Spectral Theorem for Compact Hermitian Operators

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

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.

Also see

 * Spectral Theorem for Compact Normal Operators, a more general result