Hausdorff operators on Fock Spaces

Let $μ$ be a positive Borel measure on the positive real axis. We study the integral operator $\mathcal{H}_μ(f)(z)=\int_{0}^{\infty}\frac{1}{t}f\left(\frac{z}{t}\right)\,dμ(t),\quad z\in \mathbb{C}\,,$ acting on the Fock spaces $F^{p}_α$, $p\in [1,\infty],\,α>0$. Its action is easily seen to be a coefficient multiplication by the moment sequence $μ_n= \int_{1}^{\infty}\frac{1}{t^{n+1}}\,dμ(t) .$ 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 $\mathcal{H}_μ$ on every $F^{p}_α,\,p\in (1,\infty )$. In addition, we completely characterize the Schatten class membership of $\mathcal{H}_μ$.

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