Sequences of the Projection-valued Measures and Functional Calculi

Abstract

Let a pair be functional calculus, where is a homomorphism from the space of the measurable functions on into the space of all linear bounded operators on a reflexive Banach space . We define a norm of functional calculus by , the convergence of the sequence of functional calculi is a convergence relative to this norm. We study the correspondence between sequences of spectral decompositions, well-bounded operators defined on the reflexive Banach space , and their correspondence with the theory of functional calculus for such operators. In this article, we establish that if a sequence of the projection-valued measures strongly converges to then the sequence of the functional calculi converges to the functional calculus Results of the article can be employed in the modern extensions of the quantum theory and theory of quantum information.