Operators preserving disjointness on rearrangement invariant spaces
Let X and Y be two rearrangement invariant spaces on a measure space (\(\Omega\),\(\Sigma\),\(\mu)\) with a finite, nonatomic measure \(\mu\). We show that if there exists a non-zero order continuous disjointness preserving operator T:X\(\to Y\), then \(X\subseteq Y\). [Let us remark incedentally that the proof depends on the multiplicative representation of disjointness preserving operators obtained by the author in Indag. Math. 45, 265-279 (1983; Zbl 0527.47025). This result has many corollaries of which we mention two: 1) If \(T:L_ p(\Omega,\Sigma,\mu)\to L_ q(\Omega,\Sigma,\mu)\) \((0<p<q\leq \infty)\) preserves disjointness, then \(T\equiv 0\). 2) If X and Y are rearrangement invariant Banach function spaces with strictly monotone norms and T is a positive isometry from X onto Y, then \(X=Y\).
order continuous disjointness preserving operator
scientific article; zbMATH DE number 4129476