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Near Invariance of The Dual Compressed Shift

Arup Chattopadhyay, Supratim Jana

TL;DR

This work analyzes the nearly dual compressed shift-invariant subspaces of the orthogonal complement $K_ heta^ot$ of model spaces in the Hardy space $H^2$, introducing a nearly $U_ heta$-invariant framework and deriving a concrete structure via Hitt's algorithm. It provides a detailed classification when the inner function satisfies $ heta_0=0$, showing that any nontrivial nearly $U_ heta$-invariant subspace of the form $ ext{M}= ext{M}_-oxplus ext{M}_+$ decomposes into explicit Beurling-type components, with $ ext{M}_-$ and $ ext{M}_+$ drawn from $ar{H_0^2}$ and $ heta H^2$ respectively, and extending these ideas to the case $ heta_0 eq0$ where invariant subspace forms for $U_ heta$ and $U_ heta^*$ are identified. The paper further broadens the scope by applying the near-invariance concept to operators like $T^*_{B_n}=T_{ar{B_n}}$ for finite Blaschke products, obtaining a Hitt-type description and linking to nearly $T^{-1}$-invariant subspaces. Overall, the results deepen the understanding of near-invariance in model spaces, connect to truncated Toeplitz operator theory, and provide algorithmic descriptions that may inform further studies of invariant subspace structures in related settings.

Abstract

We present the notion of the nearly dual compressed shift-invariant subspaces of the orthogonal complement of the model space and obtain their structure using Hitt's algorithm \cite{DH}.

Near Invariance of The Dual Compressed Shift

TL;DR

This work analyzes the nearly dual compressed shift-invariant subspaces of the orthogonal complement of model spaces in the Hardy space , introducing a nearly -invariant framework and deriving a concrete structure via Hitt's algorithm. It provides a detailed classification when the inner function satisfies , showing that any nontrivial nearly -invariant subspace of the form decomposes into explicit Beurling-type components, with and drawn from and respectively, and extending these ideas to the case where invariant subspace forms for and are identified. The paper further broadens the scope by applying the near-invariance concept to operators like for finite Blaschke products, obtaining a Hitt-type description and linking to nearly -invariant subspaces. Overall, the results deepen the understanding of near-invariance in model spaces, connect to truncated Toeplitz operator theory, and provide algorithmic descriptions that may inform further studies of invariant subspace structures in related settings.

Abstract

We present the notion of the nearly dual compressed shift-invariant subspaces of the orthogonal complement of the model space and obtain their structure using Hitt's algorithm \cite{DH}.
Paper Structure (7 sections, 16 theorems, 27 equations)

This paper contains 7 sections, 16 theorems, 27 equations.

Table of Contents

  1. Abstract
  2. Keywords
  3. Mathematics Subject Classification (2020)
  4. Introduction and Preliminaries
  5. Facts on invariance of the dual compressed shift
  6. Main Results
  7. Near invariance of some other operators

Key Result

Lemma 2.1

For $h\in K_\theta^\perp,$ we have where $\theta_0$ stands for both $\theta(0)$ (when $\theta$ is considered to be analytic on $\mathbb{D}$) and the $0^{th}$ Fourier coefficient (when $\theta$ is considered as a boundary function) of the inner function $\theta.$ Note that, for a Hilbert space $\mathcal{H}$, and for $f, g\in \mathcal{

Theorems & Definitions (29)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Lemma 2.1: Lemma 2.1 and 2.2 CG1
  • Theorem 2.2
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • Theorem 2.5
  • proof
  • ...and 19 more