Complex symmetric monomial Toeplitz operators on the unit ball

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