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

In this paper, we completely characterize the reducing subspaces for T phi a on weighted Hardy space Script capital H omega 2D2 under three assumptions on omega, where phi a=zk+az over bar l, k,l & ISIN;N2, k & NOTEQUAL;l, and a & ISIN;0,1. It is shown that the coefficient a & ISIN;0,1 does not affect the reducing subspaces for T phi a. We also prove that, for every delta > 0, weighted Dirichlet space D delta 2D2 is a weighted Hardy space which satisfies these assumptions. As an application, we describe the reducing subspaces for T phi a on D delta 2D2 and get the structure of commutant algebra V*T phi a.