I asked this question to the math.stackexchange but couldn't get an answer.
Let $A$ be a closed convex process from $R^n$ to $R^n$, $I$ be the identity map, $\lambda$ be a real number, and $k$ be a positive integer. It is obvious that $A - \lambda I$ and $(A - \lambda I)^{-1}$ are closed but I cannot find a way to prove that $(A - \lambda I)^{-k}$ is closed. This result is stated or used without a proof in this book (Theorem 2.14) and this book (pp. 176).
If we can assume that the domain or the range of $A$ is bounded, the proof is staightforward but not in other case. Then, I tried to use a condition found from Theorem 39.8 of Rockafellar's Convex Analysis:
Let $A$ and $B$ be convex processes from $R^n$ to $R^m$ and from $R^m$ to $R^p$, respectively. If $A$ and $B$ are closed and $\mathrm{ri}(\mathrm{range}B^*)$ meets $\mathrm{ri}(\mathrm{dom}A^*)$, then $BA$ is closed.
but couldn't make any progress. Does anybody know where I can find the proof of it?
