Commuting Toeplitz Operators on the 2-analytic Bergman Space

Abstract

On the classical Bergman space, Toeplitz operators with radial symbols are diagonal and those operators commute. However, on the n-analytic Bergman space \(A_{n}^{2}(\mathbb D)\) when n ≥ 2, the case is different. In this paper, our focus is on the problem of commuting Toeplitz operators with quasiho-mogeneous symbols, specifically in the context of the function space \(A_{2}^{2}(\mathbb D)\). We show a kind of block matrice expression of Toeplitz operators on \(A_{2}^{2}(\mathbb D)\). Based on the block expression, we give several important properties. Our results indicate that in some cases, two Toeplitz operators are commutative if and only if both operators are analytic or differ by a constant multiple.