Measurable functional calculi and spectral theory in the reflexive Banach spaces

Abstract

In this paper, we consider the spectral theory. More precisely, we study the spectral families and their corresponding operators on reflexive Banach spaces. Assume [Formula: see text] is a well-bounded operator on reflexive Lebesgue spaces then the operator [Formula: see text] is a scalar-type spectral operator. We establish that if a weak spectral family [Formula: see text] is concentrated on [Formula: see text] then there exists a linear well-bounded operator [Formula: see text] on the reflexive Banach space [Formula: see text], such that [Formula: see text] holds for all [Formula: see text] and [Formula: see text]. We prove that by assuming [Formula: see text] is a functional calculus on the measurable space [Formula: see text], then there are a semi-finite measure space [Formula: see text] and solitary operator [Formula: see text], and an injective pointwise continuous *-homomorphism [Formula: see text], such that [Formula: see text] holds for any operator of the multiplication [Formula: see text] that multiplies by function [Formula: see text].