Toeplitz Operators on

ﬁnite and inﬁnite dimensional spaces

with associated

∗Ψ -Fr´echet Algebras

Dissertation

zur Erlangung des Grades

”Doktor

der Naturwissenschaften”

am Fachbereich ”Physik, Mathematik und Informatik”

der Johannes Gutenberg-Universit¨at

in Mainz

Wolfram Bauer

geb. in Wiesbaden

Mainz, den 13. Juni 2005Tag der mu¨ndlichen Pru¨fung: Freitag, 25. November 2005.

Date of the oral examination:

(D77) Dissertation, Johannes Gutenberg-Universit¨at MainzSummary

The present thesis is a contribution to the multi-variable theory of Bergman and Hardy

Toeplitz operators on spaces of holomorphic functions over ﬁnite and inﬁnite dimensional

domains. In particular, we focus on certain spectral invariant Fr´echet operator algebrasF

closely related to the local symbol behavior of Toeplitz operators inF.

Wesummarizeresultsduetotheauthorsof[79]and[107]ontheconstructionofΨ -and0

∗Ψ -algebras in operator algebras and corresponding scales of generalized Sobolev spaces

using commutator methods, generalized Laplacians and strongly continuous group actions.

2 nInthecaseoftheSegal-BargmannspaceH (C ,)ofGaussiansquareintegrableentire

n n nfunctions onC we determine a class of vector-ﬁelds Y(C ) supported in cones C ⊂C .

nFurther, we require that for any ﬁnite subset V ⊂ Y(C ) the Toeplitz projection P is a

smooth element in the Ψ -algebra constructed by commutator methods with respect to0

∗V. As a result we obtain Ψ - and Ψ -operator algebras F localized in cones C. It is an0

nimmediate consequence that F contains all Toeplitz operators T with f bounded onCf

and smooth with bounded derivatives of all orders in a neighborhood ofC.

2 nThere is a natural unitary group action on H (C ,) which is induced by weighted

n ∗shifts and unitary groups onC . We examine the corresponding Ψ -algebrasA of smooth

∗elements in Toeplitz-C -algebras. Among other results suﬃcient conditions on the symbol

˜f for T to belong toA are given in terms of estimates on its Berezin-transform f.f

Local aspects of the Szeg¨o projectionP on the Heisenbeg group and the correspondings

Toeplitz operatorsT with symbolf are studied. In this connection we apply a result duef

to Nagel and Stein [117] which states that for any strictly pseudo-convex domain Ω the

1 1projection P is a pseudodiﬀerential operator of exotic type ( , ).s 2 2

The second part of this thesis is devoted to the inﬁnite dimensional theory of Bergman

and Hardy spaces and the corresponding Toeplitz operators. We give a new proof of a

result observed by Boland [24], [25] and Waelbroeck [141]. Namely, that the space of all

holomorphic functionsH(U) on an open subset U of aDFN-space (dual Fr´echet nuclear

space) is a FN-space (Fr´echet nuclear space) equipped with the compact open topology.

Using the nuclearity of H(U) we obtain Cauchy-Weil-type integral formulas for closed

∞subalgebrasA inH (U), the space of all bounded holomorphic functions on U, whereA

separatespoints. Further, weprovetheexistenceofHardyspacesofholomorphicfunctions

onU correspondingtotheabstractShilovboundaryS ofAandwithrespecttoasuitableA

boundary measure Θ onS .A

Finally, for a domain U in a DFN-space or a polish spaces we consider the sym-

metrizations of measures on U by suitable representations of a group G in the groups

of homeomorphisms on U. In particular, in the case where leads to Bergman spaces of

holomorphic functions onU, the groupG is compact and the representation is continuous

we show that deﬁnes a Bergman space of holomorphic functions on U as well. Thiss

pleads to unitary group representations ofG onL - and Bergman spaces inducing operator

algebras of smooth elements related to the symmetries of U.Contents

Introduction 5

1 Fr´echet algebras with spectral invariance 17

1.1 Fr´echet algebras generated by closed derivations . . . . . . . . . . . . . . . 21

1.2 Operator algebras by commutator methods . . . . . . . . . . . . . . . . . . 23

1.3 Smooth elements by C -group action . . . . . . . . . . . . . . . . . . . . . 290

1.4 Projections of algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2 Cone localization of the Segal Bargmann projection 39

