A matrix acting between Fock spaces

If ${\mathcal{H}}_{\nu }={({\nu }_{n,k})}_{n,k\ge 0}$ $\mathcal{H}_{\nu}=(\nu _{n,k})_{n,k\geq 0}$ is the matrix with entries ${\nu }_{n,k}={\int }_{[0,\mathrm{\infty })}\frac{t^{n+k}}{n!}d\nu (t)$ $\nu _{n,k}=\int _{[0,\infty )}\frac{ t^{n+k}}{n!}\,d\nu (t)$ , where ν is a nonnegative Borel measure on the interval $[0,\mathrm{\infty })$ $[0,\infty )$ , the matrix ${\mathcal{H}}_{\nu }$ $\mathcal{H}_{\nu}$ acts on the space of all entire functions $f(z)={\sum }_{n=0}^{\mathrm{\infty }}{a}_n{z}^n$ $f(z) =\sum_{n=0}^{\infty} a_{n} z^{n}$ and induces formally the operator in the following way: ${\mathcal{H}}_{\nu }(f)(z)=\sum _{n=0}^{\mathrm{\infty }}\left(\sum _{k=0}^{\mathrm{\infty }}{\nu }_{n,k}{a}_k\right)z^n.$ \( \mathcal{H}_{\nu}(f) (z)=\sum_{n=0}^{\infty} \Biggl(\sum_{k=0}^{\infty}\nu _{n,k}a_{k}\Biggr)z^{n}. \) In this paper, for $0<p\le \mathrm{\infty }$ $0< p\leq \infty $ , we classify for which measures the operator ${\mathcal{H}}_{\nu }(f)$ $\mathcal{H}_{\nu}(f)$ is well defined on $F^p$ $F^{p}$ and also gets an integral representation, and among them we characterize those for which ${\mathcal{H}}_{\nu }$ $\mathcal{H}_{\nu}$ is a bounded (resp., compact) operator between $F^p$ $F^{p} $ and $F^{\mathrm{\infty }}$ $F^{\infty }$ .