Spectral Theorem for Bounded Self-Adjoint Operators

Every bounded self-adjoint operator can be represented as multiplication by a bounded real function. The representation need not be an eigenvector expansion: a bounded observable can have entirely continuous spectrum. The bounded theorem is also the starting point for constructions of the unbounded spectral theorem.

Helpful background. Bounded Operators supplies norm continuity and adjoints. Familiarity with continuous functions on compact sets, finite Borel measures, and $L^2$ completion is assumed.

The bounded spectral representation

Let $A=A^*\in\mathcal B(\mathcal H)$. There is a unique projection-valued measure $E_A$ supported on the compact set $\sigma(A)\subseteq[-\|A\|,\|A\|]$ such that

$$A=\int_{\sigma(A)}\lambda\,dE_A(\lambda).$$

The integral is defined on every vector. For every bounded Borel $f$, spectral integration defines $f(A)$, and continuous $f$ satisfy

$$\|f(A)\|=\max_{\lambda\in\sigma(A)}|f(\lambda)|.$$

The measure’s additivity is in the strong operator topology. Its full measure-theoretic construction and event interpretation belong to Projection-Valued Measures. This page explains the route from elementary bounded-operator algebra to a multiplication representation.

The theorem extends to bounded normal operators, with a PVM on a compact subset of $\mathbb C$. General bounded operators need not have this representation; a nontrivial Jordan block is already a finite-dimensional counterexample.

From polynomials to continuous functions

Polynomials are initially unambiguous:

$$p(A)=\sum_{k=0}^n c_k A^k.$$

A key lemma is the norm identity $\|p(A)\|=\max_{\sigma(A)}|p|$. One route proves it from polynomial spectral mapping together with the equality of norm and spectral radius for normal operators. The latter follows from the $C^*$ identity and the spectral-radius formula, rather than assuming an eigenbasis.

By uniform polynomial approximation on the compact real spectrum, each continuous $f$ can be approximated by polynomials $p_n$. The norm identity makes $p_n(A)$ a Cauchy sequence and defines

$$f(A)=\lim_{n\to\infty}p_n(A)$$

in operator norm. The result is independent of the approximating sequence. Multiplication and complex conjugation pass to the limit, producing a unital isometric $*$-homomorphism from $C(\sigma(A))$ into $\mathcal B(\mathcal H)$.

Positivity follows because a nonnegative continuous $f$ has a continuous square root: $f(A)=\sqrt f(A)^*\sqrt f(A)\geq0$. This is the bridge from operator algebra to measures.

A cyclic vector turns the operator into multiplication

A vector $\Omega$ is cyclic for $A$ when vectors $p(A)\Omega$ span a dense subspace. For normalized $\Omega$, the functional

$$L(f)=\langle\Omega,f(A)\Omega\rangle$$

is positive and satisfies $L(1)=1$. The Riesz representation theorem gives a probability measure $\mu$ on $\sigma(A)$ with $L(f)=\int f\,d\mu$.

Define initially

$$W\bigl(p(A)\Omega\bigr)=p.$$

This is well-defined in $L^2(\mu)$ because

$$\|p(A)\Omega\|^2 =\langle\Omega,|p|^2(A)\Omega\rangle =\int|p(\lambda)|^2\,d\mu(\lambda).$$

Polynomial functions are dense in $L^2(\mu)$, and cyclicity supplies density on the operator side. Thus $W$ extends to a unitary map $\mathcal H\to L^2(\mu)$. Multiplication of polynomials shows

$$WAW^{-1}=M_\lambda.$$

The spectral projections now arise concretely as $E_A(\Delta)=W^{-1}M_{\mathbf1_\Delta}W$. Their strong additivity follows from dominated convergence in $L^2$.

This proves the cyclic case once the functional-calculus lemmas and the two approximation theorems have been supplied. It is not an argument that an arbitrary vector is cyclic.

General operators require several cyclic pieces

For any $\Omega$, the closed span of $p(A)\Omega$ is invariant under $A$. Since $A$ is self-adjoint, its orthogonal complement is invariant too, so the subspace is reducing.

Decomposing $\mathcal H$ into orthogonal cyclic reducing subspaces and applying the construction to each yields a direct sum of multiplication representations. In a separable Hilbert space only countably many nonzero orthogonal summands are needed. The repeated spectral channels carry multiplicity.

Uniqueness of the PVM follows because its integrals against continuous functions must equal the already determined continuous calculus. Uniqueness of scalar measures, followed by polarization of matrix elements, fixes every projection.

This is a proof architecture with its substantive ingredients identified. It shows why compactness of the spectrum, positivity, and cyclic decomposition enter the theorem, rather than treating the theorem as formal diagonalization.

A bounded observable with no eigenvectors

On $L^2([0,1],dx)$ take $A=M_x$. The constant function $1$ is cyclic because polynomials are dense in this $L^2$ space. The preceding construction returns Lebesgue measure and the original multiplication representation.

An eigenvector at $\lambda$ would be supported on $\{x=\lambda\}$, a null set, so none exists. Nevertheless $\sigma(A)=[0,1]$. Compactness of the spectrum is not compactness of the operator: this $A$ is bounded but not compact.

Exercises

Let $A$ be a Hermitian matrix with an eigenvalue of multiplicity two. Can one vector be cyclic for $A$?

Solution

No. In that eigenspace, every $p(A)\Omega$ is a scalar multiple of the single projection of $\Omega$ onto the eigenspace. Those vectors cannot span a two-dimensional eigenspace. More cyclic channels, or additional commuting observables, are needed to resolve the multiplicity.

For bounded self-adjoint $A$ and continuous $f$, explain why the construction is stable under uniform approximation but not automatically under pointwise approximation.

Solution

The norm identity gives $\|f_n(A)-f(A)\|=\|f_n-f\|_{C(\sigma(A))}$. Uniform convergence therefore controls the operator norm. Pointwise convergence lacks that bound; with a uniform bound it gives strong convergence through spectral dominated convergence, but need not give norm convergence.

References

• B. C. Hall, Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, Springer, 2013, doi:10.1007/978-1-4614-7116-5, Chapters 7–8 — statements and proofs of the bounded self-adjoint spectral theorem.
• R. B. Melrose, Functional Analysis, MIT lecture notes, 2020, course reading page — bounded-operator and spectral preliminaries.
• G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., American Mathematical Society, 2014, author’s book page — spectral representations and functional calculus.