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Linear isometries on the annulus: description and spectral properties

Isabelle Chalendar, Lucas Oger, Jonathan R. Partington

TL;DR

The paper advances the understanding of linear isometries on spaces of holomorphic functions by establishing that, on annuli symmetric to the unit circle, all linear isometries of $\mathrm{Hol}(\mathbb{A})$ are rotations or inversions, realizable as $T_{\alpha,\beta}$ or $S_{\alpha,\beta}$. It develops a robust framework connecting isometries to weighted composition operators via seminorms $\|\cdot\|_{\infty,n}$ and leverages Hadamard-type results and unimodular boundary factorisations to describe the admissible maps on the annulus. A central contribution is the precise spectral description of these isometries: for $S_{\alpha,\beta}$ the spectrum is limited to $\{\alpha, -\alpha\}$ with infinite-dimensional eigenspaces, while for $T_{\alpha,\beta}$ the spectrum depends on whether $\beta$ is periodic; in the aperiodic case the spectrum can strictly contain the point spectrum, as demonstrated by Diophantine constructions. The results couple structural rigidity with nuanced spectral phenomena, enriching the theory of composition-operator isometries on non-simply connected domains and offering new factorisation tools for unimodular boundary values.

Abstract

We give a complete characterisation of the linear isometries of ${\rm Hol}(Ω)$, where $Ω$ is the half-plane, the complex plane or an annulus centered at 0 and symmetric to the unit circle. Moreover, we introduce new techniques to describe the holomorphic maps on the annulus that preserve the unit circle, and we finish by proving results about the spectra of the linear isometries on the annulus.

Linear isometries on the annulus: description and spectral properties

TL;DR

The paper advances the understanding of linear isometries on spaces of holomorphic functions by establishing that, on annuli symmetric to the unit circle, all linear isometries of are rotations or inversions, realizable as or . It develops a robust framework connecting isometries to weighted composition operators via seminorms and leverages Hadamard-type results and unimodular boundary factorisations to describe the admissible maps on the annulus. A central contribution is the precise spectral description of these isometries: for the spectrum is limited to with infinite-dimensional eigenspaces, while for the spectrum depends on whether is periodic; in the aperiodic case the spectrum can strictly contain the point spectrum, as demonstrated by Diophantine constructions. The results couple structural rigidity with nuanced spectral phenomena, enriching the theory of composition-operator isometries on non-simply connected domains and offering new factorisation tools for unimodular boundary values.

Abstract

We give a complete characterisation of the linear isometries of , where is the half-plane, the complex plane or an annulus centered at 0 and symmetric to the unit circle. Moreover, we introduce new techniques to describe the holomorphic maps on the annulus that preserve the unit circle, and we finish by proving results about the spectra of the linear isometries on the annulus.
Paper Structure (8 sections, 15 theorems, 61 equations)

This paper contains 8 sections, 15 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. First results
  3. Algebraic characterisation of composition operators
  4. From distances to seminorms
  5. Three useful characterisations
  6. Main result
  7. Description of unimodular functions on the unit circle
  8. Spectral study of linear isometries on the annulus

Key Result

Lemma 2.1

Let $L : \mathop{\rm Hol}\nolimits(\mathbb{A}) \to \mathbb{C}$ be non-zero, linear, continuous and multiplicative. Then there exists $z_0 \in \mathbb{A}$ such that for all $f \in \mathop{\rm Hol}\nolimits(\mathbb{A})$, $L(f) = f(z_0)$.

Theorems & Definitions (26)

  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Lemma 2.3: COP
  • Lemma 2.4: COP
  • Theorem 3.1
  • Corollary 3.2
  • proof
  • Theorem 3.3
  • proof
  • ...and 16 more