The spectrum of $S$ is a countable set?

Question

Suppose $K:[0,1]\times[0,1] \rightarrow \mathbb{C}$ be a continuous function. Let us define $S:L^2[0,1]\rightarrow L^2[0,1]$ by $$(Sf)(x)=f(x)+ \int^1_0K(x,y)f(y)dy$$ Is the spectrum of $S$ a countable set? Why?

Answer 1

Let $Tf(x)=\int_0^{1} K(x,y) f(y)dy$. As is well-known this a compact operator. Non-zero points in its spectrum consists of at most countably many eigen values (converging to $0$ of there are in finitely many of them). These are standard facts about compact operators. Now just note that $S=T+I$ so $\sigma (S) =\{\lambda +1 : \lambda \in \sigma (T)\}$.