Some classes of Banach spaces and complemented subspaces of operators (Q1714432)

Let $X$ be a Banach space. In this paper, the author continues her work (see [Acta Math. Hung. 157, No. 1, 63--79 (2019; Zbl 1438.46021)] and the review by this reviewer) on establishing equality of certain classes of operators and deciding on the question of complementability when the inclusions are proper. \par For $1 < p < \infty$, let $LC_p(X,\ell_{\infty})$ denote the space of operators that map weakly $p$-summable sequences to norm null sequences. When $X$ does not have the $p$-$L$-limited property, it is shown that the space of weakly compact operators $W(X,\ell_{\infty})$ is not complemented in $LC_p(X,\ell_{\infty})$.