Kernel of Linear Transformation is Orthocomplement of Image of Adjoint/Corollary
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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:
- $\ker A = \paren {\Img A}^\perp$
$\blacksquare$