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Indexed by:Journal Papers

Date of Publication:2021-06-05

Journal:ACTA MATHEMATICA SINICA-ENGLISH SERIES

Volume:37

Issue:5

Page Number:805-824

ISSN No.:1439-8516

Key Words:Submodules; Beurling type theorem; compression operators; Fredholmness

Abstract:Let H-2(D-2) be the Hardy space over the bidisk D-2, and let M-phi = [(z - phi(w))(2)] be the submodule generated by (z - phi(w))(2), where phi(w) is a function in H-infinity(w). The related quotient module is denoted by N phi=H-2(D-2)circle minus M phi. In the present paper, we study the Fredholmness of compression operators S-z, S-w on N-phi. When phi(w) is a nonconstant inner function, we prove that the Beurling type theorem holds for the fringe operator F-w on [(z - w)(2)] circle minus z[(z - w)(2)] and the Beurling type theorem holds for the fringe operator F-z on M-phi circle minus wM(phi) if phi(0) = 0. Lastly, we study some properties of F-w on [(z - w(2))(2)] circle minus z[(z - w(2))(2)].