x, y ∈ H and k ≥ 1. Observe also that k 

an−1 T |T |n(α+β)−1 = T

n=1

 k 

 an−1 |T |(n−1)(α+β) |T |(α+β)−1 ,

n=1 k 

|an−1 | |T |2nα =

n=1

and k 

 k 

 |an−1 | |T |2(n−1)α |T |2α

n=1

 k    ∗ 2nβ  ∗ 2(n−1)β  ∗ 2β T  |an−1 | T  = |an−1 | T 

n=1

n=1

for any k ≥ 1. Therefore, by (3.10) we have the inequality   k 2       (n−1)(α+β) (α+β)−1 |T | an−1 |T | x, y   T   n=1 

 k   |an−1 | |T |2(n−1)α |T |2α x, x ≤ n=1

×

 k 

(3.11)

   ∗ 2(n−1)β  ∗ 2β T  y, y |an−1 | T 

n=1

for any x, y ∈ H and k ≥ 1.   n

n

|an | |T |2α and From (3.7) we have that the series ∞ an |T |α+β , ∞ n=0 n=0  ∗ 2β n

∞ are convergent in B (H ) and taking the limit over k → ∞ in n=0 |an | |T | (3.11) we deduce the desired result in (3.8).  Corollary 3.3 (Dragomir [4]) With the assumptions of Theorem 3.2 we have the norm inequality      α+β  α+β−1 2  2α  2α 2β  ∗ 2β T f |T | ≤ f A |T | |T | |T | T f A T ∗  and the numerical radius inequality

(3.12)

3.2 Functional Generalizations

65

      1    2β  ∗ 2β  T w T f |T |α+β |T |α+β−1 ≤ f A |T |2α |T |2α + f A T ∗  . 2 (3.13) Proof The proof of (3.12) follows by (3.8) on taking the supremum over x, y ∈ H with x = y = 1. By the inequality (3.8) we also have   α+β  α+β−1   T f |T | |T | x, x  1/2    1/2   ∗ 2β  ∗ 2β T  x, x ≤ f A |T |2α |T |2α x, x f A T        1   2α  2α 2β  ∗ 2β T f A |T | |T | + f A T ∗  x, x ≤ 2 for any x ∈ H. Taking the supremum over x = 1 we deduce the desired inequality (3.13).  Remark 3.4 If we take f (z) = 1, then we get from (3.8) the Furuta’s inequality (F). If we take β = α in (3.8), then we get   2α  2α−1 2  T f |T | |T | x, y        2α  ∗ 2α  T  y, y ≤ f A |T |2α |T |2α x, x f A T ∗ 

(3.14)

provided α ≥ 21 and T 2α < R. In particular, we have       |T f (|T |) x, y|2 ≤  f A (|T |) |T | x, x f A T ∗  T ∗  y, y

(3.15)

for any T ∈ B (H ) with T  < R. Remark 3.5 If we take β = 1 − α with α ∈ [0, 1] in (3.8) then we get the following generalization of Kato’s inequality (3.1) |T f (|T |) x, y|2        2(1−α)  ∗ 2(1−α) T  ≤ f A |T |2α |T |2α x, x f A T ∗  y, y

(3.16)

for any x, y ∈ H and T ∈ B (H ) with T 2α , T 2(1−α) < R. The following result concerning two functions also holds:

∞ n n Theorem 3.6 (Dragomir [4]) Let f (z) = ∞ n=0 an z and be g (z) = n=0 bn z be two functions defined by power series with real coefficients and both of them convergent on the open disk D (0, R) ⊂ C, R > 0. Let T ∈ B (H ), α, β ≥ 0 with α + β ≥ 1 and z, u ∈ C such that

66

3 Generalizations of Furuta’s Type

|z|2 , |u|2 , T 2α , T 2β < R,

(3.17)

then we have the inequality      T f (z |T |α ) g u |T |β |T |α+β−1 x, y 2 (3.18)   2   2    2α  2α    ∗ 2β  ∗ 2β T  y, y ≤ f A |z| g A |u| f A |T | |T | x, x g A T  for any x, y ∈ H. Proof Since α, β ≥ 0 with α + β ≥ 1, then for any n, m ≥ 1 natural numbers we also have that nα + mβ ≥ 1. From Furuta’s inequality (F) written for nα + mβ ≥ 1 we have for any natural numbers n ≥ 1 and m ≥ 1 the following power inequality 1/2        T |T |nα+mβ−1 x, y  ≤ |T |2nα x, x 1/2 T ∗ 2mβ y, y ,

(3.19)

where x, y ∈ H. If we multiply this inequality with the positive quantities |an−1 | |z|n−1 and |bm−1 | |u|m−1 , use the triangle inequality and the Cauchy–Bunyakowsky–Schwarz discrete inequality we have successively:  k l       n−1 m−1 nα+mβ−1 an−1 z bm−1 u x, y  T |T |   

(3.20)

n=1 m=1

≤

k  l 

  |an−1 | |z|n−1 |bm−1 | |u|m−1  T |T |nα+mβ−1 x, y 

n=1 m=1

≤

k 

l   1/2  1/2  2mβ |an−1 | |z|n−1 |T |2nα x, x |bm−1 | |u|m−1 T ∗  y, y

n=1

≤

 k  

×

|an−1 | |z|

2(n−1)

1/2 k 

n=1 l 

m=1

1/2

|an−1 | |T |

2nα

x, x

n=1

1/2 |bm−1 | |u|

2(m−1)

m=1

for any x, y ∈ H and k ≥ 1, l ≥ 1.

l 

m=1

 2mβ |bm−1 | T ∗  y, y

1/2

3.2 Functional Generalizations

67

Observe also that k  l 

  an−1 z n−1 bm−1 u m−1 T |T |nα+mβ−1 x, y

(3.21)

n=1 m=1

 k   l    n−1 m−1 α+β−1 (n−1)α (m−1)β |T | |T | |T | = T an−1 z bm−1 u x, y n=1

m=1

for any x, y ∈ H and k ≥ 1, l ≥ 1. Making use of (3.20) and (3.21) we get   k   l        n−1 m−1 α+β−1 (n−1)α (m−1)β |T | |T | |T | an−1 z bm−1 u x, y   T   n=1

 ≤  ×

k 

m=1

|an−1 | |z|2(n−1)

1/2  k 

n=1 l 



|bm−1 | |u|

2(m−1)

m=1

(3.22)

|an−1 | |T |2(n−1)α |T |2α x, x

n=1

1/2 

1/2

l 

1/2   ∗ 2(m−1)β  ∗ 2β T  y, y |bm−1 | T 

m=1

for any x, y ∈ H and k ≥ 1, l ≥ 1. From (3.17) we have that the series ∞  n=0

an z n |T |nα ,

∞ 

bm u m |T |mβ ,

m=0

∞ 

|an | |T |2nα

n=0

and ∞ 

 2mβ |bm | T ∗ 

m=0



∞ 2n 2m are convergent in B (H ) and the series ∞ are conn=0 |an | |z| and m=0 |bm | |u| vergent in R and then, by taking the limit over k → ∞ and l → ∞ in (3.22) we deduce desired result (3.18).  Remark 3.7 The above inequality (3.18) can provide various particular instances of interest. For instance, if we take g = f and z = u in Theorem 3.6 then we get      T f (z |T |α ) f z |T |β |T |α+β−1 x, y 2         2β  ∗ 2β T  y, y ≤ f A2 |z|2 f A |T |2α |T |2α x, x f A T ∗ 

(3.23)

68

3 Generalizations of Furuta’s Type

for any x, y ∈ H. Also if we take f (z) = 1 in (3.23), then we get Furuta’s inequality (F). Corollary 3.8 (Dragomir [4]) With the assumptions of Theorem 3.6 we have the norm inequality   T f (z |T |α ) g u |T |β |T |α+β−1 2           2β  ∗ 2β ≤ f A |z|2 g A |u|2 f A |T |2α |T |2α g A T ∗  T

(3.24)

and the numerical radius inequality     w T f (z |T |α ) g u |T |β |T |α+β−1 (3.25)       1   2   2 1/2 2β  ∗ 2β T ≤ f A |z| g A |u| f A |T |2α |T |2α + g A T ∗  . 2 Proof The inequality (3.24) follows from (3.18) by taking the supremum over x, y ∈ H with x = y = 1. Now, from (3.18) we also have      T f (z |T |α ) g u |T |β |T |α+β−1 x, x  1/2     1/2   ∗ 2β  ∗ 2β 1/2   2α  2α T  x, x ≤ f A |z|2 g A |u|2 g A T  f A |T | |T | x, x       1   2   2 1/2   2α  2α 2β  ∗ 2β f A |z| g A |u| f A |T | |T | + g A T ∗  T x, x ≤ 2 for any x ∈ H. Taking the supremum over x ∈ H with x = 1 we get the desired result (3.25).  Remark 3.9 Special cases of (3.18) can be obtained if we take β = α or β = 1 − α with α ∈ [0, 1] as in Remarks 3.4 and 3.5. The details are omitted.

3.3 Some Examples Utilising the inequality (3.8) and the power series representations of some elementary functions as above, we have   2 −1   |T |α+β−1 x, y   T 1 H ± |T |α+β      2β −1  ∗ 2β −1 T  y, y |T |2α x, x ≤ 1 H − |T |2α 1 H − T ∗  and

(3.26)

3.3 Some Examples

69

  2 −1  α+β−1   |T | x, y   T ln 1 H ± |T |α+β    −1  2α |T | x, x ≤ ln 1 H − |T |2α      ∗ 2β −1  ∗ 2β     T y, y × ln 1 H − T

(3.27)

for T ∈ B (H ), α, β ≥ 0 with α + β ≥ 1 and such that T  < 1 and for any x, y ∈ H. From (3.8) we also have the inequalities      T sin |T |α+β |T |α+β−1 x, y 2          2β  ∗ 2β T ≤ sinh |T |2α |T |2α x, x sinh T ∗  y, y ,

(3.28)

     T cos |T |α+β |T |α+β−1 x, y 2          2β  ∗ 2β T ≤ cosh |T |2α |T |2α x, x cosh T ∗  y, y ,

(3.29)

     T exp |T |α+β |T |α+β−1 x, y 2          2β  ∗ 2β ≤ exp |T |2α |T |2α x, x exp T ∗  T y, y

(3.30)

and

for T ∈ B (H ), α, β ≥ 0 with α + β ≥ 1 and for any x, y ∈ H. Utilizing the inequality (3.18) we have      T exp z |T |α + u |T |β |T |α+β−1 x, y 2 (3.31)        2      2β  ∗ 2β ≤ exp |z| + |u|2 exp |T |2α |T |2α x, x exp T ∗  T y, y and      T sin (z |T |α ) cos u |T |β |T |α+β−1 x, y 2 (3.32)       2   2   2α  2α 2β  ∗ 2β T ≤ sinh |z| cosh |u| sinh |T | |T | x, x cosh T ∗  y, y for T ∈ B (H ), α, β ≥ 0 with α + β ≥ 1 and for any z, u ∈ C, x, y ∈ H. By the same inequality (3.18) we also get  2  −1   −1 |T |α+β−1 x, y  1 H ± u |T |β  T (1 H ± z |T |α )     −1 −1 ∗ 2β |T |2α x, x 1 H − |T ∗ |2β |T | y, y 1 H − |T |2α    ≤ 1 − |z|2 1 − |u|2

(3.33)

70

3 Generalizations of Furuta’s Type

for T ∈ B (H ), z, u ∈ C with T  , |z| , |u| < 1, α, β ≥ 0 with α + β ≥ 1 and for any x, y ∈ H.

3.4 More Functional Inequalities We can state the following corollary of Furuta’s inequality (F) for the numerical radius w of an operator V ∈ B (H ), namely w (V ) = supx=1 |V x, x|, which satisfies the following basic inequalities 1 V  ≤ w (V ) ≤ V  . 2 Corollary 3.10 Let T ∈ B (H ) and α, β ≥ 0 with α + β ≥ 1. Then we have  2β   1 w T |T |α+β−1 ≤ |T |2α + T ∗  . 2

(3.34)

In particular, we also have  2α   1 w T |T |2α−1 ≤ |T |2α + T ∗  , 2 for any α ≥

1 2

(3.35)

and, as a special case, w (T |T |) ≤

1 2  ∗ 2 |T | + T . 2

(3.36)

Proof We have from (F) for any x ∈ H that 1/2        T |T |α+β−1 x, x  ≤ |T |2α x, x 1/2 T ∗ 2β x, x  1  2α  ∗ 2β  |T | + T x, x ≤ 2

(3.37)

where α, β ≥ 0 with α + β ≥ 1. Utilising the inequality in (3.37) and taking the supremum over x ∈ H, x = 1 we get     w T |T |α+β−1 = sup  T |T |α+β−1 x, x  x=1

   2β  1 sup |T |2α + T ∗  x, x 2 x=1  2β 1 = |T |2α + T ∗  . 2 ≤



3.4 More Functional Inequalities

71

n Now, by the help of power series f (z) = ∞ n=0 an z we also have:

∞ n n Theorem 3.11 Let f (z) = ∞ n=0 an z and be g (z) = n=0 bn z be two functions defined by power series with real coefficients and both of them convergent on the open disk D (0, R) ⊂ C, R > 0. If T is a bounded linear operator on the Hilbert space H and z, u ∈ C with the property that |z|2 , |u|2 , T 2 < R,

(3.38)

|T f (z |T |) g (u |T |) x, y|2          2  2 ≤ f A |z|2 g A |u|2 f A |T |2 x, x T ∗  g A T ∗  y, y

(3.39)

then we have the inequality

for any x, y ∈ H. Proof From Furuta’s inequality (F) we have for any natural numbers n ≥ 0 and m ≥ 1 the following power inequality 1/2        T |T |n+m−1 x, y  ≤ |T |2n x, x 1/2 T ∗ 2m y, y ,

