In my functional analysis class, I have encountered the following problem
Let $H$ be a Hilbert space and $T$ a finite-rank (its range is finite-dimensional) and bounded linear operator on $H$. We are asked to show $$ \dim\ker(I-T)=\dim\ker(I-T^*)<\infty. $$
I know that the fact that $T$ is finite-rank (its range is finite-dimensional) is supposed to help, but I cannot see how to use it. I do know that $T$ is compact because it is finite-rank and maybe we need to use spectral theory, but I have no other idea how to show this problem and I am stuck. Any help is appreciated and I thank all helpers.