Analytic continuation of Toeplitz operators and commuting families of $C^*-$algebras

TL;DR

This paper studies the analytic continuation of Toeplitz operators on weighted Bergman spaces over the unit ball, establishing their commutativity, spectral properties, and connections to convolution operators and representation theory.

Contribution

It introduces a novel approach to analytic continuation of Toeplitz operators, characterizes associated $C^*$-algebras via invariant symbols, and extends the Segal-Bargmann transform in this context.

Findings

Commutativity of $C^*$-algebras generated by analytic continuation of Toeplitz operators.

Spectral formulas for Toeplitz operators as convolution operators.

Extension of the Segal-Bargmann transform to the analytic continuation setting.

Abstract

We consider the Toeplitz operators on the weighted Bergman spaces over the unit ball $\mathbb{B}^n$ and their analytic continuation. We proved the commutativity of the $C^*-$algebras generated by the analytic continuation of Toeplitz operators with a special class of symbols that satisfy an invariant property, and we showed that these commutative $C^*-$algebras with symbols invariant under compact subgroups of $SU(n,1)$ are completely characterized in terms of restriction to multiplicity free representations. Moreover, we extended the restriction principal to the analytic continuation case for suitable maximal abelian subgroups of $SU(n,1)$, we obtained the generalized Segal-Bargmann transform and we showed that it acts as a convolution operator. Furthermore, we proved that Toeplitz operators are unitarly equivalent to a convolution operator and we provided integral formulas for their spectra.