The proximal operator of a function is
$$\text{prox}_f(x) = \text{argmin}_u f(u) + \frac{1}{2}\|u-x\|_2^2.$$
Suppose I have the function that is $\text{prox}_f$. I would like a simple expression for $\text{prox}_{\lambda f}$ where $\lambda$ is a scalar. That is, I would like the proximal operator of the scaled function given the proximal operator of the function itself.
Does such an expression exist?
If $\lambda$ is positive and $f$ is convex, proper, and lower-semicontinuous, you can guarantee $\text{prox}_{\lambda f}$ is uniquely-defined. The scalar term usually appears explicitly in the formula for $\text{prox}_{\gamma f}$ e.g. see http://proximity-operator.net/index.html. In the differentiable case, $\text{prox}_{\gamma f} = x - \gamma \nabla f(x) + o(\gamma)$ as $\gamma \downarrow 0$. In general, $x = \text{prox}_{\gamma f}x + \gamma \text{prox}_{\gamma^{-1}f^*}(\gamma^{-1}x)$ where $f^*$ is the Fenchel-Legendre conjugate of $f$.
For more references, check Bauschke & Combettes' 2017 book, Section 23.3 on Resolvents (which are generalized proxes). You get results like:
$(\forall \rho \in \mathbb{R} \setminus\{0\}) \quad \text{prox}_{f}x = \rho^{-1}\text{prox}_{\rho^2 f}(\rho x)$