Compact Operators - Main Theorem

The Spectral Theorem for Compact Self-Adjoint Operators extends the diagonalization of symmetric matrices to infinite dimensions, providing a complete understanding of their structure.

TheoremSpectral Theorem for Compact Self-Adjoint Operators

Let $H$ be a separable Hilbert space and $T : H \to H$ a compact self-adjoint operator. Then:

All eigenvalues of $T$ are real
Eigenvectors corresponding to distinct eigenvalues are orthogonal
There exists an orthonormal basis $\{e_n\}$ of $H$ consisting of eigenvectors of $T$
The eigenvalues $\{\lambda_n\}$ satisfy $|\lambda_1| \geq |\lambda_2| \geq \cdots$ and $\lambda_n \to 0$
For any $x \in H$ , $T(x) = \sum_{n=1}^\infty \lambda_n \langle x, e_n \rangle e_n$

This theorem shows that compact self-adjoint operators are "diagonalizable" with respect to an orthonormal basis, just like finite-dimensional symmetric matrices.

Proof

The proof proceeds by constructively finding eigenvalues and eigenvectors:

Let $\lambda_1 = \sup_{\|x\|=1} |\langle Tx, x \rangle|$ . There exists a unit vector $e_1$ with $Te_1 = \lambda_1 e_1$
Consider $H_1 = \{e_1\}^\perp$ and restrict $T$ to $H_1$ . Since $T$ is self-adjoint, $T(H_1) \subset H_1$
Repeat the process on $H_1$ to find $\lambda_2$ and $e_2$ , then continue inductively
Compactness ensures $\lambda_n \to 0$
The span of $\{e_n\}$ is dense in $H$ , giving an orthonormal basis

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ExampleApplications

Sturm-Liouville Problems: The operator $Lu = -(p(x)u')' + q(x)u$ on $L^2[a,b]$ with appropriate boundary conditions has compact resolvent, so its eigenvalue problem has complete orthonormal solutions
Vibrating String: Normal modes of a vibrating string are eigenfunctions of a compact self-adjoint operator
Heat Equation: Solutions can be expanded in eigenfunctions: $u(x,t) = \sum c_n e^{-\lambda_n t} e_n(x)$
Quantum Mechanics: Energy eigenstates of certain Hamiltonians form complete orthonormal systems

Remark

This theorem is one of the most important results in functional analysis. It provides the theoretical foundation for Fourier series, quantum mechanics, and the solution of PDEs via separation of variables and eigenfunction expansions.

The spectral theorem transforms abstract operator theory into concrete computations with eigenvalues and eigenvectors.