Let $\mathcal{B}_{n}$ be the set of matrices $M$ in $\mathcal{M}_{n}(\mathbf{C})$ such that the sequence $\left(\left\|M^{k}\right\|_{\mathrm{op}}\right)_{k \in \mathbf{N}}$ is bounded. For $M \in \mathcal{B}_{n}$, we set $b(M) = \sup\left\{\left\|M^{k}\right\|_{\mathrm{op}}; k \in \mathbf{N}\right\}$. Let $M \in \mathcal{B}_{n}$ and $z \in \mathbf{C} \backslash \mathbb{D}$. Show that the series of matrices $\sum \frac{M^{j}}{z^{j+1}}$ converges. We will admit the following fact: let $(E, N)$ be a finite-dimensional normed vector space; if $(v_{j})_{j \in \mathbf{N}}$ is a sequence of elements of $E$ such that the series $\sum N(v_{j})$ converges, then the series $\sum v_{j}$ converges in $E$. If $m \in \mathbf{N}$, give a simplified expression for $\left(zI_{n} - M\right)\sum_{j=0}^{m} \frac{M^{j}}{z^{j+1}}$. Deduce that $$R_{z}(M) = \sum_{j=0}^{+\infty} \frac{M^{j}}{z^{j+1}}$$
Let $\mathcal{B}_{n}$ be the set of matrices $M$ in $\mathcal{M}_{n}(\mathbf{C})$ such that the sequence $\left(\left\|M^{k}\right\|_{\mathrm{op}}\right)_{k \in \mathbf{N}}$ is bounded. For $M \in \mathcal{B}_{n}$, we set $b(M) = \sup\left\{\left\|M^{k}\right\|_{\mathrm{op}}; k \in \mathbf{N}\right\}$.
Let $M \in \mathcal{B}_{n}$ and $z \in \mathbf{C} \backslash \mathbb{D}$. Show that the series of matrices $\sum \frac{M^{j}}{z^{j+1}}$ converges.\\
We will admit the following fact: let $(E, N)$ be a finite-dimensional normed vector space; if $(v_{j})_{j \in \mathbf{N}}$ is a sequence of elements of $E$ such that the series $\sum N(v_{j})$ converges, then the series $\sum v_{j}$ converges in $E$.\\
If $m \in \mathbf{N}$, give a simplified expression for $\left(zI_{n} - M\right)\sum_{j=0}^{m} \frac{M^{j}}{z^{j+1}}$.\\
Deduce that
$$R_{z}(M) = \sum_{j=0}^{+\infty} \frac{M^{j}}{z^{j+1}}$$