Spectral Measures and Spectral Integrals

Abstract

Chapter 4 provides a careful treatment of spectral measures and spectral integrals. In the first section, we define resolutions of the identity and operator-valued Stieltjes integrals by using Riemann sums. The reader may take this elementary approach as a first step into this matter. Next, spectral measures on σ-algebras are introduced and investigated. Standard examples are spectral measures defined by means of multiplication operators or orthogonal sums of projections. We prove the one-to-one correspondence between resolutions of the identity and spectral measures on the Borel algebra of the real line and the theorem on the product of commuting spectral measures. Then integrals of measurable functions with respect to some spectral measure are defined and studied. These spectral integrals are unbounded normal operators on the underlying Hilbert space. The case of an unbounded function is reduced to the bounded case by means a so-called bounding sequence, that is, an increasing sequence of sets such that its union has full spectral measure and the function is bounded on each set. Basic properties of spectral integrals (algebraic properties, spectrum, behavior under transformations of spectral measures, permutation properties) are developed in great detail. These results are applied in later chapters to the study of unbounded self-adjoint operators and their spectra.