Measure, Integration & Real Analysis, 2021a

306 Chapter 10 Linear Maps on Hilbert Spaces
By definition, isometries preserve norms. The equivalence of (a) and (b) in the
following result shows that isometries also preserve inner products.
10.60 isometries preserve inner products
Suppose T is a bounded operator on a Hilbert space V. Then the following are
equivalent:
(a) T is an isometry.
(b) 〈Tf, Tg〉 = 〈 f , g〉 for all f , g ∈ V.
(c) T ∗ T = I.
(d) {Te k } k∈Γ is an orthonormal family for every orthonormal family {e k } k∈Γ
in V.
(e) {Te k } k∈Γ is an orthonormal family for some orthonormal basis {e k } k∈Γ
of V.
Proof
If f ∈ V, then
‖Tf‖ 2 −‖f ‖ 2 = 〈Tf, Tf〉−〈f , f 〉 = 〈(T ∗ T − I) f , f 〉.
Thus ‖Tf‖ = ‖ f ‖ for all f ∈ V if and only if the right side of the equation above
is 0 for all f ∈ V. Because T ∗ T − I is self-adjoint, this happens if and only if
T ∗ T − I = 0 (by 10.46). Thus (a) is equivalent to (c).
If T ∗ T = I, then 〈Tf, Tg〉 = 〈T ∗ Tf, g〉 = 〈 f , g〉 for all f , g ∈ V. Thus (c)
implies (b).
Taking g = f in (b), we see that (b) implies (a). Hence we now know that (a), (b),
and (c) are equivalent to each other.
To prove that (b) implies (d), suppose (b) holds. If {e k } k∈Γ is an orthonormal
family in V, then 〈Te j , Te k 〉 = 〈e j , e k 〉 for all j, k ∈ Γ, and thus {Te k } k∈Γ is an
orthonormal family in V. Hence (b) implies (d).
Because V has an orthonormal basis (see 8.67 or 8.75), (d) implies (e).
Finally, suppose (e) holds. Thus {Te k } k∈Γ is an orthonormal family for some
orthonormal basis {e k } k∈Γ of V. Suppose f ∈ V. Then by 8.63(a) we have
f = ∑〈 f , e j 〉e j ,
j∈Γ
which implies that
Thus if k ∈ Γ, then
T ∗ Tf = ∑〈 f , e j 〉T ∗ Te j .
j∈Γ
〈T ∗ Tf, e k 〉 = ∑〈 f , e j 〉〈T ∗ Te j , e k 〉 = ∑〈 f , e j 〉〈Te j , Te k 〉 = 〈 f , e k 〉,
j∈Γ
j∈Γ
where the last equality holds because 〈Te j , Te k 〉 equals 1 if j = k and equals 0
otherwise. Because the equality above holds for every e k in the orthonormal basis
{e k } k∈Γ , we conclude that T ∗ Tf = f . Thus (e) implies (c), completing the proof.
Measure, Integration & Real Analysis, by Sheldon Axler