We denote by $K$ the function defined from $[0,1]^2$ to $\mathbb{R}$ by the following relation: $K(s,t) = (1-s)t$ if $0 \leq t \leq s \leq 1$ and $K(s,t) = (1-t)s$ otherwise. We denote by $T$ the application defined on $E = C([0,1], \mathbb{R})$, equipped with the norm $\|.\|_2 = \sqrt{\int_0^1 |f(x)|^2 \, dx}$, by the relation: $$\forall f \in E, \quad \forall s \in [0,1], \quad T(f)(s) = \int_0^1 K(s,t) f(t) \, dt$$ Deduce $\sigma_p(T)$. Calculate the eigenspaces $E_{\lambda} = \operatorname{Ker}(T - \lambda Id)$ associated with each element $\lambda \in \sigma_p(T)$.
We denote by $K$ the function defined from $[0,1]^2$ to $\mathbb{R}$ by the following relation:
$K(s,t) = (1-s)t$ if $0 \leq t \leq s \leq 1$ and $K(s,t) = (1-t)s$ otherwise.
We denote by $T$ the application defined on $E = C([0,1], \mathbb{R})$, equipped with the norm $\|.\|_2 = \sqrt{\int_0^1 |f(x)|^2 \, dx}$, by the relation:
$$\forall f \in E, \quad \forall s \in [0,1], \quad T(f)(s) = \int_0^1 K(s,t) f(t) \, dt$$
Deduce $\sigma_p(T)$. Calculate the eigenspaces $E_{\lambda} = \operatorname{Ker}(T - \lambda Id)$ associated with each element $\lambda \in \sigma_p(T)$.