Operator notation and the existence of solutions

23.3 OPERATOR NOTATION AND THE EXISTENCE OF SOLUTIONS
In fact, we shall be concerned with various special cases of (23.4), which are
known by particular names. Firstly, if g(x) = 0 then the unknown function y(x)
appears only under the integral sign, and (23.4) is called a linear integral equation
of the first kind. Alternatively, if g(x) = 1, so that y(x) appears twice, once inside
the integral and once outside, then (23.4) is called a linear integral equation of
the second kind. In either case, if f(x) = 0 the equation is called homogeneous,
otherwise inhomogeneous.
We can distinguish further between different types of integral equation by the
form of the integration limits a and b. If these limits are fixed constants then the
equation is called a Fredholm equation. If, however, the upper limit b = x (i.e. it
is variable) then the equation is called a Volterra equation; such an equation is
analogous to one with fixed limits but for which the kernel K(x, z) = 0 for z > x.
Finally, we note that any equation for which either (or both) of the integration
limits is infinite, or for which K(x, z) becomes infinite in the range of integration,
is called a singular integral equation.
23.3 Operator notation and the existence of solutions
There is a close correspondence between linear integral equations and the matrix
equations discussed in chapter 8. However, the former involve linear, integral relations between functions in an infinite-dimensional function space (see chapter 17),
whereas the latter specify linear relations among vectors in a finite-dimensional
vector space.
Since we are restricting our attention to linear integral equations, it will be
convenient to introduce the linear integral operator K, whose action on an
arbitrary function y is given by
b
K(x, z)y(z) dz.
(23.5)
Ky =
a
This is analogous to the introduction in chapters 16 and 17 of the notation L to
describe a linear differential operator. Furthermore, we may define the Hermitian
conjugate K† by
b
K ∗ (z, x)y(z) dz,
K† y =
a
where the asterisk denotes complex conjugation and we have reversed the order
of the arguments in the kernel.
It is clear from (23.5) that K is indeed linear. Moreover, since K operates on
the infinite-dimensional space of (reasonable) functions, we may make an obvious
analogy with matrix equations and consider the action of K on a function f as
that of a matrix on a column vector (both of infinite dimension).
When written in operator form, the integral equations discussed in the previous section resemble equations familiar from linear algebra. For example, the
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