Toeplitz operators defined by sesquilinear forms: Fock space case

The classical theory of Toeplitz operators in spaces of analytic functions deals usually with symbols that are bounded measurable functions on the domain in question. A further extension of the theory was made for symbols being unbounded functions, measures, and compactly supported distributions, all of them subject to some restrictions. In the context of a reproducing kernel Hilbert space we propose a certain framework for a `maximally possible' extension of the notion of Toeplitz operators for a `maximally wide' class of `highly singular' symbols. Using the language of sesquilinear forms we describe a certain common pattern for a variety of analytically defined forms which, besides covering all previously considered cases, permits us to introduce a further substantial extension of a class of admissible symbols that generate bounded Toeplitz operators. Although our approach is unified for all reproducing kernel Hilbert spaces, for concrete operator consideration in this paper we restrict ourselves to Toeplitz operators acting on the standard Fock (or Segal-Bargmann) space

Algebras of Toeplitz operators on the unit ball

One of the common strategies in the study of Toeplitz operators consists in selecting of various special symbol classes $S \subset L_{\infty}$ so that the properties of both the individual Toeplitz operators $T_a$, with $a \in S$, and of the algebra generated by such Toeplitz operators can be characterized. A motivation to study an algebra generated by Toeplitz operators (rather than just Toeplitz operators themselves) lies in a possibility to apply more tools, in particular those coming from the algebraic toolbox, and furthermore the results obtained are applicable not only for generating Toeplitz operators but also for a whole variety of elements of the algebra in question. To make our approach more transparent we restrict the presentation to the case of the two-dimensional unit ball $\mathbb{B}^2$. We consider various sets $S$ of symbols that are invariant under a certain subgroup of biholomorphisms of $\mathbb{B}^2$ ($\{1\}\times \mathbb{T}$ in the talk). Such an invariance permits us \emph{to lower the problem dimension} and to give a recipe, supplied by various concrete examples, on how the known results for the unit disk $\mathbb{D}$ can be applied to the study of various algebras (both commutative and non-commutative) that are generated by Toeplitz operators on the two-dimensional ball $\mathbb{B}^2$. Although we consider the operators acting on the weighted Bergman space on $\mathbb{B}^2$ with a \emph{fixed} weight parameter, the Berezin quantization effects (caused by a \emph{growing} weight parameter of the corresponding weighted Bergman spaces on the unit disk $\mathbb{D}$) have to be taken into account

Spectral theorem approach to commutative C* -algebras generated by Toeplitz operators on the unit ball: quasi-elliptic related cases

We consider commutative C* -algebras of Toeplitz operators in the weighted Bergman space on the unit ball in $\mathbb{C}^{\mathbf{n}}$. For the algebras of elliptic type we find a new representation, namely as the algebra of operators which are functions of certain collections of commuting unbounded self-adjoint operators in the Bergman space

Toeplitz operators with singular symbols in polyanalytic Bergman spaces on the half-plane

Using the approach based on sesquilinear forms, we introduce Toeplitz operator in the analytic Bergman space on the upper half-plane with strongly singular symbols, derivatives of measures. Conditions for boundedness and compactness of such operators are found. A procedure of reduction of Toeplitz operators in Bergman spaces of polyanalytic functions to operators with singular symbols in the analytic Bergman space by means of the creation-annihilation structure is elaborated, which leads to the description of the properties of the former operator

Trace class Toeplitz operators with singular symbols

We characterize the trace class membership of Toeplitz operators with distributional symbols acting on the Bergman space on the unit disk. The Berezin transform of distributions, introduced in the paper, yields a formula for the trace. Several instructive examples are also given

Uniform continuity and quantization on bounded symmetric domains

We consider Toeplitz operatorsTλfwith symbolfacting on the standard weighted Bergmanspaces over a bounded symmetric domain Ω⊂Cn.Hereλ>genus−1 is the weight parameter.The classical asymptotic relation for the semi-commutatorlimλ→∞∥∥∥TλfTλg−Tλfg∥∥∥λ=0,withf,g∈C(Bn),(∗)where Ω =Bndenotes the complex unit ball, is extended to larger classes of bounded andunbounded operator symbol-functions and to more general domains. We deal with operatorsymbols that generically are neither continuous inside Ω nor admit a continuous extension to theboundary. Letβdenote the Bergman metric distance function on Ω. We prove that(∗) remainstrue forfandgin the space UC(Ω) of allβ-uniformly continuous functions on Ω. Note that thisspace contains also unbounded functions. In case of the complex unit ball Ω =Bn⊂Cnwe showthat(∗) holds true for bounded symbols in VMO(Bn), where the vanishing oscillation insideBnismeasured with respect toβ.Atthesametime(∗) fails for generic bounded measurable symbols.We construct a corresponding counterexample using oscillating symbols that are continuousoutside of a single point in

Algebras of Toeplitz operators on the n-dimensional unit ball

We study $C^*$-algebras generated by Toeplitz operators acting on the standard weighted Bergman space $\mathcal{A}_{\lambda}^2(\mathbb{B}^n)$ over the unit ball $\mathbb{B}^n$ in $\mathbb{C}^n$. The symbols $f_{ac}$ of generating operators are assumed to be of a certain product type, see (\ref{Introduction_form_of_the_symbol}). By choosing $a$ and $c$ in different function algebras $\mathcal{S}_a$ and $\mathcal{S}_c$ over lower dimensional unit balls $\mathbb{B}^{\ell}$ and $\mathbb{B}^{n-\ell}$, respectively, and by assuming the invariance of $a\in \mathcal{S}_a$ under some torus action we obtain $C^*$-algebras $\boldsymbol{\mathcal{T}}_{\lambda}(\mathcal{S}_a, \mathcal{S}_c)$ whose structural properties can be described. In the case of $k$-quasi-radial functions $\mathcal{S}_a$ and bounded uniformly continuous or vanishing oscillation symbols $\mathcal{S}_c$ we describe the structure of elements from the algebra $\boldsymbol{\mathcal{T}}_{\lambda}(\mathcal{S}_a, \mathcal{S}_c)$, derive a list of irreducible representations of $\boldsymbol{\mathcal{T}}_{\lambda}(\mathcal{S}_a, \mathcal{S}_c)$, and prove completeness of this list in some cases. Some of these representations originate from a ``quantization effect'', induced by the representation of $\mathcal{A}_{\lambda}^2(\mathbb{B}^n)$ as the direct sum of Bergman spaces over a lower dimensional unit ball with growing weight parameter. As an application we derive the essential spectrum and index formulas for matrix-valued operators