Here again, we let H = L2(Γ\G), δH = °L2(Γ\G), and V be the orthogonal complement of °H in H.

The space H is the Hilbert direct sum of the subspace Hm (m ∈ ℤ), so we already have a spectral decomposition of H with respect to C. The list in 15.8 shows that there are at most two inequivalent irreducible unitary representations of G with the same Casimir operator, so there is in principle not much difference between the decompositions with respect to C and to G. But we want to express the latter in terms of representations.

Lemma. Let (π, E) be a unitary representation of G. Assume the existence of a Dirac sequence {αn} (n ∈ ℕ) with compact operators π(αn). Then E is a Hilbert sum of irreducible representations, with finite multiplicities.

Proof. This is rather elementary and well-known. For the convenience of the reader we reproduce one proof, following 5.8 and 5.9 in [7].

(a) We show first that a closed G-invariant subspace W ≠ 0 contains a closed G-invariant subspace that is minimal among nonzero closed G-invariant subspaces. By 13.1 and 14.2, there exists a j such that π(αj)|w ≠ 0. Let c be a nonzero eigenvalue of π(αj) in W, and let M ≠ 0 be the corresponding (finite dimensional) eigenspace in W.

Compact operators [61, X.5; 46, VI.5]. Let H be a Hilbert space (with a countable basis), ( , ) the scalar product on H, and ℒ(H) the algebra of bounded linear operators on H. If A ∈ ℒ(H) then A* denotes its adjoint – that is, the unique bounded linear operator such that (Ax, y) = (x, A*y) (x, y ∈ H). The operator A*A is self-adjoint and positive (i.e., (A*Ax, x) ≥ 0 for all x ∈ H) ; it has a unique positive square root, called the absolute value |A| of A.

The bounded operator A is compact if it transforms any bounded set into a relatively compact one. The eigenspaces of A corresponding to nonzero eigenvalues are finite dimensional. If A is compact, positive, and self-adjoint, then H has an orthonormal basis {ei} consisting of eigenvectors of A: Aei = λi,ei, with λi → 0. The operator A is compact if and only if |A| is. The compact operators obviously form an ideal in ℒ(H).

More generally, a continuous linear map A: H → H′ of H into a Hilbert space H′ is said to be compact if it transforms any bounded set into a relatively compact one or, equivalently, any bounded sequence into one containing a convergent subsequence.