Some remarks on disjointly strictly singular positive operators
Let \(E\) be a Banach lattice and \(F\) a Banach space, and let \(T\) be a bounded operator from \(E\) into \(F\). The operator \(T\) is said to be disjointly strictly singular (DSS) if there is no disjoint sequence \((x_{n})_{n}\) of non-null vectors in \(E\) such that the restriction of \(T\) to \(\text{span} [x_{n}]\) is an isomorphism. \(T\)~is called lattice strictly singular (LSS) if for every infinite-dimensional closed sublattice \(R\) of \(E\), the restriction \(T\) to \(R\) is not an isomorphism into its image. As main result, the author proves that if \(F\) is also a Banach lattice and if, in addition, \(E'\) and \(F\) are order continuous, then the following conditions are equivalent: (i) \(T\) is DSS; (ii) \(T\) is LSS; (iii) for every infinite-dimensional sublattice \(H\) in \(E\), there is an infinite-dimensional sublattice \(R \subset H\) such that the restriction \(T\) to \(R\) is a compact operator. Some results concerning operators acting in \(L^{p}\)-spaces are also given.
scientific article; zbMATH DE number 5070760