Page 141 - Linear Algebra Done Right

Self-Adjoint and Normal Operators
Proposition: If V is a complex inner-product space and T is an
7.2
operator on V such that
Tv, v = 0 129
for all v ∈ V, then T = 0.
Proof: Suppose V is a complex inner-product space and T ∈L(V).
Then
T(u + w), u + w −PT(u − w), u − w
Tu, w =
4
T(u + iw), u + iw −PT(u − iw), u − iw
+ i
4
for all u, w ∈ V, as can be verified by computing the right side. Note
that each term on the right side is of the form Tv, v for appropriate
v ∈ V.If Tv, v = 0 for all v ∈ V, then the equation above implies that
Tu, w = 0 for all u, w ∈ V. This implies that T = 0 (take w = Tu).

The following corollary is false for real inner-product spaces, as
shown by considering any operator on a real inner-product space that
is not self-adjoint.

7.3 Corollary: Let V be a complex inner-product space and let This corollary provides
T ∈L(V). Then T is self-adjoint if and only if another example of
how self-adjoint
Tv, v à R operators behave like
real numbers.
for every v ∈ V.

Proof: Let v ∈ V. Then

Tv, v − Tv, v = Tv, v −Pv, Tv
∗
= Tv, v −PT v, v
= (T − T )v, v .
∗
If Tv, v à R for every v ∈ V, then the left side of the equation above
equals 0, so (T − T )v, v = 0 for every v ∈ V. This implies that
∗
T − T ∗ = 0 (by 7.2), and hence T is self-adjoint.
Conversely, if T is self-adjoint, then the right side of the equation
above equals 0, so Tv, v = Tv, v for every v ∈ V. This implies that
Tv, v à R for every v ∈ V, as desired.