Reducing subspaces of a Toeplitz operator with some symmetric polynomial symbol on the Hardy space

This paper aims to provide a complete description of the minimal reducing subspaces of Toeplitz operators $T_p$ with symbol of the form $p(z,w) = z^k + w^l$ on the Hardy space over the bidisk. The study of reducing subspaces of Toeplitz operators has attracted considerable attention from mathematicians. In this paper, we first determine the minimal reducing subspaces of the Toeplitz operator $T_{z+w}$ and then generalize these results to the more general form $T_{z^k+w^l}$. Specifically, we show that the minimal reducing subspaces are determined. From these results, we demonstrate that the von Neumann algebra $V^{\ast}(p)$ associated with the operator $T_p$ is $\ast$-isomorphic to $M_{kl}(C) \oplus M_{kl}(C)$. The title of the submitted version has been slightly modified.