Complex symmetric monomial Toeplitz operators on the unit ball

Abstract

Monomial Toeplitz operators Tzpz¯q on the weighted Bergman space of the unit ball in Cn are natural extension of the weighted shift operators. In higher dimensions, dim ker⁡Tzpz¯q=∞ may occur for non-trivial case, thus complex symmetric phenomena will appear. In this paper, we completely characterize when the monomial Toeplitz operator Tzpz¯q on the weighted Bergman space is JA-symmetric, where JAf(z)=f(Az‾)‾ with the symmetric unitary n×n matrix A. Moreover, we also determine all possible forms of A such that Tzpz¯q is JA-symmetric. As an application, we construct the weakly closed unital commutative algebra generated by complex symmetric Toeplitz operators.