(3.40)

where x, y ∈ H. If we multiply this inequality with the positive quantities |an | |z|n and |bm−1 | |u|m−1 , use the triangle inequality and the Cauchy–Bunyakowsky–Schwarz discrete inequality we have successively:  k l         an z n bm−1 u m−1 T |T |n+m−1 x, y    

(3.41)

n=0 m=1

≤

k  l 

  |an | |z|n |bm−1 | |u|m−1  T |T |n+m−1 x, y 

n=0 m=1

≤

k 



|an | |z| |T | x, x n

n=0

≤

 k  

×

|an | |z|

2n

2n

1/2 k 

n=0 l 

l 1/2  m=1

|bm−1 | |u|

m=1

for any x, y ∈ H and k ≥ 0, l ≥ 1. Observe also that

1/2

|an | |T | x, x

n=0 1/2 2(m−1)

  1/2 2m |bm−1 | |u|m−1 T ∗  y, y

2n

l 

m=1

 2m |bm−1 | T ∗  y, y

1/2

72

3 Generalizations of Furuta’s Type k  l 

  an z n bm−1 u m−1 T |T |n+m−1 x, y

(3.42)

n=0 m=1

 k  l     n n m−1 m−1 |T | an z |T | bm−1 u = T x, y n=0

m=1

for any x, y ∈ H and k ≥ 0, l ≥ 1. Making use of (3.41) and (3.42) we get   k  l         an z n |T |n bm−1 u m−1 |T |m−1 x, y   T   n=0 m=1  k 1/2 1/2 k   2n 2n |an | |z| |an | |T | x, x ≤  ×

n=0 l 

n=0 1/2

|bm−1 | |u|

2(m−1)

m=1

(3.43)

1/2

l  ∗ 2   ∗ 2(m−1) T  |bm−1 | T  y, y m=1

for any x, y ∈ H and k ≥ 0, l ≥ 1.

n n Due to the assumption (3.38) in the theorem, we have that the series ∞ n=0 an z |T | ,

∞



∞ ∞ 2m m 2n m ∗ |T | |a | |T | |b | |T | b u , and are convergent in B (H ) m=0 m n=0 n m=0 m

∞

2n 2m are convergent in R and then, by and the series ∞ n=0 |an | |z| and m=0 |bm | |u| taking the limit over k → ∞ and l → ∞ in ( 3.43), we deduce the desired result (3.39).  Remark 3.12 The above inequality (3.39) can provide various particular instances of interest. For instance, if we take g = f in Theorem 3.11 then we get  2   T f (z |T |) x, y 

1/2     1/2  ∗ 2  ∗ 2 T  f A T  y, y ≤ f A |z|2 f A |T |2 x, x

(3.44)

for any x, y ∈ H. Also if we take g (z) = 1 in (3.39), then we get        2 |T f (z |T |) x, y|2 ≤ f A |z|2 f A |T |2 x, x T ∗  y, y

(3.45)

for any x, y ∈ H. Corollary 3.13 (Dragomir [6]) With the assumptions of Theorem 3.11 we have the norm inequality

3.4 More Functional Inequalities

T f (z |T |) g (u |T |)2         2  2 ≤ f A |z|2 g A |u|2 f A |T |2 T ∗  g A T ∗ 

73

(3.46)

and the numerical radius inequality w (T f (z |T |) g (u |T |))   1   2   2 1/2 2  2   ∗ 2 f A |z| g A |u| ≤ f A |T | + T g A T ∗  . 2

(3.47)

Proof The inequality (3.46) follows from (3.39) by taking the supremum over x, y ∈ H with x = y = 1. From (3.39) we also have the inequality |T f (z |T |) g (u |T |) x, x| 1/2 1/2   2  1/2  ∗ 2  ∗ 2     T  g A T  x, x f A |T | x, x ≤ f A |z|2 g A |u|2 1/2 1   2   2 1/2   2   ∗ 2  ∗ 2  f A |z| g A |u| f A |T | + T g A T x, x ≤ 2 for any x ∈ H, which, by taking the supremum over x = 1 produces the desired result (3.47).  The following result also holds:

∞ n Theorem 3.14 (Dragomir [6]) Let f (z) = n=0 an z be a function defined by power series with real coefficients and convergent on the open disk D (0, R) ⊂ C, R > 0. If T is a bounded linear operator on the Hilbert space H with the property that T 2 < R, then we have the inequality               T |T | f |T |2 x, y 2 ≤ |T |2 f A |T |2 x, x T ∗ 2 f A T ∗ 2 y, y

(3.48)

for any x, y ∈ H. Proof From Furuta’s inequality (F) we have for any natural numbers n ≥ 1 the power inequality 1/2        T |T |2n−1 x, y  ≤ |T |2n x, x 1/2 T ∗ 2n y, y

(3.49)

where x, y ∈ H. If we multiply this inequality with the positive quantities |an−1 |, use the triangle inequality and the Cauchy–Bunyakowsky–Schwarz discrete inequality we have successively

74

3 Generalizations of Furuta’s Type

 k       2n−1 an−1 T |T | x, y    

(3.50)

n=1

≤

k 

  |an−1 |  T |T |2n−1 x, y 

n=1

≤

k 

1/2  1/2  ∗ 2n T  y, y |an−1 | |T |2n x, x

n=1

≤

k 

1/2 |an−1 | |T | x, x 2n

n=1

k 

 2n |an−1 | T ∗  y, y

1/2

n=1

for any x, y ∈ H and k ≥ 1. Observe also that k 

an−1 T |T |2n−1 = T |T |

n=1

|an−1 | |T |2n = |T |2

n=1

k 

an−1 |T |2(n−1) ,

n=1

k 

and

k 

k 

|an−1 | |T |2(n−1)

n=1

k  2n  2   2(n−1) |an−1 | T ∗  = T ∗  |an−1 | T ∗ 

n=1

n=1

for any k ≥ 1. Therefore, by (3.50) we have the inequality

T |T |

k 

2 an−1 |T |

2(n−1)

x, y

n=1

≤ |T |

2

k  n=1

|an−1 | |T |

2(n−1)

(3.51)

  k  ∗ 2   ∗ 2(n−1) |an−1 | T  x, x T  y, y n=1

for any x, y ∈ H and k ≥ 1.

2 2n Due to the assumption R, we have that the series ∞ n=0 an |T | ,

∞ T  ∗
0. If N is a normal operator on the Hilbert space H and α, β ≥ 0 with α +β ≥ 1 with the property that N 2α , N 2β < R, then we have the inequality             f N |N |(α+β−1) x, y 2 ≤ f A |N |2α x, x f A |N |2β y, y

(3.52)

for any x, y ∈ H. Proof Utilising Furuta’s inequality written for N n we have    2      2β  2α ∗   n  n α+β−1 x, y  ≤  N n  x, x  N n  y, y  N N

(3.53)

for any x, y ∈ H. Since N is normal, then  n 2  n ∗ n N  = N N = N∗ · · · N∗N · · · N = N∗ · · · N N∗ · · · N = · · ·     = N ∗ N · · · N ∗ N = |N |2n for any natural number n, and, similarly,  n ∗ 2  ∗ n 2  ∗ 2n  N  =  N  =  N  = |N |2n for any n ∈ N.  2β These imply that |N n |2α = |N |2αn , (N n )∗  = |N |2βn and |N n |α+β−1 = |N |(α+β−1)n for any α, β ≥ 0 and for any n ∈ N. Utilising the spectral representation for Borel functions of normal operators on Hilbert spaces, see for instance [1, p. 67], we have for any α, β ≥ 0 and for any n ∈ N that  n (α+β−1)n = z n |z|(α+β−1)n d P (z) N |N | σ (N )   n = z |z|(α+β−1) d P (z) 

σ (N )

= N |N |(α+β−1)

n

,

76

3 Generalizations of Furuta’s Type

where P is the spectral measure associated to the operator N and σ (N ) is its spectrum. Therefore, the inequality (3.53) can be written as    1/2  1/2 n n n   |N |2β y, y  N |N |(α+β−1) x, y  ≤ |N |2α x, x

(3.54)

for any x, y ∈ H and for any n ∈ N. If we multiply the inequality (3.54) by |an | ≥ 0, sum over n from 0 to k ≥ 1 and utilize the Cauchy–Bunyakowsky–Schwarz discrete inequality, we have successively  k         (α+β−1) n an N |N | x, y    

(3.55)

n=0

≤

k 

   n   |an |  N |N |(α+β−1) x, y 

n=0

≤

k 

 1/2  1/2 n n |an | |N |2α x, x |N |2β y, y

n=0

≤

k 

 n |an | |N |2α x, x

n=0

1/2 k 

 n |an | |N |2β y, y

1/2

n=0

for any x, y ∈ H and for any k ≥ 1. Since N 2α , N 2β < R then N |N |(α+β−1) < R and the series ∞ 

∞   n  n |an | |N |2α , |an | |N |2β

n=0

n=0

and ∞ 

 n an N |N |(α+β−1)

n=0

are convergent in the Banach algebra B (H ). Taking the limit over k → ∞ in the inequality (3.55) we deduce the desired result from (3.52).  Corollary 3.17 (Dragomir [6]) With the assumptions of Theorem 3.16, we have the inequality       f N |N |(α+β−1) 2 ≤ f A |N |2α f A |N |2β . (3.56) Remark 3.18 If we take β = 1 − α with α ∈ [0, 1] in (3.52), then we get the following generalization of Kato’s inequality for normal operators

3.4 More Functional Inequalities

77

       | f (N ) x, y|2 ≤ f A |N |2α x, x f A |N |2(1−α) y, y

(3.57)

where x, y ∈ H and N 2α , N 2(1−α) < R.

3.5 Applications for Some Elementary Functions We can give now some examples: Example 3.19 Let x, y ∈ H. (a) If we take f (z) = sin z and g (z) = cos z in ( 3.39), then we get |T sin (z |T |) cos (u |T |) x, y|2     ≤ sinh |z|2 cosh |u|2         2 2 × sinh |T |2 x, x T ∗  cosh T ∗  y, y for any z ∈ C and T ∈ B (H ). 1 and g (z) = ln (b) If we take f (z) = ln 1+z

1 1−z

(3.58)

in (3.39), then we get

   T ln (1 H + z |T |)−1 ln (1 H − z |T |)−1 x, y 2  2 1 ≤ ln 1 − |z|2         ∗ 2 −1  2 −1 ∗ 2    × ln 1 H − |T | x, x T ln 1 H − T y, y

(3.59)

for any z ∈ C and T ∈ B (H ) with |z| < 1 and T  < 1. (c) If we take f (z) = exp(z) and g (z) = exp (z) in (3.39), then we get |T exp [(z + u) |T |] x, y|2     ≤ exp |z|2 exp |u|2         2 2 × exp |T |2 x, x T ∗  exp T ∗  y, y

(3.60)

for any z, u ∈ C and T ∈ B (H ). (d) By the inequality (3.45) we have            T sin−1 (z |T |) x, y 2 ≤ sin−1 |z|2 sin−1 |T |2 x, x T ∗ 2 y, y and

(3.61)

78

3 Generalizations of Furuta’s Type

   T tanh−1 (z |T |) x, y 2         2 ≤ tanh−1 |z|2 tanh−1 |T |2 x, x T ∗  y, y

(3.62)

for any z ∈ C and T ∈ B (H ) with |z| < 1 and T  < 1. Example 3.20 Let x, y ∈ H. 1 in (3.48), then we get (a) If we take f (z) = 1±z  2  −1   x, y   T |T | 1 H ± |T |2        2 −1  −1 2 ≤ |T |2 1 H − |T |2 x, x T ∗  1 H − T ∗  y, y

(3.63)

for any T ∈ B (H ) with T  < 1. 1 in (3.48), then we get (b) If we take f (z) = ln 1±z  2  −1   x, y   T |T | ln 1 H ± |T |2        2 −1 −1  2 ≤ |T |2 ln 1 H − |T |2 x, x T ∗  ln 1 H − T ∗  y, y

(3.64)

for any T ∈ B (H ) with T  < 1. (c) If we take f (z) = exp (z) in (3.48), then we get      T |T | exp |T |2 x, y 2        2  2 ≤ |T |2 exp |T |2 x, x T ∗  exp T ∗  y, y

(3.65)

for any T ∈ B (H ). Example 3.21 Let N be a normal operator on the Hilbert space H , α, β ≥ 0 with α + β ≥ 1 and x, y ∈ H. 1 in (3.52), then we get (a) If we take f (z) = 1±z  2 −1   x, y   1 H ± N |N |(α+β−1)     −1 −1 ≤ 1 H − |N |2α x, x 1 H − |N |2β y, y

(3.66)

provided N  < 1. In particular, we have    (1 H ± N )−1 x, y 2     −1 −1 ≤ 1 H − |N |2α x, x 1 H − |N |2(1−α) y, y ,

(3.67)

3.5 Applications for Some Elementary Functions

79

for α ∈ [0, 1]. (b) If we take f (z) = exp (z) in (3.52), then we get             exp N |N |(α+β−1) x, y 2 ≤ exp |N |2α x, x exp |N |2β y, y .

(3.68)

As a special case, we have        |exp (N ) x, y|2 ≤ exp |N |2α x, x exp |N |2(1−α) y, y ,

(3.69)

for α ∈ [0, 1].

3.6 General Vector Inequalities The following result provides a general extension for four operators of the Schwarz inequality: Theorem 3.22 (Dragomir [7]) Let A, B, C, D ∈ B (H ). Then for x, y ∈ H we have the inequality      2 |DC B Ax, y|2 ≤ A∗ |B|2 Ax, x D C ∗  D ∗ y, y .

