Reducing subspaces of non-analytic Toeplitz operators on weighted Hardy and Dirichlet spaces of the bidisk

We characterize reducing subspaces of a class of non-analytic Toeplitz operators on Hilbert spaces of holomorphic functions on the bidisk including the Hardy space and the Dirichlet space of the bidisk. These results compliment a recent characterization of the reducing subspaces for this class of non-analytic Toeplitz operators on the Bergman space of the bidisk.