Little Hankel Operators Between Vector-Valued Bergman Spaces on the Unit Ball

Abstract

In this paper, we study the boundedness and the compactness of the little Hankel operators $h_b$ with operator-valued symbols $b$ between different weighted vector-valued Bergman spaces on the open unit ball ${\mathbb{B}}_n$ in ${\mathbb{C}}^n.$ More precisely, given two complex Banach spaces $X,Y,$ and $0<p,q\le 1,$ we characterize those operator-valued symbols $b:{\mathbb{B}}_n\to \mathcal{L}(\overline{X},Y)$ for which the little Hankel operator $h_b:A_{\alpha }^p({\mathbb{B}}_n,X)⟶A_{\alpha }^q({\mathbb{B}}_n,Y),$ is a bounded operator. Also, given two reflexive complex Banach spaces $X,Y$ and $1<p\le q<\infty ,$ we characterize those operator-valued symbols $b:{\mathbb{B}}_n\to \mathcal{L}(\overline{X},Y)$ for which the little Hankel operator $h_b:A_{\alpha }^p({\mathbb{B}}_n,X)⟶A_{\alpha }^q({\mathbb{B}}_n,Y),$ is a compact operator.

v1: 40 pages