Measure, Integration & Real Analysis, 2021a

300 Chapter 10 Linear Maps on Hilbert Spaces
For real Hilbert spaces, the next result requires the additional hypothesis that T
is self-adjoint. To see that this extra hypothesis cannot be eliminated, consider the
operator T : R 2 → R 2 defined by T(x, y) =(−y, x). Then, T ̸= 0, but with the
standard inner product on R 2 ,wehave〈Tf, f 〉 = 0 for all f ∈ R 2 (which you can
verify either algebraically or by thinking of T as counterclockwise rotation by a right
angle).
10.46 〈Tf, f 〉 = 0 for all f implies T = 0
Suppose V is a Hilbert space, T ∈B(V), and 〈Tf, f 〉 = 0 for all f ∈ V.
(a) If F = C, then T = 0.
(b) If F = R and T is self-adjoint, then T = 0.
Proof
First suppose F = C. Ifg, h ∈ V, then
〈Tg, h〉 =
〈T(g + h), g + h〉−〈T(g − h), g − h〉
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+
〈T(g + ih), g + ih〉−〈T(g − ih), g − ih〉
4
as can be verified by computing the right side. Our hypothesis that 〈Tf, f 〉 = 0
for all f ∈ V implies that the right side above equals 0. Thus 〈Tg, h〉 = 0 for all
g, h ∈ V. Taking h = Tg, we can conclude that T = 0, which completes the proof
of (a).
Now suppose F = R and T is self-adjoint. Then
〈T(g + h), g + h〉−〈T(g − h), g − h〉
10.47 〈Tg, h〉 = ;
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this is proved by computing the right side using the equation
〈Th, g〉 = 〈h, Tg〉 = 〈Tg, h〉,
where the first equality holds because T is self-adjoint and the second equality holds
because we are working in a real Hilbert space. Each term on the right side of 10.47
is of the form 〈Tf, f 〉 for appropriate f . Thus 〈Tg, h〉 = 0 for all g, h ∈ V. This
implies that T = 0 (take h = Tg), completing the proof of (b).
Some insight into the adjoint can be obtained by thinking of the operation T ↦→ T ∗
on B(V) as analogous to the operation z ↦→ z on C. Under this analogy, the
self-adjoint operators (characterized by T ∗ = T) correspond to the real numbers
(characterized by z = z). The first two bullet points in Example 10.45 illustrate this
analogy, as we saw that a multiplication operator on L 2 (μ) is self-adjoint if and only
if the multiplier is real-valued almost everywhere.
The next two results deepen the analogy between the self-adjoint operators and
the real numbers. First we see this analogy reflected in the behavior of 〈Tf, f 〉, and
then we see this analogy reflected in the spectrum of T.
Measure, Integration & Real Analysis, by Sheldon Axler
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