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Volker
Posts:
3
Registered:
10/16/11


weakly compact operators
Posted:
Mar 10, 2012 12:18 PM


For two Banach spaces $X$ and $Y$, let $W(X,Y)$ denote the weakly compact operators from $X$ to $Y$ and let $V(X,Y)$ stand for the completely continuous operators from $X$ to $Y$, i.e., those operators that map weakly convergent sequences to norm convergent ones.
It is obvious that, whenever $X$, $Y$, and $Z$ are Banach spaces and $S \in W(X,Y)$ and $T \in V(Y,Z)$, then $ST$ is a compact operator from $X$ to $Z$.
Does the converse hold?
Given a bounded linear operator $T$ from $X$ to $Y$ that fails to be weakly compact, is there a completely continuous operator $S$ from $Y$ into some Banach space $Z$ such that $ST$ is not compact?
Any pertinent hints are appreciated.
Best,
Volker.



