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Hi i just have some question regarding this problem , I cannot find a counter example and neither a proof:

Let $X$ be a Banach space of infinite dimension, and $S,T\in B(X)$ (the set of bounded linear operator from $X$ to $X$) is it true that:

if $ST$ is compact then $S$ or $T$ is compact?

Thanks for your help.

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    There exist Banach spaces where the answer is yes. http://gowers.wordpress.com/2009/02/07/a-remarkable-recent-result-in-banach-space-theory/2012-11-28

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