K. 0. Friedrichs
Spectral Theory, of Operators in Hilbert Space
[S1 Springer-Verlag New York Heidelberg - Berlin 1973
K. O. Friedrichs New York University
Courant Institute of Mathematical Sciences
AMS Classification 47A05,47A10,47B25,47B40
Library of Congress Cataloging in Publication Data
Friedrichs, Kurt Otto. Spectral theory of operators in Hilbert space. (Applied mathematical sciences, v. 9). 1. Hilbert space. 2. Spectral theory (Mathematics) 3. Operator theory. 1. Title. 11. Series. QA1.A647 vol. 9 [QA322.4] 510'.8s (515'.73173-13721
All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag. -
©1973 by Springer-Verlag New York Inc.
Printed in the United States of America. ISBN 0-387.90076-4 Springer-Verlag New York Heidelberg Berlin ISBN 3-540.90076-4 Springer-Verlag Berlin Heidelberg New York
PREFACE
The present lectures intend to provide an introduction to the spectral analysis of self-adjoint operators within the framework of Hilbert space theory.
The guiding notion in this approach is that of
spectral representation.
At the same time the notion of function of
an operator is emphasized.
The formal aspects of these concepts are explained in the first two chapters.
Only then is the notion of Hilbert space introduced.
The following three chapters concern bounded, completely continuous, and non-bounded operators.
Next, simple differential operators are
treated as operators in Hilbert space, and the final chapter deals with the perturbation of discrete,and continuous spectra.
The preparation of the original version of these lecture notes was greatly helped by the assistance of P. Rejto.
Various valuable
suggestions made by him and by R. Lewis have been incorporated.
The present version of the notes contains extensive modifications, in particular in the chapters on bounded and unbounded operators.
February, 1973
K.O.F.
v
TABLE OF CONTENTS page I.
Spectral Representation 1. Three typical problems Linear space and functional representation. Linear operators
3.
Spectral representation
16
4.
Functional calculus Differential equations
22
7.
31 35 38
Adjoint operators in function spaces Orthogonality
50
11. 12.
Orthogonal projection
55
13.
Remarks about the role of self-adjoint operators in-physics
58
9.
10.
Hilbert Space Completeness
15.
First extension theorem. Fourier transformation The projection theorem Bounded forms
17. 18.
46
54
64
14.
16.
64
Ideal functions
69
98 85
91
Bounded Operators 19.
20. 21.
V.
31
Normed spaces Inner product
Inner products in function spaces -Formally self-adjoint operators
8.
IV.
12
24
Norm and Inner Product 6.
III.
1
2.
5.
II.
1
Operator inequalities, operator norm, operator convergence Integral operators
95 103
116
22.
Functions of bounded operators Spectral representation
23.
Normal and unitary operators
140
Operators with Discrete Spectra
132
143
143
26.
Operators with partly discrete spectra Completely continuous operators Completely continuous integral operators,
27.
Maximum-minimum properties of eigenvalues
157
24. 25.
vii
147 152
page VI.
28. 29. 30.
31.
VII.
Closure and adjointness Closed forms Spectral resolution of self-adjoint operators Closeable forms
Differential Operators 32. 33. 34. 35.
36.
VIII.
163
Non-Bounded Operators
163 169 174
180 186
Regular differential operators Ordinary differential operators in a semibounded domain
186
Partial differential operators Partial differential operators with boundary conditions Partial differential operators with discrete spectra
197
Perturbation of Spectra Perturbation of discrete spectra
37.
192
201 206
213 213
38.
Perturbation of continuous spectra
222
39.
Scattering
237
References
241
Index
242
viii
CHAPTER I
SPECTRAL REPRESENTATION
1.
Three Typical Problems
The problem of the spectral representation of a linear operator arises as a natural generalization of the problem of the transformation of a quadratic form to principal axes.
In this section
we shall discuss this and two well-known analogous problems in a preliminary fashion.
Example 1.
