Reducing subspaces for a class of non-analytic Toeplitz operators on the bidisk

In this paper, we completely characterize all the reducing subspaces for a class of non-analytic Toeplitz operators with symbol φ(z,w)=αzk+βw‾l, where α,β∈Cα,β∈C and αβ≠0αβ≠0. We also prove that the von Neumann algebra V⁎(φ)={Tφ,Tφ⁎}′ is abelian.