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Spectra and invariant subspaces of compressed shifts on nearly invariant subspaces

Y. Liang, J. R. Partington

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.

Spectra and invariant subspaces of compressed shifts on nearly invariant subspaces

Abstract

While the spectral properties and invariant subspaces of compressed shifts on model spaces are well understood, their behaviour on nearly -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 -invariance impacts spectral structure and invariant subspaces, bridging a gap between classical model space theory and broader function-theoretic settings.

Paper Structure

This paper contains 4 sections, 18 theorems, 100 equations.

Table of Contents

  1. Preliminaries
  2. The compressed shifts on nearly invariant subspaces
  3. Spectra of the compressed shifts on nearly invariant subspaces
  4. Invariant subspaces of the compressed shifts on nearly invariant subspaces

Key Result

Theorem 1.1

hitt The nearly $S^*$-invariant subspaces of $H^2$ have the form $\mathcal{M}=hK$, with $h\in \mathcal{M}$ of unit norm, $h(0)>0,$$h$ orthogonal to all elements of $\mathcal{M}$ vanishing at the origin, $K$ an $S^*$-invariant subspace, and the operator of multiplication by $h$ isometric from $K$ int

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2
  • proof
  • Corollary 1.3
  • Corollary 1.4
  • proof
  • Remark 1.5
  • proof
  • Proposition 2.1
  • Example 2.2
  • ...and 22 more