The Spectral Theory of Toeplitz Operators. (AM-99), Volume 99 9781400881444

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Annals of Mathematics Studies Number 99

THE SPECTRAL THEORY OF TOEPLITZ OPERATORS

BY

L. BOUTET DE MONVEL AND

V. GUILLEMIN

PRINCETON UNIVERSITY PRESS AND UNIVERSITY OF TOKYO PRESS PRINCETON, NEW JERSEY 1981

Copyright © 1981 by Princeton University Press ALL RIGHTS RESERVED

Published in Japan exclusively by University of Tokyo Press; In other parts of the world by Princeton University Press

Printed in the United States of America by Princeton University Press, Princeton, New Jersey

Library of Congress Cataloging in Publication data will be found on the last printed page of this book

T A B L E OF CONTENTS §1

Introduction

3

§2

G eneralized T oeplitz operators

11

§3

Fou rier in tegral operators of Hermite type

21

§4

The m etap lectic rep resen tation

27

§5

M etalinear and m etap lectic stru ctu res on manifolds

35

§6

Isotrop ic su b sp a ce s of sy m p lectic v ecto r s p a c e s

39

§7

The com position theorem

44

§8

The proof of Theorem 7 .5

51

§9

P u ll-b ack s, push-forwards and exterio r ten sor products

66

§10

The transport equation

77

§11

Symbolic properties of T o ep litz operators

80

§12

The tra ce formula

90

§13

Spectral properties of T oeplitz operators

99

§14

The Hilbert polynomial

110

§1 5

Some concluding remarks

123

BIBLIO G R A PH Y

1 28

Appendix: Quantized co n ta ct stru ctu res

131

v

The Spectral Theory of Toeplitz Operators

§1. Let

INTRODUCTION

W be a com p act, s tric tly pseudoconvex domain of dimension 2n

with smooth boundary.

Let

r be a smooth function on W with r > 0 on

Int W , r = 0 on dW and dr ^ 0 near dW . L e t j : (9W -> W be the in clu ­ a = j*Im (9 r. It is known that a

sion map, and co n sid er the one-form

is a

c o n ta ct form on dW ; that is a a (d a )n_1

(1.1)

is nowhere zero on dW .

Wew ill denote the form (1 .1 ) by v . A sso cia te d

with v is a m easure on

dW which we will denote by

L 2 = L 2 (d W ,v ),

H2 be the clo su re in L 2 of the s p a ce of C°°

and let

v a s w ell.

Let

functions on dW which ca n be extended to holomorphic functions on W

.

H2 is called the s p a c e of Hardy fu n ctio n s on C°°(dW)

is ca lle d a T o ep litz operator of order k if it ca n be written in the form

7TQ77 where Q is a pseudodifferential operator of order k . We will show in §2 that the s e t, tion.

J , of T oep litz operators forms a ring under com p osi­

(T h is fa ct is not at all o b viou s.)

L et

2

be the su b set = t a x ,t > 0 l

of T*dW , a

being the one form defined above.

2

is a s y m p lectic su b ­

manifold of T*dW : i.e . the re strictio n to it of the sym p lectic form on T*dW

is n on-degenerate.

Given a T o ep litz operator,

k we will denote by tr(T ) the re strictio n of a (Q ) to

T = 77Q77, of order 2

where

o ( Q) is

the leading symbol of Q . We will show in §2 that this definition is 3

4

THE S PE C T R A L THEORY OF T O E PL IT Z OPERATORS

unambiguous, i.e . if Qj

and Q2 are

tors on dW and ttQ^ tt = ttQ2 tt , then valu es on 2 .

k-th order pseudodifferential opera­ cj(Q1 ) and

cr(Q2 ) take the sam e

We will a ls o prove in §2 that a ( T ) s a tis fie s the following

ru les: (i)

a ( T 1T 2 ) = a ( T 1 ) a ( T 2 )

(ii)

a ( [ T 1 , T 2 ]) = ! a ( T 1 ) , a ( T 2 )!

(iii) If T e j k

and

ff(T ) = 0 ,

then

T e

.

(Note that in (ii) the P o isso n brack et is the in trin sic P o isso n bracket on the sym p lectic manifold, If T

PROPOSITION.

2 .)

From (i) and (iii) we will deduce

is a k-th order T o ep litz operator w hose sym bol is

now here zero , there e x is t s a (-k )-th order T o ep litz operator U su c h that TU - 77 and UT - 77 a re sm oothing operators. If T

PROPOSITION.

is a self-a d jo in t T o ep litz operator of order k > 0

and ct(T ) is ev ery w h ere p o s itiv e , then T is hounded from below and has only Let

+00

has a d is c r e te sp ectru m w hich

a s a point of accum ulation.

T be a T oeplitz operator of order one which is self-ad jo in t and

has a p o sitive symbol.

Let

< A2 < A.3 ••• be the spectrum of T . In

th is paper we will prove a number of resu lts on the asym ptotic behavior of the A-’s . T h ese re s u lts , which we will d escrib e below, are an alogu es of resu lts about e llip tic pseudodifferential operators which are w ell known (t9 ], [12], [24]).

In fa c t, more is true:

It w as shown by one of the authors

(s e e [5]) that a pseudodifferential operator can be viewed a s a s p e c ia l kind of T oep litz operator; s o , in fa c t our re su lts in clu d e the resu lts of [1 2 ], e t c ., as s p e c ia l c a s e s . An exam ple of a T oeplitz operator which has no pseudodifferential counterpart is the following.

Let X

be a non-singular p rojective variety

in C P n which is defined, in homogeneous co o rd in ates, by the equations f 1 (z 1 , ••*, z n+ i) = 0 ,

* " ' z n) = ^

^

be the in tersectio n of

the s e t of solu tions of th ese equations with the unit disk in C n+1 . Then W is a s tric tly pseudoconvex domain (e x ce p t for a singularity a t is invariant under the actio n of the c ir c le group.

0 ) which

5

§1. INTRODUCTION

(z l ’ - '- ’ z n + l) - (e l ^z l ’ - - '’ e l0 z n+ l ) • In p articu lar,

dW is a c ir c le bundle over X .

If d/dO is the in finitesim al

generator of the c ir c le group a ctio n , the operator,

(1 / V^—1 ) ( d / d d ) ,

re stricte d

to the Hardy s p a c e , is a self-ad jo in t T o ep litz operator of order one with everyw here p ositiv e symbol.

Its spectrum c o n s is ts of the n on-negative

in teg ers, the n-th integer occurring in the spectrum with m ultiplicity, p (n ), p being the Hilbert polynomial of X . We will return to th is exam ple below. L et

T and {A-i be a s above. L e t N(A) be the num ber of A^’s < A .

THEOREM 1.

( v o l ( 2 1 ) / ( 277)n)An + OCA11- 1) , an d v o l C ^ )

T h en N(A) =

b ein g the s u b s e t of 2

w here cr(T) < 1

its sy m p lec tic volum e.

