106 From Classical Mechanics to Quantum Field Theory. A Tutorial
Remark 2.2.42.
(a)It turns out thatρ(A)is alwaysopen, so thatσ(A)is alwaysclosed[8; 5;
9 ].
(b)IfAis closed and normal, in particular, ifAis either selfadjoint or uni-
tary),σr(A)=∅(e.g., see[ 5 ]). Furthermore, ifAis closed (ifA∈B(H)in
particular), λ∈ρ(A)if and only ifA−λIadmits inverse inB(H)(see (2) in
exercise 2.2.43).
(c)IfAis selfadjoint, one findsσ(A)⊂R(see (1) in exercise 2.2.43).
(d)IfAis unitary, one findsσ(A)⊂T:={eia|a∈R}(e.g., see[ 5 ]).
(e)IfU:H→His unitary andAis any operator in the complex Hilbert space
H, just by applying the definition, one findsσ(UAU†)=σ(A)and in particular,
σp(UAU†)=σp(A),σc(UAU†)=σc(A),σr(UAU†)=σr(A). (2.43)The same result holds replacingU:H→HforU:H→H′andU†forU−^1 ,
whereUis now a Hilbert space isomorphism (an isometric surjective linear map)
andH′another complex Hilbert space.
To conclude, let us mention two useful technical facts which will turn out to be
useful several times in the rest of this part. From the definition of adjoint, one
easily has forA:D(A)→Hdensely defined andλ∈C,
Ker(A†−λ∗I)=[Ran(A−λI)]⊥,
Ker(A−λI)⊂[Ran(A†−λ∗I)]⊥(2.44)
where the inclusion becomes an identity ifA∈B(H).
Exercise 2.2.43.
(1)Prove that ifAis a selfadjoint operator in the complex Hilbert spaceHthen
(i)σ(A)⊂R;
(ii)σr(A)=∅;
(iii)eigenvectors with different eigenvalues are orthogonal.Solution.Let us begin with (i). Supposeλ=μ+iν,ν= 0 and let us prove
λ∈ρ(A). Ifx∈D(A),
〈(A−λI)x,(A−λI)x〉=〈(A−μI)x,(A−μI)x〉+ν^2 〈x, x〉+iν[〈Ax, x〉−〈x, Ax〉].
The last summand vanishes forAis selfadjoint. Hence
||(A−λI)x||≥|ν|||x||.With a similar argument, we obtain
||(A−λ∗I)x||≥|ν|||x||.