A POINTWISE SPECTRUM AND REPRESENTATION OF OPERATORS

Abstract. For a self-adjoint operator A: H → H commuting with an increas-ing family of projections P = (Pt) we study the multifunction t → ΓT (t) =⋂{σI: I an open set of the topology T containing t}, where σI is the spec-trum of A on PIH. Let mP be the measure of maximal spectral type. We study the condition that ΓT is essentially a singleton, mP{t: ΓT (t) is not a singleton} = 0. We show that if T is the density topology and if mP satises the density theorem, in particular if it is absolutely continuous with respect to the Lebesgue measure, then this condition is equivalent to the fact that A is a Borel function of P. If T is the usual topology then the condition is equivalent to the fact that A is approched in norm by step functions n2N ΓT (αn)〈PInf, f〉, where the set of intervals {In: n ∈ N} covers the set where ΓT is a singleton. 1