Hausdorff operators on Fock spaces and a coefficient multiplier problem

Abstract:

Let $\mu$ be a positive Borel measure on the positive real axis. We study the integral operator \begin{equation*} \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}, \end{equation*} acting on the Fock spaces $F^{p}_{\alpha }$, $p\in [1,\infty ],\alpha >0$. Its action is easily seen to be a coefficient multiplication operator by the moment sequence \begin{equation*} \mu _n= \int _{[1,\infty )}\frac {1}{t^{n+1}} d\mu (t). \end{equation*} We prove that \begin{equation*} \|\mathcal {H}_{\mu }\|_{F^{p}_{\alpha }\to F^{p}_{\alpha }}=\int _{[1,\infty )}\frac {1}{t} d\mu (t),\quad 1\leq p\leq \infty . \end{equation*}

It turns out that $\mathcal {H}_{\mu }$ is compact on $F^{p}_{\alpha },p\in (1,\infty )$ if and only if $\mu (\{1\})=0$. In addition, we completely characterize the Schatten class membership of $\mathcal {H}_{\mu }$.