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

In this paper, we study the boundedness and the compactness of the little Hankel operators hb\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_b$$\end{document} with operator-valued symbols b between different weighted vector-valued Bergman spaces on the open unit ball Bn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {B}_n$$\end{document} in Cn.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {C}^n.$$\end{document} More precisely, given two complex Banach spaces X, Y,  and 0<p,q≤1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 < p,q \le 1,$$\end{document} we characterize those operator-valued symbols b:Bn→L(X¯,Y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b: \mathbb {B}_{n}\rightarrow \mathcal {L}(\overline{X},Y)$$\end{document} for which the little Hankel operator hb:Aαp(Bn,X)⟶Aαq(Bn,Y),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{b}: A^p_{\alpha }(\mathbb {B}_{n},X) \longrightarrow A^q_{\alpha }(\mathbb {B}_{n},Y),$$\end{document} is a bounded operator. Also, given two reflexive complex Banach spaces X, Y and 1<p≤q<∞,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1< p \le q < \infty ,$$\end{document} we characterize those operator-valued symbols b:Bn→L(X¯,Y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b: \mathbb {B}_{n}\rightarrow \mathcal {L}(\overline{X},Y)$$\end{document} for which the little Hankel operator hb:Aαp(Bn,X)⟶Aαq(Bn,Y),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{b}: A^p_{\alpha }(\mathbb {B}_{n},X) \longrightarrow A^q_{\alpha }(\mathbb {B}_{n},Y),$$\end{document} is a compact operator.