Strictly singular and disjointly strictly singular inclusions
An operator between Banach spaces is said to be strictly singular \((SS)\) if it fails to be an isomorphism on any infinite-dimensional closed subspace. An operator from a Banach lattice \(E\) into a Banach space is said to be disjointly strictly singular \((DSS)\) if there is no sequence \((x_n)\) of disjointly supported non-null vectors in \(E\) such that the restriction of the operator to the subspace generated by the sequence is an isomorphism. The present note announces several results about strict singularity and disjointly strict singularity of inclusions between r.i. spaces. The presented theorems are of two different types: General results about inclusions maps and the study of strict and disjointly strict singularity of inclusion maps between concrete r.i. spaces. For instance, for general r.i. spaces \(E\) and \(F\), the fact that \(E\subset F\) is \(SS\) is equivalent to be \(DSS\) together with the fact that either the norms of \(E\) and \(F\) are not equivalent on the span of the Rademacher functions or \(E\) not containing the closure of \(L^\infty\) in the Orlicz space \(L_M\) for \(M(t)=e^{t^2}-1\). Several conditions for the inclusions between Orlicz or Marcinkiewicz spaces to be strictly singular are presented. Finally, the authors point out how their techniques allow them to solve in the negative a question by V. D. Milman.
scientific article; zbMATH DE number 5493401