F o r pseudodifferential op erators, this theorem is due to Hormander [24]. T his theorem a ls o has an am using algeb raic-g eo m etric co ro llary , which is w ell known (se e [3 0 ]): L et X

COROLLARY. degree

is

y m b ein g a u n iv ersa l constant w hich d ep en d s

X = y m volume X ,

only on the d im en sio n , P roof.

b e a non-sin gula r co m p lex p ro jectiv e variety. T h en

m,

of X .

F o r a p rojectiv e v ariety the leading term in the Hilbert polynomial

(degree X ) / m ! t m . Now apply the theorem above to the alg eb raic exam ­

ple d escrib ed in the preceding paragraph. Let

o be the symbol of T .

sy m p lectic manifold, field, H . Since

2,

Since

o is homogeneous of degree one, H is homogeneous of

now on a s a co n ta ct v ecto r field on

T h en A ^ R p erio d ic.

cr is a real-valued function on the

there is a s s o c ia te d with it a Hamiltonian vector

degree zero ; so it a ctu a lly liv es on

THEOREM 2.

Q .E .D .

L et A

2 / R += 0.

This theorem is a ls o due to Colin de Verdiere, [10] in the

pseudodifferential setting.

8

THE S PE C T R A L THEORY OF T O E P L IT Z OPERATORS

We next give a recipe for computing p ( t ) . Since the b ich a ra c te ris tic flow on • Were v period and f asu b

a re no n ­

•r —

y l * ^ py!

1

of a

T h en

r#

( 1.5)

y,


S(R n) , j = l , - * - , p

( 2 .1 ) where

with x = ( y ,t ) a s C onsider the system of

defined by

Dj = (l/\ /= T )(d /c9yj h- y j !Dt |) |Dj

is , as u sual, the operator

L 2 -clo su re of the s p a c e of so lu tio n s:

!D^|f = M f . L e t

H be the

{f eS , Djf = 0, j = 1, ••■, p i , and let n

be the orthogonal projector of L 2 onto H . P R O P O S IT IO N 2 .1 .

T h e follow ing o scilla to ry in tegra l is the Schw artz

k ern el of n :

(2 .2 )

(2 „ )-q J ' e i L 2(R n) be the operator

. L et

(2 .7 )

R f(y, t) = (2?7)- H be a T o ep litz operator.

T h en there

e x is t s a p seu d o d ifferen tia l operator Q : L 2(R n) -* L 2(R n) su ch that Q maps H into H and the restrictio n of Q to H is

T.

The proof requires some preliminary lemmas. L em m a 2 .6 .

L e t Q : L 2 (R n) - L 2(R n ) be a p seu d o d ifferen tia l operator

w hose total sym bol v a n ish es to in fin ite order on 2 . p seu d o d ifferen tia l operators Q - 2 L jD j Proof.

L 1 , •••, Lp with the sam e property s u ch that

is sm oothing.

L e t □ = 2D *D j . The symbol of □ at (x ,< f) is ju st the square

of the d istan ce from ( x , f ) to on 2 ,

T h en there e x is t

a ( Q ) /a ( n )

order on 2 .

L et

2.

Since

a (Q ) v an ish es to infinite order

is a smooth function which a ls o van ish es to infinite PQ be a pseudodifferential operator having this function

a s its leading symbol.

We can arrange that the to ta l symbol of PQ

van ish es to infinite order on 2 .

Then Q x = Q - P Qn is of order one le s s

than Q and its to tal symbol v an ish es to infinite order on 2 . R epeating

15

§2. GEN ERALIZED T O E P L IT Z OPERATORS

the above argument with Qx , e t c ., we can recu rsiv ely co n stru ct an opera­ tor P

su ch that the to ta l symbol of P Now s e t

v an ish es to infinite order on

and Q - P n

is sm oothing.

Lj = P D * .

LEMMA 2 .7 .

T h e re e x is t operators Mj : (S(Rn) -» L 2(R n) s u c h that

2

Q .E .D .

Id = 7 7 + 2 MjDj . Proof.

See [8 ].

LEMMA 2 .8 .

G iven a T o ep litz operator,

T,

there e x is t s a p s e u d o d iffe r­

en tia l operator Q : L 2(R n) -> L 2 (R n) s u c h that o ( Q ) , re s t ric te d to 2 , is eq u a l to a ( T ) and, for a ll j = 1 , ---^p,

the total sym bol of [D j,Q ]

v a n ish es to infin ite order on 2 . Proof.

If qQ is the leading symbol of Q then the leading symbol of

[D j,Q ] is

Z jq Q where Zj

is the v e cto r field

q ( 2. 10)

Zj = (1/V~l)( V and a F ou rier integral operator F : L 2(X ) -> L 2(R n) extending 0 , su ch that F * F - I F t7 ^ F * - 77 is

is C°°

on U and

C°° on V .

It was proved by B outet de Monvel and Sjostrand in [7] that the Szego p rojector, d iscu sse d in §1, s a tis fie s th ese axiom s. of the appendix that for every com pact manifold co n e,

2,

It will be proved in §4

X and every sym p lectic

in T * X - 0 there e x is ts a T oeplitz structure on 2 .

T his

T oeplitz structure is not unique, but is in a ce rta in se n se unique up to homotopy. We will have more to say about this resu lt in §14. Let

be the image of

77

^

in L 2 ( X ) .

It follows from condition II

that 77^, maps C °°(X ) into C °°(X ) H

. By a T o ep litz operator on H^,.

we will mean an operator T : C °°(X ) PI

-> C °°(X ) fl

of the form

n ^ Q n ^ where Q : C °°(X ) -> C °°(X ) is a pseudodifferential operator. will sa y that T

is of order k if Q is of order k .

PROPOSITION 2 . 11.

S u p p o se T = 77^Qi 77^,

a re k-th order p seu d o d ifferen tia l operators. Proof.

We

i = l ,2 T h en

w here Qx and Q2

(j(Q^ )| 2 = cr(Q2 )| 2 .

It is enough to prove this for the can o n ical model.

F o r the can o n i­

c a l model, however, this follows from Proposition 2 .3 .

Q .E .D .

DEFINITION 2 .1 1 .

and

If T

is a T oeplitz operator of order k on 2

T = 77^ 77^ with Q of order k , PROPOSITION 2 .1 2 . order k -1 .

If T

then

a ( T ) = cr(Q)| S .

is of order k and ct(T) = 0

then T is of

§2. GENERALIZED T O E P L IT Z OPERATORS

Proof.

19

Again it is enough to verify this for the ca n o n ica l model.

F o r the

ca n o n ica l model, however, this follow s from P roposition 2 .5 and P ro p o si­ tion 2 .3 .

Q .E .D . L et T

PROPOSITION 2 .1 3 .

be a T o ep litz operator on 2

T h en there e x is t s a p se u d o d ifferen tia l operator, Q , and

T = Proof.

of order k s u c h that

[77^ , Q] = 0 .

By a pseudodifferential partition of unity we ca n assum e that

T = rr^Q0 7T^ where borhood.

