The relationship between the operator and its adjoint operator

Question

Let $V$ be a Hilbert space, $T: V \rightarrow V$ be a bounded linear operator, $T^*$ be the adjoint operator of $T$. Suppose that $T$ and $T^*$ have the same kernel, and the range of $T$ is $V_1$, where $V_1$ is a closed subspace of $V$. How to prove that the range of $T^*$ is $V_1$?