Measure, Integration & Real Analysis, 2021a

Self-adjoint Operators
Section 10B Spectrum 299
In this subsection, we look at a nice special class of bounded operators.
10.44 Definition self-adjoint
A bounded operator T on a Hilbert space is called self-adjoint if T ∗ = T.
The definition of the adjoint implies that a bounded operator T on a Hilbert space
V is self-adjoint if and only if 〈Tf, g〉 = 〈 f , Tg〉 for all f , g ∈ V. See Exercise 7 for
an interesting result regarding this last condition.
10.45 Example self-adjoint operators
• Suppose b 1 , b 2 ,... is a bounded sequence in F. Define a bounded operator
T : l 2 → l 2 by
T(a 1 , a 2 ,...)=(a 1 b 1 , a 2 b 2 ,...).
Then T ∗ : l 2 → l 2 is the operator defined by
T ∗ (a 1 , a 2 ,...)=(a 1 b 1 , a 2 b 2 ,...).
Hence T is self-adjoint if and only if b k ∈ R for all k ∈ Z + .
• More generally, suppose (X, S, μ) is a σ-finite measure space and h ∈L ∞ (μ).
Define a bounded operator M h ∈B ( L 2 (μ) ) by M h f = fh. Then M h ∗ = M h
.
Thus M h is self-adjoint if and only if μ({x ∈ X : h(x) /∈ R}) =0.
• Suppose n ∈ Z + , K is an n-by-n matrix, and I K : F n → F n is the operator of
matrix multiplication by K (thinking of elements of F n as column vectors). Then
(I K ) ∗ is the operator of multiplication by the conjugate transpose of K, as shown
in Example 10.10. Thus I K is a self-adjoint operator if and only if the matrix K
equals its conjugate transpose.
• More generally, suppose (X, S, μ) is a σ-finite measure space, K ∈L 2 (μ × μ),
and I K is the integral operator on L 2 (μ) defined in Example 10.5. Define
K ∗ : X × X → F by K ∗ (y, x) =K(x, y). Then (I K ) ∗ is the integral operator
induced by K ∗ , as shown in Example 10.5. Thus if K ∗ = K, or in other words if
K(x, y) =K(y, x) for all (x, y) ∈ X × X, then I K is self-adjoint.
• Suppose U is a closed subspace of a Hilbert space V. Recall that P U denotes the
orthogonal projection of V onto U (see Section 8B). We have
〈P U f , g〉 = 〈P U f , P U g +(I − P U )g〉
= 〈P U f , P U g〉
= 〈 f − (I − P U ) f , P U g〉
= 〈 f , P U g〉,
where the second and fourth equalities above hold because of 8.37(a). The
equation above shows that P U is a self-adjoint operator.
Measure, Integration & Real Analysis, by Sheldon Axler