Compute the spectrum for a operator

Question

Find the spectrum of the operator $$ \begin{split} A & \colon C[0,1] \rightarrow C[0,1] \\ & f \mapsto (Af)(x) := f(x) + \int_0^x f(t)dt \end{split} $$

P.S.: I know the spectrum of $A_1\colon f \mapsto \int_0^x f(t)dt$

Answer 1

The spectrum comes from analyzing $A-\lambda I$. Since $A = A_1+I$, you can see that $A-\lambda I = A_1-(\lambda-1)I$. It follows that $\sigma(A) = \sigma(A_1) +\{1\}$.