Reducing Subspaces of Toeplitz Operators Induced by a Class of Non-analytic Monomials over the Unit Ball

Abstract

In this paper, we describe the minimal reducing subspaces of Toeplitz operators induced by non-analytic monomials on the weighted Bergman spaces and Dirichlet spaces over the unit ball \({\mathbb{B}_2}\). It is proved that each minimal reducing subspace M is finite dimensional, and if dim M ≥ 3, then M is induced by a monomial. Furthermore, the structure of commutant algebra \(\nu ({T_{\overline w {N_z}N}}): = {\{ M_{{w^N}}^ * {M_{{z^N}}},M_{{z^N}}^ * {M_{{w^N}}}\} ^\prime }\) is determined by N and the two dimensional minimal reducing subspaces of \({T_{\overline w {N_z}N}}\). We also give some interesting examples.