7.3 Spectral theory of compact operators

Compact self-adjoint operators on Hilbert spaces have a powerful spectral theorem. This theorem guarantees an orthonormal basis of eigenvectors and real eigenvalues converging to zero, allowing for a spectral representation of the operator.

The spectral theorem enables diagonalization of compact self-adjoint operators, simplifying calculations and analysis. It also provides insights into the operator's spectrum, which consists of eigenvalues and possibly zero, with nonzero eigenvalues having finite multiplicities.

Spectral Theory of Compact Self-Adjoint Operators

Spectral theorem for compact operators

• States that for a compact self-adjoint operator $T$ on a Hilbert space $H$, there exists an orthonormal basis $\{e_n\}$ of $H$ consisting of eigenvectors of $T$
• The corresponding eigenvalues $\{\lambda_n\}$ are real numbers that converge to 0 as $n$ approaches infinity
• Provides a spectral representation of $T$ as $Tx = \sum_{n=1}^{\infty} \lambda_n \langle x, e_n \rangle e_n$ for all $x \in H$, where $\langle \cdot, \cdot \rangle$ denotes the inner product in $H$
• Proof involves showing eigenvalues are real, eigenvectors corresponding to distinct eigenvalues are orthogonal, eigenspaces are finite-dimensional, constructing orthonormal basis using Gram-Schmidt process, and establishing spectral representation using compactness of $T$

Diagonalization of self-adjoint operators

• Compact self-adjoint operators can be diagonalized using the spectral theorem
• Given orthonormal eigenvectors $\{e_n\}$ and corresponding eigenvalues $\{\lambda_n\}$, define diagonal operator $D$ by $Dx = \sum_{n=1}^{\infty} \lambda_n \langle x, e_n \rangle e_n$
• $T$ can be expressed as $T = UDU^*$, where $U$ is a unitary operator with columns $\{e_n\}$ and $U^*$ is its adjoint (conjugate transpose)
• Diagonalization simplifies computations involving powers and functions of $T$ (spectral calculus) and enables analysis of long-term behavior in dynamical systems governed by compact self-adjoint operators

Spectrum of compact operators

• The spectrum $\sigma(T)$ of a compact operator $T$ consists of eigenvalues and possibly 0
• Nonzero eigenvalues have finite multiplicities (dimensions of corresponding eigenspaces)
• For self-adjoint $T$, the spectrum is real and eigenspaces corresponding to distinct eigenvalues are orthogonal
• Nonzero spectrum elements are eigenvalues, and eigenvectors span the range of $T$
• In the self-adjoint case, eigenvectors form an orthonormal basis of the Hilbert space

Orthonormal basis of eigenvectors

• The spectral theorem guarantees the existence of a countable orthonormal basis $\{e_n\}$ of eigenvectors for a compact self-adjoint operator $T$ on a Hilbert space $H$
• The basis is countable due to finite-dimensional eigenspaces and eigenvalues converging to 0
• Hilbert space can be decomposed into a direct sum of eigenspaces $H = \bigoplus_{n=1}^{\infty} E_n$, where $E_n$ is the eigenspace corresponding to eigenvalue $\lambda_n$
• Spectral representation $Tx = \sum_{n=1}^{\infty} \lambda_n \langle x, e_n \rangle e_n$ holds for all $x \in H$, enabling efficient computation of operator powers and functions (functional calculus)