Show that the spectrum of $S$ is a countable set.

Question

Let $K:[0,1]\times[0,1] \rightarrow \mathbb{C}$ be a continuous function. Define $S:L^2[0,1]\rightarrow L^2[0,1]$ by $$(Sf)(x)=f(x)+ \int^1_0K(x,y)h(y)dy$$ How do we show that the spectrum of $S$ is a countable set? Any ideas?

Answer 1

S is an compact operator on a Hilbert space, so "the theory says" that its spectrum is countable. Maybe I am wrong, but I think it takes times to prove it. I am sure that it is proved in Functionnal Analysis and Application, by Haim Brezis.