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 \(P:L^p_{\Psi }({\mathcal {H}})\rightarrow F^p_{\Psi }({\mathcal {H}})\) $P:L_{\mathrm{\Psi }}^p(\mathcal{H})\to F_{\mathrm{\Psi }}^p(\mathcal{H})$ is bounded for all \(1\le p\le \infty \) $1\le p\le \infty$ , 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 \(F^p_{\Psi }({\mathcal {H}})(1<p<\infty )\) $F_{\mathrm{\Psi }}^p(\mathcal{H})(1<p<\infty )$ . Finally, we describe the boundedness (or compactness) of the Hankel operators \(H_G\) $H_G$ and \(H_{G^*}\) $H_{G^{\ast }}$ on \(F_{\Psi }^2({\mathcal {H}})\) $F_{\mathrm{\Psi }}^2(\mathcal{H})$ in terms of a bounded (or vanishing) mean oscillation. We also give geometrical descriptions for the operator-valued spaces \(BMO_\Psi ^2\) $BM{O}_{\mathrm{\Psi }}^2$ and \(VMO_\Psi ^2\) $VM{O}_{\mathrm{\Psi }}^2$ defined in terms of the Berezin transform.