Let $\Gamma_0$ denote the set of lower-semi-continuous convex functionals on a Hilbert space $H$. What exactly is the image of $\Gamma_0$ under the proximal operator $$ \begin{aligned} &\Gamma_0\rightarrow \{f:H \rightarrow H\}\\ & f \mapsto \left[\operatorname{argmin}_{h \in H} \|x-h\|_H^2 +\frac1{2}f(h) \right]. \end{aligned} $$ (Not its co-domain, but what characterizes it's image?)

**Necessary:**
*Here are a few necessary conditions I've noted so far.*

- The functions in its image must have a convex domain, since the Moreau envelope is convex, and $Prox_f$ associates the outputs of the Moreau envelope with elements of its domain.