Uniform Continuity and Quantization on Bounded Symmetric Domains

Wolfram Bauer, Raffael Hagger, Nikolai Vasilevski

Published 2016-11-28Version 1

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 $\Omega$.