SPECTRAL TRAJECTORIES AND DUALITY

Let E be a spectal measure on a ring of sets E,.with values in the set of projection operators in a Hilbert space H. An H-valued set function ~ on L is called a spectral trajectory controlled by the measure E if for any ~I ' ~2 E E E(~1)~(~2) = ~(6In~2) ' In other words ~ is a countably additive orthogonally scattered measure on E, controlled by E (cf. [7], [9]) • For a given locally convex space SR originating from a "generating" family R of operators on the Hilbert space H (cf. the authors ' papers [2,3,5]). It is proved that the topological dual Si is isomorphic to the space TR of R-bounded spectral traj ectories controlled by the joint spectral measure of the family R. The present paper contains a study of the duality between the spaces SR and TR, and of the topological properties of the space TR• It extends the results of the authors ' papers [2,3,4,5], which are the main references to the subject.-3