The Agler Reducing Subspace for the Operator-Valued Inner Function Over the Bidisk

In this paper, we study the Agler reducing subspace for the compressed shift on the Beurling type quotient module \(\mathcal {K}_{\Theta }\) over the bidisk, where \(\Theta \) is an operator-valued inner function. Firstly, we characterized the Agler reducing subspace when \(\Theta \) is an one variable operator-valued inner function, which is quite different with the scalar setting. Secondly, we show that the compressed shift has nontrivial Agler reducing subspaces if and only if \(\Theta \) is the product of two one variable inner fucntions(though now the order of the factors plays a role). The uniqueness of the Agler reducing subspace is also studied.