Nearly invariant subspaces for shift semigroups

TL;DR

This paper investigates nearly invariant subspaces for shift semigroups on Hilbert spaces, linking these to Hardy space model spaces and Toeplitz operators, with new examples and structural insights.

Contribution

It introduces a definition of near invariance for $C_0$-semigroups, provides new examples of nearly invariant subspaces, and explores their connections with Hardy space model spaces.

Findings

Constructed prototypical nearly invariant subspaces for shift semigroups.

Linked nearly invariant subspaces to model spaces in Hardy spaces.

Analyzed the structure of closures of certain subspaces related to model spaces.

Abstract

Let $\{T(t)\}_{t\geq0}$ be a $C_0$-semigroup on an infinite dimensional separable Hilbert space; a suitable definition of near $\{T(t)^*\}_{t\geq0}$ invariance of a subspace is presented in this paper. A series of prototypical examples for minimal nearly $\{S(t)^*\}_{t\geq0}$ invariant subspaces for the shift semigroup $\{S(t)\}_{t\geq0}$ on $L^2(0,\infty)$ are demonstrated, which have close links with nearly $T_{\theta}^*$ invariance on Hardy spaces of the unit disk for Toeplitz operators associated with an inner function $\theta$. Especially, the corresponding subspaces on Hardy spaces of the right half-plane and the unit disk are related to model spaces. This work further includes a discussion on the structure of the closure of certain subspaces related to model spaces in Hardy spaces.