Toeplitz operators and generated algebras on non-Hilbertian spaces

Abstract

In this thesis we study Toeplitz operators on spaces of holomorphic and pluriharmonic functions. The main part of the thesis is concerned with such operators on the p-Fock spaces of holomorphic functions for p ∈ [1, ∞]. We establish a notion of Correspondence Theory between symbols and Toeplitz operators, based on extended notions of convolutions as developed by Reinhard Werner, which gives rise to many important results on Toeplitz operators and the algebras they generate. Here, we find new proofs for old theorems, extending them to a larger range of values of p, and also provide entirely new results. We manage to include even the non-reflexive cases of p = 1, ∞ in our studies. Based on the notions of band-dominated and limit operators, we establish a general criterion for an operator in the Toeplitz algebra over the Fock space to be Fredholm: Such an operator is Fredholm if and only if all of its limit operators are invertible. As an example of a Toeplitz algebra over the Fock space, we study the Resolvent Algebra (in the sense of Detlev Buchholz and Hendrik Grundling) in its Fock space representation. Partially following the methods of Correspondence Theory as discussed in this thesis, we manage to extend a classical result on the boundedness of Toeplitz operators (the Berger-Coburn estimates) to the setting of p-Fock spaces. Also based on results derived from the Correspondence Theory, we discuss several new characterizations of the full Toeplitz algebra on Fock spaces, at least in the reflexive range p ∈ (1, ∞). In the last part, we discuss several results on spectral theory and quantization estimates for Toeplitz operators acting on Bergman and Fock spaces of pluriharmonic functions.