Reducing subspaces for a class of nonanalytic Toeplitz operators

Abstract

In this paper, we give a uniform characterization for the reducing subspaces for $T_{\phi }$ with the symbol $\phi \left(z\right)=z^k+{\overline{z}}^l$ ($k,l\in {\mathbb{Z}}_{+}^2$) on the Bergman spaces over the bidisk, including the known cases that $\phi (z_1,z_2)=z_1^N{z}_2^M$ and $\phi (z_1,z_2)=z_1^N+{\overline{z}}_2^M$ with $N,M\in {\mathbb{Z}}_{+}$. Meanwhile, the reducing subspaces for $T_{z^N+{\overline{z}}^M}$ ($N,M\in {\mathbb{Z}}_{+}$) on the Bergman space over the unit disk are also described. Finally, we state these results in terms of the commutant algebra ${\mathcal{V}}^{\ast }\left(\phi \right)$.