Reducing subspaces for the product of a forward and a backward operator-weighted shifts

Abstract

Abstract Let ( S 1 , S 2 ) be a pair of commuting operator-weighted shifts. We characterize the reducing subspaces of S 2 ⁎ S 1 or S 1 S 2 ⁎ as wandering subspaces with additional structures, and give a unified way to describe the reducing subspaces of Toeplitz operators induced by non-analytic monomial on weighted Hardy spaces of several variables, which including many classical function spaces, such as weighted Bergman space and Dirichlet space over the polydisk.