Spectral Theorem for Compact Hermitian Operators

Theorem
Let $H$ be a Hilbert space.

Let $T \in B_0 \left({H}\right)$ be a compact, self-adjoint operator.

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


 * $P_n P_m = P_m P_n = 0$ if $n \ne m$
 * $\displaystyle \lim_{k \to \infty} \left\Vert{T - \sum_{n=1}^k \lambda_n P_n}\right\Vert = 0$, that is, $T = \displaystyle \sum_{n=1}^\infty \lambda_n P_n$

where $P_n$ is the orthogonal projection onto the eigenspace of $\lambda_n$, and $\left\Vert{\cdot}\right\Vert$ denotes the operator norm.

Corollary 1
There exists a (possibly finite) sequence $\left({\mu_n}\right)$ of real numbers and a basis $E = \left({e_n}\right)$ for $\left({\operatorname{ker} T}\right)^\perp$ such that:


 * $\forall h \in H: Th = \displaystyle \sum_{n=1}^\infty \left\langle{h, e_n}\right\rangle_H \mu_n e_n$

Corollary 2
If $T$ has trivial kernel, then $H$ is finite dimensional.

Also see

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