Bounded Linear Operators on a Hilbert Space

imaginary terms vanish, and we find that
Self-adjoint and unitary operators 199
|〈y, Ax〉| 2 = 1
|〈x + y, A(x + y)〉 − 〈x − y, A(x − y)〉|2
16
≤ 1
16 α2 (x + y 2 + x − y 2 ) 2
= 1
4 α2 (x 2 + y 2 ) 2 ,
where we have used the definition of α and the parallelogram law. Using this result
in (8.14), we conclude that A ≤ α.
As a corollary, we have the following result.
Corollary 8.27 If A is a bounded operator on a Hilbert space then A ∗ A = A 2 .
If A is self-adjoint, then A 2 = A 2 .
Proof. The definition of A, and an application Lemma 8.26 to the self-adjoint
operator A ∗ A, imply that
A 2 = sup |〈Ax, Ax〉| = sup |〈x, A
x=1
x=1
∗ Ax〉| = A ∗ A.
Hence, if A is self-adjoint, then A 2 = A 2 .
Next, we define orthogonal or unitary operators, on real or complex spaces,
respectively.
Definition 8.28 A linear map U : H1 → H2 between real or complex Hilbert
spaces H1 and H2 is said to be orthogonal or unitary, respectively, if it is invertible
and if
〈Ux, Uy〉H2 = 〈x, y〉H1
for all x, y ∈ H1.
Two Hilbert spaces H1 and H2 are isomorphic as Hilbert spaces if there is a unitary
linear map between them.
Thus, a unitary operator is one-to-one and onto, and preserves the inner product.
A map U : H → H is unitary if and only if U ∗ U = UU ∗ = I.
Example 8.29 An n × n real matrix Q is orthogonal if Q T = Q −1 . An n × n
complex matrix U is unitary if U ∗ = U −1 .
Example 8.30 If A is a bounded self-adjoint operator, then
e iA ∞ 1
=
n! (iA)n
is unitary, since
n=0
e iA ∗ = e −iA = e iA −1 .