10 research outputs found
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 of linear operators that act
continuously on each of the Fock spaces , , and
contains all Toeplitz operators with bounded symbols. We show that compactness,
the spectrum, essential spectrum and the Fredholm index of an element of
, realized as an operator on , are independent of the
value of .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
-algebra in terms of integral kernel estimates and essential
commutants.Comment: 37 page
Commutative -invariant Toeplitz C 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 algebras on
the Fock space. In the end, we investigate the Gelfand theory of those
commutative C algebras.Comment: 26 pages; Comments are welcom
Fredholmness of Toeplitz operators on the Fock space
The Fredholm property of Toeplitz operators on the -Fock spaces on is studied. A general Fredholm criterion for arbitrary operators from the Toeplitz algebra on 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