Qq is

Conjugating

77

C°° e x ce p t in an arbitrarily sm all co n ic neigh­

^ with 77 and making use of Theorem 2 .9 we can

e a s ily find a Q ^ su ch that T - n ^ Q i77]? is smoothing and smoothing.

Now let Q 2 = 77^

Q

Q1 by a smoothing operator and

[77^ , Q j]

is

+ (1 - 77^ ) Q 1 (1 - 77^) . Q 2 differs from [77^ , Q2 ] = 0 .

F in a lly s e t

77^Qi77^.

Q = Q2 + T Q .E .D .

C O R O L L A R Y 1.

T

of order k.

T o ep litz operators form a ring under com position, and if

is a T o ep litz operator its tra n sp o se,

C O R O LL A R Y 2.

T h e sym bol,

T * , is a T o ep litz operator.

o , d e fin e d abo v e s a tis fie s the follow ing

com position ru les. (2 .1 1 )a

^ ( T xT 2 ) = a C V a O V

(2 -lD b

ff ([T j, T 2 D =

C O R O LL A R Y 3.

L et T

b e a T o ep litz operator of order k.

0.

S u p p o se

be a self-a d jo in t T o ep litz operator of order

a ( T ) > 0 ev ery w h ere on 2 .

T h en the sp ectru m of T

is

d is c r e te , bounded from below and has only + °o a s a point of a ccum ulation. P roof.

L e t Q be a pseudodifferential operator such that T = n^Qzr^ and

[Q>77^] is smoothing.

Since

Re a (Q ) > 0 everyw here. is self-ad jo in t and 775^ 772 + (1 ~

Re cr(Q) > 0 near

2

we ca n assum e

R ep lacin g Q by (Q + Q * )/2

cr(Q) everyw here p o sitiv e.

we can assum e

F in a lly

Q

Q differs from

~ n '2 ) ky a smoothing operator; so we can assu m e Q

has a ll of the above properties and a ls o commutes with n ^ . Now Q is a self-ad join t ellip tic pseudodifferential operator of p ositive degree with a symbol which is everyw here p o sitiv e; so its spectrum is d is c re te , bounded from below and has only + oo as a point of accum ulation.

The sam e is

therefore true of the restrictio n of Q to the invariant su b s p a ce ,

H ^. Q .E .D .

§3. F O U R IE R IN TE G R A L OPERA TO RS OF HERMITE T Y P E Given a smooth manifold, 2,

X , and a homogeneous isotrop ic submanifold,

of the cotan gen t bundle of X

one can define a c la s s of distributions

with wave front s e t co n cen trated on

2

having many of the properties of

the F ou rier integral distributions of Hormander [25].

Indeed when

2

is

L agran gian (i.e . maximal iso tro p ic) th ese distributions are ju st the usual Fou rier integral d istrib ution s.

T h e se d istrib u tion s, which we will define

below, are ca lle d F o u rier integral d istrib ution s of H erm ite type.

They

were first studied s y ste m a tica lly in [4] and their sym bolic properties e x ­ plored in [15].

The next few se c tio n s of this paper will be partly a review

of the m aterial in [4] and [15] and partly an elaboration of the sym bolic theory of [15] with a view to the ap p licatio n s in §13. Let

d = (r ,r j) ,

R n = Rk x R f with co o rd in ates

and let X

be an open

su b set of R n . L e t a ( x ,r , rj) be a smooth function on X x R N . We will sa y that

a

belongs to the symbol c la s s

r > 0 a ll m ulti-indices

a,

K m(k, £) if for a ll numbers

/3 and y and a ll

KCC X

there e x is ts a

co n stan t C > 0 su ch that |D“ D f D j a ( x ,r ,,) | < C|r|m- M ( l + |j/|)-r

( 3 .1 ) for a ll x e K .

N otice that if

is in Km(k,£) then a s a function of 77

a

it is rapidly d ecre a sin g and a s a function of r it behaves as a c l a s s i c a l symbol of type S™ Q. L e t a - ( x , r , 7/ ) , elem ents of H degree

mi

(k ,£ ) with

m^ in r for

i = 0, 1 , 2 , - ** , be a seq uen ce of

-» - 00, a ^ ( x ,r ,7/) being homogeneous of

|r| > > 0 . Given an elem ent,

w ill say that (3 .2 )

a (x , r , 77)

~

^ a ^ x .r ,^ ) 21

a , of K °(k , £) we

22

THE S P E C T R A L THEORY OF T O E PL IT Z OPERATORS

1 if a ( x , r , 7/ ) - ^

—,rni a - ( x ,r , rj) € J\ (k, I) for a ll i .

U n less otherw ise

j R be a non-degenerate phase function in the s e n se of Hormander, [25].

Assum e the c r itic a l s e t of /d 6l = ••• = dcf>/dON = 0 ! = ••• = rj^ = 0 , tra n sv e rsa lly .

in te rse cts the s e t,

Let

2

be the image

in T * X - 0 of this in tersectio n under the mapping (3 .3 ) 2

(x ,0 ) ^ ( x ,< 3 0 /< 3 x ) ,

-> T * X - 0 .

is a hom ogeneous, isotrop ic submanifold of T * X - 0

of dimension n-E.

C onsider now the o scilla to ry integral

(3 -4 )

La

J * a (x , t , rj/yJ |r|)ei(^ x , ^ d #

where a e Hm~N//2 (k, V) . We will prove below. P R O P O S IT IO N 3. 1.

T h e wave front s e t of (3 .4 ) is co n ta in ed in 2 .

We will say that the pair ( X x R N, 0 )

is a param etrization of 2

= rjp = 0 under the map (3 .3 ).

is the image of ^

if 2

We will a ls o prove

below. P R O P O S IT IO N 3 .2 .

If ( X x R N, 0 )

param etrizations of 2

a nd ( X x R N , 0 ' )

then for ev ery

a re two d ifferen t

p e 2 an o scilla to ry in tegra l of the

form (3 .4 ) can a lso be w ritten a s an o scilla to ry in tegra l of the form

'(x , t ', rj'/yJ \ r'\ )e^ (X’T ,77 ^dr'dr]'+ h .

f a H ere

a '£ H m (k', £ ),

m '+ N '/2 = m + N /2

and p / W F ( h ) .

23

§3 . FO U RIER IN TEGRAL OPERATORS OF HERMITE T Y P E

T o prove th e se two propositions we w ill first give an a ltern ativ e d escrip tion of the s p a c e of distrib ution s defined by (3 .4 ). L e t X = R n = k P f R xR with coord in ates x = ( t , y ) and let € = (r,r]) be the dual co o rd i­ n a te s.

Let

be the su b se t x = y = 77 = 0 of T * R n - 0 and let be

the phase function rf> (x ,t,r,rj) = rt + rjy on R n x R n . is cle a rly parametrizin g phase function for

. In [16] the following an alogu es of the

propositions above were proved. P R O P O S IT IO N 3 .1 '.

L e t 1^ b e the o scilla to ry in tegra l

(3 .5 ) T h en WF(Ib ) C

.

