Let $\Omega=[0,1]$, consider the operator $A:f \rightarrow \int_0^x f(y) dy$ where $f \in L^2(\Omega)$. Compute $||A||$. Hint: Apply the spectral theorem to $A^* A$.
I have already shown that $A: L^2(\Omega) \rightarrow L^2(\Omega)$ and the adjoint operator $A^* f(x) = \int_x^1 f(y)dy$. The spectral theorem states that there is an orthonormal basis consisting of eigenvectors of a compact and self-adjoint operator. And I know that the largest eigenvalue equals the norm of the operator, but I don't know how to compute the eigenvalue and how to make sure the found eigenvalue is the largest. Could anyone help?