Toeplitz algebras over Fock and Bergman spaces

Abstract

In this paper, we study Toeplitz algebras generated by certain class of Toeplitz operators on the $p$-Fock space and the $p$-Bergman space with $1<p<\infty$. Let BUC(${\mathbb{C}}^n$) and BUC(${\mathbb{B}}_n$) denote the collections of bounded uniformly continuous functions on ${\mathbb{C}}^n$ and ${\mathbb{B}}_n$ (the unit ball in ${\mathbb{C}}^n$), respectively. On the $p$-Fock space, we show that the Toeplitz algebra which has a translation invariant closed subalgebra of BUC(${\mathbb{C}}^n$) as its set of symbols is linearly generated by Toeplitz operators with the same space of symbols. This answers a question recently posed by Fulsche \cite{Robert}. On the $p$-Bergman space, we study Toeplitz algebras with symbols in some translation invariant closed subalgebras of BUC(${\mathbb{B}}_n)$. In particular, we obtain that the Toeplitz algebra generated by all Toeplitz operators with symbols in BUC(${\mathbb{B}}_n$) is equal to the closed linear space generated by Toeplitz operators with such symbols. This generalizes the corresponding result for the case of $p=2$ obtained by Xia \cite{Xia2015}.

38 pages

References in corpus (1)

• Correspondence theory on $p$-Fock spaces with applications to Toeplitz algebras