P R O P O S IT IO N 3 .2 '.

L e t f : T * R n - 0 -> T * R n - 0

ca n o n ica l transformation mapping

into

be a h o m ogeneous and let F

be a F ourier

in tegra l operator of order r w hose underlying ca n o n ica l transformation is f.

T h en there e x is t s a sym bol b '(r, 77) e Hm+r(k, £) s u c h that

(3 .6 )

F Ib = V + h

with h e C°° . Now let X

be an n-dim ensional manifold and

2

a homogeneous

isotrop ic submanifold of T * X - 0 of dim ensional n - £. and the point pQ : x = 0 , t = 0 ,

r = (1 , 0,

0 ),

Given

77 = 0 , in

p f I there

e x is ts a homogeneous ca n o n ica l transform ation, f , mapping a neighbor­ hood of pQ onto a neighborhood of p and mapping ( 2 Q, pQ) onto ( 2 , p ) . (See §3 of [1 5 ].) L e t F

be a zeroth order e llip tic Fo u rier integral op era­

tor with f a s its underlying ca n o n ica l transform ation and let

1^ be a

distribution of type (3 .5 ) with the support of b contained in a sm all co n ic neighborhood of 77 = 0 ,

t

= (1 , 0, •••, 0 ) . By P rop osition 3 .1 ',

FI^

has its wave-front s e t con cen trated in a sm all co n ic neighborhood of p in 2 .

Moreover, it is c le a r from ( 3 .6 ) that if { ' and F ' have the sam e

24

THE S P E C T R A L THEORY OF T O E P L IT Z OPERATORS

properties as

f and F

then there e x is ts a b'(r, rj) e Hm(k, £) su ch that

FI^ - F T ^ ' is smooth near p .

Therefore the s p a ce

Km(k, £)i is in trin sically defined modulo C °°.

iF I^ , b(r, rj) e

We will now show that FI^

is an o scillato ry integral of type 3 .3 .

L e t < £ (x ,f ) be a generating func­

tion for the can o n ica l transform ation,

f . T his means that is homoge­

neous of degree one in f , the matrix (d c f> /d x ^ j) is non-singular and f is obtained from cf> by solving the sy stem of equations (3 -7 )

= (( ty /d x jX x ', O Xj

for (x', f ' )

in terms of ( x , f )

=

G W d f iK x '.f )

near (x 0 , f 0 ) = p0 . L e t p (x ,r ) be smooth,

homogeneous of degree zero in r for

|r| > > 0 , equal to one near pQ and

supported in a .sm a ll co n ic neighborhood of pQ in

. Then by Egorov,

[1 4 ], the operator F : C ^ (R n) -> C °°(X ) defined by

is a F ou rier integral operator with f a s its underlying ca n o n ica l tra n s ­ formation.

With th is ch o ice of F

we have

(3 .8 )

By ( 3 .7 ) the image of

is ju st the s e t

771 = ... - T7£ = 0 , so

0


/d^n = 0 ; 2.

T his shows that (3 .8 ) is

C onversely if cf> is the phase

Ig is an o scilla to ry integral of the form (3 .4 ) then

Ia = FIk + h with p / W F(h) and b(r, 7/) e Mm(k, I) . (See [1 5 ].)

§3. FO U RIER IN TEGRAL OPERATORS OF HERMITE T Y P E

25

It is c le a r that the p hase function, 0 , is already fairly c lo s e to being the most gen eral type of p hase function p o ssib le. function 0 ( x , £ )

In fa c t any phase

having the following two properties can a ris e a s the

generating function of a ca n o n ica l transform ation: (3 .9 ) a ) The number of phase v ariab les of b ase v a ria b le s, (3 .9 ) b) The c r itic a l s e t:

^ , " ' , £ n is equal to the number

x 1,---,xn . dcj>/d^ = 0 is tra n sv e rsa l to the s e t

L e t us show that given an arbitrary phase function cf> we ca n modify it so that if 0 ' is the modified phase function, then cf> and ' define the sam e c la s s of o scilla to ry in tegrals and cf>' s a tis f ie s (3 .9 ).

F ir s t of a ll if

ci> has le s s than n phase v ariab les we can modify it by adding on the term

(r k+1 + •••+ r 2 )/|0|

as in §3 of Hormander [25].

Ju s t as in §3 of [25]

one can s e e that this will not change the form of the amplitude in (3 .4 ). If 0

has more than n phase v a ria b le s, the superfluous phase v ariab les

must be among the r ’s

rj’s ;

b e ca u se of the tra n sv e rsa lity condition on the

so we can reduce the number of phase variab les by applying statio n ary phase to certain of the r ’s . Again as in §3 of [25] it is e a s y to s e e that this does not a ffe c t the form of the amplitude in (3 .4 ) . to sa tis fy (3.9)^ then b e ca u s e the

rj’s

F in a lly if cf> fails

are already tra n sv e rsa l to

we

can make a linear ch ange of v ariab les of the form r j = r - - \r\ ^ ^ a ijx j > so th at (r',7]) is tra n sv e rsa l to the c r itic a l s e t.

Such a change of variab les

does not affect the symbol c l a s s e s

T his con clu d es the proof of

Hm(k ,£ ).

P rop o sition s 3 .1 and 3 .2 . Now let X

be a smooth manifold and

manifold of T * X - 0 . eralized fu n ction s, p atch ,

2

a homogeneous iso tro p ic su b ­

We will denote by Im(X , 2 )

the sp a c e of a ll gen­

u , su ch that in every su fficien tly sm all coord in ate

u can be w ritten as an o s c illa to ry integral of the form ( 3 .4 ) where

a s a tis f ie s an asym p totic exp ansion of the form (3 .2 ). an e a sy con seq u en ce of (3 .6 ).

The following is

26

THE S P E C T R A L THEORY OF T O E P L IT Z OPERATORS

THEOREM

3.4.

L et X

and

xi

be n -dim ensional m anifolds,

2

and

isotropic subm anifolds of T * X - 0 and T * X 1 - 0 re s p e c t iv e ly and f : T * X - 0 -» T *X^ - 0 a h o m ogeneous ca n o n ica l transformation m apping onto

. L et F

be a Fourier in tegra l operator of order r w hose u n d er­

lying ca n o n ica l transformation is f .

In particular if P Im(X ,

2)

2

into Im+r(X ,

T h en F

maps

Im(X ,

2)

into

is a pseudodifferential operator of order r , P maps

2) .

§4. T H E M E T A P L E C T IC R EP R ES E N T A T IO N Let

G = Sp(n) be the group of linear mappings of R 2n

the altern atin g two form

leaving fixed

^ d x - a d y •. As is well known the fundamental

group of G is infinite c y c lic so there e x is ts a unique co n nected L ie group M = Mp(n) which double co v e rs group.

