We study Toeplitz operators Tₚ on the Hardy space over the bidisk with symbols of the form p (z, w) =zᵏ+wˡ. Although the problem of describing reducing subspaces of Toeplitz operators is infinite-dimensional and analytic in nature, we show that in this case it can be completely reduced to a finite-dimensional algebraic structure. More precisely, we prove that the von Neumann algebra V^* (p) generated by Tₚ is -isomorphic to M₊₋ (C) M₊₋ (C). As a consequence, we obtain a complete classification of the minimal reducing subspaces of Tₚ, which are explicitly described in terms of polynomial data. This paper is currently under review.
Shuhei Kuwahara (Sat,) studied this question.