Measure, Integration & Real Analysis, 2021a

Section 10D Spectral Theorem for Compact Operators 331
10.108 Example an orthonormal basis of eigenvectors
Suppose V : L 2 ([0, 1]) → L 2 ([0, 1]) is the Volterra operator defined by
(V f )(x) =
The operator V is compact (see the paragraph after the proof of 10.70), but it is not
normal. Because V is compact, so is V ∗ (by 10.73). Hence V−V ∗ is compact. Also,
(V −V ∗ ) ∗ = V ∗ −V = −(V −V ∗ ). Because every operator commutes with its
negative, we conclude that V−V ∗ is a compact normal operator. Because we want
to apply the Spectral Theorem, for the rest of this example we will take F = C.
If f ∈ L 2 ([0, 1]) and x ∈ [0, 1], then the formula for V ∗ given by 10.16 shows
that
(
10.109
(V−V ∗ ) f ) ∫ x ∫ 1
(x) =2 f − f .
The right side of the equation above is a continuous function of x whose value at
x = 0 is the negative of its value at x = 1.
Differentiating both sides of the equation above and using the Lebesgue Differentiation
Theorem (4.19) shows that
( (V−V ∗ ) f ) ′ (x) =2 f (x)
for almost every x ∈ [0, 1]. Iff ∈ null(V −V ∗ ), then differentiating both sides
of the equation (V −V ∗ ) f = 0 shows that 2 f (x) =0 for almost every x ∈ [0, 1];
hence f = 0, and we conclude that V−V ∗ is injective (so 0 is not an eigenvalue).
Suppose f is an eigenvector of V−V ∗ with eigenvalue α. Thus f is in the range
of V−V ∗ , which by 10.109 implies that f is continuous on [0, 1], which by 10.109
again implies that f is continuously differentiable on (0, 1). Differentiating both
sides of the equation (V −V ∗ ) f = α f gives
2 f (x) =α f ′ (x)
for all x ∈ (0, 1). Hence the function whose value at x is e −(2/α)x f (x) has derivative
0 everywhere on (0, 1) and thus is a constant function. In other words,
∫ x
10.110 f (x) =ce (2/α)x
for some constant c ̸= 0. Because f ∈ range(V −V ∗ ),wehave f (0) =− f (1),
which with the equation above implies that there exists k ∈ Z such that
10.111 2/α = i(2k + 1)π.
Replacing 2/α in 10.110 with the value of 2/α derived in 10.111 shows that for
k ∈ Z, we should define e k ∈ L 2 ([0, 1]) by
10.112 e k (x) =e i(2k+1)πx .
Clearly {e k } k∈Z is an orthonormal family in L 2 ([0, 1]) [the orthogonality can be
verified by a straightforward calculation or by using 10.57]. The paragraph above
and the Spectral Theorem for compact normal operators (10.107) imply that this
orthonormal family is an orthonormal basis of L 2 ([0, 1]).
0
0
f .
0
Measure, Integration & Real Analysis, by Sheldon Axler