G . Mp(n) is ca lle d the m eta p lectic

It has a well-known infinite dim ensional unitary rep resen tatio n , the

so -ca lle d Segal-Shale-W eyl rep resen tatio n , which has ce rta in formal resem ­ b lan ces to the spin rep resen tation of the double covering of S O (n ). This is co n stru cted a s follow s: algeb ra with b a s is ,

Let

n be the

2n+l

dim ensional H eisenberg

x 1 , •••, x n, y 1 , y 2 , •••, yn, z , sa tisfy in g the bracket

re la tio n s : [xj.Xj]

=

( 4 .1 )

= 0

=S ijz [z,

Let

[yi ( yj ]

N be the a s s o c ia te d

anything] = 0 . 2n+l

dim ensional H eisenberg group.

By

the theorem of Stone-von Neumann there e x is ts a unique infinite dim ensional unitary rep resen tation (4 .2 )

p : N ^ tl(H )

(H being a sep arab le infinite dim ensional Hilbert s p a c e ) having the property: p(exp Az) = e ^ id , A e R . The Hilbert s p a c e fied with

H can be id en ti­

L 2(R n) . The s p a c e of C°° v e c to rs * of the rep resen tation ,

p ,

The space of C°° vectors of a unitary representation p :G -» X l(H ) of a Lie group, G, is the set of vectors v € H for which (p(g), v, w)^ is a smooth function on G for all w e H. This is the space on which the infinitesimal representation of the Lie algebra of G is defined.

27

28

THE S P E C T R A L THEORY OF T O E P L IT Z OPERATORS

is the space S(Rn) of rapidly decreasing functions; and the infinitesimal representation of n on S is described by the formulas (d p )(x i ) - \ / - l s i (d p )(Yi) = d / d s i

(4 .3 )

(dp) (z ) = y j-l id . Now G = Sp(n) a c ts on n = R 2 n ®R

by its usual actio n on R 2n and

a s the identity on R . It is cle a r that it p reserves the b rack et relation s (4 .1 ).

T herefore, it is a subgroup of the group of automorphisms of n ,

and a ls o of the group of automorphisms of N . Fo r every

g e G we get a new irreducible unitary rep resen tation of N : Pg : N -1 1 (H )

defined by p g (a) = p (g a ).

Note that p^(exp Az) = p(exp Agz) = p(exp Az) =

e ^ id ; so by the Stone-von Neumann theorem p^ is unitarily equivalent to p . Thus there e x is ts a unitary operator T l(g ): H -> H su ch that ‘U(g)p‘U(g~1 ) = p g . S ince p is irreducible li(g ) is determined uniquely e x ce p t for a m ultipli­ ca tiv e co n stan t of modulus 1.

In p articu lar, there is a function

c : G x G - S 1 su ch th at S 1 su ch that (4 -4 )

c ( g 1 ) c ( g 2 ) c ( g 1g2 r 1 = c ( g 1 , g 2 ) .

Then we can co n stru ct a unitary rep resen tation of G by settin g C Q(g) = c (g )“ 1U (g ).

It turns out that it is im possible to s a tis fy the

“ c o c y c le con d itio n ” (4 .4 ) on G , however it is p ossible to do so on the double covering of G . T herefore we get a unitary rep resen tation t

: Mp(n) -» 11(H) , the Segal-Shale-W eyl rep resen tation mentioned above.

29

§4. THE M ETA PLEC TIC REPRESENTATION

We will need a pretty e x p lic it d escrip tio n of this rep resen tation for later on.

F ir s t of a ll note that the L ie alg eb ra,

g,

of G is the sam e as

the L ie algebra of its double covering and is ju st the sy m p lectic algebra s p (n ).

In terms of the b a sis

(x,y)

the sy m p lectic algebra can be identi-

fied with the sp a c e of quadratic forms on R 2n . The id en tification is given by a s s o c ia tin g to a quadratic form cf) the linear mapping, sending ( x , y )

to ( x , y )

L^ ,

where x = d(f>/dy

( 4 .5 ) y = -dcfy/dx . (Compare with ( 3 .2 ) a b o v e .) Now write (4 .6 ) where g ^ corresponds to quadratic forms in x alon e, forms linear in x and in y se p a ra te ly , and alon e.

g Q to quadratic

g 1 to quadratic forms in y

It is ea sy to s e e that the RHS of ( 4 .6 ) is a g ra d ed alg eb ra, i.e .

>fl_! 1 = 0 , [fl0 , 0_ j ] C 0_ j , e tc . g 0 is isom orphic with the L ie a lg eb ra, given by

2

a ijXiyj

g L ( n ) , the isomorphism being

'

It turns out that the s p a c e of C ^ v e c to rs for the rep resen tation t : Mp(n)

11(H )

sen tation

p \N ^ U ( H ) ,

is the sam e a s the s p a c e of C°° v ecto rs for the repre­ that is , the Schw artz s p a c e

S(Rn).

The infini-

tesim al rep resen tation of g on S is given by the following form ulas: A) F o r A = ( 4 .7 )

f g _ j , M1 (1 ) and (Mp)^ -> Mp(W). To

prove (4 .1 5 ) it is enough to prove it for the s p e c ia l c a s e when spanned by X p - ^ X ^ ,

sin c e any two

2 ’s

2,

S ^ c o n s is ts of

a ll tempered distributions on R n such that x^u ~ ••• = x^u = 0 .

uw

is

of the sam e dimension are

con ju gated , one into the other, by S p (n ). F o r this

such distribution is of the form ^ 2 ®

2

w^ere ^ 2

*s

Every

d elta function

on R^ and uw a tempered distribution on Rn_^ . Given A e (Mp)^ let A ' and A " be its images in M£(2) and Mp(W) re sp e ctiv e ly .

From (4 .1 0 )

and (4 .1 1 ) one e a sily s e e s that r (A )(5 ^ ® u w) = Vdet A '5 ^ ® rw(A " )u w , r w being the m etap lectic rep resen tation on S'w . This e sta b lis h e s the isomorphism (4 .1 5 ). From the con ju gate linear pairing of S with S

Q .E .D . given by the Hilbert

s p a ce structure on H we get a dualized version of (4 .1 5 ):

33

§4. THE M ETA PLEC TIC REPRESENTATION

(4 .1 6 )

S -> a

REMARK.

^ (S ) ® § w .

T his map w ill play an e s s e n tia l role in the symbol ca lcu lu s

which we w ill develop in §6. We conclude this se c tio n by d iscu s s in g another analogue of (4 .1 5 ). T he sy m p lectic form, co , on R 2n is a ls o a sy m p lectic form on R 2n ®C . L e t A be a Lagran gian su b sp ace of R 2n ®C . A is ca lle d p o sitiv e d efin ite if the Hermitian form ,w ) , is positive definite on A

.

L et

v, w 6 A ,

as above be the s e t of a ll

u eS

su ch that d p (x)u = 0 , P R O P O S IT IO N 4 .2 .

an d is co n ta in ed a ll u € § A P roof.

then

If A

Vx e A .

is p o sitiv e d efin ite then

in S . M oreover if x e R 2 n ®C

is on e-d im en sio n s I

s a tis fie s

d p(x)u =

0 for

xe A .

