Hausdorff operators on Fock Spaces

Abstract

Let $\mu$ be a positive Borel measure on the positive real axis. We study the integral operator $${\mathcal{H}}_{\mu }(f)(z)={\int }_0^{\infty }\frac{1}{t}f\left(\frac{z}{t}\right)d\mu (t),z\in \mathbb{C},$$ acting on the Fock spaces $F_{\alpha }^p$, $p\in [1,\infty ],\alpha >0$. 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}_μ||_{F^{p}_α\to F^{p}_α}=\sup_{n\in\mathbb{N}}μ_n,\,\,\,\,\,1\leq p\leq \infty\,\,. \end{equation*} A little-o,condition describes the compactness of ${\mathcal{H}}_{\mu }$ on every $F_{\alpha }^p,p\in (1,\infty )$. In addition, we completely characterize the Schatten class membership of ${\mathcal{H}}_{\mu }$.

14 pages