Measurable Functional Calculus and Spectral theory

In this article, the spectral theory is considered, we study the spectral families and their correspondence to the operators on the reflexive Banach spaces; assume is a well-bounded operator on reflexive Lebesgue spaces then the operator A is a scalar type spectral operator. It is proven that if a weak spectral family is concentrated on [a,b] then there is a linear well-bounded operator on the reflexive Banach space X such that holds for all and .