Kernel of Normal Operator is Kernel of Adjoint

Theorem
Let $H$ be a Hilbert space.

Let $A \in B \left({H}\right)$ be a normal operator.

Then $\operatorname{ker}(A) = \operatorname{ker}(A^*)$, where
 * $\operatorname{ker}$ denotes kernel,
 * $A^*$ denotes the adjoint of $A$.