Nearly invariant brangesian subspaces

TL;DR

The paper addresses the problem of describing nearly invariant Brangesian subspaces that are contractively contained in reproducing kernel Hilbert spaces of analytic functions on the unit disk, extending Hitt's theorem from $H^2(\mathbb{D})$ to de Branges spaces and related RKHS settings. It develops three core frameworks: (i) a vector-valued de Branges generalization via a $Gf$-based representation with an $S^*$-invariant subspace $\mathcal{N}$ (Theorem MTH), (ii) an Era-type factorization for RKHS with an inner divisor $\phi$ (Theorem MTRKHS), and (iii) a finite-defect extension that introduces a defect space $\mathcal{F}$ of dimension $p$ and yields a model in $H^2(\mathbb{D},\ell^2(I)\oplus \mathbb{C}^p)$. The main results provide explicit constructions of isometric correspondences, norm inequalities $||h||_{\mathcal{M}}\ge ||f||_2$ (and its defect variants), and invariant-operator descriptions, while showing that equality of norms and closedness of the associated $S^*$-invariant subspaces can fail in general. By linking Hitt, Era, and Liang–Partington within the Brangesian RKHS framework, the work broadens the scope of nearly invariant subspace theory to vector-valued, inner-function-division, and finite-defect contexts with potential implications for operator theory and function theory on the disk.

Abstract

This article describes Hilbert spaces contractively contained in certain reproducing kernel Hilbert spaces of analytic functions on the open unit disc which are nearly invariant under division by an inner function. We extend Hitt's theorem on nearly invariant subspaces of the backward shift operator on $H^2(\bb D)$ as well as its many generalizations to the setting of de Branges spaces.

Theorems & Definitions (27)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Theorem 3.1
• Proposition 3.2