Let $H = L^2([0,3])$ and let $T \in B(H)$ defined by: $$Tu(t) = (1+t^2)u(t)$$ $||T||_{op} = 10$.
i) Is T compact? Justify the answer.
ii) Does $T$ have eigenvalues? Justify the answer.
If you may help me even on just one point this would be very helpfull really. Thanks!
Hints:
$i).\ $ Scale to $[0,1]$ and note that $\{e^{int}\}_{n\in \mathbb Z}$ is an orthonormal basis. Now show that the sequence defined by $f_n(t)=(1+t^2)e^{int}$ has no Cauchy subesequence.
$ii).\ $ Note that $Tf(t)-\lambda f(t)=0\Leftrightarrow (1+t^2)f(t)=\lambda f(t)$