Appendix B: Mathematical Background

Abstract

B.I Self-adjoint operators and quadratic forms In this section, we will introduce basic concepts and results on self-adjoint operators and non-negative quadratic forms on a Hilbert space. One can find detailed accounts and proofs of those subjects in Davies [26]. Also one can refer to classical textbooks, for example, Kato [81], Reed & Simon [156]. Let H be a (real or complex) separable Hilbert space with an inner product (•,•). A linear map H is called a linear operator on 'H if and only if the domain of if, denoted by Dom(if), is a dense subspace of TL and H : Dom(H)-> H. In this appendix, we assume that H is a real Hilbert space for simplicity. Definition B.I.I. Let if be a linear operator on H. (1) H is called symmetric if and only if (Hf,g) = (f,Hg) for any /, g G Dom(if). (2) if is said to be a self-adjoint operator if and only if if is symmetric and Dom(if) = {g E H : there exists h GH such that (Hf,g) = (/, A) for all / G Dom(if)}. (3) A symmetric operator H is called non-negative if (Hf, /) > 0 for all / G Dom(if). Proposition B.1.2. Let H be a non-negative self-adjoint operator. Then there exists a unique non-negative self-adjoint operator G on H such that