Fixed point sets of (firmly) nonexpansive mappings.

Question

Consider a Hilbert space $H$ with a (firmly) nonexpansive mapping $T:H\to H$. I am wondering whether there are well known conditions on $T$ or $H$ which guarantee $$\mathrm{Fix}(T)\subseteq\overline{B_{r_T}(0)}$$ for some constant $r_T$ (depending on T)? Especially the case of a finite dimensional space $H$ with a firmly nonexpansive mapping $T$ is of interest to me.

Answer 1

I am not aware of a well known condition guaranteeing this. But here are some observations:

$T$ can not be a projector onto an unbounded set, or any relaxation thereof.

let $C\subset B_{r_T}(0)$ be nonempty, closed, and convex; and let $P_C$ denote its projector. Then for any $\lambda\in[0,1[$, $T=\lambda\textrm{Id}+(1-\lambda)P_C$ satisfies your requirement.

Any operator $T$ such that $\textrm{Id}-T$ is a proximal thresholder is sufficient

Please let me know if you find an equivalence!