Measure, Integration & Real Analysis, 2021a

286 Chapter 10 Linear Maps on Hilbert Spaces
10.15 Example Volterra operator
The Volterra operator is the linear map V : L 2 ([0, 1]) → L 2 ([0, 1]) defined by
(V f )(x) =
∫ x
0
f (y) dy
for f ∈ L 2 ([0, 1]) and x ∈ [0, 1]; here dy means dλ(y), where λ is the usual
Lebesgue measure on the interval [0, 1].
To show that V is a bounded linear map from L 2 ([0, 1]) to L 2 ([0, 1]), let K be the
function on [0, 1] × [0, 1] defined by
{
1 if x > y,
K(x, y) =
0 if x ≤ y.
In other words, K is the characteristic
function of the triangle below the
Vito Volterra (1860–1940) was a
pioneer in developing functional
diagonal of the unit square. Clearly
analytic techniques to study integral
K ∈L 2 (λ × λ) and V = I K as defined
equations.
in 10.6. Thus V is a bounded linear map
from L 2 ([0, 1]) to L 2 ([0, 1]) and ‖V‖ ≤ √ 1 (by 10.8).
2
Because V ∗ = I K ∗ (by 10.9) and K ∗ is the characteristic function of the closed
triangle above the diagonal of the unit square, we see that
10.16 (V ∗ f )(x) =
∫ 1
x
f (y) dy =
∫ 1
0
f (y) dy −
∫ x
0
f (y) dy
for f ∈ L 2 ([0, 1]) and x ∈ [0, 1].
Now we can show that V ∗ is injective. To do this, suppose f ∈ L 2 ([0, 1]) and
V ∗ f = 0. Differentiating both sides of 10.16 with respect to x and using the Lebesgue
Differentiation Theorem (4.19), we conclude that f = 0. Hence V ∗ is injective. Thus
the Volterra operator V has dense range (by 10.14).
Although range V is dense in L 2 ([0, 1]), it does not equal L 2 ([0, 1]) (because
every element of range V is a continuous function on [0, 1] that vanishes at 0). Thus
the Volterra operator V has dense but not closed range in L 2 ([0, 1]).
Invertibility of Operators
Linear maps from a vector space to itself are so important that they get a special name
and special notation.
10.17 Definition operator; B(V)
• An operator is a linear map from a vector space to itself.
• If V is a normed vector space, then B(V) denotes the normed vector space
of bounded operators on V. In other words, B(V) =B(V, V).
Measure, Integration & Real Analysis, by Sheldon Axler