Uniform Continuity and Quantization on Bounded Symmetric Domains

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 $\lim_{λ\rightarrow \infty} \big{\|}T_f^λ T_g^λ -T_{fg}^λ \big{\|}=0$ with $f,g\in C(\overline{{\mathbb{B}}^n})$, where $\Omega ={\mathbb{B}}^n$ 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 $\mathrm{UC}(\Omega )$ 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 $\mathrm{VMO}({\mathbb{B}}^n)$, 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 $\Omega$.

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• Algebras of Toeplitz operators on the $n$-dimensional unit ball