Algebraic Properties of Toeplitz Operators on the Symmetrized Polydisk

Abstract

This paper is an effort to continue the legacy of the classically successful theory of Toeplitz operators on the Hardy space over the unit disk to a new domain in \({\mathbb {C}}^d\)—the symmetrized polydisk. We obtain algebraic characterizations of Toeplitz operators, analytic Toeplitz operators, compact perturbation of Toeplitz operators and dual Toeplitz operators. We then revisit the operator theory of this domain considered first in Biswas and Shyam Roy (J Funct Anal 266:6224–6255, 2014), to study the generalized Toeplitz operators and find a commutant lifting type result.