2.1 Toeplitz operators on the Segal-Bargmann space . . . . . . . . . . . . . . . 41

2.2 Commutators of P with linear vector ﬁelds . . . . . . . . . . . . . . . . . . 51

2.3 Radial symmetric vector ﬁelds. . . . . . . . . . . . . . . . . . . . . . . . . 53

2.4 Commutators of P with systems of vector ﬁelds . . . . . . . . . . . . . . . 59

2.5 Fr´echet algebras localized in cones . . . . . . . . . . . . . . . . . . . . . . . 67

3 Smooth elements in an algebra of Toeplitz operators

generated by unitary groups. 73

3.1 Smooth Toeplitz operators generated by the Weyl group action . . . . . . . 76

∗3.2 Ψ -algebras generated by the Weyl group . . . . . . . . . . . . . . . . . . . 87

3.3 Berezin Toeplitz and Gabor-Daubechies

Windowed Fourier localization operators . . . . . . . . . . . . . . . . . . . 91

3.4 Berezin Toeplitz operator and Weyl quantization . . . . . . . . . . . . . . 95

3.5 Algebras by groups of composition operators . . . . . . . . . . . . . . . . . 99

4 Fr´echet algebras by localized commutator methods

and Szeg¨o Toeplitz operators 111

4.1 Pseudodiﬀerential operators and

commutator methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.2 Localization of operator algebras

and Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.3 The Szeg¨o-projection in the theory of

pseudo-diﬀerential operators . . . . . . . . . . . . . . . . . . . . . . . . . . 125

4.4 Expansion of smooth functions on odd spheres . . . . . . . . . . . . . . . . 129

35 Gaussian measures and holomorphic functions on open

subsets of DFN-spaces. 135

5.1 Gaussian measures on Hilbert spaces . . . . . . . . . . . . . . . . . . . . . 136

5.2 Integral estimates for holomorphic functions . . . . . . . . . . . . . . . . . 139

5.3 Some topological properties ofDFN-spaces . . . . . . . . . . . . . . . . . 146

5.4 NF -measures on open sets ofDFN-spaces . . . . . . . . . . . . . . . . . 150p

5.5 The Nuclearity ofH(U) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6 The Cauchy-Weil theorem and abstract Hardy spaces

for open subsets of DFN-spaces 159

6.1 Nuclearity and generalized Bergman spaces . . . . . . . . . . . . . . . . . . 162

6.2 Grothendiecks Theorem and Nuclearity . . . . . . . . . . . . . . . . . . . . 164

6.3 Holomorphic liftings for Banach space

valued functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.4 Holomorphic liftings on an inductive nuclear

spectrum of Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

6.5 The Shilov boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

6.6 The abstract Cauchy-Weil theorem . . . . . . . . . . . . . . . . . . . . . . 175

6.7 Abstract Hardy spaces for domains with arbitrary boundary . . . . . . . . 176

7 Invariant measures for special groups of homeomorphisms

on inﬁnite dimensional spaces 185

7.1 Symmetric Borel measures on topological spaces . . . . . . . . . . . . . . . 188

7.2 Group representations and symmetric measures . . . . . . . . . . . . . . . 195

27.3 Representations of C -semi-groups on L -spaces . . . . . . . . . . . . . . . 1980

7.4 Group actions induced by symmetries . . . . . . . . . . . . . . . . . . . . . 199

27.5 Dynamical systems on L -spaces over Riemannian manifolds . . . . . . . . 205

7.6 Group action on generalized Toeplitz-algebras . . . . . . . . . . . . . . . . 207

A Appendix 211

A.1 On the topology ofDFN-spaces . . . . . . . . . . . . . . . . . . . . . . . 211

A.2 DFN-spaces of holomorphic functions . . . . . . . . . . . . . . . . . . . . 215

A.3 Holomorphy on topological spaces . . . . . . . . . . . . . . . . . . . . . . . 223

A.4 Heisenberg group and Hardy spaces over the ball . . . . . . . . . . . . . . 225

A.5 Symmetries of the ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

A.6 Cauchy-Szeg¨o projection and Hardy spaces . . . . . . . . . . . . . . . . . . 227

A.7 Cauchy-Szeg¨o projection and exotic classes . . . . . . . . . . . . . . . . . . 229

0A.8 On the symbol classS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231ρ

A.9 Hankel operators and mean oscillation . . . . . . . . . . . . . . . . . . . . 235

List of symbols 254

Index 256Introduction

The present thesis is a contribution to the construction of spectral invariant symmetric

∗ ∗Fr´echet subalgebras (Ψ -algebras) of Toeplitz-C -algebras over ﬁnite and inﬁnite dimen-

sional domains. Moreover, it is of general interest to give a notion of Bergman and Hardy

spaces and the corresponding Toeplitz operators for domains Ω in certain inﬁnite dimen-

sional nuclear spacesE. This approach only uses the nuclearity of the Fr´echet spaceH(Ω)

of all holomorphic functions on Ω equipped with the compact open topology.

