Embedding ${\ell }_{\infty }$ into the space of all Operators on Certain Banach Spaces

Abstract

We give sufficient conditions on a Banach space $X$ which ensure that ${\ell }_{\infty }$ embeds in $\mathcal{L}(X)$, the space of all operators on $X$. We say that a basic sequence $(e_n)$ is quasisubsymmetric if for any two increasing sequences $(k_n)$ and $({\ell }_n)$ of positive integers with $k_n\le {\ell }_n$ for all $n$, we have that $(e_{k_n})$ dominates $(e_{{\ell }_n})$. We prove that if a Banach space $X$ has a seminormalized quasisubsymmetric basis then ${\ell }_{\infty }$ embeds in $\mathcal{L}(X)$.

10 pages