Spectral Decompositions of Self-adjoint and Normal Operators

Abstract

Chapter 5 is devoted to the spectral decomposition of self-adjoint and normal operators. In the first section, the spectral theorem for a single bounded self-adjoint operator is proved. Then the spectral theorem for an unbounded self-adjoint operator is derived from the bounded case by using the bounded transform T(I+T ^∗ T)^−1/2. The spectral integrals with respect to the corresponding spectral measure are considered as functions of the self-adjoint operator. This functional calculus and a number of important applications (spectrum, fractional powers, Stone’s formulas) are developed. Self-adjoint operators with simple spectra are studied. For an n-tuple of strongly commuting unbounded normal operators, the spectral theorem is proved, and the joint spectrum is defined and investigated. Permutability problems involving unbounded self-adjoint or normal operators are considered, and a number of equivalent characterizations of the strong commutavitity in terms of spectral measures, resolvents, and bounded transforms are given.