Spectral Theorem for Compact Self-Adjoint Operators

Overview

The spectral theorem for compact self-adjoint operators is the infinite-dimensional analogue of the diagonalisation theorem for symmetric matrices. It states that every compact self-adjoint operator on a Hilbert space has a countable set of real eigenvalues accumulating only at 0, and the corresponding eigenvectors form a complete orthonormal system. This gives a precise spectral decomposition analogous to diagonalisation, and is the foundation for spectral theory in quantum mechanics and the study of PDEs.

Intuition

A self-adjoint operator T on a Hilbert space is like a symmetric matrix: it has real 'eigenvalues' and orthogonal 'eigenvectors'. When T is also compact (it maps bounded sets to precompact sets), the eigenvalues can only accumulate at 0, and the eigenvectors form an orthonormal basis. Every vector x can be written x = sum_n <x,e_n> e_n, and then Tx = sum_n lambda_n <x,e_n> e_n, exactly like matrix diagonalisation but in infinite dimensions.

Formal Definition

Definition

An operator T: H -> H is self-adjoint if <Tx,y> = <x,Ty> for all x,y in H. It is compact if T maps bounded sets to precompact sets (or equivalently, x_n -> x weakly implies Tx_n -> Tx strongly). The spectral theorem states: every compact self-adjoint operator T on a separable Hilbert space H has a sequence of real eigenvalues (lambda_n) with |lambda_n| -> 0, and corresponding orthonormal eigenvectors (e_n) forming a complete orthonormal system (basis). The operator is diagonalised by this system.

Te_n = \lambda_n e_n,\quad \lambda_n \in \mathbb{R},\quad |\lambda_n| \to 0

Eigenvalue equation

Tx = \sum_{n=1}^\infty \lambda_n \langle x, e_n \rangle\, e_n \quad \forall x \in H

Spectral decomposition

\|T\| = \sup_n |\lambda_n| = |\lambda_1|

Operator norm as spectral radius

f(T) = \sum_{n=1}^\infty f(\lambda_n)\langle\cdot, e_n\rangle\, e_n

Functional calculus

Notation

Notation	Meaning
$\sigma(T)$	Spectrum of T (set of lambda for which T - lambda I is not invertible)
$\sigma_p(T)$	Point spectrum (set of eigenvalues)
$T^*$	Adjoint operator: <T*x,y> = <x,Ty>
$f(T)$	Functional calculus: apply f to the eigenvalues

Theorems

Theorem 9.1: Spectral Theorem for Compact Self-Adjoint Operators

Let T be a compact self-adjoint operator on a separable Hilbert space H. Then T has countably many real eigenvalues (lambda_n) with |lambda_n| -> 0, and corresponding orthonormal eigenvectors (e_n) forming a complete orthonormal system. Tx = sum_n lambda_n <x,e_n> e_n for all x in H.

Theorem 9.2: Eigenvalues of Self-Adjoint Operators are Real

If T is self-adjoint and Te = lambda e for some e != 0, then lambda is real. Eigenvectors corresponding to distinct eigenvalues are orthogonal.

Theorem 9.3: Continuous Functional Calculus

For a compact self-adjoint T with spectral decomposition Tx = sum lambda_n <x,e_n> e_n, and any continuous function f on sigma(T): f(T)x = sum f(lambda_n) <x,e_n> e_n. The map f |-> f(T) is a continuous algebra homomorphism.

Theorem 9.4: Min-Max Theorem (Courant-Fisher)

The eigenvalues lambda_1 >= lambda_2 >= ... of a compact self-adjoint operator T (ordered by size) satisfy lambda_n = min_{dim V = n-1} max_{x perp V, ||x||=1} <Tx,x>.

Worked Examples

1
Suppose Te = lambda e with e != 0. Then lambda ||e||^2 = <lambda e, e> = <Te, e> = <e, Te> = <e, lambda e> = conj(lambda) ||e||^2.
$\lambda \|e\|^2 = \langle Te, e \rangle = \langle e, Te \rangle = \overline{\lambda}\|e\|^2$
2
Since ||e||^2 != 0, we get lambda = conj(lambda), so lambda is real.
3
Now suppose Te_1 = lambda_1 e_1 and Te_2 = lambda_2 e_2 with lambda_1 != lambda_2. Then:
$\lambda_1 \langle e_1, e_2\rangle = \langle Te_1, e_2\rangle = \langle e_1, Te_2\rangle = \lambda_2 \langle e_1, e_2\rangle$
4
Since lambda_1 != lambda_2, we get <e_1, e_2> = 0, i.e., e_1 perp e_2.

✓ Answer

Eigenvalues are real (by self-adjointness). Eigenvectors for distinct eigenvalues are orthogonal.

Practice Problems

Hardproof writing

Prove that a compact self-adjoint operator T on a Hilbert space attains its norm: there exists a unit vector e with ||Te|| = ||T||, and e is an eigenvector.

Mediumfree response

Describe how the spectral theorem for compact self-adjoint operators generalises the diagonalisation theorem from linear algebra.

MediumMultiple choice

Which property of compact self-adjoint operators ensures their eigenvalues accumulate only at 0?

Self-adjointness forces eigenvalues to be realCompactness ensures eigenvectors for distinct eigenvalues are orthogonalCompactness prevents bounded sequences of eigenvectors from having norm-convergent subsequences in different eigenspacesSelf-adjointness ensures the spectrum is contained in [-||T||, ||T||]

Common Mistakes

Common Mistake

Confusing compact operators with bounded operators.

All compact operators are bounded, but not all bounded operators are compact. Compact operators map bounded sets to precompact (totally bounded) sets; the identity on an infinite-dimensional Hilbert space is bounded but not compact.

Common Mistake

Thinking the spectral decomposition requires all eigenvalues to be nonzero.

Compact operators may have a large kernel. The spectral decomposition gives Tx = sum lambda_n <x,e_n> e_n where the sum is over nonzero eigenvalues. Elements of ker(T) contribute nothing to the sum.

Quiz

Historical Background

David Hilbert developed the spectral theory of self-adjoint operators in his study of integral equations (1906-1910), extending the classical theory of symmetric matrices. Erhard Schmidt and Friedrich Riesz contributed key results on compact operators. The modern abstract formulation was given by von Neumann in the 1920s-30s. The continuous functional calculus for self-adjoint operators was developed in the 1940s-50s.

1906
Hilbert develops spectral theory for symmetric integral operators
David Hilbert
1907
Riesz and Schmidt study compact operators (Fredholm operators)
Frédéric Riesz, Erhard Schmidt
1928
Von Neumann gives the abstract Hilbert space formulation
John von Neumann
1932
Von Neumann's book formalises spectral theory for unbounded self-adjoint operators
John von Neumann

Summary

Compact self-adjoint operators are the infinite-dimensional analogue of symmetric matrices: they have real eigenvalues accumulating only at 0.
The spectral theorem gives a complete orthonormal basis of eigenvectors, diagonalising T via Tx = sum lambda_n <x,e_n> e_n.
The continuous functional calculus defines f(T) = sum f(lambda_n) <·,e_n> e_n for continuous f on sigma(T).
Self-adjointness forces eigenvalues to be real and eigenvectors for distinct eigenvalues to be orthogonal.
Applications include quantum mechanics (Hamiltonian operators), integral equations (Sturm-Liouville theory), and the study of PDEs.

References

BookRudin, W. — Functional Analysis (2nd ed.), Chapter 12
BookConway, J. — A Course in Functional Analysis, Chapter 2
WebsiteWikipedia: Spectral theorem