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46b03 (secondary)

Bulletin of the London Mathematical Society (1)

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EMBEDDING $\ell_{\infty}$ INTO THE SPACE OF BOUNDED OPERATORS ON CERTAIN BANACH SPACES
G. ANDROULAKIS, K. BEANLAND, S. J. DILWORTH, F. SANACORY
Journal:

Bulletin of the London Mathematical Society / Volume 38 / Issue 6 / December 2006

Published online by Cambridge University Press:

19 December 2006, pp. 979-990

Print publication:

December 2006
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Sufficient conditions are given on a Banach space $X$ which ensure that $\ell_{\infty}$ embeds in ${\mathcal L}\, (X)$, the space of all bounded linear operators on $X$. A basic sequence $(e_n)$ is said to be quasisubsymmetric if for any two increasing sequences $(k_n)$ and $(\ell_n)$ of positive integers with $k_n \leq \ell_n$ for all $n$, $(e_{k_n})$ dominates $(e_{\ell_n})$. If a Banach space $X$ has a seminormalized quasisubsymmetric basis then $\ell_{\infty}$ embeds in ${\mathcal L}\, (X)$.