It is e a sy to s e e that Sp(n) a c ts tran sitiv ely on the s p a c e of

p o sitive definite Lagrangian su b sp a ce s of R 2 n ®C ; so it is enough to prove the theorem for the s p e c ia l c a s e A =K x j + v 'z l y 1 , x 2 + v/- l y 2 , - - - , xn + v '- l y n)! By (4 .3 )

uf

if

and

•

only if (^ /(9 x -+ x | )u = 0

for all

i;

so

u is a co n stan t multiple of e

—x 2 /2

. The la s t a sse rtio n we

will leave as an e x e rc is e for the reader. L et

UyY be the s e t of all A e Mp(n) su ch that the projection of A in

Sp(n) p reserv es A . It is c le a r that if A f c(A)u,

Q. E. D.

and

U f SA then r(A )u =

c(A ) being a com plex number depending only on A .

p reserves the Hilbert s p a c e structure on S ,

Since r(A )

|c(A) | = 1 . It is a ls o c le a r

34

THE S P E C T R A L THEORY OF T O E PL IT Z OPERATORS

that c (A 1A 2 ) = c (A 1 ) c ( A 2 ) ;

so

c

is a unitary c h a ra cte r of the group,

• The following analogue of Proposition 4 .1 can be proved by a straightforw ard computation (which we w ill omit). P r o p o s i t i o n 4 .3 .

For all A e

, c (A )2 = det(A|A).

In §11 we will need: P R O P O S I T I O N 4. 4.

L e t A be in Mp(n).

it s e lf.

T h en A €

.

Proof.

F o r v 6 A , we have

S u p p o se r(A ) maps S ^

into

dp(Av) = r(A )d p (v )r(A )-1 by definition of r . Therefore if r(A) p reserves a ll v e A and a ll

u e

, dp(Av)u = 0 for

. From the secon d half of Proposition 4. 2 , we

conclude that A(A) = A . An elem ent,

e , of S which is of norm one and lies in some

will be called a vacuum sta te. lies in a unique

Q .E .D .

By Proposition 4 .1 every vacuum s ta te

and the vacuum sta te in

up to a con stan t multiple of modulus one.

is determined uniquely

§5. M ETA LIN EA R AND M E T A P L E C T IC STR U C TU R ES ON MANIFOLDS We pointed out in §4 that the group G?(n) has a natural double co verin g , the m etalinear group M f(n). Its fundamental property is th at the determ inant function lifted to M£(n) has an in trin sic square root. Let X

be a smooth manifold, and let

be the b a s is bundle of X .

is a pair co n sistin g of a point x e X and a

R e c a ll that a point of B X b a sis

BX

(e i > " ’ >e n) °f T x . B X

is a principal G?(n) bundle.

By a m eta­

linear stru ctu re on X we will mean a structure co n sistin g of a ) an M£(n) bundle,

MX -> X

and b) a double co v er A f Mf and

tt: MX -> B X

su ch that n(AB) = 77(A) 77(B) for all

B e MX .

It turns out that a m etalinear stru ctu re e x is ts on X

providing a very mild

top o lo gical condition is s a tisfie d (the vanishing of the square of the first Stiefel-W hitney c l a s s ) and if X unique.

is sim ply con nected this structure is

A manifold with a prescribed m etalinear structure will from now

on be ca lle d sim ply a m etalinear manifold. A m etalinear manifold p o s s e s s e s a natural line bundle, its bundle of 1 /2

forms.

on C given by yjdet .

a ^2 ,

ca lle d

Namely co n sid er the rep resen tation of M£(n)

a ^2

is the line bundle a s s o c ia te d to MX by

by means of this rep resen tation .

Note that

ated to MX by means of the rep resen tation

a^2 ® a ^2 is the bundle a s s o c i ­ “ d e t” ; and this is ju st the

determ inant bundle or volume form bundle on X . Therefore a s e ctio n of a ^2

is the “ square ro o t” of a volume form or, more sim ply, a half-form .

35

TH E S P E C T R A L T H E O R Y O F T O E P L IT Z O P ERA T O RS

For symplectic manifolds we have a rather analogous situation to that just described. Namely let ( Z ,0 ) be a symplectic manifold with two form Q .

Let BpZ be the symplectic basis bundle of Z : an element of

BpZ is a pair consisting of a point z e Z and a basis (v i>’ “ >vn> w 1,-",wn) of Tz satisfying Q(vj, Vj) = QCwj, Wj ) = 0 , fi(vi; Wj) = S-. BpX is a principal Sp(n) bundle. By a metaplectic structure on X we w ill mean a structure consisting of a) an Mp(n) bundle, MpX -> X and b) a double covering n : MpX -» BpX such that 77(AB) = 7 7 (A ) 77(B). Mappings between manifolds, and symplectic mappings between sym­ plectic manifolds have their “ meta” counterparts. For example let X and Y be manifolds and f : X -> Y a diffeomorphism.

Let f^ be the in­

duced diffeomorphism f* : BX -> BY . If X and Y are metalinear, a metadiffeomorphism will be defined as a pair consisting of a diffeomor­ phism

f :X

Y and a M£diffeomorphism f# : MX -> MY

covering

f^ : BX -> between metaplectic manifolds are ->BY BY .. Metaplectic Metaplectic mappings mappings between metaplectic manifolds are defined the same way. There is a rather simple connection between metalinear and meta­ plectic structures on manifolds, namely: I.

If X is a metalinear manifold there is a canonical metaplectic structure on T*X .

II.

If ZZ is is aa metaplectic metaplectic manifold, manifold, every every Lagrangian Lagrangian submanifold of If submanifold of Z has a canonical metalinear metalinear structure. structure.

These two assertions should not be too surprising considering that the metalinear group sits inside the metaplectic group in simple way. For a proof of I and II, see [2]. We will need below a slightly beefed up version of II. Namely let £ be an isotropic submanifold of Z . Given x f I space to I

at x , and £ x

let £ x be the tangent

its annihilator in TXZ . Since 2 is

37

§5. META LINEAR AND M ETA PLEC TIC STRUCTURES

isotrop ic sion 2

2 X D 2 X and

2X/2

is a sym p lectic v ecto r sp a ce of dimen­

2 ( n - k ) , k being the dimension of 2 .

w hose fiber at x e S

is

The v ecto r bundle

E

over

E x = 2 X / 2 X w ill be called the sy m p lectic

normal bundle of 2 . L et

(BxS)2

be the fiber bundle over

co n sistin g of a point x e 2 b a sis bundle.

a b a sis

(v i>“ ’ >vn_k> w i >’ " ' wn -k ) It turns out that if Z

2

whose o b jects are trip les

( e 1?

of 2 X and a sym p lectic

^ x ‘ T his is a principal

is m etap lectic then ( B x S ) 2

fold coverin g by a bundle whose stru ctu re group is

G £(k )xS p (n -k ) has a four­

M£(k) x M p(n-k).

