From Classical Mechanics to Quantum Field Theory

Mathematical Foundations of Quantum Mechanics 139

if at least one of the Tk belongs to B 1 (H), the remaining ones are in

B(H),andπ:{ 1 ,...,n}→{ 1 ,...,n}is acyclicpermutation.

The trace ofT∈B 1 (H) can be computed on a basis of eigenvectors in view of the
following further result[5; 6]. Actually (d) and (e) easily follow from (a), (b), (c),
(d), and the spectral theory previously developed.

Proposition 2.3.19.LetHbe a complex Hilbert space andT†=T ∈B 1 (H).
The following facts hold.

(a)σ(T)\{ 0 }=σp(T)\{ 0 }.If 0 ∈σ(T)it may be either the unique element

ofσc(T)or an element ofσp(T).

(b)Every eigenspaceHλhas finite dimensiondλprovidedλ=0.

(c)σp(T)is made of at most countable number of reals such that

(i) 0is unique possible accumulation point;

(ii)||T||=maxλ∈σp(T)|λ|.

(d)There is a Hilbert basis of eigenvectors{xλ,a}λ∈σp(T),a=1, 2 ,...,dλ(d 0 may

be infinite) and

tr(T)=

∑

λ∈σp(T)

dλλ,

where the sum converges absolutely (and thus can be arbitrarily re-

ordered).

(e)Referring to the basis presented in (d), the spectral decomposition ofT

reads

T=

∑

λ∈σp(T)

λPλ

wherePλ=

∑

a=1, 2 ,...,dλ〈xλ,s,〉xλ,aand the sum is computed in the strong

operator topology and can be re-ordered arbitarily. The convergence holds

in the uniform topology too if the set of eigenspaces are suitably ordered

in the count.

Corollary 2.3.20.tr :B 1 (H)→Cis continuous with respect to the norm|| || 1
because|trT|≤tr|T|=||T|| 1 ifT∈B 1 (H).

Proof. IfT∈B(H), we have thepolar decompositionT=U|T|(see, e.g.,[ 5 ])
whereU∈B(H)isisometriconKer(T)⊥andKer(U)=Ker(T)=Ker(|T|)so
that, in particular||U||≤1. LetNbe a Hilbertian basis ofHmade of eigenvectors