in the sense that

N,M

Anm φn φ— ’ K ’ 0. (9.97)

m

n,m=1

9.6. SOME FUNCTIONAL ANALYSIS 271

Now the ¬nite sum

N,M

Anm φn (x)φ— (y) (9.98)

m

n,m=1

is automatically compact since it is bounded and has ¬nite-dimensional

range. (The unit ball in a Hilbert space is relatively compact ” the

space is ¬nite dimensional). Thus, Hilbert-Schmidt implies that K is

approximated in norm by compact operators. But a limit of compact

operators is compact, so K itself is compact. Thus

Hilbert-Schmidt ’ compact.

It is easy to test a given kernel to see if it is Hilbert-Schmidt (simply

use the de¬nition) and therein lies the utility of the concept.

If we have a Hilbert-Schmidt Green function g, we can reacast our di¬eren-

tial equation as an integral equation with g as kernel, and this is why the

Fredholm alternative works for a large class of linear di¬erential equations.

Example: Consider the Legendre equation operator

Lu = ’[(1 ’ x2 )u ] (9.99)

on the interval [’1, 1] with boundary conditions that u be ¬nite at the end-

√

points. This operator has normalized zero mode u0 = 1/ 2, so it does not

have an inverse. There exists, however, a modi¬ed Green function g(x, x )

that satis¬es

1

Lu = δ(x ’ x ) ’ . (9.100)

2

It is

11

g(x, x ) = ln 2 ’ ’ ln(1 + x> )(1 ’ x< ), (9.101)

22

where x> is the greater of x and x , and x< the lesser. We may verify that

1 1

|g(x, x )|2 dxdx < ∞, (9.102)

’1 ’1

so g is Hilbert-Schmidt and therefore the kernel of a compact operator. The

eigenvalue problem

Lun = »n un (9.103)

can be recast as as the inetgral equation

1

µn u n = g(x, x )un (x ) dx (9.104)

’1

272 CHAPTER 9. INTEGRAL EQUATIONS

with µn = »’1 . The compactness of g guarantees that there is a complete

n

set of eigenfunctions (these being the Legendre polynomials Pn (x) for n > 0)

having eigenvalues µn = 1/n(n+1). The operator g also has the eigenfunction

P0 with eigenvalue µ0 = 0. This example provides the justi¬cation for the

claim that the “¬nite” boundary conditions we adopted for the Legendre

equation in chpater 8 give us a self adjoint operataor.

Note that K(x, y) does not have to be bounded for K to be Hilbert-

Schmidt.

Example: The kernel

1

|x|, |y| < 1

K(x, y) = , (9.105)

(x ’ y)±

1

is Hilbert-Schmidt provided ± < 2 .

Example: The kernel

1 ’m|x’y|

x, y ∈ R

K(x, y) = e , (9.106)

2m

is not Hilbert-Schmidt because |K(x ’ y)| is constant along the the lines

x ’ y = constant, which lie parallel to the diagonal. K has a continuous

spectrum consisting of all real numbers less than 1/m2 . It cannot be compact,

therefore, but it is bounded, and K = 1/m2 .

9.6.2 Closed Operators

One motivation for our including a brief account of functional analysis is that

the astute reader will have realized that some of the statements we have made

in earlier chapters appear inconsistent. We have asserted in chapter 2 that no

signi¬cance can be attached to the value of an L2 function at any particular

point ” only integrated averages matter. In later chapters, though, we have

happily imposed boundary conditions that require these very functions to

take speci¬ed values at the endpoints of our interval. In this section we will

resolve this paradox. The apparent contradiction is intimately connected

with our imposing boundary conditions only on derivatives of lower order

than than that of the di¬erential equation, but understanding why this is so

requires some analytical language.

Di¬erential operators L are never continuous. We cannot deduce from

un ’ u that Lun ’ Lu. Di¬erential operators can be closed however. A

9.6. SOME FUNCTIONAL ANALYSIS 273

closed operator is one for which whenever a sequence un converges to a limit

u and at the same time the image sequence Lun also converges to a limit

f , then u is in the domain of L and Lu = f . The name is not meant to

imply that the domain of de¬nition is closed, but instead that the graph of

L ” this being the set {u, Lu} considered as a subset of L2 [a, b] — L2 [a, b] ”

contains its limit points and so is a closed set.

i) The property of being closed is desirable because a closed operator has

a closed null-space: Suppose L is closed and we have a sequence such

that Lzn = 0, and zn ’ z. Then z is in the domain of L and Lz = 0.

A closed null-space is necessary prerequisite to satisfying the Fredholm

alternative.

ii) A deep result states that a closed operator de¬ned on a closed domain

is bounded. Since they are always unbounded, the domain of a closed

di¬erential operator can never be a closed set.

An operator may not be closed but may be closable, in that we can make it

closed by including additional functions in its domain. The essential require-

ment for closability is that we never have two sequences un and vn which

converge to the same limit, w, while Lun and Lvn both converge, but to

di¬erent limits. Closability is equivalent to requiring that if un ’ 0 and

Lun converges, then Lun converges to zero.

Example: Let L = d/dx. Suppose that un ’ 0 and Lun ’ f . If • is a

smooth L2 function that vanishes at 0, 1, then

dun

1 1 1

dx = ’ lim

•f dx = lim • φ un dx = 0. (9.107)

dx

n’∞ 0 n’∞

0 0

Here we have used the continuity of the inner product (a property that follows

from the Cauchy-Schwarz-Bunyakovsky inequality) to justify the interchange

the order of limit and integral. By the same arguments we used when dealing

with the calculus of variations, we deduce that f = 0. Thus d/dx is closable.

If an operator is closable, we may as well add the extra functions to its

domain and make it closed. Let us consider what closure means for the

operator

d

, D(L) = {y ∈ C 1 [0, 1] : y (0) = 0}.

L= (9.108)

dx

Here, in ¬xing the derivative at the endpoint, we are imposing a boundary

condition of higher order than we ought.

Consider a sequence of di¬erentiable functions ya which have vanishing

274 CHAPTER 9. INTEGRAL EQUATIONS

derivative at x = 0, but tend in L2 to a function y whose derivative is non-

zero at x = 0.

y

ya