Spectral Theory

Abstract

In this section we utilize the knowledge of measure theory at our disposal to prove other versions of the Spectral Theorem for bounded self-adjoint operators—other than the resolution of the identity version given in Chapter 7. Although it is quite possible to generate a spectral measure from the resolution of the identity corresponding to a given self-adjoint operator, we prefer here to prove a more sophisticated spectral measure version of the Spectral Theorem making use of an elegant functional calculus version of the Spectral Theorem. Many of the results of this section will be utilized and duplicated in the next section, where we deal with unbounded operators. Throughout this section, H denotes a complex Hilbert space.