Toeplitz and Hankel Operators on a Vector-valued Bergman Space

Abstract

In this paper, we derive certain algebraic properties of Toeplitz and Hankel operators defined on the vector-valued Bergman spaces $L_a^{2, mathbb{C}^n}(mathbb{D})$, where $mathbb{D}$ is the open unit disk in $mathbb{C}$ and $ngeq 1.$ We show that the set of all Toeplitz operators $T_{Phi}, Phiin L_{M_n}^{infty}(mathbb{D})$ is strongly dense in the set of all bounded linear operators ${mathcal L}(L_a^{2, mathbb{C}^n}(mathbb{D}))$ and characterize all finite rank little Hankel operators.