Suppose a quadratic form in
n
real variables
Ell." ..'En
is given as the expression n
I+laaa,Eaca, o,
in which the coefficients numbers
Ea
I
are real numbers.
anal = aa,a
The
n
may be regarded as the components of a vector
a)
with respect to a coordinate system in an n-dimensional Euclidean space.
Then the problem is to rotate this coordinate 'system so that
the quadratic form assumes the simple form
4(
)
n
n
J,-aaa,E,E., _ l
1a4 Ti
G
Here
nl,...,nn
2
11=1
are the components of the vector.
to the new system, connected with the coordinates transformation given by-linear relations
n (1.2)
a
u°luupnu
1
with respect Ea
through a
n
nu = aIlvuaEa
The requirement that the new coordinate system be obtained through rotation from the original system is expressed by the condition that the square of the magnitude of the vector is the sum of the squares of the coordinates with respect to each coordinate system:
2
=
2
u=1 u
a=1 a
or,...,an
The numbers
entering identity (1.1) are called
"eigen-values" of the quadratic form
Q(-}
since this form
assumes these values for the unit vectors of the new coordinate system. These are the vectors for which all components except one which equals
1.
nl,...,nq
Specifically, we denote by
equal
H u
)
0
the
vector with the new coordinates
nu
Here we have employed the "Kronecker symbol"
duu, = 0
The
g
for
u # u'.
coordinates of the vector
6uu = 1. H(u)
lire a =
uou
as
seen from (1.2), H(U) - (uua).
(1.4)
The vectors
H(u)
are also called unit "eigen-vectors" of
the quadratic form; any multiple
cH(
30 0
of such a vector will be
called "eigen-vector".
Before indicating how one could establish a transformation
2
(1.2), (1.2)* such that identities (1.1), (1.3) hold we shall assume
that there is such a transformation and derive various consequences from this assumption.
If the relation (1.3) holds for all vectors
it also
(1)
holds for the linear combination
+ cl
c
of any two
(1)
vectors
with arbitrary coefficients
Identify-
c,cl.
ing the mixed terms in the relation
(cnv + cln(l))2,
(CEO + c1Eal))2
in which
nQ
and
n( l)
are the new coordinates of
-
and
(1)
, we obtain the relation
E EoE(1) v _ Q
(1.3)'
n u
'n(i)
vu
_
which is thus seen to hold for all vectors
n
the following
stands for a
Here and in
n J*.
E ,
aml
u
(1)' ,
V=1
In a similar manner one derives from formula (1.1) the identity
E
'
aununr(1)
a,a
Thus the identities (1.1), (1.3) concerning quadratic forms imply corresponding identities concerning the corresponding "bilinear forms". We can draw further conclusions from relations (1.3)', (1.1)'.
To this end let us take the eigenvector relations.
H(u )
gince, by (1.4), the c-coordinates of
and the n-coordinates of
H(u)
are
(1.3)"
3
duu,
,
-
for
(1)
in these H(u)
we find
are
uQu
(1.1)"
having replaced
u'
by
= a
c'u
0J
Expressing
p.
u
n
'
nu
in terms of the
E
by
on both sides we ob-
(1.2)* and identifying the coefficients of tain the identities
(1.3)' 1
1
aCC,u
v uC.
C'u = au
Relation (1.3)" allows one to determine the transformation matrix
(uCU)
.once its inverse
is known and vice versa.
(vuC)
In-
serting this relation into (1.1)" we obtain the important formula
(1.5)
I'M CC,uC,u = auu0u,
which we Pall-interpret presently. In an extensive investigation of transformations to principal
axes one must shift the emphasis from the quadratic form to the This operator, A, transforms the vector
operator associated with it.
41,...,tn
with the components components
1,a0C,E0,
.
into the vector, A
, with the
Thus
a
A
(1.6)
-
=
(
I
a$
aaa,EC,). I
What is the effect of this operator in terms of the new coordinates
nu?
To find this out we express
by (1.2) and determine the
nu-coordinate of (
The result is
4
in terms of the
EC,
1,a aaa'EC') C
nu
by (1.2)*.