Consider now the m etap lectic rep resen tation of Mp(n-k) on the Schw artz s p a c e

S and the rep resen tation of M?(k) on the com plex num­

bers given by y jd e t. The ten sor product of th e se two rep resen tation s is a rep resen tation of Mf(k) x Mp(n-k) on C ® S . vector bundle on 2

L e t S p in (2) be the

a s s o c ia te d with this rep resen tation .

S p in (2) will be

called the b undle of s y m p lectic sp in o rs a s s o c ia t e d to 2 . The fiber of S p in (2) is an infinite dim ensional v ecto r s p a c e ; th ere­ fore, a certain amount of cau tion must be e x e rcise d when applying standard v ecto r bundle operations to it.

F o r exam ple how do we define

a smooth

s e ctio n of S p in (2 )?

U is a coord in ate patch on 2

on which

Suppose

the sy m p lectic normal bundle is triv ial.

Then a se c tio n of S p in (2) on U

is ju st a map s : U -> S (R n~k) which to e ach z e U a s s o c ia t e s a rapidly d e cre a sin g function s ( z , x ) the variab les

x = ( x 1 ,• ••>xn_k ) . We will c a ll

sim ultaneously smooth in both If Z = T * X - 0 ,

x and z

s

smooth if s ( z , x )

in

is

v a ria b le s.

then there is an actio n of R+ on Z

given by

( a , x , f ) -» ( x , a £ ) for a ( R+ , x ( X , f f T * . T o get an induced actio n of R+ on S p in (2) we first ob serve that the m etap lectic rep resen tation ca n be extended to a rep resen tation of Mp(n) x R+ by lettin g

R + a c t triv ially .

Now the

38

THE S P E C T R A L THEORY OF T O E PL IT Z OPERATORS

action of R+ on T *X - 0 does not p reserve the sym p lectic stru ctu re, but it is conform ally sy m p le ctic; that is ,

a f R+ maps

T^x ^

to T^x a(

by a linear mapping which is the com posite of a sym p lectic isomorphism and the conform al mapping yja Id . This means that with the convention fixed on above we get an induced actio n of R^ on S p in (S ). We will henceforth denote by Sm( 2 )

the s p a ce of smooth se ctio n s of S p in (2)

which are homogeneous of order m .

§6. ISOTROPIC SU BSPA C ES OF SYM PLEC T IC V EC TO R SPA C ES In this se c tio n we will develop the a lg eb raic tools n e ce ssa ry to formulate our main resu lts in §7. rem arks. V©W.

L et

We w ill begin with some elem entary

V and W be v e cto r s p a c e s and let V

We w ill think of F

as a co lle ctio n of p airs,

be a su b sp a ce of (v, w),

with v in

V and w in W , i.e . a s a “ re la tio n ” in the s e t th eo retic s e n s e . Given a su b sp ace,

S,

of W we w ill denote by r ° 2

the s e t

! v e V , 3 w e I , (v,w)fFi . (If f : W -> V is a map and F = graph 2

with re sp e ct .to f . )

f ,then T ° 2

We will denote by T -1

is the usual image

of

the sp a ce

l ( w, v ) f W©V, ( v , w ) f T i . P R O P O S IT IO N 6. 1.

L et

2^

a nd r^-

and F

in W* and V* ® W*

Proof.

F ir s t suppose that T = graph

Let f * ; V * ^ W *

re s p e c t iv e ly .

be the tran sp o se of

< f*u , w > = 0 , Vw e 2

2*^ and

T h en

(r°2 )^ = F ^ ° 2 ^ .

f , f : W -> V being a linear map. f . Then

u e

° 2-^

u e (f* )_ 1 2^-

< u, fw > = 0 , Aw e 2 < ^ > u e (f ° 2 ) ^ ; so

in this c a s e the a ss e rtio n is true. being a linear mapping.

b e the annihilator s p a c e s of 2

N ext suppose

T = (graph f )_1 , f : V ->W

By the previous argument, with

f by f* , (f * ° 2 ^ ) ^ = f - 1 ( 2 ) ; so

2

rep laced by

° 2^- = f * ° 2^- = f - 1 ( 2 ) ^ =

( r ° 2 )^ . F in a lly the gen eral c a s e can be factored into a com position of the two above c a s e s by means of the diagram

39

40

THE S P E C T R A L T H EO RY OF T O E P U T Z OPERATORS

r

w

V

where a and /3 are the projection of T onto V and W respectively Q.E.D. From now on we will suppose that V and W are symplectic vector spaces. Then by means of the symplectic forms on V and W we can identify V with V *, W with W*

and V®W with V*®W*. An im-

portant special case of the result above is the following P R O P O S IT IO N

6.2. Let r

be a Lagrangian subspace of V ® W .

if 2 is a subspace of W , T ° Proof. If r

= (r °2)^ .

is Lagrangian then T =

COROLLARY.

Then

If £ is isotropic, r ° £

in Proposition 6.1.

Q.E.D.

is isotropic and if 2 is

Lagrangian T o 2 is Lagrangian. Assume, as above, that T is a Lagrangian subspace of V®W.

Let

a be the projection of V®W onto W. PR O P O SIT IO N

(6.1)

Proof. a(T) = r

6.3. a(r) is a co-isotropic subspace of W and :(r)^ = iw €W, (0, w )(V\ . a\ l o V; so a (r)^ = T 1o{0l by Proposition 6.2. Q.E.D.

Let 2 be an isotropic subspace of W and set (6 .2 )

u 0 - a(r)^ n s = jw € 2, (o, w) crs

and

(6.3)

U 1 =a(r)^ fl

= lw r ° S . Thus, on the fiber above r , we have an object which is an element of Spin(F ° S )f tensored with a density along the fiber. By Proposition 7.1, the fiber is compact; so we can integrate the density over the fiber and we are left with an element a ' ( r) of S p in (r° S )f . It is clear that o ' depends smoothly on r ; so all we have left to check is that it has the right degree of homogeneity. To do so we must show that the map (7.10) has the right degree of homogeneity as p varies along a ray in F . Going back to the proof of Proposition 6.5, consider the map 6.10: (7.11) This map maps w e (U j)p

into w J

, where 12 is the symplectic

form on T * Y . Since 1) is homogeneous of degree one, (7.11) is homogeneous of degree one. Therefore, the induced map on

(7.12)

-

A^W/Uj)*

is homogeneous of degree,

=

A ^ U j)®

forms:

A - 1/2( W q )

dim(U^ )p . The volume form, Qn ,

n = dim Y , is homogeneous of degree n; so the trivializing map a-

1/\Wq) -> C sending

into 1 is homogeneous of degree n/2 .

Composing this with the map (7.12) we get a map (7.13) which is homogeneous of degree: t- h Icm

dim UI +

.

In the proof of Proposition 7.2 we showed that dim U1 = e ; so dim U^- =

48

THE S P E C T R A L THEORY OF T O E PL IT Z OPERATORS

dim W - e = 2 dim Y - e . Thus the degree of homogeneity of (7 .1 3 ) is -i-(d im Y - e ) .