∗The concept of Ψ -algebras is closely related to local aspects in operator theory. To

give an idea of the kind of results we prove let us state the following theorems which can

nbe found in chapter 2 and 6. Let be a Gaussian measure onC and ﬁx open cones

n ∗ Δ 2 nC ⊂C inC . We show how to obtain a rich variety of Ψ -algebras Ψ inL(L (C ,))1 2 ∞

containing the Segal-Bargmann Toeplitz projection P and localized in C in the following1

sense (cf. Theorem 2.5.3 and Proposition 2.5.2):

∞ n n ∞Theorem1Leth∈L (C ) such thatsupp(h)⊂C \C or let us assume thath∈C (C ),2 2b

Δ 2 nthen T ∈ Ψ , where T g :=P(hg) for all g∈L (C ,).h h∞

ΔHence the algebra Ψ is invariant under perturbations by Toeplitz operators T withh∞

symbols supported outside ofC and related to the regularity of h restricted toC . In our2 2

constructions above we apply quite general ideas which were suggested by the authors of

[79] and do not depend on the ﬁnite dimension of the underlying domains. Based on a

combination of results in [25], [121] and [141] we prove that for any open subset U of a

DFN-space E (topological dual of a Fr´echet nuclear space) there is a notion of Bergman

2 ∞space H (U,). Moreover, let A be a closed subalgebra of the Banach algebra H (U) of

bounded holomorphic functions on U which separates points and has the abstract Shilov

boundaryS . TheninadditionwecanprovetheexistenceofaHardyspaceofholomorphicA

functions on U (cf. Theorem 6.7.1) :

Theorem 2 Let , ∈ MF (U) be measures where F = H(U) (cf. Deﬁnition 5.4.1).1 2 2

Assume that there is a diagram

J J1 2∞ 2 2H (U)⊃A −−−→ H (U, ) −−−→ H (U, )1 2

where J are continuous embeddings and J is nuclear. Then there is a Hardy spacei 2

2 2H (S ,Θ) containing A which admits a quasi-nuclear embedding into H (U, ).A 2

We want to remark that our construction of Hardy spaces seems to be intrinsic and it

requires no assumption on the boundary of U. For instance note that via biholomorphic6 Introduction

equivalence for each simply connected domainU in the complex plane the Banach algebra

¯ofholomorphicfunctionsonU withcontinuousextensionstoU leadstoaclosedsubalgebra

∞ofH (D) separating points where D denotes the open unit disc.

In the inﬁnite dimensional setting the symmetries of the domains U resp. S can beA

∗ ∗used to obtain Ψ -algebras of smooth elements in Toeplitz C -algebras (cf. chapter 7).

There is an extended theory on (locally) spectral invariant Fr´echet algebras subsequent

to[79]. Applicationsariseinthestructuralanalysisofcertainalgebrasofpseudodiﬀerential

operators, complex analysis, analytic perturbation theory of Fredholm operators, non-

abelian cohomology or for analyzing isomorphisms of abelian groups in K-theory. As for

a more detailed list and the consequences that follow for the algebras given in the present

thesiswerefertoRemark1.0.2. Weconsiderourworktobeaﬁrststeptomaketheabstract

∗theory of Ψ -algebra applicable to algebras of Bergman and Hardy Toeplitz operators.

Toeplitz operators and related algebras on classical spaces such as the Hardy space on

the 1-torus or the Bergman space on the unit disc are well-understood and they play an

important role in operator theory. In fact, there are many applications to function theory,

integral equations and control theory. In the higher dimensional setting, the analysis of

multi-variable Toeplitz operators is more complicated and is less well-known. Here the

∗geometry of the underlying space is closely related to the associated ToeplitzC -algebras.

Forseveralclassesofdomainswithnotnecessarilysmoothboundarythespectralbehavior,

∗index theory and solvability of C -algebras is examined in [140],[83].