[va.uO.I]nMP u
By virtue of the relation (1.5), the expression ?n the bracket equals
vu auau,au6uu, since (1.2) and (1.2)* imply
E vuauau, = 6uu,.
coordinate of A_ is simply
Hence the n
aunu.
Thus we have found that in the new coordinate system, in has the component
which the vector the components
T\, the vector
A -
has
aunu.
This fact leads to a different formulation of the problem of transformation to principal axes.. Instead of requiring that the
quadratic form
Q
should become a sum or difference of squares, as
given by (1.1), we may require that in terms of the new coordinates the given operator
A
should simply consist in multiplying each coordinate
by a number, called an "eigen-value" of this operator.
This formula-
tion will lead directly to the notion of "spectral representation". To explain the significance of the property of the operator A
just derived let us apply this operator on the vector
n-components are all zero except plies that the n-components of component, which is
aun)J .
nu
AH(u)
0.
H(u), whose
Our property evidently im-
are all zero except the uth
An obvious consequence of this fact is the
relation
AH(u) -auH(u).
(1.5)'.
Thus, when applied to the vector multiplier with the value au
and the vector
of the operator
H(u)
au.
H(u), the operator
A
acts like a
It is for this reason that the number
are called an "eigen-value" and "eigen-vector"
A.
5
Note, in view of (1.4), that equation (1.5) is nothing but the expression of equation (1.5)' in terms of c-coordinates. So far we have derived a number of properties from the assumption that transformation of the form
Q, or the operator
A, is
What about the problem of proving that there is such a
possible.
transformation?
One possible approach to doing this starts with
equation (1.5).
Writing
&a, in place of
u0
and
a
in place of
au, this equation takes the form
(1.5)"
a&a
which shows that all'vectors - - H(u) satisfy the same equation.
,
and eigen-value
a = ap
Once o% is chosen this equation may be
regarded as a homogeneous system of equations for the n unknowns C n'
The conditiah that this system have a solution other than
Elf...'En - 0
is that its determinant vanishes:
det(aao, - a600,) = 0.
(1.7)
This condition may be regarded as an equation of the nth degree for
al it can be used to determine the eigen-values
Having found this, vectors - - H(u) satisfy equation (1.5)".
a = a
can be found whose components
Moreover, it is possible to find n such
eigen-vectors which have all the properties discussed and whose &component
uou
are the coefficients of the derived transformation to
principal axes.
We shall not follow up this approach since it is not suitable for extension to problems involving a space of infinitely many dimensions.
6
In our second problem we begin.with the transformation
Problem 2.
to prindipal axes; only later on we shall interpret the vectors having the direction of principal axes as eigen-vectors of an associated operator.
We consider complex-valued functions variable
s
it.
need be defined only for
O(s)
Of these functions we assume at present that they have
continuous derivatives up to the second order. functions
of the real
which are periodic with the period V , so that
Actually, therefore, the functions -n < s <
0(s)
0(s)
As is well-known, such
admit a uniformly convergent Fourier expansion
eisunu
0(s) u
in which the summation runs over all integers
u
The Fourier coefficients
nu
u,
integer.
are given by the formula
_n
The analogy of these two formulas with the formula (1.2) and (1.2)* is apparent.
There is even an analogue with formula 11.10), viz, n
(1.10)
2n 11#(s)12ds _n
I
m
in12.
We may try to push the analogy further and consider the values
.7
of"the function integral
F
for
-Ti
< s <
as well as the values of
it
for
as the coordinates, with respect to different coordinate
p
systems, of a single entity, a "vector"
(P.
Formula (1.10) then exIn doing so we
presses the square of the "magnitude" of this vector. have no trouble in introducing unit vectors ordinate system as vectors for which u' ¢ u.
n
The functions
Ti,, = 1
H(u)
and
for the new coflu,
0
for
associated with these vectors are the
0(s)
"eigen-functions"
(u) (s)
as seen from (1.9).
= elus,
We do have some trouble, however, in trying to
introduce unit vectors with respect to an original coordinate system. What to do about this difficulty will also be discussed in Section 2. At present we simply say that the function
n
is the representative
with respect to one coordinate system, while the
of the vector sequence
0(s)
is its representative with respect to another coordinate
u
system, never mind that these coordinate systems are not defined in terms of unit-vectors.
