Analysis of Toeplitz Operators with $BMO^1_{\alpha}$ operator-valued symbols on $\ell^2-$Valued Bergman Spaces

As a class of compact operators on the $\ell^2-$valued Bergman space $A^2_\alpha (\mathbb B_n, \ell^2)$ on the unit ball $\mathbb B_n,$ we study Toeplitz operators with $BMO^1_\alpha (\mathbb B_n, \mathcal L(\ell^2))$ operator-valued symbols. First, we describe a method of restriction to a finite dimension which allows us to apply earlier results of Rahm and Wick; then we exhibit an explicit example of a compact Toeplitz operator on $A^2_\alpha (\mathbb B_n, \ell^2).$ Secondly, we apply two sufficient conditions for compactness established by Rahm in infinite dimension. The first condition is in terms of the Toeplitz algebra $\mathcal T_{L^\infty_{fin}},$ the second one is in terms of sufficiently localized operators and is implied by the first condition. To get the second condition, we additionally assume that the symbol and its adjoint belong to $BMO^1_\alpha (\mathbb B_n, \mathcal L(\ell^2, \ell^1)).$ Finally, inspired by Xia and Sadeghi-Zorboska, we ask the question of the validity of the reverse implication.