Toeplitz operators on finite and infinite dimensional spaces with associated psi *-Fréchet algebras

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 Frechet operator algebras F closely related to the local symbol behavior of Toeplitz operators in F. We summarize results due to B. Gramsch et.al. on the construction of Psi_0- and Psi^*-algebras in operator algebras and corresponding scales of generalized Sobolev spaces using commutator methods, generalized Laplacians and strongly continuous group actions. In the case of the Segal-Bargmann space H^2(C^n,m) of Gaussian square integrable entire functions on C^n we determine a class of vector-fields Y(C^n) supported in complex cones K. Further, we require that for any finite subset V of Y(C^n) the Toeplitz projection P is a smooth element in the Psi_0-algebra constructed by commutator methods with respect to V. As a result we obtain Psi_0- and Psi^*-operator algebras F localized in cones K. It is an immediate consequence that F contains all Toeplitz operators T_f with a symbol f of certain regularity in an open neighborhood of K. There is a natural unitary group action on H^2(C^n,m) which is induced by weighted shifts and unitary groups on C^n. We examine the corresponding Psi^*-algebra A of smooth elements in Toeplitz-C^*-algebras. Among other results sufficient conditions on the symbol f for T_f to belong to A are given in terms of estimates on its Berezin-transform. Local aspects of the Szegö projection P_s on the Heisenbeg group and the corresponding Toeplitz operators T_f with symbol f are studied. In this connection we apply a result due to Nagel and Stein which states that for any strictly pseudo-convex domain U the projection P_s is a pseudodifferential operator of exotic type (1/2, 1/2). 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 and Waelbroeck. Namely, that the space of all holomorphic functions H(U) on an open subset U of a DFN-space (dual Frechet nuclear space) is a FN-space (Frechet nuclear space) equipped with the compact open topology. Using the nuclearity of H(U) we obtain Cauchy-Weil-type integral formulas for closed subalgebras A in H_b(U), the space of all bounded holomorphic functions on U, where A separates points. Further, we prove the existence of Hardy spaces of holomorphic functions on U corresponding to the abstract Shilov boundary S_A of A and with respect to a suitable boundary measure on S_A. Finally, for a domain U in a DFN-space or a polish spaces we consider the symmetrizations m_s of measures m on U by suitable representations of a group G in the group of homeomorphisms on U. In particular,in the case where m leads to Bergman spaces of holomorphic functions on U, the group G is compact and the representation is continuous we show that m_s defines a Bergman space of holomorphic functions on U as well. This leads to unitary group representations of G on L^p- and Bergman spaces inducing operator algebras of smooth elements related to the symmetries of U.

Representation theorems for indefinite quadratic forms and applications

Diese Arbeit widmet sich den Darstellungssätzen für symmetrische indefinite (das heißt nicht-halbbeschränkte) Sesquilinearformen und deren Anwendungen. Insbesondere betrachten wir den Fall, dass der zur Form assoziierte Operator keine Spektrallücke um Null besitzt. Desweiteren untersuchen wir die Beziehung zwischen reduzierenden Graphräumen, Lösungen von Operator-Riccati-Gleichungen und der Block-Diagonalisierung für diagonaldominante Block-Operator-Matrizen. Mit Hilfe der Darstellungssätze wird eine entsprechende Beziehung zwischen Operatoren, die zu indefiniten Formen assoziiert sind, und Form-Riccati-Gleichungen erreicht. In diesem Rahmen wird eine explizite Block-Diagonalisierung und eine Spektralzerlegung für den Stokes Operator sowie eine Darstellung für dessen Kern erreicht. Wir wenden die Darstellungssätze auf durch (grad u, h() grad v) gegebene Formen an, wobei Vorzeichen-indefinite Koeffzienten-Matrizen h() zugelassen sind. Als ein Resultat werden selbstadjungierte indefinite Differentialoperatoren div h() grad mit homogenen Dirichlet oder Neumann Randbedingungen konstruiert. Beispiele solcher Art sind Operatoren die in der Modellierung von optischen Metamaterialien auftauchen und links-indefinite Sturm-Liouville Operatoren.