(3.70)

The equality case holds in (3.70) if the vectors B Ax and C ∗ D ∗ y are linearly dependent in H. Proof The Schwarz inequality in the Hilbert space H states that for any u, v ∈ H we have the inequality |u, v|2 ≤ u2 v2 (3.71) with equality if and only if the vectors u and v are linearly dependent in H. Now, if we take u = B Ax and v = C ∗ D ∗ y then we have   u2 = B Ax, B Ax = B ∗ B Ax, Ax     = A∗ B ∗ B Ax, x = A∗ |B|2 Ax, x ,     v2 = C ∗ D ∗ y, C ∗ D ∗ y = CC ∗ D ∗ y, D ∗ y      2 = DCC ∗ D ∗ y, y = D C ∗  D ∗ y, y and     u, v = B Ax, C ∗ D ∗ y = C B Ax, D ∗ y = DC B Ax, y .

80

3 Generalizations of Furuta’s Type



Utilising (3.71) we deduce the desired result (3.70).

Corollary 3.23 (Dragomir [7]) The Furuta inequality (F) for α, β ≥ 0 with α +β ≥ 1 is a particular case of (3.70). Proof Let T = U |T | be the polar decomposition of an operator T, where U is partial isometry and the kernel N (U ) = N (|T |). If we take D = U, C = |T |β , B = 1 H and A = |T |α then we have DC B A = U |T |β |T |α = U |T | |T |α+β−1 = T |T |α+β−1 , A∗ |B|2 A = |T |α |T |α = |T |2α and  2β  2 D C ∗  D ∗ = U |T |2β U ∗ = T ∗  , 

which by (3.70) implies the desired inequality (F). The following similar result also holds

Corollary 3.24 (Dragomir [7]) For any operator T ∈ B (H ) and any α, β ≥ 1 we have the inequality         T |T |β−1 T |T |α−1 x, y 2 ≤ |T |2α x, x T ∗ 2β y, y ,

(3.72)

where x, y ∈ H. Proof Let T = U |T | be the polar decomposition of an operator T, where U is partial isometry and the kernel N (U ) = N (|T |). If we take D = U, C = |T |β , B = U and A = |T |α then we have DC B A = U |T |β U |T |α = U |T | |T |β−1 U |T | |T |α−1 = T |T |β−1 T |T |α−1 , A∗ |B|2 A = |T |α U ∗ U |T |α = |T |α−1 |T | U ∗ U |T | |T |α−1 = |T |α−1 T ∗ T |T |α−1 = |T |α−1 |T |2 |T |α−1 = |T |2α and  2β  2 D C ∗  D ∗ = U |T |2β U ∗ = T ∗  , which by (3.70) implies the desired inequality (3.72).



Remark 3.25 The above inequality (3.72) contains some nice particular inequalities as follows:

3.6 General Vector Inequalities

81

  2   2   2α    T |T |α−1 x, y  ≤ |T |2α x, x T ∗  y, y ,

(3.73)

for α ≥ 1 producing for α = 1 the result   2       T x, y 2 ≤ |T |2 x, x T ∗ 2 y, y ,

(3.74)

and for α = 2 the result

        (T |T |)2 x, y 2 ≤ |T |4 x, x T ∗ 4 y, y ,

(3.75)

for any x, y ∈ H. If we take α = 1 in (3.72), then we get         T |T |β−1 T x, y 2 ≤ |T |2 x, x T ∗ 2β y, y ,

(3.76)

for any β ≥ 1 and if we take β = 1 then we also get   2       T |T |α−1 x, y 2 ≤ |T |2α x, x T ∗ 2 y, y ,

(3.77)

for any x, y ∈ H. Corollary 3.26 (Dragomir [7]) For any operator T ∈ B (H ) and any γ, δ ≥ 0 we have the inequality   γ 2       |T | T |T |δ x, y 2 ≤ |T |2δ+2 x, x T ∗ |T |γ 2 y, y ,

(3.78)

where x, y ∈ H. Proof If we take D = |T |γ , C = T, B = T and A = |T |δ then we have DC B A = |T |γ T 2 |T |δ , A∗ |B|2 A = |T |δ |T |2 |T |δ = |T |2δ+2 and  2  2 D C ∗  D ∗ = |T |γ T ∗  |T |γ = |T |γ T T ∗ |T |γ 2  ∗  ∗ 2  = |T |γ T |T |γ T =  |T |γ T  = T ∗ |T |γ  , which by (3.70) implies the desired inequality (3.78).



82

3 Generalizations of Furuta’s Type

Remark 3.27 The particular case γ = δ = 1 provides the inequality         |T | T 2 |T | x, y 2 ≤ |T |4 x, x T ∗ |T |2 y, y ,

(3.79)

for any x, y ∈ H. We also have Corollary 3.28 (Dragomir [7]) For any operator T ∈ B (H ) and any γ, δ ≥ 0 we have the inequalities  γ+δ+2 2     |T | x, y  ≤ |T |2δ+2 x, x |T |2γ+2 y, y

(3.80)

and  2      2 2 2    |T |γ T ∗  |T |δ x, y  ≤ T ∗ |T |δ  x, x T ∗ |T |γ  y, y

(3.81)

where x, y ∈ H. Proof If we take D = |T |γ , C = T ∗ , B = T and A = |T |δ then we have DC B A = |T |γ T ∗ T |T |δ = |T |γ |T |2 |T |δ = |T |γ+δ+2 , A∗ |B|2 A = |T |δ |T |2 |T |δ = |T |2δ+2 and  2 D C ∗  D ∗ = |T |γ |T |2 |T |γ = |T |2γ+2 which by (3.70) implies the desired inequality (3.80). The dual choice D = |T |γ , C = T, B = T ∗ and A = |T |δ gives  2 DC B A = |T |γ T ∗  |T |δ ,   2 2 A∗ |B|2 A = |T |δ T ∗  |T |δ = T ∗ |T |δ  and   2 2  2 D C ∗  D ∗ = |T |γ T ∗  |T |γ = T ∗ |T |γ  , which by (3.70) produces (3.81).



Remark 3.29 If we take δ = γ in (3.81), then we get  2      2 2 2    |T |γ T ∗  |T |γ x, y  ≤ T ∗ |T |γ  x, x T ∗ |T |γ  y, y

(3.82)

3.6 General Vector Inequalities

83

where x, y ∈ H. The following corollary also holds Corollary 3.30 (Dragomir [7]) For any operator T ∈ B (H ) and any β ≥ 0 we have the inequalities

and

   2      2β   ∗ β  T x, y  ≤ |T |2 x, x T T ∗  T ∗ y, y T T

(3.83)

      T |T |β T x, y 2 ≤ |T |2 x, x T |T |2β T ∗ y, y

(3.84)

where x, y ∈ H. Proof Let T = U |T | be the polar decomposition of an operator T, where U is partial isometry and the kernel N (U ) = N (|T |). If we take D = U, C = |T | |T ∗ |β , B = U and A = |T | then we have  β  β DC B A = U |T | T ∗  U |T | = T T ∗  T, A∗ |B|2 A = |T | U ∗ U |T | = T ∗ T = |T |2 and  2  β  β D C ∗  D ∗ = U CC ∗ U ∗ = U |T | T ∗  T ∗  |T | U ∗  2β = T T ∗  T ∗ , which by (3.70) produces (3.83). Now, if we take D = U, C = |T |β+1 , B = U and A = |T | then we have DC B A = U |T |β+1 U |T | = T |T |β T, A∗ |B|2 A = |T |2 and  2 D C ∗  D ∗ = U CC ∗ U ∗ = U |T |β+1 |T |β+1 U ∗ = T |T |2β T ∗ .  Remark 3.31 The case β = 1 produces from the inequalities (3.83) and (3.84) the simple results    ∗        T T  T x, y 2 ≤ |T |2 x, x T 2 T ∗ 2 y, y (3.85)

84

3 Generalizations of Furuta’s Type

    4 |T |T | T x, y|2 ≤ |T |2 x, x T ∗  y, y

and

(3.86)

for any x, y ∈ H.

3.7 Norm and Numerical Radius Inequalities We can state the following corollary of Furuta’s inequality for the numerical radius w of an operator V ∈ B (H ), namely w (V ) = supx=1 |V x, x|, which satisfies the following basic inequalities 1 V  ≤ w (V ) ≤ V  . 2

(3.87)

Theorem 3.32 (Dragomir [7]) Let A, B, C, D ∈ B (H ). Then we have  2 DC B A2 ≤ A∗ |B|2 A D C ∗  D ∗

(3.88)

and for any r ≥ 1 wr (DC B A) ≤

r   2 r 1  ∗ A |B|2 A + D C ∗  D ∗ . 2

(3.89)

Proof Taking the supremum over x, y ∈ H with x = y = 1 in (3.70) we have DC B A2 = ≤

sup

x=y=1

sup

x=y=1

|DC B Ax, y|2 

    2 A∗ |B|2 Ax, x D C ∗  D ∗ y, y

      2 = sup A∗ |B|2 Ax, x sup D C ∗  D ∗ y, y x=1

y=1

 2 = A∗ |B|2 A D C ∗  D ∗ and the inequality (3.88) is proved. By taking x = y in (3.70) and utilising the increasing monotonicity of the power means for two positive numbers, we have for any r ≥ 1 that

3.7 Norm and Numerical Radius Inequalities

85



1/2    2 A∗ |B|2 Ax, x D C ∗  D ∗ x, x     ∗ A |B|2 Ax, x + D |C ∗ |2 D ∗ x, x ≤ 2  r  r 1/r A∗ |B|2 Ax, x + D |C ∗ |2 D ∗ x, x ≤ 2

|DC B Ax, x| ≤

(3.90)

for any x ∈ H. Now, utilising Hölder-McCarthy inequality P x, xr ≤ P r x, x , x ∈ H, x = 1 that holds for any positive operator P and any power r ≥ 1 we have 

r  r A∗ |B|2 Ax, x + D |C ∗ |2 D ∗ x, x 2     r r  2 ∗ A |B| A x, x + D |C ∗ |2 D ∗ x, x ≤ 2 

 r r  2 ∗ A |B| A + D |C ∗ |2 D ∗ x, x = 2

(3.91)

for any x ∈ H, x = 1. By making use of (3.90) and (3.91) we get the inequality of interest

 |DC B Ax, x| ≤ r

A∗ |B|2 A

r

  r + D |C ∗ |2 D ∗ x, x 2

(3.92)

for any x ∈ H, x = 1. Finally, by taking the supremum over x ∈ H, x = 1 in (3.92) we deduce the desired result (3.89).  The above theorem has a number of particular cases for one operator that are of interest: Corollary 3.33 (Dragomir [7]) 1. Let T ∈ B (H ) , r ≥ 1 and α, β ≥ 0 with α +β ≥ 1. Then we have  2βr   1 (3.93) wr T |T |α+β−1 ≤ |T |2αr + T ∗  . 2 In particular, we also have  2αr   1 wr T |T |2α−1 ≤ |T |2αr + T ∗  , 2 for any α ≥

1 2

and wr (T |T |) ≤

1 2r  ∗ 2r . |T | + T 2

(3.94)

(3.95)

86

3 Generalizations of Furuta’s Type

2. For any operator T ∈ B (H ) , r ≥ 1 and any α, β ≥ 1 we have the inequality  2βr   1 wr T |T |β−1 T |T |α−1 ≤ |T |2αr + T ∗  . 2

(3.96)

In particular, we also have wr



T |T |α−1

2

≤

1 2αr  ∗ 2αr |T | + T , 2

(3.97)

for any α ≥ 1 which provides the result  2r   1 wr T 2 ≤ |T |2r + T ∗  . 2

(3.98)

3.For any operator T ∈ B (H ) , r ≥ 1 and any β ≥ 0 we have the inequalities    1 r    β 2β wr T T ∗  T ≤ |T |2r + T T ∗  T ∗ 2

(3.99)

r      1 2β wr T |T |β T ≤ |T |2r + T T ∗  T ∗ . 2

(3.100)

and

In particular, we have    r     1 2 wr T T ∗  T ≤ |T |2r + T 2 T ∗ 2

(3.101)

and wr (T |T | T ) ≤

1 2r  ∗ 4r . |T | + T 2

(3.102)

Chapter 4

Trace Inequalities

In this chapter, after recalling some fundamental facts on Hilbert–Schmidt operators, trace operators and some properties of traces of such operators, we present a trace version of Kato’s inequality. Some natural functionals associated to this inequality and their superadditivity and monotonicity are established. Several inequalities for sequences of operators and power series of operators are given as well.

4.1 Trace of Operators Let (H, ·, ·) be a complex Hilbert space and {ei }i∈I an orthonormal basis of H. We say that A ∈ B (H ) is a Hilbert–Schmidt operator if 

 Aei 2 < ∞.

(4.1)

i∈I

  It is well know that, if {ei }i∈I and f j j∈J are orthonormal bases for H and A ∈ B (H ) then       A f j 2 =  A∗ f j 2 Aei 2 = (4.2) i∈I

j∈I

j∈I

showing that the definition (4.1) is independent of the orthonormal basis and A is a Hilbert–Schmidt operator iff A∗ is a Hilbert–Schmidt operator. Let B2 (H ) the set of Hilbert–Schmidt operators in B (H ) . For A ∈ B2 (H ) we define  1/2  2 Aei  A2 := (4.3) i∈I

for {ei }i∈I an orthonormal basis of H. This definition does not depend on the choice of the orthonormal basis. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2019 S. S. Dragomir, Kato’s Type Inequalities for Bounded Linear Operators in Hilbert Spaces, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-030-17459-0_4

87

88

4 Trace Inequalities

Using the triangle inequality in l 2 (I ) , one checks that B2 (H ) is a vector space and that ·2 is a norm on B2 (H ) , which is usually called in the literature as the Hilbert–Schmidt norm. Denote the modulus of an operator A ∈ B (H ) by |A| := (A∗ A)1/2 . Because |A| x = Ax for all x ∈ H, A is Hilbert–Schmidt iff |A| is Hilbert– Schmidt and A2 = |A|2 . From (4.2) we have that if A ∈ B2 (H ) , then A∗ ∈ B2 (H ) and A2 =  A∗ 2 . The following theorem collects some of the most important properties of Hilbert– Schmidt operators: Theorem 4.1 We have (i) (B2 (H ) , ·2 ) is a Hilbert space with inner product A, B2 :=



Aei , Bei  =



i∈I

B ∗ Aei , ei



(4.4)

i∈I

and the definition does not depend on the choice of the orthonormal basis {ei }i∈I ; (ii) We have the inequalities  A ≤ A2 (4.5) for any A ∈ B2 (H ) and AT 2 , T A2 ≤ T  A2

(4.6)

for any A ∈ B2 (H ) and T ∈ B (H ) ; (iii) B2 (H ) is an operator ideal in B (H ) , i.e. B (H ) B2 (H ) B (H ) ⊆ B2 (H ) ; (iv) B f in (H ) , the space of operators of finite rank, is a dense subspace of B2 (H ); (v) B2 (H ) ⊆ K (H ), where K (H ) denotes the algebra of compact operators on H. If {ei }i∈I an orthonormal basis of H, we say that A ∈ B (H ) is trace class if A1 :=



|A| ei , ei  < ∞.

