Spectrum of an operator

Question

Let $X=C([0,1],\mathbb{R})$ the Banach space of continuous real functions in $[0,1]$ equipped with the supremum norm. We define the operator $A$ for each $x\in X$ by $$(Ax)(t)=\int_0 ^t x(s)ds, \ \ \ \forall t\in \mathbb{R}.$$ I know that it is a compact operator. How can we find its spectrum $\sigma(A)$ ?

Answer 1

Every nonzero spectral value of a compact operator is an eigenvalue.

So one method to find the spectrum of $A$ is to determine its eigenvalues. (There aren't many.)

What does an identity

$$\lambda x(t) = \int_0^t x(s)\,ds$$

for all $t\in [0,1]$ imply about $x$?