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When is it possible to prove that a compact operator $T: V \to V$ where $V$ is a Banach space is also differentiable? Fréchet differentiable?

PS: There is a further information which might help. My operator $T$ associates to each vector field $j$ a vector field $b$ solution of a certain boundary value problem. I will write the whole set of equations if asked to.

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    Thank you for the information.2011-12-05

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Reposting comment as answer, since it seems to be what the OP was looking for:

If $V$ is any Banach space, then $T$ is differentiable as soon as it is continuous. This is very easy to prove straight from the definition.

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    @Ali: that's usually the case, yes.2011-12-05