(4.7)

i∈I

The definition of A1 does not depend on the choice of the orthonormal basis {ei }i∈I . We denote by B1 (H ) the set of trace class operators in B (H ). The following proposition holds: Proposition 4.2 If A ∈ B (H ), then the following are equivalent: (i) A ∈ B1 (H ) ; (ii) |A|1/2 ∈ B2 (H ) ; (ii) A (or |A|) is the product of two elements of B2 (H ) .

4.1 Trace of Operators

89

The following properties are also well known: Theorem 4.3 With the above notations: (i) We have   A1 =  A∗ 1 and  A2 ≤  A1

(4.8)

for any A ∈ B1 (H ) ; (ii) B1 (H ) is an operator ideal in B (H ) , i.e. B (H ) B1 (H ) B (H ) ⊆ B1 (H ) ; (iii) We have B2 (H ) B2 (H ) = B1 (H ) ; (iv) We have  A1 = sup {| A, B2 | | B ∈ B2 (H ) |, B ≤ 1} ; (v) (B1 (H ) , ·1 ) is a Banach space. (iv) We have the following isometric isomorphisms B1 (H ) ∼ = B (H ) , = K (H )∗ and B1 (H )∗ ∼ where K (H )∗ is the dual space of K (H ) and B1 (H )∗ is the dual space of B1 (H ) . We define the trace of a trace class operator A ∈ B1 (H ) to be tr (A) :=



Aei , ei  ,

(4.9)

i∈I

where {ei }i∈I an orthonormal basis of H. Note that this coincides with the usual definition of the trace if H is finite-dimensional. We observe that the series (4.9) converges absolutely and it is independent from the choice of basis. The following result collects some properties of the trace: Theorem 4.4 We have (i) If A ∈ B1 (H ) then A∗ ∈ B1 (H ) and

 tr A∗ = tr (A);

(4.10)

(ii) If A ∈ B1 (H ) and T ∈ B (H ) , then AT, T A ∈ B1 (H ) and tr (AT ) = tr (T A) and |tr (AT )| ≤ A1 T  ; (iii) tr (·) is a bounded linear functional on B1 (H ) with tr = 1;

(4.11)

90

4 Trace Inequalities

(iv) If A, B ∈ B2 (H ) then AB, B A ∈ B1 (H ) and tr (AB) = tr (B A) ; (v) B f in (H ) is a dense subspace of B1 (H ) . Utilising the trace notation we obviously have that  







A, B2 = tr B ∗ A = tr AB ∗ and A22 = tr A∗ A = tr |A|2 for any A, B ∈ B2 (H ). For the theory of trace functionals and their applications the reader is referred to [31].

4.2 Trace Inequalities via Kato’s Result We start with the following result: Theorem 4.5 (Dragomir [8]) Let T ∈ B (H ) . (i) If for some α ∈ (0, 1) we have |T |2α , |T ∗ |2(1−α) ∈ B1 (H ) , then T ∈ B1 (H ) and we have the inequality  2(1−α) 

|tr (T )|2 ≤ tr |T |2α tr T ∗ ;

(4.12)

(ii) If for some α ∈ [0, 1] and an orthonormal basis {ei }i∈I the sum 

 1−α T ei α T ∗ ei 

i∈I

is finite, then T ∈ B1 (H ) and we have the inequality |tr (T )| ≤



 1−α T ei α T ∗ ei  .

(4.13)

i∈I

  Moreover, if the sums i∈I T ei  and i∈I T ∗ ei  are finite for an orthonormal basis {ei }i∈I , then T ∈ B1 (H ) and we have |tr (T )| ≤ inf

α∈[0,1]

  i∈I

 1−α T ei  T ∗ ei  α



 ≤ min

 i∈F

   ∗   T ei  , T ei . i∈F

(4.14) Proof (i) Assume that α ∈ (0, 1) . Let {ei }i∈I be an orthonormal basis in H and F a finite part of I. Then by Kato’s inequality (K) we have



 1/2  1/2  ∗ 2(1−α)

T T ei , ei  ≤ |T ei , ei | ≤ |T |2α ei , ei ei , ei .

i∈F i∈F i∈F (4.15)

4.2 Trace Inequalities via Kato’s Result

91

By Cauchy–Buniakovski–Schwarz inequality for finite sums we have 

|T |2α ei , ei

1/2 1/2  ∗ 2(1−α)

T ei , ei

i∈F

≤

  

|T |

i∈F

=

2α

ei , ei

1/2 2

(4.16)

1/2   1/2 2 1/2   2(1−α) ∗

T ei , ei i∈F

 

1/2    1/2   2(1−α) 2α ∗

T |T | ei , ei ei , ei .

i∈F

i∈F

Therefore, by (4.15) and (4.16) we have

 1/2  



 1/2    2(1−α)

2α ∗

T ei , ei  ≤ |T | ei , ei T ei , ei

i∈F

i∈F

(4.17)

i∈F

for any finite part F of I. ∗ 2(1−α) If  for some α ∈ (0, 1) we have |T |2α , |T ∈ B1 (H ) , then the | 2α ∗ 2(1−α) ei , ei are finite and by (4.17) we have sums i∈I |T | ei , ei and i∈I |T | that i∈I T ei , ei  is also finite and we have the inequality (4.12). (ii) Assume that α ∈ [0, 1] . Let {ei }i∈I be an orthonormal basis in H and F a finite part of I. Utilising McCarthy’s inequality for the positive operator P, namely  β P x, x ≤ P x, xβ , that holds for β ∈ [0, 1] and x ∈ H, x = 1, we have  2α  α |T | ei , ei ≤ |T |2 ei , ei and

   1−α

T ∗ 2(1−α) ei , ei ≤ T ∗ 2 ei , ei

for any i ∈ I. Making use of (4.15) we have



 1/2  1/2  ∗ 2(1−α)

T T ei , ei  ≤ |T ei , ei | ≤ |T |2α ei , ei ei , ei (4.18)

i∈F i∈F i∈F (1−α)/2  α/2  ∗ 2

T ei , ei |T |2 ei , ei ≤ i∈F

=

 i∈F

=

 i∈F

T ∗ T ei , ei

α/2 

T T ∗ ei , ei

 1−α T ei α T ∗ ei  .

(1−α)/2

92

4 Trace Inequalities

Utilizing Hölder’s inequality for finite sums and p = α1 , q = 

  

=

we also have

 1−α T ei α T ∗ ei 

i∈F

≤

1 1−α

i∈F

 i∈F

α 1/α

(T ei  )

α  T ei 

(4.19)

α  1−α   1−α 1/(1−α) ∗ T ei  i∈F

1−α

  T ∗ ei 

.

i∈F

Since all the series involved in (4.18) and (4.19) are convergent, then we get



  1−α

T ei , ei  ≤ T ei α T ∗ ei 

i∈I i∈I  1−α α     ∗   T ei  T ei ≤ i∈I

(4.20)

i∈I

for any α ∈ [0, 1] . Taking the infimum over α ∈ [0, 1] in (4.20) produces

 

   

α  ∗ 1−α T ei , ei  ≤ inf T ei  T ei

α∈[0,1]

i∈I i∈F 1−α  α     ∗ T ei  T ei  ≤ inf α∈[0,1]



= min

i∈F



(4.21)

i∈F

   ∗   T ei  , T ei .

i∈F

i∈F

 Corollary 4.6 (Dragomir [8]) Let T ∈ B (H ) . (i) If we have |T | , |T ∗ | ∈ B1 (H ) , then T ∈ B1 (H ) and we have the inequality

 |tr (T )|2 ≤ tr (|T |) tr T ∗ ; (ii) If for an orthonormal basis {ei }i∈I the sum T ∈ B1 (H ) and we have the inequality |tr (T )| ≤

 i∈I

√

(4.22) T ei  T ∗ ei  is finite, then

 T ei  T ∗ ei . i∈I

(4.23)

4.2 Trace Inequalities via Kato’s Result

93

Corollary 4.7 (Dragomir [8]) Let N ∈ B (H ) be a normal operator. If for some α ∈ (0, 1) we have |N |2α , |N |2(1−α) ∈ B1 (H ) , then N ∈ B1 (H ) and we have the inequality 



|tr (N )|2 ≤ tr |N |2α tr |N |2(1−α) . (4.24) In particular, if |N | ∈ B1 (H ) , then N ∈ B1 (H ) and |tr (N )| ≤ tr (|N |) .

(4.25)

The following result also holds. Theorem 4.8 (Dragomir [8]) Let T ∈ B (H ) and A, B ∈ B2 (H ) . (i) For any α ∈ [0, 1] we have |A∗ |2 |T |2α , |B ∗ |2 |T ∗ |2(1−α) and B ∗ T A ∈ B1 (H ) and  

∗  2

tr AB T ≤ tr A∗ 2 |T |2α tr B ∗ 2 T ∗ 2(1−α) ; (4.26) (ii) We also have

∗  2

tr AB T 



 2  2 2  ≤ min tr |B|2 tr A∗ |T |2 , tr |A|2 tr B ∗ T ∗ .

(4.27)

Proof (i) Let {ei }i∈I be an orthonormal basis in H and F a finite part of I. Then by Kato’s inequality (K) we have   2(1−α)  |T Aei , Bei |2 ≤ |T |2α Aei , Aei T ∗ (4.28) Bei , Bei for any i ∈ I. This is equivalent to 1/2

 ∗  

B T Aei , ei ≤ A∗ |T |2α Aei , ei 1/2 B ∗ T ∗ 2(1−α) Bei , ei

(4.29)

for any i ∈ I. Using the generalized triangle inequality for the modulus and the Cauchy– Bunyakowsky–Schwarz inequality for finite sums we have from (4.29) that

 

∗ (4.30) B T Aei , ei

i∈F  

B ∗ T Aei , ei ≤ i∈F

≤



A∗ |T |2α Aei , ei

1/2 

1/2

2(1−α) B ∗ T ∗ Bei , ei

i∈F

≤

   i∈F

A∗ |T |2α Aei , ei

1/2 2

1/2

94

4 Trace Inequalities

×

  

1/2

2(1−α) B ∗ T ∗ Bei , ei

i∈F

=

 

A∗ |T |2α Aei , ei

2 1/2

1/2  



i∈F

1/2



2(1−α) B ∗ T ∗ Bei , ei

i∈F

for any F a finite part of I. Let α ∈ [0, 1] . Since A, B ∈ B2 (H ) , then A∗ |T |2α A, B ∗ |T ∗ |2(1−α) B and ∗ B T A ∈ B1 (H ) and by (4.30) we have

∗  



tr B T A ≤ tr A∗ |T |2α A

1/2

 1/2 2(1−α) tr B ∗ T ∗ B .

(4.31)

Since, by the properties of trace we have



 tr B ∗ T A = tr AB ∗ T ,   



2 tr A∗ |T |2α A = tr A A∗ |T |2α = tr A∗ |T |2α and

  2(1−α) 2 2(1−α) , B = tr B ∗ T ∗ tr B ∗ T ∗

then by (4.31) we get (4.26). (ii) Utilising McCarthy’s inequality [29] for the positive operator P 

P β x, x ≤ P x, xβ

that holds for β ∈ (0, 1) and x ∈ H, x = 1, we have 

P β y, y ≤ y2(1−β) P y, yβ

(4.32)

for any y ∈ H. Let {ei }i∈I be an orthonormal basis in H and F a finite part of I. From (4.32) we have  α  2α |T | Aei , Aei ≤ Aei 2(1−α) |T |2 Aei , Aei and

  1−α 

T ∗ 2(1−α) Bei , Bei ≤ Bei 2α T ∗ 2 Bei , Bei

for any i ∈ I. Making use of the inequality (4.28) we get  1−α  α 2 |T Aei , Bei |2 ≤  Aei 2(1−α) |T |2 Aei , Aei Bei 2α T ∗ Bei , Bei

4.2 Trace Inequalities via Kato’s Result

95

 1−α  α 2 = Bei 2α |T |2 Aei , Aei Aei 2(1−α) T ∗ Bei , Bei and taking the square root we get   1−α  α 2 2 |T Aei , Bei | ≤ Bei α |T |2 Aei , Aei 2 Aei 1−α T ∗ Bei , Bei

(4.33)

for any i ∈ I. Using the generalized triangle inequality for the modulus and the Hölder’s inequal1 we get from (4.33) that ity for finite sums and p = α1 , q = 1−α

 

∗ B T Aei , ei

i∈F  

B ∗ T Aei , ei ≤ i∈F

≤



(4.34)

  1−α  α 2 2 Bei α |T |2 Aei , Aei 2 Aei 1−α T ∗ Bei , Bei

i∈F

 α   2 α2 1/α α Bei  |T | Aei , Aei ≤ i∈F

 1/(1−α) 1−α   1−α  2 1−α ∗ 2  Aei  T × Bei , Bei i∈F

=

 



Bei  |T | Aei , Aei 2

21

α  

i∈F

  21 2 Aei  T ∗ Bei , Bei

1−α .

i∈F

By Cauchy–Bunyakowsky–Schwarz inequality for finite sums we also have 

1/2  1/2     2 21 2 2 Bei  |T | Aei , Aei ≤ Bei  |T | Aei , Aei

i∈F

i∈F

=

i∈F

 

|B| ei , ei 2

1/2  

i∈F

1/2

∗

A |T | Aei , ei 2



i∈F

and 

 1/2     21  1/2    2 2 ∗ 2 ∗

T Bei , Bei Aei  T Bei , Bei ≤ Aei 

i∈F

i∈F

=

i∈F

 

|A| ei , ei

i∈F

2

1/2   i∈F

1/2



2 B T ∗ Bei , ei ∗

96

4 Trace Inequalities

and by (4.34) we obtain

 

∗ B T Aei , ei

i∈F  α/2  α/2   2 ∗ 2 |B| ei , ei ≤ A |T | Aei , ei i∈F

×

i∈F

(4.35)

(1−α)/2 

 

|A|2 ei , ei

i∈F





(1−α)/2



2 B ∗ T ∗ Bei , ei

i∈F

for any F a finite part of I. Let α ∈ [0, 1] . Since A, B ∈ B2 (H ) , then A∗ |T |2 A and B ∗ |T ∗ |2 B ∈ B1 (H ) and by (4.35) we get

∗  2

tr AB T



 α  2  ∗ ∗ 2 1−α 

tr |A| tr B T B ≤ tr |B|2 tr A∗ |T |2 A α 



 2  2 2 1−α tr |A|2 tr B ∗ T ∗ = tr |B|2 tr A∗ |T |2 .

