Strictly singular and strictly co-singular inclusions between symmetric sequence spaces
This article is devoted to an extensive study of strict singularity and strict co-singularity of inclusions between symmetric sequence spaces. First, the special cases of Lorentz and Marcinkiewicz spaces are characterized in terms of the underlying weight sequences. Then, for given symmetric sequence spaces \(E \not=\ell_1\) and \(F \not= c_0, \ell_\infty\) such that \(E \hookrightarrow F\), an intermediate symmetric sequence space \(G\) such that \(E \hookrightarrow G \hookrightarrow F\) and both inclusions are not strictly singular is constructed. The proof uses certain properties of a suitable right shift operator defined on a block basis with constant coefficients. This leads to characterizations of the sequence spaces \(\ell_1\) and \(c_0\) within the class of all symmetric sequence spaces in terms of strictly singular inclusions, respectively. In the concluding section, the authors prove sufficient and necessary conditions for the strict singularity of inclusions involving inclusion indices, Boyd indices and associated fundamental functions of the symmetric sequence spaces involved, respectively. Eventually, strict co-singularity of inclusions is considered, leading to a broad extension of a result by \textit{A. Defant}, \textit{M. Mastyło} and the reviewer [North-Holland Math. Stud. 189, 43--60 (2001; Zbl 1025.47014)] on the strict singularity of inclusions \(E \hookrightarrow \ell_p\), where \(E \not= \ell_p\) is a \(p\)-concave symmetric sequence space.
scientific article; zbMATH DE number 2075635
scientific article