Uniform Continuity and Quantization on Bounded Symmetric Domains

TL;DR

This paper extends the semi-commutator relation for Toeplitz operators on weighted Bergman spaces over bounded symmetric domains to broader classes of symbols, including unbounded and discontinuous functions, using the Bergman metric.

Contribution

It generalizes the classical semi-commutator relation to uniformly continuous functions and VMO symbols on bounded symmetric domains, including unbounded and discontinuous cases.

Findings

Semi-commutator relation holds for uniformly continuous symbols on the domain.

The relation is valid for VMO symbols in the unit ball.

Counterexamples show failure for generic bounded measurable symbols.

Abstract

We consider Toeplitz operators $T_f^{\lambda}$ with symbol $f$ acting on the standard weighted Bergman spaces over a bounded symmetric domain $\Omega\subset \mathbb{C}^n$. Here $\lambda > genus-1$ is the weight parameter. The classical asymptotic semi-commutator relation $\lim_{\lambda \rightarrow \infty} \big{\|}T_f^{\lambda} T_g^{\lambda} -T_{fg}^{\lambda} \big{\|}=0$ with $f,g \in C(\overline{\mathbb{B}^n})$, where $\Omega=\mathbb{B}^n$ denotes the complex unit ball, is extended to larger classes of bounded and unbounded operator symbol-functions and to more general domains. We deal with operator symbols that generically are neither continuous inside $\Omega$ (Section 4) nor admit a continuous extension to the boundary (Section 3 and 4). Let $\beta$ denote the Bergman metric distance function on $\Omega$. We prove that the semi-commutator relation remains true for $f$ and $g$ in the space ${\rm UC}(\Omega)$ of all $\beta$-uniformly continuous functions on $\Omega$. Note that this space contains also unbounded functions. In case of the complex unit ball $\Omega=\mathbb{B}^n \subset \mathbb{C}^n$ we show that the semi-commutator relation holds true for bounded symbols in ${\rm VMO}(\mathbb{B}^n)$, where the vanishing oscillation inside $\mathbb{B}^n$ is measured with respect to $\beta$. At the same time the semi-commutator relation fails for generic bounded measurable symbols. We construct a corresponding counterexample using oscillating symbols that are continuous outside of a single point in $\Omega$.