Separately Radial and Radial Toeplitz Operators on the Unit Ball and Representation Theory

TL;DR

This paper demonstrates a representation theoretic approach to analyzing Toeplitz operators with radial symbols on the unit ball, providing a shorter and clearer method compared to previous analytic techniques.

Contribution

It introduces a purely representation theoretic method for studying Toeplitz operators, simplifying and clarifying the derivation of their unitary equivalence to multiplication operators.

Findings

Representation theoretic approach is effective for Toeplitz operators.

Simplifies the derivation of unitary equivalence.

Provides explicit formulas for operators.

Abstract

We study Toeplitz operators with separately radial and radial symbols on the weighted Bergman spaces on the unit ball. The unitary equivalence of such operators with multiplication operators on $\ell^2$ spaces was previously obtained by analytic methods elsewhere (see the references). We prove that the same constructions can be performed with a purely representation theoretic approach to obtain the same conclusions and formulas. However, our method is shorter, more elementary and more elucidating.