Kernel of Normal Operator is Kernel of Adjoint

Theorem

Let $H$ be a Hilbert space.

Let $A \in \map B H$ be a normal operator.

Then:

$\ker A = \ker A^*$

where:

$\ker$ denotes kernel

$A^*$ denotes the adjoint of $A$.

Proof

Let $x \in H$ be arbitrary.

Then:

\(\ds x\)

\(\in\)

\(\ds \ker A\)

\(\ds \leadstoandfrom \ \ \)

\(\ds A x\)

\(=\)

\(\ds \mathbf 0_H\)

Definition of Kernel of Linear Transformation

\(\ds \leadstoandfrom \ \ \)

\(\ds \gen {A x, A x}\)

\(=\)

\(\ds 0\)

Definition of Inner Product

\(\ds \leadstoandfrom \ \ \)

\(\ds \gen {A^*Ax, x}\)

\(=\)

\(\ds 0\)

Definition of Adjoint Linear Transformation

\(\ds \leadstoandfrom \ \ \)

\(\ds \gen {A A^* x, x}\)

\(=\)

\(\ds 0\)

Definition of Normal Operator

\(\ds \leadstoandfrom \ \ \)

\(\ds \gen {x, AA^*x}\)

\(=\)

\(\ds 0\)

Definition of Inner Product

\(\ds \leadstoandfrom \ \ \)

\(\ds \gen {A^*x, A^* x}\)

\(=\)

\(\ds 0\)

Definition of Adjoint Linear Transformation

\(\ds \leadstoandfrom \ \ \)

\(\ds A^*x\)

\(=\)

\(\ds \mathbf 0_H\)

Definition of Inner Product

\(\ds \leadstoandfrom \ \ \)

\(\ds x\)

\(\in\)

\(\ds \ker A^*\)

Definition of Kernel of Linear Transformation

Hence, by definition of set equality:

$\ker A = \ker A^*$

$\blacksquare$