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 Ψ p ( H ) → F Ψ p ( H ) is bounded for all 1 ≤ p ≤ ∞ , 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 Ψ p ( H ) ( 1 < p < ∞ ) . Finally, we describe the boundedness (or compactness) of the Hankel operators H G and H G ∗ on F Ψ 2 ( H ) in terms of a bounded (or vanishing) mean oscillation. We also give geometrical descriptions for the operator-valued spaces B M O Ψ 2 and V M O Ψ 2 defined in terms of the Berezin transform.