Taking the infimum over α ∈ [0, 1] we get (4.27).

(4.36)



Corollary 4.9 (Dragomir [8]) Let T ∈ B (H ) and A, B ∈ B2 (H ) . We have |A∗ |2 |T |, |B ∗ |2 |T ∗ | and B ∗ T A ∈ B1 (H ) and  

∗  2

tr AB T ≤ tr A∗ 2 |T | tr B ∗ 2 T ∗ .

(4.37)

Corollary 4.10 (Dragomir [8]) Let N ∈ B (H ) be a normal operator and A, B ∈ B2 (H ) . (i) For any α ∈ [0, 1] we have |A∗ |2 |N |2α , |B ∗ |2 |N |2(1−α) and B ∗ N A ∈ B1 (H ) and  

∗  2

tr AB N ≤ tr A∗ 2 |N |2α tr B ∗ 2 |N |2(1−α) . (4.38) In particular, we have |A∗ |2 |N |, |B ∗ |2 |N | and B ∗ N A ∈ B1 (H ) and  

∗  2

tr AB N ≤ tr A∗ 2 |N | tr B ∗ 2 |N | .

(4.39)

(ii) We also have

∗  2

tr AB N 



  2  2 ≤ min tr |B|2 tr A∗ |N |2 , tr |A|2 tr B ∗ |N |2 .

(4.40)

4.2 Trace Inequalities via Kato’s Result

97

Remark 4.11 Let α ∈ [0, 1] . By replacing A with A∗ and B with B ∗ in (4.26) we get 

∗

 

tr A BT 2 ≤ tr |A|2 |T |2α tr |B|2 T ∗ 2(1−α) (4.41) for any T ∈ B (H ) and A, B ∈ B2 (H ) . If in this inequality we take A = B, then we get 



2  2 

tr |B| T ≤ tr |B|2 |T |2α tr |B|2 T ∗ 2(1−α)

(4.42)

for any T ∈ B (H ) and B ∈ B2 (H ) . If in (4.41) we take A = B ∗ , then we get  

2  2



tr B T ≤ tr B ∗ 2 |T |2α tr |B|2 T ∗ 2(1−α)

(4.43)

for any T ∈ B (H ) and B ∈ B2 (H ) . Also, if T = N , a normal operator, then (4.42) and (4.43) become

and

2  2 



tr |B| N ≤ tr |B|2 |N |2α tr |B|2 |N |2(1−α)

(4.44)





2  2 

tr B N ≤ tr B ∗ 2 |N |2α tr |B|2 |N |2(1−α) ,

(4.45)

for any B ∈ B2 (H ) .

4.3 Some Functional Properties Let A ∈ B2 (H ) and P ∈ B (H ) with P ≥ 0. Then Q := A∗ P A ∈ B1 (H ) with Q ≥ 0 and writing the inequality (4.42) for B = (A∗ P A)1/2 ∈ B2 (H ) we get 



∗  

tr A P AT 2 ≤ tr A∗ P A |T |2α tr A∗ P A T ∗ 2(1−α) , which, by the properties of trace, is equivalent to 



 

tr P AT A∗ 2 ≤ tr P A |T |2α A∗ tr P A T ∗ 2(1−α) A∗ ,

(4.46)

where T ∈ B (H ) and α ∈ [0, 1] . For a given A ∈ B2 (H ) , T ∈ B (H ) and α ∈ [0, 1] , we consider the functional σ A,T,α defined on the cone B+ (H ) of nonnegative operators on B (H ) by  

σ A,T,α (P) := tr P A |T |2α A∗



 − tr P AT A∗ .

1/2



2(1−α) ∗ 1/2 tr P A T ∗ A

98

4 Trace Inequalities

The following theorem collects some fundamental properties of this functional. Theorem 4.12 (Dragomir [8]) Let A ∈ B2 (H ) , T ∈ B (H ) and α ∈ [0, 1] . (i) For any P, Q ∈ B+ (H ) we have σ A,T,α (P + Q) ≥ σ A,T,α (P) + σ A,T,α (Q) (≥ 0) ,

(4.47)

namely, σ A,T,α is a superadditive functional on B+ (H ) ; (ii) For any P, Q ∈ B+ (H ) with P ≥ Q we have σ A,T,α (P) ≥ σ A,T,α (Q) (≥ 0) ,

(4.48)

namely, σ A,T,α is a monotonic nondecreasing functional on B+ (H ) ; (iii) If P, Q ∈ B+ (H ) and there exist the constants M > m > 0 such that M Q ≥ P ≥ m Q then Mσ A,T,α (Q) ≥ σ A,T,α (P) ≥ mσ A,T,α (Q) (≥ 0) . Proof (i) Let P, Q ∈ B+ (H ). On utilizing the elementary inequality 1/2 2 1/2

2 c + d2 a + b2 ≥ ac + bd, a, b, c, d ≥ 0 and the triangle inequality for the modulus, we have σ A,T,α (P + Q)

2(1−α) ∗ 1/2 

 1/2  = tr (P + Q) A |T |2α A∗ tr (P + Q) A T ∗ A



 − tr (P + Q) AT A∗ 

 1/2 = tr P A |T |2α A∗ + Q A |T |2α A∗ 

2(1−α) ∗

2(1−α) ∗ 1/2 × tr P A T ∗ A + Q A T ∗ A



 ∗ ∗ − tr P AT A + Q AT A 



 1/2 = tr P A |T |2α A∗ + tr Q A |T |2α A∗ 

2(1−α) ∗ 1/2

2(1−α) ∗  × tr P A T ∗ A + tr Q A T ∗ A





 − tr P AT A∗ + tr Q AT A∗

2(1−α) ∗ 1/2 

 1/2  ≥ tr P A |T |2α A∗ tr P A T ∗ A

2(1−α) ∗ 1/2  1/2  

tr Q A T ∗ + tr Q A |T |2α A∗ A



  − tr P AT A∗ − tr Q AT A∗ = σ A,T,α (P) + σ A,T,α (Q) and the inequality (4.47) is proved.

(4.49)

4.3 Some Functional Properties

99

(ii) Let P, Q ∈ B+ (H ) with P ≥ Q. Utilising the superadditivity property we have σ A,T,α (P) = σ A,T,α ((P − Q) + Q) ≥ σ A,T,α (P − Q) + σ A,T,α (Q) ≥ σ A,T,α (Q) and the inequality (4.48) is obtained. (iii) From the monotonicity property we have σ A,T,α (P) ≥ σ A,T,α (m Q) = mσ A,T,α (Q) and a similar inequality for M, which prove the desired result (4.49).



Corollary 4.13 (Dragomir [8]) Let A ∈ B2 (H ) , T ∈ B (H ) and α ∈ [0, 1] . If P ∈ B (H ) is such that there exist the constants M > m > 0 with M1 H ≥ P ≥ m1 H , then we have    1/2

  

2α ∗ 1/2 ∗ 2(1−α) ∗ ∗

(4.50) tr A T A − tr AT A M tr A |T | A

2(1−α) ∗ 1/2



 1/2   ≥ tr P A |T |2α A∗ tr P A T ∗ A − tr P AT A∗    1/2



  2α ∗ 1/2 ∗ 2(1−α) ∗ ∗

. ≥ m tr A |T | A tr A T A − tr AT A For a given A ∈ B2 (H ) , T ∈ B (H ) and α ∈ [0, 1] , if we take P = |V |2 with V ∈ B (H ) , we have

 

 σ A,T,α |V |2 = tr |V |2 A |T |2α A∗



 − tr |V |2 AT A∗ 

 = tr V ∗ V A |T |2α A∗



 − tr V ∗ V AT A∗

1/2



2(1−α) ∗ 1/2 tr |V |2 A T ∗ A

1/2



2(1−α) ∗ 1/2 tr V ∗ V A T ∗ A

2(1−α) 1/2 

 1/2  ∗ ∗ = tr A∗ V ∗ V A |T |2α tr A V V A T ∗



 − tr A∗ V ∗ V AT

2(1−α) 1/2 

 1/2  = tr (V A)∗ V A |T |2α tr (V A)∗ V A T ∗



 − tr (V A)∗ V AT

2(1−α) 1/2



 1/2   = tr |V A|2 |T |2α tr |V A|2 T ∗ − tr |V A|2 T .

Assume that A ∈ B2 (H ) , T ∈ B (H ) and α ∈ [0, 1] .

100

4 Trace Inequalities

If we use the superadditivity property of the functional σ A,T,α we have for any V, U ∈ B (H ) that 

  1/2 

 2(1−α) 1/2 tr |V A|2 + |U A|2 |T |2α tr |V A|2 + |U A|2 T ∗ (4.51)



  − tr |V A|2 + |U A|2 T

2(1−α) 1/2



 1/2   ≥ tr |V A|2 |T |2α tr |V A|2 T ∗ − tr |V A|2 T

2(1−α) 1/2



 1/2   + tr |U A|2 |T |2α tr |U A|2 T ∗ − tr |U A|2 T (≥ 0) . Also, if |V |2 ≥ |U |2 with V, U ∈ B (H ) , then 

2(1−α) 1/2

 (4.52) tr |V A|2 T ∗ − tr |V A|2 T  

2(1−α) 1/2



 1/2  ≥ tr |U A|2 |T |2α tr |U A|2 T ∗ − tr |U A|2 T (≥ 0) .



 tr |V A|2 |T |2α

1/2

If U ∈ B (H ) is invertible, then 1   x ≤ U x ≤ U  x for any x ∈ H, U −1  which implies that 1 2 2   1 H ≤ |U | ≤ U  1 H . U −1 2 Utilising (4.50) we get U 2

  2  tr |A| |T |2α

1/2

 

2(1−α) 1/2 2  tr |A|2 T ∗ − tr |A| T

(4.53)

2(1−α) 1/2



 1/2   ≥ tr |U A|2 |T |2α tr |U A|2 T ∗ − tr |U A|2 T   

2(1−α) 1/2 2   2  1 2α 1/2 ≥ tr |A|2 T ∗ − tr |A| T . 2 tr |A| |T | U −1 

4.4 Inequalities for Sequences of Operators For n ≥ 2, define the Cartesian products B (n) (H ) := B (H ) × ... × B (H ), (n) B2(n) (H ) := B2 (H ) × ... × B2 (H ) and B+ (H ) := B+ (H ) × ... × B+ (H ) where B+ (H ) denotes the convex cone of nonnegative selfadjoint operators on H, i.e. P ∈ B+ (H ) if P x, x ≥ 0 for any x ∈ H.

4.4 Inequalities for Sequences of Operators

101

(n) Proposition 4.14 (Dragomir [8]) Let P = (P1 , ..., Pn ) ∈ B+ (H ), T = (n) (n) (T1 , ..., Tn ) ∈ B (H ), A = (A1 , ..., An ) ∈ B2 (H ) and z = (z 1 , ..., z n ) ∈ Cn with n ≥ 2. Then

 n  2



∗ z k Pk Ak Tk Ak

tr

k=1   n   n  

2(1−α) |z k | Pk Ak |Tk |2α A∗k tr |z k | Pk Ak Tk∗ A∗k ≤ tr k=1

(4.54)

k=1

for any α ∈ [0, 1] . Proof Using the properties of modulus and the inequality (4.46) we have

 n 



∗ z k Pk Ak Tk Ak

tr

k=1

n

n







 



|z k | tr Pk Ak Tk A∗k = z k tr Pk Ak Tk A∗k ≤

k=1

≤

n 

k=1



 |z k | tr Pk Ak |Tk |2α A∗k

1/2



2(1−α) ∗ 1/2 tr Pk Ak Tk∗ Ak .

k=1

Utilizing the weighted discrete Cauchy–Bunyakovsky–Schwarz inequality we also have n 



 |z k | tr Pk Ak |Tk |2α A∗k

1/2



2(1−α) ∗ 1/2 tr Pk Ak Tk∗ Ak

k=1

≤

 n  k=1

×

 n 



 |z k | tr Pk Ak |Tk |2α A∗k

=

1/2

 1/2 

∗ 2(1−α) ∗ 1/2 2 |z k | tr Pk Ak Tk Ak

k=1

 n 

 1/2 2



|z k | tr Pk Ak |Tk |

2α

A∗k

1/2  n  

k=1

which imply the desired result (4.54).