As a model space for quantum mechanical operators, I.E. Segal and V. Bargmann

2 n ninventedaspaceF :=H (C ,)ofentirefunctionsonC squareintegrablewithrespectton

aGaussianmeasureandcanonicallyisomorphictotheFockspace[5],[133]. Fundamental

conceptsasthecreationandannihilationoperatorscanberepresentedasToeplitzoperators

onF andtherehavebeenfarreachinginvestigationsofthecorrespondingoperatortheoryn

by L.A. Coburn and C.A. Berger [21], [22], [23], [35], [36], [37] see also the results in [11],

[12], [92], [145].

From a physical point of view, it is signiﬁcant to extend the number of freedom cor-

responding to the complex dimension n to inﬁnity. There also have been approaches on

Toeplitz operators over inﬁnite dimensional domains, motivated by quantum ﬁeld theory

[93], [90], [14]. J. Janas and K. Rudol have generalized the notion of Toeplitz operators

non the Segal-Bargmann space F by replacingC by a separable complex Hilbert spacen

H and by an inﬁnite dimensional Gaussian measure on H. Some new phenomena arise,

which have no counterpart in the case of F but also create diﬃculties in the analysis ofn

[93], [90] and [14] which come from the measure theory on H. A second model, which

is referred to in the literature as the inductive limit approach, only uses a so-called quasi

Gaussian measure on H with non-nuclear correlation and it is based on Segal’s model of

the Fock space [132].

We focus on both, the multi-variable and the inﬁnite dimensional theory of Bergman

and Hardy Toeplitz operators. On the one hand it is of interest in operator theory to

∗construct subalgebras ofC algebras inL(H) whereH is a Hilbert space which are related

tolocalpropertiesofitselementsandactingonscalesofSobolevspaces. Inthisconnection

∗the notion of a Ψ-algebra and in particular of a Ψ -algebra has attached great importanceW. Bauer 7

(cf. the list of recent results at the beginning of chapter 1.) On the other hand, motivated

by quantum mechanics, representation theory and inﬁnite dimensional holomorphy we

focus on the theory of Bergman and Hardy spaces over inﬁnite dimensional domains. As

for the area of Toeplitz operators our main interest is the following:

∗(I) The construction of spectral invariant Fr´echet algebras and Ψ -algebras in particular

∗inToeplitzC -algebrasforBergmanandHardyToeplitzoperatorsT relatedtolocalf

properties of the symbol f.

(II) Bergman and Hardy spaces over certain inﬁnite dimensional spaces related to the

symmetry of the underlying domain and corresponding Toeplitz operators.

We describe what is meant by (I) and (II) above. In general, by passing from a ring

∗ ∞of operators to its closure (a Banach or C -algebra) one looses local C -properties such

as pseudo- or micro-locality. Hence in order to keep control of the local behavior and,

motivated by the ring of zero-order pseudodiﬀerential operators, the notion of a Ψ-algebra

A in a Banach algebra B was introduced (cf. [69]). In particular, if A is symmetric in a

∗ ∗C -algebra B it is referred to as a Ψ -algebra. Essential in the deﬁnition is the notion of

spectral invariance or invariance under holomorphic functional calculus:

−1 −1A∩B =A

−1 −1whereA (resp. B ) denotes the group of invertible elements [17], [69], [79], [130], [139].

A spectrally invariant algebra A sometimes is called full or alge`bre plaine, the pair of

algebras (A,B) is said to be a Wiener-pair following ideas due to Bourbaki, Naimark and

Waelbroeck [27], [118], [142], [143]. As an immediate consequence, a Ψ-algebra A has an

−1opengroupA whichingeneralisnotthecaseforanarbitraryFr´echetalgebra. Moreover,

we mention that the inversion inA is continuous and it is a useful property of the stability

∗that countable intersections of Ψ- resp. Ψ - algebras are of this type again.

The concept of extracting the notion of spectral invariance from the operator theory

andfocusingonanabstractΨ-algebrashasbeensuccessful. Anextensivestudyonspectral

invariant Fr´echet algebras starting with [69] led to many applications and among others

we mention perturbation theory and homotopy theory of Fredholm functions or the holo-

morphic functional calculus of L. Waelbroeck.

