grandes-ecoles 2022 Q4

grandes-ecoles · France · mines-ponts-maths2__pc Not Maths
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.
Assume that $n \geq 2$. Indicate, with justification, a matrix $M$ in $\mathcal{M}_{n}(\mathbf{C})$, upper triangular, such that $\sigma(M) \subset \mathbb{D}$, but not belonging to $\mathcal{B}_{n}$.
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.

Assume that $n \geq 2$. Indicate, with justification, a matrix $M$ in $\mathcal{M}_{n}(\mathbf{C})$, upper triangular, such that $\sigma(M) \subset \mathbb{D}$, but not belonging to $\mathcal{B}_{n}$.