I'm trying to find the spectrum of the operator $T: C[0,1] \to C[0,1]$ given by:
$$T(f)(t) = f(0) + \int_0 ^{t} f(s) ds$$
I can show that $0$ is contained in the approximate point spectrum with $f_n(t) = t^n$ and also that $1$ is in the point spectrum with eigenvector $f(t) = \exp(t)$.
I'm trying to show that in fact $\sigma_p(T) = \{1\}$. If we look at the subspace of differentiable functions then it's quite clear that the only eigenvalue is 1 (as the eigenfunction satisfies a differential equation) however I'm unsure how to extend this argument to the whole space?
Thanks for any help