Toeplitz and Hankel Operators on Vector-Valued Fock-Type Spaces

In this paper, we study some characterizations of the Toeplitz and Hankel operators with positive operator-valued function as symbol on the vector-valued Fock-type spaces. We first discuss that the Bergman projection P:L Psi p(H)-> F Psi p(H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P:L<^>p_{\Psi }({\mathcal {H}})\rightarrow F<^>p_{\Psi }({\mathcal {H}})$$\end{document} is bounded for all 1 <= p <=infinity\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le p\le \infty $$\end{document}, and obtain the duality of the vector-valued Fock-type spaces. Second, using operator-valued Carleson conditions, we give a complete characterization of the boundedness and compactness of the Toeplitz operators on F Psi p(H)(1<p<infinity)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F<^>p_{\Psi }({\mathcal {H}})(1<p<\infty )$$\end{document}. Finally, we describe the boundedness (or compactness) of the Hankel operators HG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_G$$\end{document} and HG & lowast;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{G<^>*}$$\end{document} on F Psi 2(H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_{\Psi }<^>2({\mathcal {H}})$$\end{document} in terms of a bounded (or vanishing) mean oscillation. We also give geometrical descriptions for the operator-valued spaces BMO Psi 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$BMO_\Psi <^>2$$\end{document} and VMO Psi 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$VMO_\Psi <^>2$$\end{document} defined in terms of the Berezin transform.