1/2

∗ 2(1−α) ∗  |z k | tr Pk Ak Tk Ak ,

k=1



Remark 4.15 If we take Pk = 1 H for any k ∈ {1, ..., n} in (4.54), then we have the simpler inequality

102

4 Trace Inequalities

 n  2



z k |Ak |2 Tk

tr

k=1   n   n  

2 2α 2 ∗ 2(1−α) |z k | |Ak | |Tk | |z k | |Ak | Tk tr ≤ tr k=1

(4.55)

k=1

provided that T = (T1 , ..., Tn ) ∈ B (n) (H ), A = (A1 , ..., An ) ∈ B2(n) (H ) , α ∈ [0, 1] and z = (z 1 , ..., z n ) ∈ Cn . We consider the functional for n-tuples of nonnegative operators P = (n) (P1 , ..., Pn ) ∈ B+ (H ) as follows:   n 1/2  2α ∗ Pk Ak |Tk | Ak σA,T,α (P) := tr

(4.56)

k=1

  n 1/2  n 

 

∗ 2(1−α) ∗

∗ × tr Pk Ak Tk Ak − tr Pk Ak Tk Ak ,

k=1

k=1

where T = (T1 , ..., Tn ) ∈ B (n) (H ), A = (A1 , ..., An ) ∈ B2(n) (H ) and α ∈ [0, 1] . Utilising a similar argument to the one in Theorem 4.12 we can state: Proposition 4.16 (Dragomir [8]) Let T = (T1 , ..., Tn ) ∈ B (n) (H ), A = (A1 , ..., An ) ∈ B2(n) (H ) and α ∈ [0, 1] . (n) (i) For any P, Q ∈ B+ (H ) we have σA,T,α (P + Q) ≥ σA,T,α (P) + σA,T,α (Q) (≥ 0) ,

(4.57)

(n) namely, σA,T,α is a superadditive functional on B+ (H ) ; (n) (ii) For any P, Q ∈ B+ (H ) with P ≥ Q, namely Pk ≥ Q k for all k ∈ {1, ..., n} we have σA,T,α (P) ≥ σA,T,α (Q) (≥ 0) , (4.58) (n) namely, σA,B is a monotonic nondecreasing functional on B+ (H ) ; (n) (iii) If P, Q ∈ B+ (H ) and there exist the constants M > m > 0 such that MQ ≥ P ≥ mQ then

MσA,T,α (Q) ≥ σA,T,α (P) ≥ mσA,T,α (Q) (≥ 0) .

(4.59)

If P = ( p1 1 H , ..., pn 1 H ) with pk ≥ 0, k ∈ {1, ..., n} then the functional of real nonnegative weights p = ( p1 , ..., pn ) defined by

4.4 Inequalities for Sequences of Operators

103

  n 1/2  σA,T,α (p) := tr pk |Ak |2 |Tk |2α

(4.60)

k=1

  n 1/2  n 

 

2(1−α)

pk |Ak |2 Tk∗ − tr pk |Ak |2 Tk × tr

k=1

k=1

has the same properties as in Theorem 4.12. Moreover, we have the simple bounds ⎛  1/2 n  |Ak |2 |Tk |2α max { pk } ⎝ tr

k∈{1,...,n}

(4.61)

k=1

  n 1/2  n  ⎞

 

2(1−α)

|Ak |2 Tk∗ × tr − tr pk |Ak |2 Tk ⎠

k=1

k=1

  n 1/2   n 1/2  

2 2α 2 ∗ 2(1−α) ≥ tr pk |Ak | |Tk | pk |Ak | Tk tr k=1

k=1

 n 



2 − tr pk |Ak | Tk

k=1 ⎛  1/2 n  2 2α |Ak | |Tk | ≥ min { pk } ⎝ tr k∈{1,...,n}

k=1

  n 1/2  n  ⎞

 

2(1−α)

|Ak |2 Tk∗ × tr − tr pk |Ak |2 Tk ⎠ .

k=1

k=1

4.5 Inequalities for Power Series of Operators We have the following trace inequalities:  n Theorem 4.17 (Dragomir [8]) Let f (λ) := ∞ n=1 αn λ be a power series with complex coefficients and convergent on the open disk D (0, R) , R > 0. Let N ∈ B (H ) 2α 2(1−α) ∈ B1 (H ) with be a normal If for  operator.

2(1−α)  some α ∈ (0, 1) we have |N | , |N | 2α < R, then we have the inequality tr |N | , tr |N | 





|tr ( f (N ))|2 ≤ tr f a |N |2α tr f a |N |2(1−α) .

(4.62)

Proof Since N is a normal operator, then for any natural number k ≥ 1 we have



k 2α

N = |N |2αk and N k 2(1−α) = |N |2(1−α)k .

104

4 Trace Inequalities

By the generalized triangle inequality for the modulus we have for n ≥ 2

 n

 n n







k 



k |αk | tr N k . αk N = αk tr N ≤

tr



k=1

k=1

(4.63)

k=1

If for some α ∈ (0, 1) we have |N |2α , |N |2(1−α) ∈ B1 (H ), then by Corollary 4.7 we have N ∈ B1 (H ). Now, since N , |N |2α , |N |2(1−α) ∈ B1 (H ) then any natural power of these operators belong to B1 (H ) and by (4.24) we have

k  2 



tr N ≤ tr |N |2αk tr |N |2(1−α)k ,

(4.64)

for any natural number k ≥ 1. Making use of (4.64) we have n 

n

 



1/2 2(1−α)k 1/2 |αk | tr N k ≤ |αk | tr |N |2αk . tr |N |

k=1

(4.65)

k=1

Utilising the weighted Cauchy–Bunyakovsky–Schwarz inequality for sums we also have n 



1/2 2(1−α)k 1/2 |αk | tr |N |2αk tr |N |

k=1

≤

 n 



1/2 2 |αk | tr |N |2αk

1/2

k=1

×

 n  k=1

 =

n 

(4.66)



1/2 2 |αk | tr |N |2(1−α)k

1/2

1/2  n 1/2 

2αk 

2(1−α)k  |αk | tr |N | |αk | tr |N | .

k=1

k=1

Making use of (4.63), (4.65) and (4.66) we get the inequality

 n  n  2   n 

  

k 2αk 2(1−α)k |αk | |N | |αk | |N | αk N ≤ tr tr

tr

k=1

k=1

(4.67)

k=1

for any n ≥ 2. 



Due to the fact that tr |N |2α , tr |N |2(1−α) < R it follows by (4.24) that tr (|N |) < R and the operator series

4.5 Inequalities for Power Series of Operators ∞ 

αk N k ,

k=1

∞ 

105

|αk | |N |2αk and

k=1

∞ 

|αk | |N |2(1−α)k

k=1

are convergent in the Banach space B1 (H ) . Taking the limit over n → ∞ in (4.67) and using the continuity of the tr (·) on  B1 (H ) we deduce the desired result (4.62). 

Example 4.18 (a) If we take in f (λ) = (1 ± λ)−1 − 1 = ∓λ (1 ± λ)−1 , |λ| < 1 then we get from (4.62) the inequality





tr N (1 H ± N )−1 2

−1  2(1−α)

−1  ≤ tr |N |2α 1 H − |N |2α tr |N | , 1 H − |N |2(1−α)

(4.68)

2α provided that N ∈ B (H ) is a normal and 

operator  for α ∈ (0, 1) we have |N | , 2(1−α) 2α 2(1−α) |N | < 1. ∈ B1 (H ) with tr |N | , tr |N | (b) If we take in (4.62) f (λ) = exp (λ) − 1, λ ∈ C then we get the inequality

 

 



|tr (exp (N ) − 1 H )|2 ≤ tr exp |N |2α − 1 H tr exp |N |2(1−α) − 1 H , (4.69) provided that N ∈ B (H ) is a normal operator and for α ∈ (0, 1) we have |N |2α , |N |2(1−α) ∈ B1 (H ) . The following result also holds:  n Theorem 4.19 (Dragomir [8]) Let f (λ) := ∞ n=0 αn λ be a power series with complex coefficients and convergent on the open disk D (0, R) , R > 0. If T ∈ B (H ) , ∗ double A ∈ B2 (H ) are normal operators

2 that  commute, i.e. T A = AT and T A = 2 2α 2(1−α) ∗ < R for some α ∈ [0, 1] , then A T and tr |A| |T | , tr |A| |T |

2  2 





tr f |A| T ≤ tr f a |A|2 |T |2α tr f a |A|2 |T |2(1−α) .

(4.70)

Proof From the inequality (4.55) we have

 n  2

 2

αk Ak T k

tr

k=0   n   n  

k 2 k 2α

k 2 k 2(1−α) |αk | A T |αk | A T tr . ≤ tr k=0

(4.71)

k=0

2α

2 Since A and T are normal operators, then Ak = |A|2k , T k = |T |2αk and

k 2(1−α)

T = |T |2(1−α)k for any natural number k ≥ 0 and α ∈ [0, 1].

106

4 Trace Inequalities

Since T and A double commute, then is easy to see that k

k

|A|2k T k = |A|2 T , |A|2k |T |2αk = |A|2 |T |2α and

k

|A|2k |T |2(1−α)k = |A|2 |T |2(1−α)

for any natural number k ≥ 0 and α ∈ [0, 1] . Therefore (4.71) is equivalent to

 n  2



k

2 αk |A| T

tr

k=0  n   n   

2

2   2α k 2(1−α) k |αk | |A| |T | |αk | |A| |T | ≤ tr tr , k=0

(4.72)

k=0

for any natural number n ≥

. 

1and α ∈ [0, 1] Due to the fact that tr |A|2 |T |2α , tr |A|2 |T |2(1−α) < R it follows by (4.55)

2  for n = 1 that tr |A| T < R and the operator series ∞  k=1

αk N , k

∞  k=1

|αk | |N |

2αk

and

∞ 

|αk | |N |2(1−α)k

k=1

are convergent in the Banach space B1 (H ) . Taking the limit over n → ∞ in (4.72) and using the continuity of the tr (·) on  B1 (H ) we deduce the desired result (4.70). Example 4.20 (a) If we take f (λ) = (1 ± λ)−1 , |λ| < 1 then we get from (4.70) the inequality



−1 

2

tr 1 H ± |A|2 T

−1 

−1  tr 1 H − |A|2 |T |2(1−α) , ≤ tr 1 H − |A|2 |T |2α

(4.73)

provided

) , A ∈ B2 (H ) are normal operators that double commute

that T ∈ B (H and tr |A|2 |T |2α , tr |A|2 |T |2(1−α) < 1 for α ∈ [0, 1] . (b) If we take in (4.70) f (λ) = exp (λ), λ ∈ C then we get the inequality





 



tr exp |A|2 T 2 ≤ tr exp |A|2 |T |2α tr exp |A|2 |T |2(1−α) ,

(4.74)

provided that T ∈ B (H ) and A ∈ B2 (H ) are normal operators that double commute and α ∈ [0, 1] .

4.5 Inequalities for Power Series of Operators

107

 ∞ j j Theorem 4.21 (Dragomir [8]) Let f (z) := ∞ j=0 p j z and g (z) := j=0 q j z be two power series with nonnegative coefficients and convergent on the open disk D (0, R) , R > 0. If T ∈ B (H

) , A ∈ B2 (H ) are normal operators that double commute and tr |A|2 |T |2α , tr |A|2 |T |2(1−α) < R for α ∈ [0, 1] , then 

 1/2  2 tr f |A| |T |2α + g |A|2 |T |2α 

 1/2 

× tr f |A|2 |T |2(1−α) + g |A|2 |T |2(1−α)

2 

 − tr f |A| T + g |A|2 T 

 1/2  2  1/2 ≥ tr f |A|2 |T |2α tr f |A| |T |2(1−α)



 − tr f |A|2 T 

 1/2  2  1/2 + tr g |A|2 |T |2α tr g |A| |T |2(1−α)



 − tr g |A|2 T (≥ 0) .

(4.75)

Moreover, if p j ≥ q j for any j ∈ N, then, with the above assumptions on T and A, we have  2  1/2  2  1/2 tr f |A| |T |2α tr f |A| |T |2(1−α)



 − tr f |A|2 T 

 1/2  2  1/2 ≥ tr g |A|2 |T |2α tr g |A| |T |2(1−α)



 − tr g |A|2 T (≥ 0) .

(4.76)

The proof follows in a similar way to the proof of Theorem 4.19 by making use of the superadditivity and monotonicity properties of the functional σA,T,α (·) . We omit the details. Example 4.22 Now, observe that if we take f (λ) = sinh λ =

∞  n=0

and g (λ) = cosh λ =

1 λ2n+1 (2n + 1)!

∞  n=0

1 2n λ (2n)!

then f (λ) + g (λ) = exp λ = for any λ ∈ C.

∞  1 n λ n! n=0

108

4 Trace Inequalities

If T ∈ B (H ) , A ∈ B2 (H ) are normal operators that double commute and α ∈ [0, 1] , then by (4.75) we have  1/2 

 1/2 

tr exp |A|2 |T |2(1−α) tr exp |A|2 |T |2α



 − tr exp |A|2 T 

 1/2 

 1/2 ≥ tr sinh |A|2 |T |2α tr sinh |A|2 |T |2(1−α)



 − tr sinh |A|2 T 

 1/2 

 1/2 + tr cosh |A|2 |T |2α tr cosh |A|2 |T |2(1−α)



 − tr cosh |A|2 T (≥ 0) .

(4.77)

 ∞ 1 n 1 1 n Now, consider the series 1−λ = ∞ n=0 λ , λ ∈ D (0, 1) and ln 1−λ = n=1 n λ , 1 λ ∈ D (0, 1) and define pn = 1, n ≥ 0, q0 = 0, qn = n , n ≥ 1, then we observe that for any n ≥ 0 we have pn ≥ qn . If T ∈ B (H ) ,A ∈ B2 (H ) are normal  operators that double commute, α ∈ [0, 1] and tr |A|2 |T |2α , tr |A|2 |T |2(1−α) < 1, then by (4.76) we have 

−1 1/2 

−1 1/2 tr 1 H − |A|2 |T |2(1−α) (4.78) tr 1 H − |A|2 |T |2α



 −1

− tr 1 H − |A|2 T



−1 1/2 

−1 1/2 ≥ tr ln 1 H − |A|2 |T |2α tr ln 1 H − |A|2 |T |2(1−α)



−1 

− tr ln 1 H − |A|2 T

(≥ 0) .

