×

On a question of Pełczyński about strictly singular operators. (English) Zbl 1244.46002

It was proved by A. Pełczyński [Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 13, 31–36, 37–41 (1965; Zbl 0138.38604)] that if the conjugate operator \(T^*\) is strictly singular then \(T\) is strictly cosingular. The converse implication is true in some special cases: if \(T:X\to Y\) is strictly singular and \(X\) is reflexive then \(T^*\) is strictly cosingular. Pełczyński conjectures that \(T\) weakly compact and strictly singular implies \(T^*\) strictly cosingular. This conjecture was disproved by M. González [Quaest. Math. 28, No. 1, 37–38 (2005; Zbl 1065.47020)].
In the paper under review the authors investigate this question in the special case when \(T\) is a quotient map. They prove new positive statements as well as exhibit new examples of weakly compact strictly singular quotient maps with dual not strictly cosingular. The authors also obtain new examples of super-strictly singular quotient maps and show that the strictly singular quotient maps in Kalton-Peck sequences are not super-strictly singular.

MSC:

46B03 Isomorphic theory (including renorming) of Banach spaces
47B07 Linear operators defined by compactness properties
46B08 Ultraproduct techniques in Banach space theory
46B10 Duality and reflexivity in normed linear and Banach spaces
PDFBibTeX XMLCite
Full Text: DOI