The reducibility of compressed shifts on a class of quotient modules over the bidisk

Abstract

In this paper, we show that, for the rational inner function $\theta (z,w)=\frac{zw+\overline{b}w+\overline{c}z+\overline{d}}{1+bz+cw+dzw}$, $S_z$ is reducible on the quotient module ${\mathcal{K}}_{\theta }=H^2\ominus \theta H^2$ over the bidisk if and only if $\theta$ is the product of two one-variable inner functions.