Let 𝒜 be a family commuting
selfadjoint of (normal) operators in a complex (not necessarily separable) Hilbert
space H. A natural triplet ϕ ⊂ H ⊂ ϕ′ is described, such that (1) 𝒜 possesses a
complete system of joint generalized eigenvectors in ϕ′; (2) the joint generalized point
spectrum of 𝒜 essentially coincides with the joint spectrum of 𝒜; (3) the generalized
point spectra, generalized spectra and spectra essentially coincide for all A ∈𝒜; (4)
the simultaneous diagonalization of 𝒜 in H by means of its spectral measure
extends to ϕ′. Also the multiplicity of the joint generalized eigenvectors of 𝒜 is
discussed.
