From Classical Mechanics to Quantum Field Theory

108 From Classical Mechanics to Quantum Field Theory. A Tutorial

rI)ψ||. Therefore

||(Xm−rI)−^1 ||≥

1

||(Xm−rI)ψ||

For every fixed >0, it is simply constructedψ ∈D(Xm) with||ψ||=1and
||(Xm−rI)ψ||< . Therefore (Xm−rI)−^1 cannot be bounded and thusr∈
σc(Xm). In view of (e) in remark 2.2.42, we also conclude that

σ(Pm)=σc(Pm)=R, (2.46)

just because the momentum operatorPmis related to the position one by means
of a unitary operator given by the Fourier-Plancherel operatorFˆas discussed in
(2) of example 2.2.39.

2.2.5 Spectralmeasures

Definition 2.2.45.LetHbe a complex Hilbert space.P∈B(H)is calledorthog-
onal projectorwhenPP=P andP†=P.L(H)denotes the set of orthogonal
projectors ofH.

We have the well known relation between orthogonal projectors and closed sub-
spaces[8; 5]

Proposition 2.2.46. IfP ∈L(H),thenP(H)is a closed subspace. IfH 0 ⊂H
is a closed subspace, there exists exactly oneP ∈L(H)such thatP(H)=H 0.
Finally,I−P∈L(H)and it projects ontoH⊥ 0 (e.g., see[ 5 ]).

We can now state one of the most important definitions in spectral theory.

Definition 2.2.47.LetHbe a complex Hilbert space andΣ(X)aσ-algebra over
X.Aprojection-valued measure (PVM)(also known asresolution of the
identity)onX,P,isamapΣ(X)E→PE∈L(H)such that:

(i)PX=I;

(ii)PEPF=PE∩F;

(iii)IfN⊂Nand{Ek}k∈N⊂Σ(X)satisfiesEj∩Ek=∅fork=j,then

∑

j∈N

PEjx=P∪j∈NEjx for everyx∈H.

(IfNis infinite, the sum on the left hand side of (iii) is computed referring

to the topology ofH.)