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

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Indexed by:Journal Papers

Date of Publication:2020-01-01

Journal:JOURNAL OF FUNCTIONAL ANALYSIS

Included Journals:SCIE

Volume:278

Issue:1

ISSN No.:0022-1236

Key Words:Beurling type quotient module; Compressed shift operator; Reducibility

Abstract: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.

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