A pointwise spectrum and representation of operators
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Date
1998
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AMER MATHEMATICAL SOC
Abstract
For a self-adjoint operator A : H --> H commuting with an increasing family of projections P = (P-t) we study the multifunction t --> Gamma(T) (t) = boolean AND {sigma(I) : I an open set of the topology T containing t}, where sigma(I) is the spectrum of A on PIH, Let m(p) be the measure of maximal spectral type. We study the condition that Gamma(T) is essentially a singleton, m(p){t : Gamma(T) (t) is not a singleton} = 0. We show that if T is the density topology and if m(p) satisfies 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 approached in norm by step functions Sigma(n epsilon N) Gamma(T) (alpha(n)) [P(In)f, f], where the set of intervals {I-n : n epsilon N} covers the set where Gamma(T) is a singleton.