Toeplitz operators on finite and infinite dimensional spaces with associated {_Y63*-Fréchet [Psi*-Fréchet] algebras [Elektronische Ressource] / Wolfram Bauer

Sujets

• Mathematics

Informations

• Publié par: johannes_gutenberg-universitat_mainz
• Publié le: 01 janvier 2006
• Nombre de lectures: 12
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

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 ∞