Proximal Operator image of convex functionals

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.
• There should be a square norm in the definition of the proximal map. – cheyp Feb 4 at 8:51
• @cheyp Thanks, for pointing that out :) – AIM_BLB Feb 4 at 8:53
• I do not believe that your first condition is necessary: If $f$ is the indicator function of $\{0\}$, then the proximal operator is $x \mapsto 0$ and this is not invertible. Second, the domain of a proximal operator is always $H$, i.e., it is trivially convex. – gerw Feb 4 at 9:57
• True, that's a good example. I'll edit the question. – AIM_BLB Feb 4 at 10:45
• The prox equals identity minus the gradient of the Moreau envelope of $f$ (which is a strongly convex function). Does that help? – Dirk Feb 4 at 14:04

I am not sure that it is going to work but it seems to me that this is the Legendre transform: $$\|x-h\|^2+\frac{1}{2}f(h)=\|x\|^2-2\Re\langle x,h\rangle+\|h\|^2+\frac{1}{2}f(h)$$ (Where we see $$H$$ as a $$\mathbb{R}$$ linear space and $$\Re\langle .,.\rangle$$ is the real scalar product. So if we call $$h_\min =\text{argmin} (-\Re\langle 2x,h\rangle+\|h\|^2+\frac{1}{2}f(h))$$ and $$-g(2x)=\min_h (-\Re\langle 2x,h\rangle+\|h\|^2+\frac{1}{2}f(h)) \\= -\Re\langle 2x,h_\min\rangle+\|h_\min\|^2+\frac{1}{2}f(h_\min)$$Then $$g$$ is the Legendre transform of $$h\rightarrow \|h\|^2+\frac{1}{2}f(h)$$. And there is the formula $$h_\min =\partial_x g$$ So I thing one should first caracterize the Legendre transform of the set $$[\|h\|^2+\frac{1}{2}f(h), f\in \Gamma_0]$$ and take its differential.