Reducing Subspaces for Toeplitz Operator T z 1 k 1 z 2 k 2 + a z ¯ 1 l 1 z ¯ 2 l 2 on the Weighted Hardy Space over the Bidisk

In this paper, we completely characterize the reducing subspaces for T φ a on weighted Hardy space ℋ ω 2 D 2 under three assumptions on ω , where φ a = z k + a z ¯ l , k , l ∈ ℕ 2 , k ≠ l , and a ∈ 0,1 . It is shown that the coefficient a ∈ 0,1 does not affect the reducing subspaces for T φ a . We also prove that, for every δ > 0 , weighted Dirichlet space D δ 2 D 2 is a weighted Hardy space which satisfies these assumptions. As an application, we describe the reducing subspaces for T φ a on D δ 2 D 2 and get the structure of commutant algebra V ∗ T φ a .