= Solution
For a <proper convex function> $E$ that is <lower semicontinuous> on a <Hilbert space>, its <proximal operator> at scale $\gamma>0$ is
$$
\boxed{\operatorname{prox}_{\gamma E}(z)=\arg\min_x\left\{\frac12\|x-z\|^2+\gamma E(x)\right\}=(I+\gamma\partial E)^{-1}z.}
$$
The minimizer is unique because the objective is <strongly convex>; existence follows from the closed proper convex functional and the coercive quadratic, using an affine lower bound. The <subgradient optimality condition> is $0\in x-z+\gamma\partial E(x)$, which is exactly the displayed <resolvent of a monotone operator> relation. Setting $\gamma=1$ gives the unscaled proximity or resolvent operator requested here. This is a nonlinear resolvent of the <subdifferential>, distinct from the spectral resolvent of a linear operator.
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