# oc.optimization and control – Proximal operator of composition with norm squared on a Hilbert space

Let $$(X,Sigma,mu)$$ be a finite measure space and let $$L^2(mu,mathbb{R}^d)$$ denote the Bochner space of strongly measurable functions taking values in $$mathbb{R}^d$$. Let
begin{aligned} T &:L^2(mu,mathbb{R}^d)rightarrow L^2(mu,mathbb{R})\ & fmapsto |f|_2, end{aligned}
where $$|cdot|_2$$ is the Euclidean norm on $$mathbb{R}^d$$. Fix a $$phiinGamma_0(L^2(mu,mathbb{R}))$$.

Is there a known expression for
$$operatorname{Prox}_{Tcirc phi}$$
in terms of $$operatorname{Prox}_{phi}$$, $$phi$$, and $$T$$?