A matrix acting between Fock spaces

Abstract

Abstract If H ν = ( ν n , k ) n , k ≥ 0 $\mathcal{H}_{\nu}=(\nu _{n,k})_{n,k\geq 0}$ is the matrix with entries ν n , k = ∫ [ 0 , ∞ ) t n + k n ! d ν ( 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 , ∞ ) $[0,\infty )$ , the matrix H ν $\mathcal{H}_{\nu}$ acts on the space of all entire functions f ( z ) = ∑ n = 0 ∞ a n z n $f(z) =\sum_{n=0}^{\infty} a_{n} z^{n}$ and induces formally the operator in the following way: H ν ( f ) ( z ) = ∑ n = 0 ∞ ( ∑ k = 0 ∞ ν n , k a k ) 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 ≤ ∞ $0< p\leq \infty $ , we classify for which measures the operator H ν ( 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 H ν $\mathcal{H}_{\nu}$ is a bounded (resp., compact) operator between F p $F^{p} $ and F ∞ $F^{\infty }$ .

Keywords

• Fock spaces
• Matrices
• Fock Carleson measure