grandes-ecoles 2014 Q2d

grandes-ecoles · France · x-ens-maths__psi Invariant lines and eigenvalues and vectors Compute eigenvalues of a given matrix

We now work in the space of bounded real sequences $F = l^{\infty}(\mathbb{N})$ equipped with the norm $$\|u\|_{\infty} = \sup_{n \in \mathbb{N}} |u_n|$$ We denote by $S$, respectively $V$, the left shift application: $(Su)_n = u_{n-1}$ if $n \geq 1$ and $(Su)_0 = 0$, respectively the right shift: $(Vu)_n = u_{n+1}$ if $n \geq 0$.
Calculate the point spectrum of $S$ and $V$ in $F$.

We now work in the space of bounded real sequences $F = l^{\infty}(\mathbb{N})$ equipped with the norm
$$\|u\|_{\infty} = \sup_{n \in \mathbb{N}} |u_n|$$
We denote by $S$, respectively $V$, the left shift application: $(Su)_n = u_{n-1}$ if $n \geq 1$ and $(Su)_0 = 0$, respectively the right shift: $(Vu)_n = u_{n+1}$ if $n \geq 0$.

Calculate the point spectrum of $S$ and $V$ in $F$.

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