The Beurling-Lax-Halmos Theorem for almost invariant subspaces

Release time:2025-12-13

Journal:Submitted

Abstract:In this paper we establish the Beurling-Lax-Halmos Theorem for almost invariant subspaces. Let $S_{E}$ be the shift operator on vector-valued Hardy space $H_{E}^{2}.$ The Beurling-Lax-Halmos Theorem identifies the invariant subspaces of $S_{E}.$ Recently, almost invariant subspaces of $S_{E}^{\ast}$ are studied by the approach of nearly invariant subspaces with finite defect in the scalar-valued case ($\dim E=1$) by Chalendar-Gallardo-Partington and in vector-valued case ($\dim E<\infty$) by Chalendar-Chevrot-Partington, Chattopadhyay-Das-Pradhan, and independently by O'Loughlin. It was also known that when $\dimE<\infty,$ a closed subspace $M$ of $H_{E}^{2}$ is $S_{E}$-almost invariant if and only if $M$ is $S_{E}^{\ast}$-almost invariant. A simple example shows a $S_{E}$-invariant subspace is not $S_{E}^{\ast}$-almost invariant when $\dim E=\infty$. In this paper, we characterize $S_{E}^{\ast}% $-almost invariant subspaces by a direct approach. Our result simplifies previous results when $\dim E<\infty,$ and more importantly, our result holds when $\dim E=\infty.$

Co-author:Caixing Gu, In Sung Hwang, Woo Young Lee, Pan Ma

Indexed by:Journal paper

Discipline:Natural Science

First-Level Discipline:Mathematics

Document Type:J

Translation or Not:no