Discrete and continuous spectrum of a unitary representation

Definition

Let $U$ be a strongly continuous unitary representation of a locally compact group $G$ on a complex Hilbert space $\mathcal H$ . Its discrete-spectrum subspace $\mathcal H_{\mathrm d}$ is the closed linear span of all irreducible closed $U(G)$ -invariant subspaces of $\mathcal H$ . Its continuous-spectrum subspace is

\mathcal H_{\mathrm c}=\mathcal H_{\mathrm d}^{\perp}.

Both subspaces are invariant, and $U=U_{\mathrm d}\oplus U_{\mathrm c}$ . This orthogonal decomposition is canonical under unitary intertwiners. The representation has purely discrete spectrum when $\mathcal H_{\mathrm d}=\mathcal H$ , and purely continuous spectrum when $\mathcal H_{\mathrm d}=0$ .

Direct sums and direct integrals

The discrete part can be organized as a Hilbert direct sum of irreducible representations, with repetitions recording discrete multiplicity. By construction, the continuous part contains no nonzero irreducible subrepresentation. In type I harmonic analysis it may nevertheless decompose into irreducibles through a direct integral over a nonatomic measure. This is why “continuous” does not mean “indecomposable.”

Standard examples

The regular representation of a compact group is purely discrete: the Peter–Weyl theorem decomposes it as a Hilbert sum of finite-dimensional irreducibles Folland, chapter on compact groups. The translation representation of $\mathbb R$ on $L^2(\mathbb R)$ is purely continuous; Fourier transformation diagonalizes translations over the continuum of characters, but no character occurs as an $L^2$ invariant line.

Conventions and scope

For separable representations of type I groups, the direct-integral formulation and the distinction between atomic and nonatomic spectral measure are treated in Folland, §7.4.

References

Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: the chapter on compact groups and §7.4 on direct-integral decomposition.
Jacques Dixmier, $C^*$ -Algebras, North-Holland, 1977. Publisher record. Relevant: §18.8, disintegration and decomposition of group representations.