In another respect we have no trouble in pushing the analogy further.
It is easy to find an operator whose eigen-values are just
the numbers
u
and whose eigen-vectors are just the vectors
H(u).
This is the differential operator
d
l ds
(1.12)
acting on the functions
0(s).
presentative of a vector
Regarding the function
we regard the operator
representative of an operator - denoted by
0(s)
-id/ds
as reas
M - acting on such vectors.
By virtue of the strong differentiability assumptions we have made on the functions
0
we may differentiate relation (1.9) 8
under
the summation sign and carry out integration by parts on
Thus
(1.9)*.
we obtain the relations
(1.13)
elsp un
-i ds 0(s) p rn
e 's"(-i as 0(s))ds.
unu = Zn
(1.13)'
1
-n
They show that, in terms of n-components, application of the operator M
just means multiplying each component np
particular, if the vector
4
In
P.
is specialized to be the vector
whose n-components all vanish except
(1.14)
by the number
H(p)
na, we have
MH(u) = uH(u)
Therefore, the number vector H(p)
p
is an eigen-value of the operator
M
and the
is the associated eigen-vector.
In terms of the eigen-functions
yl0(s) = e1su, which corre-
spond to the eigen-vectors H(u), relation (1.14) assumes the form
-1 ds 0(p) (s)
(1.14)'
=
u0'(p) (s)
.
One of the major differences between the situation in the present and the first example is the fact that now the spectrum consists of infinitely many eigen-values, instead of a finite number, as in the first example.
Problem 3.
Iri the third problem, which we proceed to discuss now, the
notion of spectrum as such is affected.
This example is concerned
with the Fourier integral in contrast with the Fourier series. Again we consider complex-valued functions variable
s
0(s)
of a real
which now is to range over the whole s-axis, -- < 9
s
< -.
We again assume, for convenience, that the functions tinuous derivatives up to the second '0 order. the functions tend to zero as
IsI - m
have con-
0(s)
Also, we require that
sufficiently rapidly.
We
adopt the customary condition
(1.15)
J 10(s)Ids < - together with 0(s) i 0 as IsI i
supplementing it by the condition w (1.15)'
J
d9 I$
P(s)Ids < m together with
(s) i 0 as IsI i cis
Actually, for our present purposes it does not matter much in which way the function
0(s)
Functions
is required to die out "sufficiently rapidly,
0(s)
which behave in the manner described admit
a Fourier integral representation m
(1.16)
0(s) - f eisun(N)du
in terms of a continuous "transform"
n() s
(1.16)*
The functions
i(s)
and
n(u), given by
Je80(s)ds.
.n(v)
are also connected through the
relation
(1.17)
J
Ltu) I2ds
du.
Again'we may try tc interpret the values of the function those of
n(1)
as components of a "vector". 0
old and a new coordinate system.
0(s) and
with respect to-an
But, this time we cannot introduce
10
unit coordinate vectors with respect to either coordinate system. There is no function n(u)
0(s)
among those considered, whose transform
is different from zero for only one value of
the function
In particular,
jd(u)(s) = else is not one of these functions; its trans-
If the transform
form is not defined in a proper sense. eisul
u.
y ¢ u'
were defined it would vanish for
nu,(u)
of
and be infinite for
u=u'. In spite of this awkwardness it would be possible to enlarge the class of functions so as to include those that may serve as coordinate unit vectors in the present problem.
We shall eventually
discuss this extension but not do so in the development gf the basic theory; for, either the simplicity or the completeness of the general theory would be destroyed.
notion of function
We shall employ a generalization of the
(in Chapter III); but a rather restricted one.
We shall be guided by the requirement that the "space integrals" entering formula (1.17) remain finite for the generalized functions. In the first two chapters we shall confine ourselves to describing the notion of transformation to principal axes in terms which do not require an extension of the notion of function. Returning to our third problem we maintain that the differential operator
d
ds plays essentially. the same role as in the second example, except that
this operator now applies to functions defined all over the s-axis. Let
n(p)
be the transform of the function
the transform of the function
-i ds 0(s)
0(s).
