In a Hilbert space, does every nonexpansive mapping have a fixed point?

Question

In a Hilbert space, does every nonexpansive mapping have a fixed point?

For example rotations or mirroring are nonexpansive, but the zero vector is a fixed point, hence the set of fixed points is nonempty. With this logic I can come up with a bunch of examples of nonexpansive mappings, which have nonempty fixed point sets. I cannot come up with an example where both conditions hold, but I believe that the answer on the question is NO.

Thanks in advance!

Answer 1

If by non-expansive you mean $\|Tx-Ty\| \leq \|x-y\|$ then a translation by any non-zero vector ($Tx=x+x_0$) is an obvious example.