The reducibility of compressed shifts on Beurling type quotient modules over the bidisk

摘要

In this paper, we study the compressed shift operator S-z1, on the Beurling-type quotient module k(theta) of Hardy space H-2(D-2) over the bidisk. Firstly, we give a necessary and sufficient condition such that S z , has nontrivial pure isometry reducing subspace. As an application, we show that S-z1, has Agler reducing subspaces if and only if theta is the product of two one variable inner functions. Secondly, for a rational inner function with degree (n, 1), we show that S-z1, is reducible on k(theta) if and only if S-z1, has Agler reducing subspaces. Furthermore, we study the case when the rational inner functions have degree (n, 2), and this case is quite different from that the degree of theta is (n, 1). (C) 2019 Elsevier Inc. All rights reserved.