Abstract
In this paper, we study the compressed shift operator Sz1 on the Beurling-type quotient module Kθ of Hardy space H2(D2) over the bidisk. Firstly, we give a necessary and sufficient condition such that Sz1 has nontrivial pure isometry reducing subspace. As an application, we show that Sz1 has Agler reducing subspaces if and only if θ is the product of two one variable inner functions. Secondly, for a rational inner function with degree (n,1), we show that Sz1 is reducible on Kθ if and only if Sz1 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 θ is (n,1).
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