I would like to know the spectral radius of the operator $T_k$ from $C[0,1] \to C[0,1]$ : $$T_k x (t)= \int_0^1 k(t,s) x(s) ds$$ where $k(x,y)\colon [0,1]^2 \to \mathbb C$ is continuous.
And also although I know that $Tf(x)=\int_0^1f(x)dx$ is compact, I am not able to follow that $T_k$ is compact. Any hints and ideas ? Thanks!
To show compactness, we can use Arzela-Ascoli's theorem. Let $B$ the set of continuous functions on $[0,1]$ of norm $\leqslant 1$. We have to show equi-continuity of $T_k(B)$. It follows from the fact that $k$ is uniformly continuous on $[0,1]^2$.
For the spectral radius, show by induction that $$\lVert T_k^p\rVert\leqslant\frac{\lVert k\rVert_\infty^p}{p!},$$ the spectral radius formula and this thread.