Toeplitz Operators on Dirichlet-Type Space of Unit Ball

Abstract

We construct a function u in L2Bn, dV which is unbounded on any neighborhood of each boundary point of Bn such that Toeplitz operator Tu is a Schatten p-class 0<p<∞ operator on Dirichlet-type space DBn, dV. Then, we discuss some algebraic properties of Toeplitz operators with radial symbols on the Dirichlet-type space DBn, dV. We determine when the product of two Toeplitz operators with radial symbols is a Toeplitz operator. We investigate the zero-product problem for several Toeplitz operators with radial symbols. Furthermore, the corresponding commuting problem of Toeplitz operators whose symbols are of the form ξku is studied, where k ∈ Zn, ξ ∈ ∂Bn, and u is a radial function.