11.1 Spectral Theorem for Self-Adjoint Operators

The Spectral Theorem for Self-Adjoint Operators is a game-changer in linear algebra. It gives us a way to break down complex operators into simpler parts, like taking apart a puzzle to see how it works.

This theorem lets us understand self-adjoint operators by looking at their spectrum and spectral measure. It's like having a special lens that reveals the hidden structure of these operators, making them easier to work with and understand.

Spectral Theorem for Self-Adjoint Operators

Statement and Properties of the Spectral Theorem

• The Spectral Theorem states that if A is a self-adjoint operator on a Hilbert space H, then there exists a unique spectral measure E on the Borel subsets of R such that A can be represented as the integral of the identity function with respect to E
• The spectral measure E is a projection-valued measure, meaning that for each Borel set B, E(B) is an orthogonal projection on H
• The family of projections {E(B)} satisfies the properties of a measure (countable additivity, normalization, and monotonicity)
• The Spectral Theorem provides a canonical form for self-adjoint operators, analogous to the diagonalization of symmetric matrices in finite-dimensional spaces (eigendecomposition)

Spectrum and Spectral Measure

• The spectrum of a self-adjoint operator A, denoted by σ(A), is the set of all λ ∈ R such that A - λI does not have a bounded inverse, where I is the identity operator
• The Spectral Theorem implies that the spectrum of a self-adjoint operator is real and consists of the support of the spectral measure E
• The spectral measure E assigns a projection E(B) to each Borel set B ⊂ R, which corresponds to the part of the operator A associated with the spectrum in B
• The spectral projections E(B) provide a resolution of the identity operator: ∫R dE(λ) = I

Spectral Decomposition of Self-Adjoint Operators

Representation as an Integral

• The spectral decomposition of a self-adjoint operator A is the representation of A as an integral with respect to its spectral measure E: A = ∫R λ dE(λ)
• The spectral decomposition can be understood as a continuous analog of the eigendecomposition of a symmetric matrix, where the eigenvalues are replaced by the spectrum and the eigenvectors are replaced by the spectral projections
• The spectral projections E(B) correspond to the eigenspaces of A associated with the eigenvalues in the Borel set B
• Example: If A has a discrete spectrum with eigenvalues {λi} and corresponding eigenprojections {Pi}, then A = ∑i λi Pi

Functional Calculus

• The spectral decomposition allows for the functional calculus of self-adjoint operators, where functions of A can be defined by integrating the function with respect to the spectral measure: f(A) = ∫R f(λ) dE(λ)
• The functional calculus provides a way to extend the notion of applying a function to an operator, similar to applying a function to a matrix by applying it to its eigenvalues
• Example: The square root of a positive self-adjoint operator A can be defined as √A = ∫R √λ dE(λ), where √λ is the usual square root function on R
• The functional calculus is a powerful tool for studying the properties of self-adjoint operators and their relationships with other operators

Diagonalization of Self-Adjoint Operators

Diagonalizability and Purely Atomic Spectrum

• Diagonalization of a self-adjoint operator A means finding an orthonormal basis of the Hilbert space H consisting of eigenvectors of A
• The Spectral Theorem implies that a self-adjoint operator A is diagonalizable if and only if its spectral measure E is purely atomic, i.e., the spectrum of A consists only of eigenvalues
• In the case of a diagonalizable self-adjoint operator, the spectral decomposition takes the form A = ∑i λi Pi, where {λi} are the eigenvalues of A and {Pi} are the orthogonal projections onto the corresponding eigenspaces
• Example: A compact self-adjoint operator on a Hilbert space has a purely discrete spectrum and is therefore diagonalizable

Diagonalization Process

• To diagonalize a self-adjoint operator, one needs to find its eigenvalues and eigenvectors, which can be done by solving the eigenvalue equation Ax = λx and using the Spectral Theorem to construct the spectral projections
• The eigenvalues of a self-adjoint operator are real, and the corresponding eigenvectors can be chosen to form an orthonormal basis of the Hilbert space
• The spectral projections Pi are constructed as the orthogonal projections onto the eigenspaces corresponding to each eigenvalue λi
• Diagonalization simplifies the study of self-adjoint operators and their functions, as it allows for the reduction of the operator to a multiplication operator on a direct sum of eigenspaces

Spectrum and Eigenspaces of Self-Adjoint Operators

Decomposition of the Spectrum

• The spectrum of a self-adjoint operator A is a closed subset of R and can be decomposed into three disjoint parts: the point spectrum (eigenvalues), the continuous spectrum, and the residual spectrum (which is always empty for self-adjoint operators)
• Eigenvalues of a self-adjoint operator have finite multiplicity, and the corresponding eigenspaces are orthogonal to each other
• The continuous spectrum of a self-adjoint operator consists of those λ ∈ R for which A - λI has a dense range but is not surjective. The spectral measure E is continuous (non-atomic) on the continuous spectrum
• Example: The position operator in quantum mechanics has a purely continuous spectrum, while the Hamiltonian of a bound system has a discrete spectrum (eigenvalues) and possibly a continuous spectrum above a certain energy threshold

Properties of Eigenspaces

• The spectral projections E(B) associated with disjoint Borel sets B are orthogonal to each other, and their ranges (the corresponding eigenspaces) are orthogonal subspaces of H
• The eigenspaces of a self-adjoint operator corresponding to distinct eigenvalues are orthogonal, and the direct sum of all the eigenspaces is dense in H
• The multiplicity of an eigenvalue λ is the dimension of the corresponding eigenspace, which is equal to the trace of the spectral projection E({λ})
• Example: In quantum mechanics, the eigenspaces of the Hamiltonian operator correspond to the energy levels of the system, and the eigenvectors represent the stationary states of the system
• The orthogonality of eigenspaces allows for the decomposition of the Hilbert space into a direct sum of invariant subspaces under the action of the self-adjoint operator