New York Journal of Mathematics
Volume 16 (2010) 489-505


Kei Ji Izuchi, Kou Hei Izuchi, and Yuko Izuchi

Wandering subspaces and the Beurling type theorem. II

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Published: November 22, 2010
Keywords: Beurling type theorem, Hardy space over the bidisk, invariant subspace, wandering subspace, fringe operator
Subject: Primary 47A15, 32A35; Secondary 47B35.

Let H2 be the Hardy space over the bidisk. Let φ(w) be a nonconstant inner function. We denote by [z-φ(w)] the smallest invariant subspace for both operators Tz and Tw containing the function z-φ(w). Aleman, Richter and Sundberg showed that the Beurling type theorem holds for the Bergman shift on the Bergman space. It is known that the compression operator Sz on H2⊝[z-w] is unitarily equivalent to the Bergman shift, so the Beurling type theorem holds for Sz on H2⊝[z-w]. As a generalization, we shall show that the Beurling type theorem holds for Sz on H2⊝[z-φ(w)]. Also we shall prove that the Beurling type theorem holds for the fringe operator Fw on [z-w]⊝z[z-w] and for Fz on [z-φ(w)]⊝w[z-φ(w)] if φ(0)=0.


The first author is partially supported by Grant-in-Aid for Scientific Research (No.21540166), Japan Society for the Promotion of Science.

Author information

Kei Ji Izuchi:
Department of Mathematics, Niigata University, Niigata 950-2181, Japan

Kou Hei Izuchi:
Department of Mathematics, Korea University, Seoul 136-701, Korea

Yuko Izuchi:
Aoyama-shinmachi 18-6-301, Niigata 950-2006, Japan