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Topic: weakly compact operators
Replies: 1   Last Post: Mar 13, 2012 10:15 AM

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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.

Date Subject Author
3/10/12 Volker
3/13/12 Philip Brooker