Kernel of Linear Transformation is Orthocomplement of Image of Adjoint/Corollary

Corollary to Kernel of Linear Transformation is Orthocomplement of Image of Adjoint

Let $\HH$ be a Hilbert space.

Let $\map \BB \HH$ denote the set of normal operators on $\HH$.

Let $A \in \map \BB \HH$ be a normal operator.

Then:

$\ker A = \paren {\Img A}^\perp$

where:

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

$\ker A$ is the kernel of $A$

$\Img A$ is the image of $A$

$\perp$ signifies orthocomplementation

Proof

From Kernel of Linear Transformation is Orthocomplement of Image of Adjoint, we have:

$\ker A = \paren {\Img {A^\ast} }^\perp$

From Kernel of Normal Operator is Kernel of Adjoint, we then have:

$\ker A = \ker A^\ast$

and so:

$\ker A^\ast = \paren {\Img {A^\ast} }^\perp$

Substituting $A^\ast$ for $A$ we obtain:

$\ker A^{\ast \ast} = \paren {\Img {A^{\ast \ast} } }^\perp$

From Adjoint is Involutive, we can conclude:

$\ker A = \paren {\Img A}^\perp$

$\blacksquare$