Matrix multiplication and composition of operators on the direct sum of an infinite sequence of Banach spaces

Abstract

Let [Efr ] be a Banach space with a normalized, 1-unconditional basis. Each operator on the [Efr ]-direct sum of a sequence ([Xfr ]_i)_i∈ℕ of Banach spaces corresponds to an infinite matrix. We study whether this correspondence is multiplicative, in which case we say that matrix multiplication works. We prove that matrix multiplication works if at least one of the following two conditions is satisfied:

(i) for each i ∈ ℕ, each operator from [Xfr ]_i to [Efr ] is compact;

(ii) the basis of [Efr ] is shrinking and, for each i ∈ ℕ, each operator from [Efr ] to [Xfr ]_i is compact.

In the case where [Efr ] is either c₀ or [lscr ]_p, where 1 [les ] p < ∞, the converse also holds.