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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.

• A map in the image must be invertible, since $Prox_f = (I + \partial f)^{-1}$ is single-valued.
• 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.

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.

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.

• A map in the image must be invertible, since $Prox_f = (I + \partial f)^{-1}$ is single-valued.

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.

• A map in the image must be invertible, since $Prox_f = (I + \partial f)^{-1}$ is single-valued.
• 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.

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 $$ \begin{aligned} &\Gamma_0\rightarrow \{f:H \rightarrow H\}\\ & f \mapsto \left[\operatorname{argmin}_{h \in H} \|x-h\| +\frac1{2}f(h) \right]. \end{aligned} $$ (Not its co-domain, but what characterizes it's image?)

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.

• A map in the image must be invertible, since $Prox_f = (I + \partial f)^{-1}$ is single-valued.