A matrix acting between Fock spaces

Abstract

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