Thenj we maintain
is the function
un(u).
This is immediately verified by differentiation under the integral sign in (1.16) or integration
in (1.16)*.
Thus, if we regard the function
11
n(u)
as the representative
of a function
a(s), we find that the function
is the repre-
This description will lead to the
-i ds a(s).
sentative of function
on(p)
notion of "spectral representation".
Although we no longer can speak of eigen-values and eigenvectors or -functions in the proper sense, we recognize that the values
U
The set
play a role analogous to that of an eigen-value.
of all these values
u
is again called the spectrum of the operator
M, but this spectrum is said to be continuous in contrast to the spectra of the first two examples, which are called discrete. "improper eigen-functions"
euse
The
will not be regarded as functions
on which the operator may aot; these functions will be relegated to the role they play in the transformation formulas (1.16) and (1.16)'.
In spite of these differences between the cases with discrete and continuous spectrum the notion of spectral representation can be formulated in general fashion so as to include all cases.
In order
to do this we first recall the notions of "linear space" and "linear operator".
2.
Linear Space and Functional Representation.
Linear Operators.
In this section we shall give a brief resume of the basic facts of linear spaces as much as we need them for our purposes. A linear space
CS
consists of elements, called "vectors",
for which linear combinations are defined.
of vectors and every pair
c,c1
That means, to every pair
of numbers a vector denoted
by
CO + c10(1)
is assigned in such a way that the customary rules of algebraic operations hold; that is, the operation "addition" should be commutative and associative, and the operation "multiplication" by a number should
12
be distributive and associative.
There should be a "zero" vector
0,
such that
00 = 0,
4 + 0 = @,
co = 0.
At present we leave open whether the coefficients
may be
c
complex or should be real; i.e., at present we consider complex and real linear spaces together. to complex spaces.
Eventually we shall confine ourselves
When we speak of a "space" in the following we
shall always mean a "linear space".
We say a space 6 has finite dimension* if it contains a
_(n)
(1)
finite number, n say, of vectors _ vector'
6 can be written as a linear combination of them:
in
0
_ &ly (1) +... +&n-
(2.1)
We, then say, the
of the space n
vectors.
system.
such that every
Ch
n
vectors
is exactly
We then say the
The numbers
-
span the space
C .
The dimension
if it cannot be spanned by less than
n n
gl,...,gn
with respect to this system.
(n)
vectors
generate a coordinate
are called the components of
If 6 is real the components
0 g
are
real, otherwise they may be complex.
A space which cannot be spanned by a finite number of vectors has "infinite dimension".
The manifold of vectors
0
considered in
,our second example is such a space of infinite dimension (for proof see a later section). The number of components
nP
with respect to
the "new" coordinate system is infinite, specifically, a denumerable set of numbers.
The set of "components"
0(s)
with respect to the
original description is non-denumerable, since the set of all numbers
We prefer simply to say "finite dimension" instead of "a finite number of dimensions", although the latter expression corresponds with the original meaning of the term dimension. 13
between
s
-n
and
For this reason we do
is non- denumerable.
n
not simply say the dimension of the space is the cardinal number of the set of components. Let functions
(S
be any space and let
0
of
are represented by the functions of the
c
and the functions
linear combination cp(s) + cic(1)(s) and
We then say the
if there is a one-to-one correspondence of the vectors
(4] Ct3
be a space consisting of
E(s),,a "function space" for short.
vectors of the space space
[R]
01)(s).
0.
0(1)
and
The functions
of the vectors
is represented by the function
cO + c10(1)-
provided
in such a way that the
of
c(s)
F(s)
are represented by
U(s)
will be called the "representers"
Note that the required one-to-one character of
the correspondence guarantees that representation of the
0
c(s)
by functions
implies
= 0
0 = 0.
A
c(s) will be referred to as
a "functional representation"; it will symbolically be expressed by the formula
<---+ The double arrow is to indicate the one-to-one character of the correspondence.
