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Trace ideal criteria for generalized integration operators

Georgios Nikolaidis

TL;DR

This work characterizes when the generalized Volterra operator $T_{g,a}$ lies in the Schatten classes $S^p(A^2_\alpha)$ on Hardy and Bergman spaces. By introducing the index $\lambda$ from the order-$n$ operator and employing hyperbolic geometry, Littlewood–Paley theory, and a Luecking-type framework, the authors prove that $T_{g,a}\in S^p(A^2_\alpha)$ if and only if the symbol $g$ belongs to the Besov space $\mathcal{B}_p$ whenever $p>\frac{1}{n-\lambda}$; for $0<p\le\frac{1}{n-\lambda}$, the operator is zero. The argument carefully handles term cancellations and reduces Schatten-norm estimates to Besov regularity of $g$, extending the classical $T_g$ results (the case $\lambda=n-1$) to the broader family $T_{g,a}$ on both Hardy and Bergman spaces. The results sharpen the understanding of factorization-type operators in spaces of analytic functions and provide explicit, sharp criteria for Schatten-class membership in terms of symbol smoothness.

Abstract

For $g \in \operatorname{Hol}(\mathbb D)$, we study the class of generalized integration operators $T_{g,a}$, acting on Hardy and Bergman spaces of the unit disc in the complex plane. This class of integral operators were introduced to study factorization theorems in the Hardy spaces of the unit disc. We completely characterize the space of symbols g, for which $T_{g,a}$ belongs to the Schatten-von Neumann ideals of the Hardy and Bergman spaces.

Trace ideal criteria for generalized integration operators

TL;DR

This work characterizes when the generalized Volterra operator lies in the Schatten classes on Hardy and Bergman spaces. By introducing the index from the order- operator and employing hyperbolic geometry, Littlewood–Paley theory, and a Luecking-type framework, the authors prove that if and only if the symbol belongs to the Besov space whenever ; for , the operator is zero. The argument carefully handles term cancellations and reduces Schatten-norm estimates to Besov regularity of , extending the classical results (the case ) to the broader family on both Hardy and Bergman spaces. The results sharpen the understanding of factorization-type operators in spaces of analytic functions and provide explicit, sharp criteria for Schatten-class membership in terms of symbol smoothness.

Abstract

For , we study the class of generalized integration operators , acting on Hardy and Bergman spaces of the unit disc in the complex plane. This class of integral operators were introduced to study factorization theorems in the Hardy spaces of the unit disc. We completely characterize the space of symbols g, for which belongs to the Schatten-von Neumann ideals of the Hardy and Bergman spaces.
Paper Structure (8 sections, 7 theorems, 70 equations)

This paper contains 8 sections, 7 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hyperbolic geometry
  4. Hilbert Spaces of analytic functions
  5. Schatten-von Neumann ideals
  6. Some further lemmas
  7. Proof of Theorem \ref{['Theorem of paper Schatten class membership']} for
  8. Proof of Theorem \ref{['Theorem of paper Schatten class membership']} for

Key Result

Theorem 1.1

Let $\alpha\geq -1$, $n\in\mathbb{N}$, $a=(a_0,\dots,a_{n-1})\in\mathbb{C}^n\setminus\{\mathbf{0}\}$, $\lambda=\max\{k\colon a_k\neq 0\}$, $g\in \mathop{\mathrm{Hol}}\nolimits(\mathbb{D})$, and $0<p<\infty$.

Theorems & Definitions (9)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • proof : Proof of Theorem \ref{['Theorem of paper Schatten class membership']} for $2\leq p<\infty$
  • proof : Proof of Theorem \ref{['Theorem of paper Schatten class membership']} for