Hausdorff operators on Fock Spaces

TL;DR

This paper investigates the properties of Hausdorff operators on Fock spaces, establishing their norm, compactness criteria, and Schatten class membership based on the measure a, providing a comprehensive operator analysis.

Contribution

It provides a complete characterization of the norm, compactness, and Schatten class membership of Hausdorff operators on Fock spaces, extending operator theory in complex analysis.

Findings

Operator norm equals the supremum of the moment sequence.

Compactness characterized by the decay of the moment sequence.

Schatten class membership determined by the summability of the moment sequence.

Abstract

Let $\mu$ be a positive Borel measure on the positive real axis. We study the integral operator $$ \mathcal{H}_{\mu}(f)(z)=\int_{0}^{\infty}\frac{1}{t}f\left(\frac{z}{t}\right)\,d\mu(t),\quad z\in \mathbb{C}\,, $$ acting on the Fock spaces $F^{p}_{\alpha}$, $p\in [1,\infty],\,\alpha >0$. Its action is easily seen to be a coefficient multiplication by the moment sequence $$ \mu_n= \int_{1}^{\infty}\frac{1}{t^{n+1}}\,d\mu(t) . $$ We prove that \begin{equation*} ||\mathcal{H}_{\mu}||_{F^{p}_{\alpha}\to F^{p}_{\alpha}}=\sup_{n\in\mathbb{N}}\mu_n,\,\,\,\,\,1\leq p\leq \infty\,\,. \end{equation*} A little-o,condition describes the compactness of $\mathcal{H}_{\mu}$ on every $F^{p}_{\alpha},\,p\in (1,\infty )$. In addition, we completely characterize the Schatten class membership of $\mathcal{H}_{\mu}$.