Toeplitz algebras over Fock and Bergman spaces

TL;DR

This paper investigates Toeplitz algebras on p-Fock and p-Bergman spaces, showing they are generated by Toeplitz operators with symbols in certain translation invariant subalgebras of bounded uniformly continuous functions, extending previous results.

Contribution

It demonstrates that Toeplitz algebras on these spaces are linearly generated by Toeplitz operators with symbols in specific subalgebras, answering open questions and generalizing known results.

Findings

Toeplitz algebra on p-Fock space generated by translation invariant symbols

Toeplitz algebra on p-Bergman space equals the linear span of Toeplitz operators with such symbols

Generalizes previous results for p=2 to all 1<p<∞

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}.