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

Laustsen, Niels Jakob (2001) Matrix multiplication and composition of operators on the direct sum of an infinite sequence of Banach spaces. Mathematical Proceedings of the Cambridge Philosophical Society, 131 (1). pp. 165-183. ISSN 0305-0041

Abstract

Let script F sign be a Banach space with a normalized, 1-unconditional basis. Each operator on the script F sign-direct sum of a sequence (xi)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 xi to script F sign is compact; (ii) the basis of script F sign is shrinking and, for each i ∈ ℕ, each operator from script F sign to xi is compact. In the case where script F sign is either c0 or ℓp, where 1 ≤ p < ∞, the converse also holds.