volterra operator consists the single point 0?

Question

It is a question on Lax's book functional analysis: Suppose the kernel of the integral operator:

$Kf(s)a$ = $\int_{0}^{s} K(s,t) $ $f(t)$ $ dt$

is a continuous function of s, t in $t\leq s$.

1. Show that $K$ maps $C[0,1]$ to $C[0,1]$

2. Show that the spectrum of $K$ consists of the single point $0$

   It is easy to show first one with the definition of continuous. I am just wondering how could I get the second one. Which also means the $spectrum $ consists of the single point zero. Thanks !