Reducing Subspaces for Toeplitz Operator Tz1k1z2k2+az¯1l1z¯2l2 on the Weighted Hardy Space over the Bidisk

Abstract

In this paper, we completely characterize the reducing subspaces for Tφa on weighted Hardy space ℋω2D2 under three assumptions on ω, where φa=zk+az¯l, k,l∈ℕ2, k≠l, and a∈0,1. It is shown that the coefficient a∈0,1 does not affect the reducing subspaces for Tφa. We also prove that, for every δ>0, weighted Dirichlet space Dδ2D2 is a weighted Hardy space which satisfies these assumptions. As an application, we describe the reducing subspaces for Tφa on Dδ2D2 and get the structure of commutant algebra V∗Tφa.