Spectra and invariant subspaces of compressed shifts on nearly invariant subspaces

TL;DR

This paper explores the spectral properties and invariant subspaces of compressed shifts on nearly invariant subspaces, extending classical results to a broader, less restrictive setting using advanced operator theory techniques.

Contribution

It provides a complete characterization of spectra and invariant subspaces for compressed shifts on nearly $S^*$-invariant subspaces, a generalization of classical model space results.

Findings

Characterization of point and whole spectra for these operators

Identification of invariant subspace structures under weaker invariance conditions

Bridging classical model space theory with broader function-theoretic contexts

Abstract

While the spectral properties and invariant subspaces of compressed shifts on model spaces are well understood, their behaviour on nearly $S^*$-invariant subspaces, a natural generalization with weaker structural constraints, remains largely unexplored. These operators are closely related to the Clark-type unitary operators, yet differ from them in several ways. In this paper, we completely characterize the point spectrum, whole spectrum and invariant subspace structure for such compressed shifts by unitary equivalence, using the Frostman shift, Crofoot transform, and Sz.-Nagy--Foias theory. Our results reveal how the relaxation of $S^*$-invariance impacts spectral structure and invariant subspaces, bridging a gap between classical model space theory and broader function-theoretic settings.