Kernel of the closure of operators and finite dimensionality

Question

Let $H_1$ and $H_2$ be Hilbert spaces, and let $T:H_1 \to H_2$ be a densely defined unbounded operator with finite dimensional kernel. If we assume that $T$ is closable, with closure $S$, when can one conclude that the kernel of $S$ is also finite dimensional? For example, will this work if $T$ is essentially self-adjoint?