Invariance and near invariance for non-cyclic shift semigroups

TL;DR

This paper characterizes subspaces of the Hardy space that are invariant or nearly invariant under certain non-cyclic shift operators and their adjoints, extending classical results to higher order shifts.

Contribution

It provides a comprehensive description of invariant and nearly invariant subspaces for higher order shift operators and Toeplitz operators induced by Blaschke products.

Findings

Characterization of subspaces invariant under $S^2$ and $S^{2k+1}$.

Identification of nearly invariant subspaces under $(S^2)^*$ and $(S^{2k+1})^*$.

Extension to higher order shifts and Toeplitz operators with Blaschke products.

Abstract

This paper characterises the subspaces of $H^2(\mathbb D)$ simultaneously invariant under $S^2 $ and $S^{2k+1}$, where $S$ is the unilateral shift, then further identifies the subspaces that are nearly invariant under both $(S^2)^*$ and $(S^{2k+1})^*$ for $k\geq 1$. More generally, the simultaneously (nearly) invariant subspaces with respect to $(S^m)^*$ and $(S^{km+\gamma})^*$ are characterised for $m\geq 3$, $k\geq 1$ and $\gamma\in \{1,2,\cdots, m-1\},$ which leads to a description of (nearly) invariant subspaces with respect to higher order shifts. Finally, the corresponding results for Toeplitz operators induced by a Blaschke product are presented. Techniques used include a refinement of Hitt's algorithm, the Beurling--Lax theorem, and matrices of analytic functions.