Bounded Operators and Their Spectra

Operators are the observables and symmetries of quantum theory. This page develops the algebra of bounded operators on a Hilbert space — adjoints and the special classes (self-adjoint, unitary, normal, compact) — and the spectrum, the infinite-dimensional replacement for the set of eigenvalues. It builds on hilbert-spaces.md; the spectral theorem that diagonalizes these operators is spectral-theorem.md.

The adjoint

For a bounded operator $T$ on a Hilbert space $H$ , the adjoint $T^{†}$ (mathematicians: $T^{*}$ ) is the unique bounded operator with $⟨ T^{†} x, y ⟩ = ⟨ x, T y ⟩ for all x, y .$ It exists by Riesz representation, satisfies $(ST)^{†} = T^{†} S^{†}$ , $T^{††} = T$ , and the C*-identity $∥ T^{†} T ∥ = ∥ T ∥^{2}$ — the property that makes $B (H)$ a C*-algebra (operator-algebras.md).

Special classes

The adjoint sorts operators into the physically important classes:

Class	Definition	Role
self-adjoint	$T = T^{†}$	observables (real spectrum)
unitary	$U^{†} U = U U^{†} = 1$	symmetries / dynamics (preserve inner product)
normal	$T T^{†} = T^{†} T$	diagonalizable (spectral theorem applies)
projection	$P = P^{2} = P^{†}$	measurements (project onto a subspace)
positive	$⟨ x, T x ⟩ \geq 0$	densities, $T = S^{†} S$

Self-adjoint and unitary operators are both normal; normality is exactly the condition for the spectral theorem to diagonalize $T$ . Unitary operators are the norm-preserving bijections — the Hilbert-space isometries — and represent both symmetries and time evolution ( $e^{- i H t}$ ).

The spectrum

In infinite dimensions "eigenvalue" is too narrow. The resolvent set of $T$ is ${λ : (T - λ 1)^{- 1} exists and is bounded}$ , and the spectrum $σ (T)$ is its complement. The spectrum is always non-empty, compact, and contained in ${∣ λ ∣ \leq ∥ T ∥}$ . It splits into three pieces:

Point spectrum $σ_{p}$ : $T - λ$ not injective — genuine eigenvalues with eigenvectors in $H$ (discrete/bound states).
Continuous spectrum $σ_{c}$ : $T - λ$ injective with dense but not closed range — no eigenvector in $H$ , but "approximate" ones (scattering states; position/momentum). This is the piece with no finite- dimensional analogue, and the reason for rigged Hilbert space.
Residual spectrum $σ_{r}$ : range not dense (empty for normal operators).

Key facts: a self-adjoint operator has real spectrum; a unitary operator has spectrum on the unit circle; a projection has spectrum $\subseteq {0, 1}$ . These constraints are why observables (self-adjoint) yield real measurement values.

Compact operators

An operator is compact if it maps bounded sets to relatively compact ones — equivalently, a norm-limit of finite-rank operators. Compact operators are the "closest to finite-dimensional":

Spectral theorem for compact self-adjoint operators. A compact self-adjoint operator has a discrete spectrum: an orthonormal basis of eigenvectors with real eigenvalues $λ_{n} \to 0$ . Thus $T = \sum_{n} λ_{n} ∣ e_{n} ⟩ ⟨ e_{n} ∣$ .

This is the exact infinite-dimensional analogue of diagonalizing a Hermitian matrix, and it is why Hamiltonians with compact resolvent (e.g. a particle in a box, the harmonic oscillator) have discrete spectra and honest eigenfunctions. The general (non-compact) case — position, momentum, free Hamiltonian — needs the full spectral theorem.

References

Reed & Simon, Methods of Modern Mathematical Physics I, Ch. VI.
Rudin, Functional Analysis, Ch. 12–13.
Hall, Quantum Theory for Mathematicians, Ch. 6–7.