98 From Classical Mechanics to Quantum Field Theory. A Tutorial
because selfadjoint. (b) impliesA†=B. That is, every selfadjoint extension ofA
coincides withA†.
Another elementary though important result, helping understand why in QM ob-
servables are very often described by selfadjoint operators which are unbounded
and defined in proper subspaces, is the following proposition (see (c) in remark
2.2.60).
Theorem 2.2.29(Hellinger-Toepliz theorem). LetA be a selfadjoint operator
in the complex Hilbert spaceH. Ais bounded if and only ifD(A)=H(thus
A∈B(H)).
Proof. AssumeD(A)=H.AsA=A†we haveD(A†)=H.SinceA†is closed,
Theorem 2.2.24 implies theA†(=A) is bounded. Conversely, ifA=A†is bounded,
sinceD(A) is dense, we can continuously extend it to a bounded operatorA 1 :
H→H. That extension, by continuity, trivially satisfies〈A 1 x|y〉=〈x|A 1 y〉for
allx, y∈HthusA 1 is symmetric. SinceA†=A⊂A 1 ⊂A† 1 , (a) in Proposition
2.2.28 impliesA=A 1.
Let us pass to focus on unitary operators. The relevance of unitary operators is
evident from the following proposition where it is proven that they preserve the
nature of operators with respect to the Hermitian conjugation.
Proposition 2.2.30.LetU:H→Hbe a unitary operator in the complex Hilbert
spaceHandAanother operator inH. Prove thatUAU†with domainU(D(A))
(resp. U†AU with domainU†(D(A))) is symmetric, selfadjoint, essentially self-
adjoint, unitary, normal if Ais respectively symmetric, selfadjoint, essentially
selfadjoint, unitary, normal.
Proof. SinceU†is unitary whenU is and (U†)† =U, it is enough to estab-
lish the thesis for UAU†. First of all, notice thatD(UAU†)=U(D(A)) is
dense ifD(A) is dense sinceUis bijective and isometric andU(D(A)) =Hif
D(A)=HbecauseUis bijective. By direct inspection, applying the definition of
adjoint operator, one sees that (UAU†)†=UA†U†andD((UAU†)†)=U(D(A†)).
Now, if Ais symmetricA ⊂A†which impliesUAU† ⊂UA†U† =(UAU†)†
so thatUAU†is symmetric as well. IfA is selfadjointA = A† which im-
plies UAU† = UA†U† =(UAU†)† so thatUAU† is selfadjoint as well. If
A is essentially selfadjoint it is symmetric and (A†)†=A†,sothatUAU†is
symmetric andU(A†)†U†=UA†U†that is (UA†U†)†=UA†U†which means
((UAU†)†)†=(UAU†)†so thatUA†U†is essentially selfadjoint. IfAis unitary,
we haveA†A=AA†=I so thatUA†AU†=UAA†U†=UU†which, since