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