Operators preserving disjointness on rearrangement invariant spaces

Yuri A. Abramovich

Vol. 148 (1991), No. 2, 201–206

DOI: 10.2140/pjm.1991.148.201

Abstract

Let X and Y be two rearrangement invariant spaces on a measure space (Ω,Σ,μ) with a finite, nonatomic measure μ. We show that if there exists a non-zero order continuous disjointness preserving operator T : X → Y , then X ⊆ Y . This result has many consequences. For example, if T : L_p(Ω,Σ,μ) → L_q(Ω,Σ,μ) (0 < p < q ≤∞) preserves disjointness, then T ≡ 0.

Mathematical Subject Classification 2000

Primary: 47B60

Secondary: 46E30, 47B38

Milestones

Received: 18 July 1988

Revised: 19 January 1990

Published: 1 April 1991