The theory of spectral
operators, when applied to eigenfunction expansions, covers the unconditionally
convergent case. However, by perturbing certain spectral differential operators, J.
Schwartz has obtained differential operators which are not spectral but whose
eigenfunctions span the whole space. In this paper we show how new norms can be
constructed so as to make these perturbed differential operators spectral. This we
achieve by showing that such operators have an underlying generalized spectral
measure (as defined by V. È. Ljance) and that every generalized spectral measure is
essentially a C-spectral measure.