Chapter 5

Integral Inequalities

In this chapter, after recalling some fundamental facts on Bochner integral for measurable functions with values in Banach spaces, we provide an integral version of Kato’s inequality. Several Norm and Numerical Radius inequalities with applications for the Operator Exponential are also given.

5.1 Some Facts on Bochner Integral Let F (B; E, A, μ) be the linear space of functions x (t), t ∈ E, with values in a real or complex Banach space B, given on a measurable space (E, A, μ) endowed with a countably-additive scalar measure μ on a σ -algebra A of subsets of E. A function x0 ∈ F is called simple if can be defined as, see [32] ⎧ xi ∈ B, t ∈ Ai ∈ A, μ (Ai ) < ∞, i ∈ {1, ..., n} ⎪ ⎪ ⎨ Ak ∩ A j = ∅, k = j, k, j ∈ {1, ..., n} , x0 (t) := ⎪ ⎪ ⎩ n Ai , n ∈ N. 0, t ∈ E \ ∪i=1 A function x ∈ F is called strongly measurable if there exists a sequence {xn } of simple functions with xn − x → 0 almost-everywhere with respect to the measure μ on E. As a consequence of this, the scalar function x is A-measurable. For the simple function x0 ∈ F as above we define the integral by  x0 (t) dμ (t) := E

n 

x i μ ( Ai ) .

i=1

A function x ∈ F is said to be Bochner integrable if it is strongly measurable and if for some approximating sequence {xn } of simple functions we have  x (t) − xn (t) dμ (t) = 0. lim n→∞

E

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2019 S. S. Dragomir, Kato’s Type Inequalities for Bounded Linear Operators in Hilbert Spaces, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-030-17459-0_5

109

110

5 Integral Inequalities

The Bochner integral of such a function over a set A ∈ A is defined as   x (t) dμ (t) = lim χ A (t) xn (t) dμ (t) , n→∞

A

E

where χ A is the characteristic function of A, and the limit is understood in the sense of strong convergence in the Banach space E. This limit exists, and is independent of the choice of the approximation sequence of simple functions. It is well-known that, for a strongly-measurable function to be Bochner integrable it is necessary and sufficient for the norm of this function to be integrable, i.e.  x (t) dμ (t) < ∞. A

The set of Bochner-integrable functions forms a vector subspace L (B; E, A, μ) of F (B; E, A, μ), and the Bochner integral is a linear operator on this subspace. Some fundamental properties of Bochner integrals are as follows [32], 1. For any x ∈ L (B; E, A, μ) we have the norm inequality       x (t) dμ (t) ≤ x (t) dμ (t) .   A

A

2. Bochner integral is a countably-additive μ-absolutely continuous set-function on the σ -algebra A, i.e.  ∞ ∪i=1 Ai

x (t) dμ (t) =

∞   i=1

x (t) dμ (t) Ai

if Ai ∈ A, μ ( Ai ) < ∞, i ∈ N, Ak ∩ A j = ∅, k = j, k, j ∈ N, and      x (t) dμ (t) → 0 if μ ( A) → 0,   A

uniformly over A ∈ A. 3. If xn ∈ F, xn → x almost-everywhere with respect to the measure μ on A ∈ A, if xn  ≤ f almost-everywhere with respect to μ on A, and if A f (t) dμ (t) < ∞, then x ∈ L (B; E, A, μ) and   xn (t) dμ (t) → x (t) dμ (t) . A

A

4. The space is complete with respect to the norm  x :=

x (t) dμ (t) . A

5.1 Some Facts on Bochner Integral

111

5. If T is a closed linear operator from a Banach space X into a Banach space Y and if x ∈ L (X ; E, A, μ) and T x ∈ L (Y ; E, A, μ), then  x (t) dμ (t) .



 T x (t) dμ (t) = T A

A

If T is bounded, the condition T x ∈ L (Y ; E, A, μ) is automatically satisfied.

5.2 Applications of Kato’s Inequality In this section we consider a measurable space (E, A, μ) and operator-valued μmeasurable functions E  t −→ Vt ∈ B (H ), where B (H ) denotes the Banach algebra of all bounded linear operators on a complex Hilbert space (H ; ·, ·). We can the adjoint function by Vt∗ := (Vt )∗ and the modulus function by |V |t := define ∗ (Vt ) Vt , t ∈ E. Theorem 5.1 (Dragomir [5]) Let V(·) : E → B (H ) and p : E → [0, ∞) be μmeasurable functions on E and such that p |V |2(·) and p |V ∗ |2(·) are Bochner integrable on E. Then we have the inequality  p (t) | Vt x, y|2 dμ (t) E



≤ E

p (t) |V |2t

(5.1)

 α   1−α  ∗ 2   dμ (t) x, x p (t) V t dμ (t) y, y E

for any x, y ∈ H with x = y = 1 and α ∈ [0, 1]. Proof Let t ∈ E. If we write Kato’s inequality for the operator Vt we have     ∗ 2(1−α) V  | Vt x, y|2 ≤ |V |2α x, x y, y t t

(5.2)

for any x, y ∈ H . By Hölder-McCarthy inequality P r x, x ≤ P x, xr that holds for any positive operator P, for any x ∈ H with x = 1 and any power r ∈ (0, 1) we have   2 α |V |2α t x, x ≤ |Vt | x, x

(5.3)

   1−α   V ∗ 2(1−α) y, y ≤ V ∗ 2 y, y

(5.4)

 and

t

for any x, y ∈ H with x = y = 1.

t

112

5 Integral Inequalities

On making use of (5.2)–(5.4) we get   α   1−α 2 2 | Vt x, y|2 ≤ V t x, x V ∗ t y, y

(5.5)

for any x, y ∈ H with x = y = 1 and t ∈ E. Now, if we multiply the inequality (5.5) by p (t) ≥ 0, integrate over dμ (·) on E and use the scalar weighted Hölder inequality we have  p (t) | Vt x, y|2 dμ (t)    α   1−α 2 2 ≤ p (t) V t x, x V ∗ t y, y dμ (t) E α

  α 1/α ≤ p (t) |V |2t x, x dμ (t) E

E



1−α  1−α 1/(1−α)  ∗ 2   p (t) V y, y dμ (t)

×

t

E

 =

E

α  1−α       2 ∗ 2    p (t) V t x, x dμ (t) p (t) V t y, y dμ (t) E

 =

E

 α   1−α  2  2 p (t) V t dμ (t) x, x p (t) V ∗ t dμ (t) y, y E

for any x, y ∈ H with x = y = 1, and the proof is complete.



Remark 5.2 The inequality (5.1) becomes for y = x the following simpler result that is useful for deriving numerical radius inequalities:  p (t) | Vt x, x|2 dμ (t) E

(5.6)

 α   1−α  2  2 p (t) V t dμ (t) x, x p (t) V ∗ t dμ (t) x, x E E  

     ∗ 2  2 ≤ p (t) α V t + (1 − α) V t dμ (t) x, x



≤

E

for any x ∈ H with x = 1. Remark 5.3 In addition to the assumptions of Theorem 5.1, if the values of the function V(·) are normal operators for μ-almost every (a.e.) t ∈ E, i.e., |V |2t = |V ∗ |2t for μ-a.e. t ∈ E we have  p (t) | Vt x, y|2 dμ (t) (5.7) E

5.2 Applications of Kato’s Inequality

113



 ≤ E

p (t) |V |2t

dμ (t) x, x

α  E

p (t) |V |2t

 1−α dμ (t) y, y

for any x, y ∈ H with x = y = 1 and α ∈ [0, 1]. This inequality implies the simpler result



 p (t) | Vt x, x|2 dμ (t) ≤ E

E

  p (t) |V |2t dμ (t) x, x

(5.8)

for any x ∈ H with x = 1. From a different perspective, we can state the following result as well: Theorem 5.4 (Dragomir [5]) Let V(·) : E → B (H ) and p : E → [0, ∞) be ∗ 2(1−α) μ-measurable functions on E and such that p |V |2α are Bochner (·) and p |V |(·) integrable on E for some α ∈ [0, 1]. Then we have the inequality  p (t) | Vt x, y| dμ (t) E



≤ E

(5.9)

 1/2   1/2  2α  2(1−α) p (t) V t dμ (t) x, x p (t) V ∗ t dμ (t) y, y E

for any x, y ∈ H . In particular, we have  p (t) | Vt x, x| dμ (t) E

(5.10)

 1/2   1/2  2α  ∗ 2(1−α)     ≤ p (t) V t dμ (t) x, x p (t) V t dμ (t) x, x E E  

    2(1−α)  1 2α dμ (t) x, x ≤ p (t) V t + V ∗ t 2 E



for any x ∈ H . Proof Let t ∈ E. If we write Kato’s inequality for the operator Vt we have   1/2   1/2 2α V ∗ 2(1−α) y, y | Vt x, y| ≤ V t x, x t

(5.11)

for any x, y ∈ H . Now, by multiplying the inequality (5.11) with p (t) ≥ 0, integrate over dμ (·) on E and use the weighted Cauchy–Bunyakovsky–Schwarz integral inequality we get  p (t) | Vt x, y| dμ (t) E

114

5 Integral Inequalities



  1/2   1/2 2α V ∗ 2(1−α) y, y p (t) V t x, x dμ (t) t

≤ E

 ≤ E

1/2  1/2       2α 2(1−α) p (t) V t x, x dμ (t) p (t) V ∗ t y, y dμ (t) E

 =

E

 1/2   1/2  2α  ∗ 2(1−α)     p (t) V t dμ (t) x, x p (t) V t dμ (t) y, y E

for any x, y ∈ H . The second inequality in (5.10) follows by the arithmetic mean-geometric mean inequality. The proof is complete.  Remark 5.5 The symmetric case for powers, namely the case α = interest since will produce the simpler result

1 2

in (5.9) is of

 p (t) | Vt x, y| dμ (t) E



≤ E

(5.12)

 1/2   1/2    ∗     p (t) V t dμ (t) x, x p (t) V t dμ (t) y, y E

for any x, y ∈ H and provided that p |V |(·) and p |V ∗ |(·) are Bochner integrable on E. If in this inequality we take y = x, then we get  p (t) | Vt x, x| dμ (t) E



≤ E

(5.13)

 1/2   1/2     p (t) V t dμ (t) x, x p (t) V ∗ t dμ (t) x, x E

for any x ∈ H . Moreover, if the values of the function V(·) are normal operators for μ-a.e. t ∈ E, then the inequality (5.12) becomes  p (t) | Vt x, y| dμ (t) E



≤ E

(5.14)

 1/2   1/2     p (t) V t dμ (t) x, x p (t) V t dμ (t) y, y E

for any x, y ∈ H , while the inequality (5.13) becomes 

  p (t)  V t x, x  dμ (t) ≤ E

for any x ∈ H .

 E

      p (t) V t dμ (t) x, x

(5.15)

5.3 Norm and Numerical Radius Inequalities

115

5.3 Norm and Numerical Radius Inequalities Let p : E → [0, ∞) be a μ-measurable function on E and such that E p (t) dμ (t) = 1. For V(·) : E → B (H ) a μ-measurable function on E and such that p |V |2(·) is Bochner integrable on E, we define the s-2- p-semi-norm by 1/2 p (t) | Vt x, y|2 dμ (t)



  V(·) 

s, p,2

:=

sup

x=y=1

E

and the s-2- p-semi-numerical radius by 

1/2 p (t) | Vt x, x| dμ (t) .





ws, p,2 V(·) := sup

2

x=1

E

If we consider the Banach space L2 , p (E, B (H ) , μ) of all functions V(·) : E → B (H ) that are μ-measurable on E and such that   V(·) 

1/2 p (t) Vt 2 dμ (t) < ∞,

 p,2

:= E

we observe that · p,2 and w (·) p,2 defined on L2 , p (E, B (H ) , μ) are nonnegative, absolute homogeneous and satisfy the triangle inequality on this space. If we consider the norm on L2 , p (E, B (H ) , μ) induced by the numerical radius on B (H ), i.e. 1/2

   p (t) w 2 (Vt ) dμ (t) w p,2 V(·) := E

then by taking into account the well known numerical radius-norm inequalities 1 T  ≤ w (T ) ≤ T  , T ∈ B (H ) 2

(5.16)

we observe that the norms · p,2 and w p,2 (·) will preserve the inequalities (5.16). Utilising the properties of the supremum, we also observe that   V(·) 

s, p,2

      ≤ V(·)  p,2 and ws, p,2 V(·) ≤ w p,2 V(·)

(5.17)

for any V(·) ∈ L2 , p (E, B (H ) , μ). We have the following result. Theorem 5.6 (Dragomir [5]) For any V(·) ∈ L2 , p (E, B (H ) , μ) and α ∈ [0, 1] we have the inequalities

116

5 Integral Inequalities

 2      p (t) Vt dμ (t) ≤ V(·) 2 (5.18)   s, p,2 E  α  1−α   2      V  dμ (t)  p (t) V ∗ 2 dμ (t) ≤ p (t)     t t E

E

and  p (t) Vt dμ (t)

 w2

(5.19)

E

  2 ≤ ws, p,2 V(·)

≤

⎧  α  1−α ⎨  E p (t) |V |2t dμ (t)  E p (t) |V ∗ |2t dμ (t) , ⎩ 

E

   p (t) α |V |2t + (1 − α) |V ∗ |2t dμ (t) .

