Toeplitz operators and generated algebras on non-Hilbertian spaces

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

A Wiener algebra for Fock space operators

We introduce an algebra $\mathcal W_t$ of linear operators that act continuously on each of the Fock spaces $F_t^p$, $1 \leq p \leq \infty$, and contains all Toeplitz operators with bounded symbols. We show that compactness, the spectrum, essential spectrum and the Fredholm index of an element of $\mathcal W_t$, realized as an operator on $F_t^p$, are independent of the value of $p$.Comment: 15 page

Quantum harmonic analysis for polyanalytic Fock spaces

We develop the quantum harmonic analysis framework in the reducible setting and apply our findings to polyanalytic Fock spaces. In particular, we explain some phenomena observed in arXiv:2201.10230 and answer a few related open questions. For instance, we show that there exists a symbol such that the corresponding Toeplitz operator is unitary on the analytic Fock space but vanishes completely on one of the true polyanalytic Fock spaces. This follows directly from an explicit characterization of the kernel of the Toeplitz quantization, which we derive using quantum harmonic analysis. Moreover, we show that the Berezin transform is injective on the set of of Toeplitz operators. Finally, we provide several characterizations of the $\mathcal{C}_1$-algebra in terms of integral kernel estimates and essential commutants.Comment: 37 page

Commutative GG-invariant Toeplitz C∗^\ast algebras on the Fock space and their Gelfand theory through Quantum Harmonic Analysis

We discuss the notion of spectral synthesis for the setting of Quantum Harmonic Analysis. Using these concepts, we study subalgebras of the full Toeplitz algebra with certain invariant symbols and their commutators. In particular, we find a new class of commutative Toeplitz C$^\ast$ algebras on the Fock space. In the end, we investigate the Gelfand theory of those commutative C$^\ast$ algebras.Comment: 26 pages; Comments are welcom

Fredholmness of Toeplitz operators on the Fock space

The Fredholm property of Toeplitz operators on the $p$-Fock spaces $F_\alpha^p$ on $\mathbb{C}^n$ is studied. A general Fredholm criterion for arbitrary operators from the Toeplitz algebra $\mathcal{T}_{p,\alpha}$ on $F_\alpha^p$ in terms of the invertibility of limit operators is derived. This paper is based on previous work, which establishes corresponding results on the unit balls $\mathbb{B}^n$