The Essential Norm of Operators on $$\ell ^{2}$$ ℓ 2 -Valued Bergman-Type Function Spaces

Abstract

In this paper, conditions—some sufficient and some necessary and sufficient—for an operator to be compact on \(\ell ^{2}\)-valued Bergman-type function spaces. This paper generalizes many well-known results about classical function spaces to their \(\ell ^{2}\)-valued versions. In particular, we characterize the compact operators acting on weighted \(\ell ^{2}\)-valued Bergman spaces on the unit ball, the unit polydisc and, more generally to weighted \(\ell ^{2}\)-valued Fock spaces.