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

TL;DR

This paper characterizes the boundedness and compactness of little Hankel operators with operator-valued symbols between various weighted vector-valued Bergman spaces on the unit ball in complex space.

Contribution

It provides new characterizations of boundedness and compactness for operator-valued little Hankel operators between vector-valued Bergman spaces, extending previous scalar results.

Findings

Characterization of bounded little Hankel operators for 0 < p,q ≤ 1.

Characterization of compact little Hankel operators for 1 < p ≤ q < ∞.

Results apply to operator-valued symbols between Banach spaces.

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 $0 < p,q \leq 1,$ we characterize those operator-valued symbols $b: \mathbb{B}_{n}\rightarrow \mathcal{L}(\overline{X},Y)$ for which the little Hankel operator $h_{b}: A^p_{\alpha}(\mathbb{B}_{n},X) \longrightarrow A^q_{\alpha}(\mathbb{B}_{n},Y),$ is a bounded operator. Also, given two reflexive complex Banach spaces $X,Y$ and $1 < p \leq q < \infty,$ we characterize those operator-valued symbols $b: \mathbb{B}_{n}\rightarrow \mathcal{L}(\overline{X},Y)$ for which the little Hankel operator $h_{b}: A^p_{\alpha}(\mathbb{B}_{n},X) \longrightarrow A^q_{\alpha}(\mathbb{B}_{n},Y),$ is a compact operator.