Spectral Theorem for Compact Hermitian Operators

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Theorem

Let $\HH$ be a Hilbert space.

Let $T \in \map {B_0} \HH$ be a compact Hermitian operator.


Then there exists a (possibly finite) sequence $\sequence {\lambda_n}$ of distinct nonzero eigenvalues of $T$ such that:

$(1): \quad P_n P_m = P_m P_n = 0$ if $n \ne m$
$(2): \quad \ds \lim_{k \mathop \to \infty} \norm {T - \sum_{n \mathop = 1}^k \lambda_n P_n} = 0$, that is, $T = \ds \sum_{n \mathop = 1}^\infty \lambda_n P_n$

where:

$P_n$ is the orthogonal projection onto the eigenspace of $\lambda_n$
$\norm {\, \cdot \,}$ denotes the norm on bounded linear operators.


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Corollary 1

There exists a (possibly finite) sequence $\sequence {\mu_n}$ of real numbers and a basis $E = \sequence {e_n}$ for $\paren {\ker T}^\perp$ such that:

$\forall h \in H: T h = \ds \sum_{n \mathop = 1}^\infty \innerprod h {e_n}_\HH \mu_n e_n$


Corollary 2

If $T$ has trivial kernel, then $\HH$ is finite dimensional.


Proof


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Also see


Sources

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