Page 157 - Linear Algebra Done Right

Positive Operators
You should verify that every orthogonal projection is positive. For
another set of examples, look at the proof of 7.11, where we showed
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that if T ∈L(V) is self-adjoint and α, β ∈ R are such that α < 4β, 145
2
then T + αT + βI is positive.
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An operator S is called a square root of an operator T if S = T.
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For example, if T ∈L(F ) is defined by T(z 1 ,z 2 ,z 3 ) = (z 3 , 0, 0), then
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the operator S ∈L(F ) defined by S(z 1 ,z 2 ,z 3 ) = (z 2 ,z 3 , 0) is a square
root of T.
The following theorem is the main result about positive operators. The positive operators
Note that its characterizations of the positive operators correspond to correspond, in some
characterizations of the nonnegative numbers among C. Specifically, sense, to the numbers
a complex number z is nonnegative if and only if it has a nonnegative [0, ∞), so better
square root, corresponding to condition (c) below. Also, z is nonnega- terminology would call
tive if and only if it has a real square root, corresponding to condition these nonnegative
(d) below. Finally, z is nonnegative if and only if there exists a complex instead of positive.
number w such that z = ¯ ww, corresponding to condition (e) below. However, operator
theorists consistently
7.27 Theorem: Let T ∈L(V). Then the following are equivalent: call these the positive
operators, so we will
(a) T is positive; follow that custom.
(b) T is self-adjoint and all the eigenvalues of T are nonnegative;

(c) T has a positive square root;
(d) T has a self-adjoint square root;
(e) there exists an operator S ∈L(V) such that T = S S.
∗
Proof: We will prove that (a) ⇒ (b) ⇒ (c) ⇒ (d) ⇒ (e) ⇒ (a).
First suppose that (a) holds, so that T is positive. Obviously T is
self-adjoint (by the definition of a positive operator). To prove the other
condition in (b), suppose that λ is an eigenvalue of T. Let v be a nonzero
eigenvector of T corresponding to λ. Then

0≤ Tv, v
= λv, v
= λ v, v ,

and thus λ is a nonnegative number. Hence (b) holds.
Now suppose that (b) holds, so that T is self-adjoint and all the eigen-
values of T are nonnegative. By the spectral theorem (7.9 and 7.13),