Proof By the Cauchy–Bunyakovsky–Schwarz integral inequality and the inequality (5.1) we have    2      ≤ x, y p V dμ p (t) | Vt x, y|2 dμ (t) (5.20) (t) (t) t   E E  α   1−α

  2  ∗ 2     p (t) V t dμ (t) x, x p (t) V t dμ (t) y, y ≤ E

E

for any x, y ∈ H with x = y = 1 and α ∈ [0, 1]. Taking the supremum over x = y = 1 we have    2   p (t) Vt dμ (t) x, y  sup  x=y=1 E  p (t) | Vt x, y|2 dμ (t) ≤ sup



x=y=1



≤

sup

 ×

E



x=1

E

 sup

y=1

E

p (t) |V |2t

  α dμ (t) x, x

 1−α  ∗ 2 p (t) V  dμ (t) y, y t

and since sup

x=y=1

             p (t) Vt dμ (t) x, y  =  p (t) Vt dμ (t)  , E

E

(5.21)

5.3 Norm and Numerical Radius Inequalities

 sup

x=y=1

E

117

 2 p (t) | Vt x, y|2 dμ (t) = V(·) s, p,2 ,

and

 sup

x=1

E

p (t) |V |2t

 

  2 dμ (t) x, x = w p (t) |V |t dμ (t) E     2  =  p (t) |V |t dμ (t)  E

while

 sup

y=1

E

 

   2  2 p (t) V ∗ t dμ (t) y, y = w p (t) V ∗ t dμ (t) E

    ∗ 2    V = p dμ (t) (t)   t E





since the operators E p (t) |V |2t dμ (t) and E p (t) |V ∗ |2t dμ (t) are selfadjoint, then we deduce from (5.21) the desired result (5.18). From the inequality (5.6) we also have   2    dμ (t)  p V x, x (t) t   E  ≤ p (t) | Vt x, x|2 dμ (t) E ⎧   α   1−α ⎨ E p (t) |V |2t dμ (t) x, x |V ∗ |2t dμ (t) x, x , E p (t) ≤     ⎩  |V |2t + (1 − α) |V ∗ |2t dμ (t) x, x E p (t) α for any x ∈ H with x = 1. Taking the supremum over x = 1 we deduce the desired inequality (5.19). The details are omitted.  Remark 5.7 Since by the integral triangle inequality for the norm we have         p (t) |V |2 dμ (t) ≤ p (t) |V |2t  dμ (t) t   E E  2 = p (t) Vt 2 dμ (t) = V(·)  p,2 E

and            ∗ 2   p (t) V ∗ 2 dμ (t) ≤ p (t)  V t  dμ (t)   t E

E

118

5 Integral Inequalities

 = E

 2 p (t) Vt 2 dμ (t) = V(·)  p,2 ,

then we have from (5.18) the following sequence of inequalities 2     p (t) Vt dμ (t)   E  2 ≤ V(·) s, p,2  α  1−α     ∗ 2  2     |V | V ≤ p dμ p dμ (t) (t)  (t) (t) t   t  E  E        2  + (1 − α)  p (t) V ∗ 2 dμ (t) |V | ≤ α p dμ (t) (t) t     t E

(5.22)

E

 2 ≤ V(·)  p,2

for any V(·) ∈ L2 , p (E, B (H ) , μ) and α ∈ [0, 1]. From (5.19) we also have  p (t) Vt dμ (t)

 w

2

(5.23)

E

  2 ≤ ws, p,2 V(·)

≤

⎧  α  1−α ⎨  E p (t) |V |2t dμ (t)  E p (t) |V ∗ |2t dμ (t) ⎩ 

E

   p (t) α |V |2t + (1 − α) |V ∗ |2t dμ (t)

        ∗ 2   2 2        ≤ α  p (t) |V |t dμ (t) + (1 − α)  p (t) V t dμ (t)  ≤ V(·) p,2 E

E

for any V(·) ∈ L2 , p (E, B (H ) , μ) and α ∈ [0, 1]. Now, we can consider the following Banach space L1 , p (E, B (H ) , μ) of all functions V(·) : E → B (H ) that are μ-measurable on E and such that   V(·) 

 p,1

p (t) Vt  dμ (t) < ∞.

:= E

We can consider in this space the following numerical radius 





w p,1 V(·) :=

p (t) w (Vt ) dμ (t) E

5.3 Norm and Numerical Radius Inequalities

119

and taking into account the inequality (5.16) we can state that      1 V(·)  ≤ w p,1 V(·) ≤ V(·)  p,1 p,1 2

(5.24)

for any V(·) ∈ L1 , p (E, B (H ) , μ). We can define the s-1- p-semi-norm by  p (t) | Vt x, y| dμ (t)



  V(·) 

s, p,1

:=

sup

x=y=1

E

and the s-1- p-semi-numerical radius by 







p (t) | Vt x, x| dμ (t) ,

ws, p,1 V(·) := sup

x=1

E

where V(·) ∈ L1 , p (E, B (H ) , μ). Utilising the supremum properties we also have   V(·) 

s, p,1

      ≤ V(·)  p,1 and ws, p,1 V(·) ≤ w p,1 V(·)

(5.25)

for any V(·) ∈ L1 , p (E, B (H ) , μ). More related results are incorporated in the following theorem. ∗ 2(1−α) Theorem 5.8 (Dragomir [5]) If |V |2α belong to L1 , p (E, B (H ) , μ) (·) and |V |(·) for some α ∈ [0, 1], then we have

     p (t) Vt dμ (t)  

(5.26)

E

  ≤ V(·) s, p,1  1/2  1/2   2α    ∗ 2(1−α)         ≤  p (t) V t dμ (t)  p (t) V t dμ (t)  E

E

and  p (t) Vt dμ (t)

 w E

  ≤ ws, p,1 V(·)

(5.27)

120

5 Integral Inequalities

≤

⎧  1/2    ∗ 2(1−α) ⎪  p (t) |V |2α dμ (t)1/2  ⎪ |V | p dμ (t) (t)   , t ⎨ E t E     ⎪ ⎪  ∗ 2(1−α) ⎩1 dμ (t) . p (t) |V |2α t + |V |t 2  E

Proof By the modulus properties and the inequality (5.10) we have        p (t) Vt dμ (t) x, y   E ≤ p (t) | Vt x, y| dμ (t) E

 ≤ E

 1/2   1/2  2α  ∗ 2(1−α)     p (t) V t dμ (t) x, x p (t) V t dμ (t) y, y , E

for any x, y ∈ H . Taking the supremum over x = y = 1 we deduce (5.26). The second inequality follows by (5.10) and the details are omitted.



Remark 5.9 Since by the integral triangle inequality for the norm we have             2α  2α   p (t) V 2α dμ (t) ≤ V dμ = p p (t) V t  dμ (t) (t) (t)     t t E

and

E

E

           ∗ 2(1−α)   p (t) V ∗ 2(1−α) dμ (t) ≤ V p (t)   dμ (t)   t t E E  2(1−α) = p (t) Vt∗  dμ (t) E

then by (5.26) we have the following sequence of inequalities      p (t) Vt dμ (t)   E

  ≤ V(·) s, p,1  1/2  1/2   2α    ∗ 2(1−α)         ≤  p (t) V t dμ (t)  p (t) V t dμ (t)  E

1/2  1/2 2(1−α) p (t) Vt  dμ (t) p (t) Vt  dμ (t)

 ≤

2α

E

1 ≤ 2

E



E

  p (t) Vt 2α + Vt 2(1−α) dμ (t) E

(5.28)

5.3 Norm and Numerical Radius Inequalities

121

∗ 2(1−α) provided that |V |2α belong to L1 , p (E, B (H ) , μ) for some α ∈ [0, 1]. (·) and |V |(·) Under the same assumptions we also have

 p (t) Vt dμ (t)

 w

(5.29)

E

  ≤ ws, p,1 V(·)

≤

⎧  1/2    ∗ 2(1−α) ⎪  p (t) |V |2α dμ (t)1/2  ⎪ |V | p dμ (t) (t)   t ⎨ E t E     ⎪ ⎪  ∗ 2(1−α) ⎩1 dμ (t) p (t) |V |2α t + |V |t 2  E

⎧  1/2  1/2 2α ⎪ Vt 2(1−α) dμ (t) ⎨ E p (t) Vt  dμ (t) E p (t) ≤   ⎪ ⎩ 1 p (t)  ∗ 2(1−α)   dμ (t)  |V |2α t + |V |t 2 E 1 ≤ 2



  p (t) Vt 2α + Vt 2(1−α) dμ (t) . E

Remark 5.10 The case α = lowing inequalities

1 2

is of interest since it generates from (5.29) the fol-

  ws, p,1 V(·)

≤

⎧  1/2  1/2 ⎨  E p (t) |V |t dμ (t)  E p (t) |V ∗ |t dμ (t) ⎩1  2

≤

(5.30)

E

   p (t) |V |t + |V ∗ |t dμ (t)

 ⎧ ⎨ V(·)  p,1 ⎩1 2

E

  p (t)  |V |t + |V ∗ |t  dμ (t)

for any V(·) ∈ L1 , p (E, B (H ) , μ).

  ! ≤ V(·)  p,1

122

5 Integral Inequalities

From (5.26), we also have for α = triangle inequality for norm:

1 2

the following refinement of the integral

     p (t) Vt dμ (t)   E   ≤ V(·) s, p,1

(5.31)

 1/2  1/2         V  dμ (t)  p (t) V ∗  dμ (t) ≤ p (t)     t t E

≤

E

             1   p (t) V  dμ (t) +  p (t) V ∗  dμ (t) ≤ V(·)      t t p,1 2 E E

for any V(·) ∈ L1 , p (E, B (H ) , μ). Moreover, if the values of the function V(·) are normal operators for μ-a.e. t ∈ E, then the inequality (5.31) becomes                 p (t) Vt dμ (t) ≤ V(·)  ≤  p (t) |V |t dμ (t)   ≤ V(·) p,1 .  s, p,1 E

(5.32)

E

5.4 Applications for the Operator Exponential It is known that if U and V are commuting operators, then the operator exponential function exp : B (H ) → B (H ) given by exp (T ) :=

∞  1 n T n! n=0

satisfies the property exp (U ) exp (V ) = exp (V ) exp (U ) = exp (U + V ) . Also, if A is invertible and a, b ∈ R with a < b then  b   exp (t A) dt = A−1 exp (b A) − exp (a A) . a

5.4 Applications for the Operator Exponential

123

We observe that if the values of the function V(·) are normal operators for μ-a.e. t ∈ E, i.e., |V |2t = |V ∗ |2t for μ -a.e. t ∈ E, then we have from (5.7) that    2    p (t) Vt dμ (t) x, y  (5.33)  E  α   1−α

 p (t) |V |2t dμ (t) x, x p (t) |V |2t dμ (t) y, y ≤ E

E

for any x, y ∈ H with x = y = 1 and α ∈ [0, 1]. If N is a normal operator, then for any real number t we have      |exp(t N )|2 = (exp (t N ))∗ (exp (t N )) = exp t N ∗ (exp (t N )) = exp t N ∗ + N . Proposition 5.11 (Dragomir [5]) Let N be an invertible normal operator and such that N ∗ + N is also invertible. Then for any a, b ∈ R with a < b we have  −1   2 N exp (bN ) − exp (a N ) x, y  (5.34) α          −1 x, x exp b N ∗ + N − exp a N ∗ + N ≤ N∗ + N  1−α   −1      exp b N ∗ + N − exp a N ∗ + N × N∗ + N y, y for any x, y ∈ H with x = y = 1 and α ∈ [0, 1]. Proof Follows from (5.33) applied for E = [a, b], p (t) = dμ (t) = dt, the usual Lebesgue measure.

1 , b−a

Vt = exp(t N ) and 

If S is an invertible selfadjoint operator, then from (5.34) we get for any a, b ∈ R with a < b that  −1   2 S exp (bS) − exp (aS) x, y 

(5.35)

 α 1  −1  S ≤ exp (2bS) − exp (2aS) x, x 2   1−α  × S −1 exp (2bS) − exp (2aS) y, y for any x, y ∈ H with x = y = 1 and α ∈ [0, 1]. In particular, we have from (5.35) that  −1   2 S exp (S) − I x, y 

(5.36)

 α    1−α 1  −1  S ≤ exp (2S) − I x, x S −1 exp (2S) − I y, y 2 for any x, y ∈ H with x = y = 1 and α ∈ [0, 1]. If we use the inequality (5.10) for functions V(·) that are normal operators for μ-a.e. t ∈ E, then we have

124

5 Integral Inequalities

        (5.37) x, y p V dμ (t) (t) t   E

  1/2   1/2 2(1−α) 2α ≤ p (t) |V |t dμ (t) x, x p (t) |V |t dμ (t) y, y E

E

for any x, y ∈ H and α ∈ [0, 1]. Utilising the inequality (5.37) we can state: Proposition 5.12 (Dragomir [5]) With the assumptions of Proposition 5.11, we have  −1    N (5.38) exp (bN ) − exp (a N ) x, y   1/2          −1 ≤ α −1 N ∗ + N exp bα N ∗ + N − exp aα N ∗ + N x, x   −1 × (1 − α)−1 N ∗ + N        1/2 × exp b (1 − α) N ∗ + N − exp a (1 − α) N ∗ + N y, y for any x, y ∈ H and α ∈ (0, 1). If S is an invertible selfadjoint operator, then from (5.38) we get for any a, b ∈ R with a < b that  −1    S exp (bS) − exp (aS) x, y 

(5.39)

 −1   1/2 1 ≤ √ S exp (2bαS) − exp (2aαS) x, x 2 α (1 − α)  1/2  −1  exp (2b (1 − α) S) − exp (2a (1 − α) S) y, y × S for any x, y ∈ H and α ∈ (0, 1). In particular, we have from (5.39) that  −1    S exp (S) − I x, y 

(5.40)

 −1   1/2 1 ≤ √ S exp (2αS) − I x, x 2 α (1 − α)  1/2  −1  exp (2 (1 − α) S) − I y, y × S for any x, y ∈ H and α ∈ (0, 1). The interested reader may state the corresponding norm inequalities. However the details are omitted.

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