Alternative proofs of Mandrekar's theorem

TL;DR

This paper provides two new proofs of Mandrekar's theorem, characterizing invariant subspaces of the Hardy space on the bidisc as Beurling type under doubly commuting shifts, using Toeplitz operators and reproducing kernels.

Contribution

It introduces novel proofs of Mandrekar's theorem, expanding understanding of invariant subspaces in Hardy spaces through operator and kernel methods.

Findings

Two alternative proofs of Mandrekar's theorem are presented.

The proofs utilize Toeplitz operator properties and reproducing kernel techniques.

The characterizations of invariant subspaces are clarified through these methods.

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.