From this one e a sily s e e s that the map (7 .1 0 ) has the

sam e degree of homogeneity.

(Compare (6 .1 2 ) and ( 6 .1 4 ) .)

Q .E .D .

As a first ap plication of (7 .9 ) we will define symbols for the d istribu­ tio n s,

I

o

f

§3.

We re ca ll that a generalized function, Im(X , 2 )

u , on X

if it is lo cally e x p re ssib le by an o scilla to ry integral of the form

a (x , r, n /y J f)e lct>(x ’T’r1^drdr)

(7 .1 4 )

with < £ : X x R N -> R a defining phase function of 2 N = k+£.

belongs to the s p a c e

and a e Km -N //2(k,£),

(See ( 3 .4 ) .) T o a tta ch symbols to th e se distributions we w ill

have to make some additional assum ptions about (7 .1 4 ):

(I)

The amplitude in (7 .1 4 ) admits an asym ptotic expansion of the form (3 .2 ).

(II)

The order of homogeneity of the terms in (3 .2 ) are either integer or ha If-integer.

(HI)

X

is a m etalinear manifold and

i.e .

u is a generalized half-form,

u is lo cally e x p re ssib le by an o scilla to ry integral of the

form a ( x, r , rj/yJf)e1(&(x ’T

(7.14/

dr d77^ yjdx

the term in paren th eses being a s before. We equip the projection map 77 : X x R N -> X with the structure of a mor­ phism of half-forms by requiring that 77*\/dx = \/dx dr d rj. L e t V be the conormal bundle of graph 77 in T * ( X x X x R n) .

Then the pull-back map,

77*, on half-forms is a Fou rier integral operator a s s o c ia te d with the can o n ical relation ,

T . Its symbol is a half-form Qp on T .

If we

49

§7. THE COMPOSITION THEOREM

identify T (x,£)

with 77* T * X

and u se co o rd in ates

are the cotan gen t co o rd in ates on T * X

(x ,

, 6 ) on T where

and 0 = (r, 77) then

O p = y/ dxd^dO . L e t d T * (X x RN) be the map, (x , 6 ) -> (d of the s e t

rj^ = ••• =

isotrop ic submanifold of T * ( X x R N). 2^

= 0.

Then

is an

It is not hard to s e e that F

in te rse ct tra n sv e rsa lly in T * ( X x R N) and that

and

2 = T ° 2 ^ • (The

secon d statem en t is ju st another way of say in g that 0 phase function for £ . )

2^

2^

is a defining

C onsider now on 2 ^ fl (\r\ = 1 ) the sy m p lectic

spinor v = \/dx dr a

( 7 .1 5 )

o

(x , r, rj)yjdr]

where a m (x , r , r j ) is the leading term in the asym ptotic expansion (3 .2 ). o E xtend v to a sy m p lectic spinor on a ll of 2 ^ by requiring it to have degree of homogeneity D E F IN I T IO N 7. 3.

m-N/2.

We will define the sym bol,

S an isotropic su b b u n d le of

N|S w here Ng = ( T g S ) 1 / ^ ! . Assume U is invariant under the a ction of the hom otheties

( x , f ) -> ( x , A f ) on N .

e x is t s a neighborhood, subm anifold,

2 ,

0 ,

of s Q in 2

of T * X - 0

T h en for ev ery s Q e S there a nd a hom ogeneous isotropic

s u ch that 2

D 2H 0

and for a ll s e S P l O ,

T s r / T s S = Us F o r the proof of Lemma 8.1 s e e §3 of [15]. 0

be a neighborhood of s Q in T * X - 0

To prove Lemma 8 .2 , let

and f a function on 0

with the

following properties (i) (ii)

£ 0 . o f = 0 on 2 H (3 . dfs

(iii) f is homogeneous of degree one. (iv)

F o r all s exp t H f ,

conormal bundle of of 2 ,

S,

and let Uj

be the image of U °

in the symplectic

2 j . Argue by induction with

in place

U.

Q .E .D .

As a first step in proving Theorem 7 .5 we will show that we can , with­ out loss of generality, make some additional restrictions on 2 besides the conditions (7 .4). assume F -> T * Y - 0 arbitrary V

and

2

F i r s t of all we will show that we can

is an im bedding.

To se e this suppose we are given

satisfying the conditions (7.4).

sider the tensor product,

and T

u®£,

If u e IS(Y, 2 ) co n ­

as a generalized half-form on Y x Y x X .

We will show below (§9) that this belongs to Ir+s(Y x Y x X , I x T ) diagonal in T * Y x T * Y x T * X . map and n : Y x X -> X operator tt^A *

Let A : Y x X ^ Y x Y x X

the trivial fibering.

Then

near the

be the diagonal

Ku = 77^ * ^ ® £ ) .

is a Fourier integral operator from C ^ ( Y x Y x X )

The

into

C q ( X ) a s s o c ia t e d with the can o nical relation = l((y, r], y, -rj, x, O , ( x , - f )), ((x, £ ) e T * X - 0, (y, 17)^ T * Y - 0 ) ! . Letting

^

= SxT, -> T ^ Y l

addition

and

2 X sa tisfy all the hypotheses (7 .4 ) and in

is in jective, with Y 1 = Y x Y x X .

From now on we will assume manifold of T * Y - 0 .

T,

in Theorem 7 .5 , is an imbedded sub­

By Proposition 6 .3 this imbedded submanifold is

c o -isotropic.

By condition d) of ( 7 .4 ) T

submanifold,

F,

(TpO^- fl (T 2 ) ^

of

2.

F o r e a ch

and

p e F , let

in ( T p S ) ^ / T p S .

2

in tersect cleanly in a

Up be the image of

In §7 we showed that the assignment

p -> Up defines a vector bundle U -> F , called the e x c e s s bundle. We will now show that both

2

and

U can be chosen to have very simple

forms. As in §2 we will write variables

(r ,r 7) .

R

V Q = R x R with variables

(t, y) and dual

By Lemmas 8.1 and 8.2 we can find a germ of homogene­

ous can o nical transformation f : ( T * Y , p) -> T * R n mapping 2

onto the

isotropic manifold t = y = 77 = 0 and mapping the vector subbundle U of the symplectic conormal bundle of 2

into a bundle over f ( F ) which is

53

§8. THE PROOF OF THEOREM 7.5

spanned at each point by d / d y , d /d y x . If K^ is an F . 1.0 . , a s s o c i ­ ated with the ca n o n ical transformation f , which is elliptic near p then, K^u. and

replacing u by

K by K°K^

in Theorem 4 . 1 , we are reduced

to proving Theorem 7 .5 with the additional hypotheses a)

The manifold, 0 = t = y = rj,

Y,

is

R n and

£

in T * R n .

(8 .1 ) b) T -> T * R n is an imbedding and T a submanifold F c)

is the isotropic manifold,

and

2

in terse ct cleanly in

of S .

The e x c e s s bundle,

U -> F , is the following bundle:

If p € F ,

the fiber of Up is just the image of the vectors d / d y 1 , d /d y 2 ,