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

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