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Spectra of infinitesimal generators of composition semigroups on weighted Bergman spaces induced by doubling weights

Ruishen Qian, Fanglei Wu, Hasi Wulan

TL;DR

The paper addresses the problem of describing the spectrum of the infinitesimal generator $\Gamma$ of a composition semigroup on the doubling-weighted Bergman space $A^p_\omega$ by elliptic semigroups. It combines a spectral mapping theorem with detailed spectra of composition operators, and establishes an eventual norm continuity framework to enable SMT-based spectral descriptions. Key contributions include explicit descriptions of the spectrum and point spectrum in terms of Koenigs data and spiral-like growth of the Koenigs image, a Carleson-measure–based criterion for eventual norm continuity, a resolvent representation $R(\lambda,\Gamma)$ when the Denjoy-Wolff point is $0$, and the introduction of the operator $R_h$ whose boundedness/compactness on $A^p_\omega$ is tied to the smoothness class of $\log\frac{h(z)}{z}$. These results extend known Hardy and standard Bergman space findings to doubling-weight settings and provide concrete tools for analyzing resolvents and spectral properties of composition semigroups.

Abstract

Suppose $(C_t)_{t\geq0}$ is the composition semigroup induced by a one-parameter semigroup $(\varphi_t)_{t\geq0}$ of analytic self-maps of the unit disk. The main purpose of the paper is to investigate the spectrum of the infinitesimal generator of $(C_t)_{t\geq0}$ acting on the weighted Bergman space induced by doubling weights, provided $(\varphi_t)_{t\geq0}$ is elliptic. The method applied is a certain spectral mapping theorem and a characterization of the spectra of certain composition operators. Eventual norm-continuity of $(C_t)_{t\geq0}$ also plays an important role, which can be depicted in terms of studying the difference of two distinct composition operators. As a byproduct, we also characterize a certain compact integral operator that is closely related to the resolvent of the infinitesimal generator of $(C_t)_{t\geq0}$.

Spectra of infinitesimal generators of composition semigroups on weighted Bergman spaces induced by doubling weights

TL;DR

The paper addresses the problem of describing the spectrum of the infinitesimal generator of a composition semigroup on the doubling-weighted Bergman space by elliptic semigroups. It combines a spectral mapping theorem with detailed spectra of composition operators, and establishes an eventual norm continuity framework to enable SMT-based spectral descriptions. Key contributions include explicit descriptions of the spectrum and point spectrum in terms of Koenigs data and spiral-like growth of the Koenigs image, a Carleson-measure–based criterion for eventual norm continuity, a resolvent representation when the Denjoy-Wolff point is , and the introduction of the operator whose boundedness/compactness on is tied to the smoothness class of . These results extend known Hardy and standard Bergman space findings to doubling-weight settings and provide concrete tools for analyzing resolvents and spectral properties of composition semigroups.

Abstract

Suppose is the composition semigroup induced by a one-parameter semigroup of analytic self-maps of the unit disk. The main purpose of the paper is to investigate the spectrum of the infinitesimal generator of acting on the weighted Bergman space induced by doubling weights, provided is elliptic. The method applied is a certain spectral mapping theorem and a characterization of the spectra of certain composition operators. Eventual norm-continuity of also plays an important role, which can be depicted in terms of studying the difference of two distinct composition operators. As a byproduct, we also characterize a certain compact integral operator that is closely related to the resolvent of the infinitesimal generator of .
Paper Structure (4 sections, 20 theorems, 134 equations)

This paper contains 4 sections, 20 theorems, 134 equations.

Table of Contents

  1. Introduction and main results
  2. Spectra of composition operators on $A^p_\omega$
  3. Eventually norm continuous composition semigroup
  4. A novel integral operator on $A^p_\omega$

Key Result

Theorem 1

Let $1\leq p<\infty$ and $\omega\in\widehat{\mathcal{D}}$. Suppose $(\varphi_t)_{t\geq0}$ is a semigroup of analytic self-maps of $\mathbb{D}$ with Denjoy-Wolff point $b\in\mathbb{D}$, infinitesimal generator $G$, and associated Koenigs function $h$. Denote by $\Gamma$ the infinitesimal generator of

Theorems & Definitions (36)

  • Theorem 1
  • Proposition 2
  • Corollary 3
  • Theorem 4
  • Theorem 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 26 more