Alternative proofs of Mandrekar's theorem

2022/09/29 by Linus Bergqvist, Bergqvist, Linus
Mathematics · #32A10 #32A35 #46E22 #Advanced Harmonic Analysis Research #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2209.14747

openalex publication_date 2022/09/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We present two alternative proofs of Mandrekar's theorem, which states that an invariant subspaces of the Hardy space on the bidisc is of Beurling type precisely when the shifts satisfy a doubly commuting condition. The first proof uses properties of Toeplitz operators to derive a formula for the reproducing kernel of certain shift invariant subspaces, which can then be used to characterize them. The second proof relies on the reproducing property in order to show that the reproducing kernel at the origin must generate the entire shift invariant subspace.

Related

• A Beurling Theorem for almost-invariant subspaces of the shift operator
• Beurling Criteria for Reproducing Kernels
• A Beurling Theorem for Generalized Hardy Spaces on a Multiply Connected Domain
• Variations on a theme of Beurling
• Nearly invariant subspaces and kernels of Toeplitz operators on the Hardy space over the bidisk