Various methods of generating Ψ-algebras can be found in [79] and in part they are

described in chapter 1 of this thesis. In our constructions later on we use commutator

methodswithvectorﬁeldssupportedinsuitablesets,generalizedLaplacians[44]andunitary

group actions [28], [79], [107]. Let us describe the concept of commutator methods. It was

0 1shownbyR.Beals[17]thattheH¨ormanderclassesΨ ofpseudodiﬀerentialoperators canρ,δ

completelybecharacterizedbyconditionsoniteratedcommutatorswiththemultiplications

M and the derivatives ∂ of all orders. Using this result, it was shown by R. Beals [17]x xj j

0and ﬁnally by J. Ueberberg [139] and E. Schrohe [130] (cf. [26]) that the classes Ψ areρ,δ

2 n 0 ∗spectral invariant in L(H) where H :=L (R ,v) and so Ψ is a Ψ -algebra in the senseρ,δ

10≤δ≤ρ≤ 1 and δ < 18 Introduction

of [79]. Moreover, as a result in [40] there are similar descriptions using commutators with

smooth vector ﬁelds. The authors of [79] pointed out how to generalize this method to an

abstract setting in order to construct Ψ-algebrasA in certain subalgebras of the bounded

operators on a Hilbert space H. These methods are using commutator characterizations

fom begin on and so spectral invariance is an immediate consequence. Let V be a ﬁnite

set of densely deﬁned closed operators on H. Then, roughly speaking, for an operator

Va∈L(H) to be inA =: Ψ we require that all the iterated commutators∞h i

[a,V ], V , , V ∈V (0.0.1)1 2 j

are well-deﬁned on a suitable dense subspace in H and admit extensions to bounded op-

erators on H. Additionally, we remark that in the case where all operators V ∈ V are

∗symmetric, A becomes a Ψ -algebra; associated to V there is a scale of abstract Sobolev

Vspaces which is preserved by the operators in Ψ .∞

As for question (I) we start our investigation with Toeplitz operators T on the Segal-f

2 n nBargmann space H (C ,) of -square integrable entire functions onC introduced by

I.E. Segal and V. Bargmann [5], [133] (for the theory of Toeplitz operators: [21], [22], [23],

[37], [55], [56],[91], [92], [135]). Here f is an admissible measurable symbol and denotes

2 n 2 na Gaussian measure. With the orthogonal projection P from L (C ,) onto H (C ,)

the (in general unbounded) operator T is deﬁned by T (g) := P(fg) provided that thef f

2 n n nproduct fg is in L (C ,). We determine a subspace Y(C ) in the class W(C ) of all

n 2 nsmooth vector ﬁelds onC and a suitable dense subspaceZ ⊂L (C ,) with the following

properties:

n(P1) For any ﬁnite systemV ⊂Y(C ) of vector ﬁelds, with the Toeplitz projection a :=P

and V ∈ V all the iterated commutators in (0.0.1) are well-deﬁned on Z and theyj

2 nadmit (unique) extensions to bounded operators on L (C ,).

n(P2) LetA be an open set in the complex unit sphere∂B inC . Then there exist vector-2n

n n −1ﬁelds 0 =V ∈Y(C ) supported in C :={z∈C \{0} :zkzk ∈A}.A

As a ﬁrst step, we will prove that the spaceY of all vector ﬁelds with constant coeﬃ-1

ncientsfullﬁls(P1)withZ :=C (C ). Now,inadditionwewanttofulﬁll(P2)andsowehavec

to enlargeY . As it is indicated by Example 2.3.1 one has to be careful. It turns out that1 P

n ¯the boundedness of commutators ofP with a smooth vector ﬁeldV = {a ∂ +b ∂ }j j j jj=1

is closely related to the oscillation of the coeﬃcients a and b at inﬁnity. We prove thatj j

nradial extensions toC of smooth functions on the complex sphere (which are cut oﬀ at 0)

form a class of admissible coeﬃcients a for V. Moreover, we can choose b to be smoothj j

nwith bounded derivatives of all orders. This ﬁnally leads to a spaceY(C ) of vector ﬁelds

with (P1) and (P2) which, up-to a cut oﬀ at 0, contains Y as well as the derivatives1

n∂ ∈W(C ) tangential to the complex sphere ∂B .ϕ 2nj

As a consequence in our theory of operator algebras and with the notations above we

V 2 n nconclude that P ∈ Ψ ⊂ L(L (C ,) ) for all ﬁnite sets V ⊂ Y(C ) and the Toeplitz∞

projection P is said to be smooth with respect to V. Let C be the cone over A ⊂ ∂BA 2n

6