infinite-dimensional analysis problem

Dear specialists in the infinite-dimensional analysis,
As I know there exists a continuous fixed point free involution f:X->X in an infinite-dimensional Banach space X ( f(f(x))=x for all x \in X ).
I would like to know does there exist not only continuous but also infinitely many times differentiable or, moreover, analytic involution f:X->X without fixed points where X is an infinite-dimensional Banach or Hilbert space.
If you know something about it write please.
Thank you very much.

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