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Cyclic nearly invariant subspaces for semigroups of isometries

Yuxia Liang, Jonathan R. Partington

Abstract

In this paper, the structure of the nearly invariant subspaces for discrete semigroups generated by several (even infinitely many) automorphisms of the unit disc is described. As part of this work, the near $S^*$-invariance property of the image space $C_\varphi({\rm ker\, } T)$ is explored for composition operators $C_\varphi$, induced by inner functions $\varphi$, and Toeplitz operators $T$. After that, the analysis of nearly invariant subspaces for strongly continuous multiplication semigroups of isometries is developed with a study of cyclic subspaces generated by a single Hardy class function. These are characterised in terms of model spaces in all cases when the outer factor is a product of an invertible function and a rational (not necessarily invertible) function. Techniques used include the theory of Toeplitz kernels and reproducing kernels.

Cyclic nearly invariant subspaces for semigroups of isometries

Abstract

In this paper, the structure of the nearly invariant subspaces for discrete semigroups generated by several (even infinitely many) automorphisms of the unit disc is described. As part of this work, the near -invariance property of the image space is explored for composition operators , induced by inner functions , and Toeplitz operators . After that, the analysis of nearly invariant subspaces for strongly continuous multiplication semigroups of isometries is developed with a study of cyclic subspaces generated by a single Hardy class function. These are characterised in terms of model spaces in all cases when the outer factor is a product of an invertible function and a rational (not necessarily invertible) function. Techniques used include the theory of Toeplitz kernels and reproducing kernels.
Paper Structure (6 sections, 31 theorems, 104 equations)

This paper contains 6 sections, 31 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Near invariance for discrete semigroups and related questions
  3. Nearly $\{T_{\psi_1^m\psi_2^n}^{-1}:\;m,n\in \mathbb{N}_0\}$ invariant subspaces
  4. Properties of $C_\varphi(\mathop{\rm ker}\nolimits T)$ for inner functions $\varphi$ and Toeplitz kernels $\mathop{\rm ker}\nolimits T$.
  5. the subspace $N(1/(1+s)^{n+1})$ in $H^2(\mathbb{C}_+)$
  6. the subspace $N(g)$ with a rational outer function $g$

Key Result

Theorem 1.1

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

Theorems & Definitions (54)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 2.1
  • proof
  • Corollary 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Remark 2.5
  • Proposition 2.6
  • proof
  • ...and 44 more