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}

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