Here is a lemma that I find myself googling regularly since I always forget it’s exact form.

Lemma 1 Let ${A}$ be a monotone operator, ${\lambda>0}$ and denote by ${R_{\lambda A} = (I+\lambda A)^{-1}}$ the resolvent of ${\lambda A}$. Then it holds that

$\displaystyle \begin{array}{rcl} R_{\lambda A^{-1}}(x) = x - \lambda R_{\lambda^{-1}A}(\lambda^{-1}x). \end{array}$

Proof: We start with the left hand side ${y = R_{\lambda A^{-1}}(x) = (I+\lambda A^{-1})^{-1} x}$ and deduce

$\displaystyle \begin{array}{rcl} x &\in& y + \lambda A^{-1}y\\ \iff \frac{x-y}{\lambda} &\in& A^{-1}y\\ \iff y &\in& A(\frac{x-y}{\lambda})\\ \iff x &\in& A(\frac{x-y}{\lambda}) + x-y\\ \iff \frac{x}{\lambda} &\in& \frac{1}{\lambda}A(\frac{x-y}{\lambda}) + \frac{x-y}{\lambda}\\ \iff \frac{x-y}{\lambda} & = &(I + \lambda^{-1}A)^{-1}(\lambda^{-1}x)\\ \iff x - \lambda (I+\lambda^{-1}A)^{-1}(\lambda^{-1}x) & = & y. \end{array}$

$\Box$

I do not know any official name of this, but would call it Moreau’s identity which is the name of the respective statement for proximal operators for convex functions ${f}$ and ${g}$:

$\displaystyle \begin{array}{rcl} \mathrm{prox}_{\lambda f^{*}}(x) = x - \lambda\mathrm{prox}_{\lambda^{-1}f}(\lambda^{-1}x). \end{array}$

The version for monotone operators is Proposition 23.18 in the first edition of Bauschke and Combette’s book “Convex Analysis and Monotone Operator Theory in Hilbert Spaces”.