Hausdorff operators on Fock spaces and a coefficient multiplier problem

Abstract

Let μ \mu be a positive Borel measure on the positive real axis. We study the integral operator H μ ( f ) ( z ) = ∫ ( 0 , ∞ ) 1 t f ( z t ) d μ ( t ) , z ∈ C , \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 α p F^{p}_{\alpha } , p ∈ [ 1 , ∞ ] , α > 0 p\in [1,\infty ],\alpha >0 . Its action is easily seen to be a coefficient multiplication operator by the moment sequence μ n = ∫ [ 1 , ∞ ) 1 t n + 1 d μ ( t ) . \begin{equation*} \mu _n= \int _{[1,\infty )}\frac {1}{t^{n+1}} d\mu (t). \end{equation*} We prove that ‖ H μ ‖ F α p → F α p = ∫ [ 1 , ∞ ) 1 t d μ ( t ) , 1 ≤ p ≤ ∞ . \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 H μ \mathcal {H}_{\mu } is compact on F α p , p ∈ ( 1 , ∞ ) F^{p}_{\alpha },p\in (1,\infty ) if and only if μ ( { 1 } ) = 0 \mu (\{1\})=0 . In addition, we completely characterize the Schatten class membership of H μ \mathcal {H}_{\mu } .