6.4 Spectral theorem for compact self-adjoint operators

The Spectral Theorem for Compact Self-Adjoint Operators is a powerful tool in functional analysis. It shows that these operators have an orthonormal basis of eigenvectors, with real eigenvalues converging to zero if infinite.

This theorem allows us to represent compact self-adjoint operators in a canonical form, diagonalize them, and solve eigenvalue problems. It's crucial for understanding operator behavior and properties in Hilbert spaces.

Spectral Theorem for Compact Self-Adjoint Operators

Spectral theorem for compact operators

• States that if $T$ is a compact self-adjoint operator on a Hilbert space $H$, then $H$ has an orthonormal basis consisting of eigenvectors of $T$
• Eigenvalues of $T$ are real and converge to 0 if there are infinitely many ($\lambda_1, \lambda_2, ...$)
• $T$ can be represented as $Tx = \sum_{n=1}^{\infty} \lambda_n \langle x, e_n \rangle e_n$, where $\lambda_n$ are the eigenvalues and $e_n$ are the corresponding eigenvectors
• Provides a canonical form for compact self-adjoint operators ($T = \sum_{n=1}^{\infty} \lambda_n \langle \cdot, e_n \rangle e_n$)
• Allows for the diagonalization of compact self-adjoint operators ($U^{*}TU = D$, where $D$ is diagonal)
• Enables the solution of eigenvalue problems involving compact self-adjoint operators ($Tx = \lambda x$)
• Establishes a connection between the operator's properties and its eigenvalues and eigenvectors ($\lambda_n \to 0$, $\{e_n\}$ orthonormal basis)

Orthonormal basis of eigenvectors

• Let $T$ be a compact self-adjoint operator on a Hilbert space $H$
• Consider the eigenvalue problem $Tx = \lambda x$
• Eigenvalues are real since $T$ is self-adjoint ($\langle Tx, y \rangle = \langle x, Ty \rangle$)
• Eigenvectors corresponding to distinct eigenvalues are orthogonal ($\langle e_i, e_j \rangle = 0$ for $i \neq j$)
• If $\lambda \neq 0$ is an eigenvalue, then the eigenspace $E_{\lambda} = \{x \in H : Tx = \lambda x\}$ is finite-dimensional because $T - \lambda I$ is compact and has a finite-dimensional null space
• Choose an orthonormal basis for each eigenspace $E_{\lambda}$ ($\{e_1, ..., e_k\}$ for $E_{\lambda}$)
• The union of these bases forms an orthonormal set in $H$ ($\{e_1, e_2, ...\}$)
• If $H$ is not spanned by the eigenvectors, consider the orthogonal complement of the span of eigenvectors ($H \ominus \text{span}\{e_1, e_2, ...\}$)
  • $T$ restricted to this complement is compact and self-adjoint, so 0 is the only possible eigenvalue
• Conclude that the union of the orthonormal bases for the eigenspaces forms an orthonormal basis for $H$

Spectral decomposition of operators

• Let $T$ be a compact self-adjoint operator on a Hilbert space $H$
• Find the eigenvalues $\lambda_n$ and corresponding orthonormal eigenvectors $e_n$ of $T$
• Express $T$ as $Tx = \sum_{n=1}^{\infty} \lambda_n \langle x, e_n \rangle e_n$, which is the spectral decomposition of $T$
• The spectral decomposition can be truncated if there are finitely many eigenvalues ($Tx = \sum_{n=1}^{N} \lambda_n \langle x, e_n \rangle e_n$)
• Provides a way to represent the operator in terms of its eigenvalues and eigenvectors ($T = \sum_{n=1}^{\infty} \lambda_n \langle \cdot, e_n \rangle e_n$)
• Useful for understanding the behavior and properties of the operator ($Tx = \sum_{n=1}^{\infty} \lambda_n \langle x, e_n \rangle e_n$)

Applications of spectral theorem

• Given a compact self-adjoint operator $T$, use the spectral theorem to find the eigenvalues and eigenvectors of $T$ and express $T$ in terms of its spectral decomposition
• Diagonalization:
  1. Let $\{\lambda_n\}$ be the eigenvalues and $\{e_n\}$ be the corresponding orthonormal eigenvectors of $T$
  2. Define $U: H \to H$ by $Ue_n = e_n$ (identity on the basis)
  3. Then $U^{*}TU = D$, where $D$ is a diagonal operator with $\lambda_n$ on the diagonal
• Solving eigenvalue problems:
  1. Use the spectral decomposition to express the eigenvalue problem $Tx = \lambda x$ as $\sum_{n=1}^{\infty} \lambda_n \langle x, e_n \rangle e_n = \lambda x$
  2. Solve for the eigenvalues and eigenvectors using the properties of the spectral decomposition ($\lambda = \lambda_n$, $x = e_n$)
• Approximation of compact self-adjoint operators ($T \approx \sum_{n=1}^{N} \lambda_n \langle \cdot, e_n \rangle e_n$)
• Analysis of integral equations with compact self-adjoint integral operators ($Tx(s) = \int_a^b K(s,t)x(t)dt$)