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 $A$.

Proof
Let $x \in H$ be arbitrary. Then:

Hence, by definition of set equality:


 * $\operatorname{ker} A = \operatorname{ker} A^*$