Products of Toeplitz Operators on a Vector Valued Bergman Space

TL;DR

This paper characterizes when Toeplitz products with matrix symbols are bounded or invertible on vector valued Bergman spaces, extending scalar results using Berezin transform techniques.

Contribution

It provides necessary and sufficient conditions for boundedness and invertibility of Toeplitz products with matrix symbols on vector valued Bergman spaces, generalizing scalar cases.

Findings

Derived conditions for boundedness of Toeplitz products with matrix symbols.

Characterized invertibility of Toeplitz products via Berezin transform.

Extended scalar results to vector valued Bergman spaces.

Abstract

We give a necessary and a sufficient condition for the boundedness of the Toeplitz product $T_FT_{G^*}$ on the vector valued Bergman space $L_a^2(\mathbb{C}^n)$, where $F$ and $G$ are matrix symbols with scalar valued Bergman space entries. The results generalize existing results in the scalar valued Bergman space case. We also characterize boundedness and invertibility of Toeplitz products $T_FT_{G^*}$ in terms of the Berezin transform, generalizing results found by Zheng and Stroethoff for the scalar valued Bergman space.