Uniform Continuity and Quantization on Bounded Symmetric Domains

Abstract

We consider Toeplitz operators $T_f^λ$ with symbol $f$ acting on the standard weighted Bergman spaces over a bounded symmetric domain $Ω\subset \mathbb{C}^n$. Here $λ> genus-1$ is the weight parameter. The classical asymptotic semi-commutator relation $\lim_{λ\rightarrow \infty} \big{\|}T_f^λ T_g^λ -T_{fg}^λ \big{\|}=0$ with $f,g \in C(\overline{\mathbb{B}^n})$, where $Ω=\mathbb{B}^n$ 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 $Ω$ (Section 4) nor admit a continuous extension to the boundary (Section 3 and 4). Let $β$ denote the Bergman metric distance function on $Ω$. We prove that the semi-commutator relation remains true for $f$ and $g$ in the space ${\rm UC}(Ω)$ of all $β$-uniformly continuous functions on $Ω$. Note that this space contains also unbounded functions. In case of the complex unit ball $Ω=\mathbb{B}^n \subset \mathbb{C}^n$ 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 $β$. 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 $Ω$.