grandes-ecoles 2014 Q4h

grandes-ecoles · France · x-ens-maths__psi Matrices Matrix Norm, Convergence, and Inequality

With $\mathcal{L}^2(H) = \left\{T \in \mathcal{L}(H), \quad \|T\|_2 < +\infty\right\}$ where $\|T\|_2 = \sum_{i=0}^{+\infty} \|T(b_i)\|^2$ for any Hilbert basis $B = (b_i)_{i \in \mathbb{N}}$ of $H = l^2(\mathbb{N})$,
We consider $L$ and $U$ two operators in $\mathcal{L}(H)$. Show that if $L \in \mathcal{L}^2(H)$, then so is $UL$.

With $\mathcal{L}^2(H) = \left\{T \in \mathcal{L}(H), \quad \|T\|_2 < +\infty\right\}$ where $\|T\|_2 = \sum_{i=0}^{+\infty} \|T(b_i)\|^2$ for any Hilbert basis $B = (b_i)_{i \in \mathbb{N}}$ of $H = l^2(\mathbb{N})$,

We consider $L$ and $U$ two operators in $\mathcal{L}(H)$. Show that if $L \in \mathcal{L}^2(H)$, then so is $UL$.

Paper Questions

Q1a Q1b Q1c Q1d Q1e Q2a Q2b Q2c Q2d Q2e Q3a Q3b Q3c Q3d Q3e Q4a Q4b Q4c Q4d Q4e Q4f Q4g Q4h Q4i