From Classical Mechanics to Quantum Field Theory

Mathematical Foundations of Quantum Mechanics 117

More precise statements based on the notion ofessential rangecan be found in[ 5 ].
It turns out that, forAselfadjoint andf:σ(A)→Cmeasurable,z∈σ(f(A))
if and only ifP(A)(Ez)=0for some open setEzz.Nowz∈σ(f(A))is in
σp(f(A))iffP(A)(f−^1 (z))=0or it is inσc(f(A))iffP(A)(f−^1 (z)) = 0.
(b) It is fundamental to stress that in QM, (2.58) permits us to adopt the
standard operational approach on observablesf(A)as the observable whose set of
possible values is (the closure of) the set of realsf(a)whereais a possible value
ofA.

Proposition 2.2.61.A selfadjoint operator is bounded (and its domain coincide
to the wholeH)ifandonlyifσ(A)is bounded.

Proof. It essentially follows from (2.50), restricting the integration space toX=
σ(A). In fact, ifσ(A) is bounded and thus compact it being closed, the continuous
functionid:σ(A)λ→λis bounded and (2.50) implies thatA=

∫

σ(A)iddP

(A)

is bounded and the inequality holds

||A||≤sup{|λ||λ∈σ(A)}. (2.60)

In this case, it also holdsD(A)=Δid=H.
If, conversely,σ(A) is not bounded, we can find a sequenceλn∈σ(A) with
|λn|→∞asn→+∞. With the help of (c) and (d) in Theorem 2.2.55, it is
easy to construct vectorsxnwith||xn||=0and xn∈PB(A(λ)n)(H)whereB(λn):=
[λn− 1 ,λn+ 1]. (2.50) implies

||Axn||^2 ≥||xn||^2 inf

z∈B(λn)

|id(z)|^2

Since infz∈B(λn)|id(z)|^2 →+∞,wehavethat||Axn||/||xn||is not bounded and
A, in turn, cannot be bounded. In this case, sinceA=A†, Theorem 2.2.29 entails
thatD(A)isstrictlyincluded inH.

It is possible to prove[ 5 ]that (2.60) can be turned into an identity whenA∈B(H)
also ifAis not selfadjoint but only normal

||A||=sup{|λ||λ∈σ(A)}, (2.61)

This is the well knownspectral radius formula,thespectral radiusofA∈B(H)
being, by definition, the number in the right hand side.
(d)The result stated in (c) explains the reason why observablesAin QM are
very often represented by unbounded selfadjoint operators.σ(A)is the set of values
of the observableA. When, as it happens very often, that observable is allowed