Let $T=int_{0}^{x}f(y)dt$. Find eigenvalues and range of $T+T^*$

Consider $T=int_{0}^{x}f(y)dt$ as a map from $L^2[0,M]$ to $L^2[0,M]$ find eigenvalues and range of $T+T^*$

I am pretty sure the solution is simple but I would like to make sure I am right.

By Fubini it is easy to see that $T^*=int_{x}^{M}f(y)dy$ Thus $T+T^*=int_{0}^{M}f(y)dy$ Hence the range are all constant functions and the only eigenvalue is $0$ as for any constant $anot=aM$ ($M>0$)