The algebra generated by Toeplitz operators with sums of coordinate powers on the Hardy space over the bidisk

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.