Separately radial and radial Toeplitz operators on the unit ball and representation theory

Abstract

We study Toeplitz operators with radial and separately 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 in Grudsky et al. (J Oper Theory 49(2):325–346, 2003) and Quiroga-Barranco and Vasilevski (Integral Equ Oper Theory 59(1):67–98, 2007), respectively. 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.