92 From Classical Mechanics to Quantum Field Theory. A Tutorial
particular the topological dual ofXX′=B(X,C)withX complex normed
space, is always complete sinceCis complete.
B(H) is more strongly a unitalC∗-algebra, if we introduce the notion of adjoint
of an operator. To this end, we have the following general definition concerning
also unbounded operators defined on non-maximal domains.
Definition 2.2.14. LetAbe a densely defined operator in the complex Hilbert
spaceH.DefinethesubspaceofH,
D(A†):={y∈H|∃zy∈Hs.t.〈y,Ax〉=〈zy,x〉∀x∈D(A)}.The linear mapA†:D(A†)y→zyis called theadjointoperator ofA.
Remark 2.2.15.
(a)Above,zyis uniquely determined byy,sinceD(A)is dense. If bothzy,zy′
satisfy 〈y,Ax〉=〈zy,x〉and〈y,Ax〉=〈zy′,x〉,then〈zy−z′y,x〉=0for every
x∈D(A). Taking a sequenceD(A)xn→zy−z′y, we conclude that||zy−zy′||=0.
Thuszy=zy′. The fact thaty→zyis linear can immediately be checked.
(b)By construction, we immediately have that
〈A†y,x〉=〈y,Ax〉 forx∈D(A)andy∈D(A†)and also
〈x, A†y〉=〈Ax, y〉 forx∈D(A)andy∈D(A†),if taking the complex conjugation of the former identity.
Exercise 2.2.16. Prove thatD(A†) can equivalently be defined as the set (sub-
space) ofy∈Hsuch that the linear functionalD(A)x→〈y,Ax〉is continuous.
Solution. It is a simple application of Riesz’ lemma, after having uniquely
extendedD(A)x→〈y,Ax〉to a continuous linear functional defined onD(A)=
Hby continuity.
Remark 2.2.17.
(a)IfAis densely defined andA†is also densely defined thenA⊂(A†)†.The
proof immediately follows form the definition of adjoint.
(b)IfAis densely defined andA⊂BthenB†⊂A†. The proof immediately
follows the definition of adjoint.