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

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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.

Item Type:
Journal Article
Journal or Publication Title:
Mathematical Proceedings of the Cambridge Philosophical Society
Uncontrolled Keywords:
/dk/atira/pure/subjectarea/asjc/2600
Subjects:
ID Code:
160250
Deposited By:
Deposited On:
05 Oct 2021 14:55
Refereed?:
Yes
Published?:
Published
Last Modified:
06 Oct 2021 08:29