From Classical Mechanics to Quantum Field Theory

138 From Classical Mechanics to Quantum Field Theory. A Tutorial

Remark 2.3.16. Notice that, above,T†T is selfadjoint andσ(T†T)∈[0,+∞),
so that

√

T†Tis well defined as a function ofT†T.

Trace class operators enjoy several remarkable properties[5; 6]. Hereweonly
mention the ones relevant for these lecture notes.

Proposition 2.3.17. Let Hbe a complex Hilbert space, B 1 (H)satisfying the
following properties.

(a)IfT∈B 1 (H)andN⊂His any Hilbertian basis, then (2.75) holds and

thus

||T|| 1 :=

∑

z∈N

〈z,|T|z〉

is well defined.

(b)B 1 (H)is a subspace of B(H)which is moreover a two-sided∗-ideal,

namely

(i)AT, T A∈B 1 (H)ifT∈B 1 (H)andA∈B(H);

(ii)T†∈B 1 (H)ifT∈B 1 (H).

(c)|| || 1 is a norm onB 1 (H)making it a Banach space and satisfying

(i)||TA|| 1 ≤||A|| ||T|| 1 and||AT|| 1 ≤||A|| ||T|| 1 ifT ∈B 1 (H)and

A∈B(H);

(ii)||T|| 1 =||T†|| 1 ifT∈B 1 (H).

(d)IfT∈B 1 (H),thetraceofT,

tr T:=

∑

z∈N

〈z,Tz〉∈C

is well defined, does not depend on the choice of the Hilbertian basisN

and the sum converges absolutely (so can be arbitrarily re-ordered).

Remark 2.3.18.
(1)Obviously we havetr|T|=||T|| 1 ifT∈B 1 (H).
(2)The trace just possesses the properties one expects from the finite dimen-
sional case. In particular,[5; 6],

(i)it is linear onB 1 (H);

(ii)tr T†=tr TifT∈B 1 (H);

(iii)the trace satisfies thecyclic property,

tr(T 1 ···Tn)=tr(Tπ(1)···Tπ(n)) (2.76)