Spectrum of Self-Adjoint Bounded Linear Operator is Real and Closed
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It has been suggested that this page be renamed. In particular: Spectrum of Bounded Self-Adjoint Operator is Real To discuss this page in more detail, feel free to use the talk page. |
Theorem
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $T : \HH \to \HH$ be a bounded self-adjoint operator.
Let $\map \sigma T$ be the spectrum of $T$.
Then:
- $\map \sigma T \subseteq \R$
Proof 1
This follows from:
- Spectrum of Self-Adjoint Densely-Defined Linear Operator is Real and Closed
- Spectrum of Bounded Linear Operator equal to Spectrum as Densely-Defined Linear Operator
$\blacksquare$
Proof 2
For all $\phi \in \HH$ and $\lambda := a + i b \in \C$:
\(\ds \norm {\paren {A - \lambda I} \phi}_\HH^2\) | \(=\) | \(\ds \norm {\paren {A - a I} \phi}_\HH^2 + b^2 \norm {\phi}_\HH^2\) | ||||||||||||
\(\text {(1)}: \quad\) | \(\ds \) | \(\ge\) | \(\ds b^2 \norm {\phi}_\HH^2\) |
Let $b \ne 0$.
In view of $(1)$, both $A - \lambda I$ and $A - \overline \lambda I$ are injective.
Moreover:
\(\ds \paren {\Img {A - \lambda I} }^\perp\) | \(=\) | \(\ds \paren {\Img {A - \overline \lambda I}^\ast }^\perp\) | as $A - \lambda I = \paren {A - \overline \lambda I}^\ast$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \ker \paren {A - \overline \lambda I}\) | Kernel of Linear Transformation is Orthocomplement of Image of Adjoint | |||||||||||
\(\ds \) | \(=\) | \(\ds \set 0\) | as $A - \overline \lambda I$ is injective |
By Linear Subspace Dense iff Zero Orthocomplement, $\Img {A - \lambda I}$ is dense in $\HH$.
Now applying $(1)$ again, we can conclude that $A - \lambda I$ is invertible.
That is, $\lambda \not \in \map \sigma T$.
This needs considerable tedious hard slog to complete it. In particular: more details To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Finish}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |