Beurling type theorem on the Bergman space via the Hardy space of the bidisk

Abstract

ABSTRACT. In this paper, by lifting the Bergman shift as the compression of an isometry on a subspace of the Hardy space of the bidisk, we give a proof of the Beurling type theorem on the Bergman space of Aleman, Richter and Sundberg [1] via the Hardy space of the bidisk.

Wandering subspaces and the Beurling type Theorem I

“…The following theorem is just a rewriting of an identity given in the proof of Theorem 3.1 in [11]. Here we give a simpler proof.…”

“…This result reveals the inside of the structure of invariant subspaces of the Bergman space and becomes a fundamental theorem in the function theory on L 2 a [4,6]. Later, different proofs of the Beurling type theorem are given in [8][9][10][11]. In [10], Shimorin proved the following theorem.…”

“…As an application of this theorem, Shimorin gave a simpler proof of the Aleman, Richter, and Sundberg theorem. In [11], Sun and Zheng gave another proof of this theorem. Their idea was to lift up the Bergman shift as the compression of a commuting pair of isometries on the subspace of the Hardy space over the bidisk.…”

“…Our idea of the proof comes from the one given by Sun and Zheng [11] essentially. Our proof is just rewriting their proof in the most elementary way.…”

Quasi-wandering Subspaces in the Bergman Space

“…In 1996, Aleman, Richter, and Sundberg [1] proved that [I BI] L 2 a = I. Different proofs of this theorem are given in [12,13]. The purpose of this paper is to study quasi-wandering subspaces in L 2 a .…”