Bounded inverse for a closed range closed operator

Question

I am dealing with an unbounded operator $T$ on an Hilbert space $H$. I am interested in proving that it has a bounded inverse $T^{-1}$. I managed to prove that the operator is closed and self-adjoint. By the Closed Range theorem I have that $\mathcal{R}(T)=\ker(T)^{\perp}$. If $\ker T$ would have been trivial then $T^{-1}$ would have been defined on the whole space and by the Closed Graph Theorem it had to bounded. Unfortunately $\ker T \neq \{0\}$, but I know that $\dim \ker < \infty$. Is it still possible to have a bounded inverse? Thanks in advance for the answers.