The subscript is to indicate the specific space of
representing functions involved. Evidently, the vectors functional representations.
0
of a space
8 admit different
For instance, the vectors
third example are represented by functions
0
of the
n(u); thus we may write
(-----) { n { u)) . Originally, these vectors were given as functions
p(s); giving
vectors as functions automatically establishes a functional representation.
Instead of writing
0 t 0 '' 0(s), we prefer to write simply
14
in such a case.
The vectors of the second example were also given as functions, 4
with respect to the new coordinate system they were repre-
sented by infinitely many numbers, nu, u = 0, ±1, ±2,...
Here
.
may be regarded as a function of the variable
u, never mind that
this variable was restricted to the integers.
Thus we may write
4)
(
n )
n
In our first example the vectors
@
nu
(nu}. were given with the aid of their
with respect to the original coordinate system.
components
We may just as well say they were given through a particular representation and write accordingly
4 E--->
{
Q}
Note that the set of components of a vector of a representer, a function of the variable
or simply 4 _
(
Q}
then plays the role
0
a
which runs over the
numbers i,2,...,n. Next we discuss the notion of operator.
A linear operator is
a transformation, or mapping, which transforms vectors
spacer
into vectors
A4)
in
IS
in the
0
in such a way that the relation
A(c4) + cl(P(1)) = cA9 + C1AO(1)
(2.2)
holds for all
c,cI
and all
We shall always omit the
qualification "linear"; that is to say whenever we say "operator" we mean "linear operator".
Such an operator need not be defined in the whole space For example, if operator
d/ds
Cr
is the space of continuous functions
6 A
the
is defined only in the subspace of functions with a
continuous derivative. an operator
Cs(s)
Ch.
The manifold of vectors
A is defined is called its "domain"
is a linear space.
0
:n
CAA; by definition
Instead of saying that a vector 15
for which
CB
4)
belongs
CAA, we shall frequently simply say that
to the space A, or that
is applicable to
A
An operator
A
"admits"
0
0.
S of finite dimension,
definbd in a space
e
n, is represented by a matrix, when the vectors
.
by components, $ the components
are represented
$
,
For, let for
Eo,. = 1, o = 0
O(a
)
be the vector with
a # a', and denote by ,
ala,...,ana,
the
n
components of the vector
AO(a
.
Then, by
(2.1) and (2.2) we have
and hence
Am = E ( a
Thus the representation
Clearly, the operator When the vector
a'
implies the representation
¢ =
is represented by the matrix
A $
is represented by functions
continuous variable, the operator
A
(ace,). O(s)
of a
will be represented by a
"functional operator" which may be an integral operator or a differential operator such as nature.
3.
or it may be of a more involved
i d/ds
Later on we shall study such functional operators in detail.
Spectral Representation.
We are now in a position to say what is meant by a "spectral" representation of an operator space
A
defined in a subspace
S.
16
6A of a
First we explain the notion of simple spectral representation. It consists in the (linear) representation of the vectors space
B
by certain functions
n(a)- of a real* variable
such a way that the vector M, for function
(3.1)
an(a).
'
in the
(P
a
in
CSA, corresponds to the
in
In formulas:
' n ) {n(a)}
for all
9
in
for all
(P
in
(
implies
(3.1) A
AQ
(an, (a) }
Thus the operator
A
is simply represented by multiplication by the
independent variable of the functions The variable
a
domain of the functions which the functions
(aA.
n(a).
will be called the "spectral" variable; the n(a), i.e., the set
n(a)
S
of values
a
for
are defined will -- temporarily -- be
called the "spectrum" of the operator
A.
Note that this domain may
consist of the whole a-axis or it may be part of it.
In fact, it may
consist just of distinct points, as in our first and second examples. The notion of "spectrum" is introduced here with reference to a spectral representation.
Later on, in Chapters V and VI we shall define
this term differently, independently of such a representation. Let us ask which role eigen-values and eigen-vectors play in connection with a spectral representation. a'
to which there is a vector
For the representer
n'(a)
H' # 0 of
H'
in
Eigen-values are numbers
eA
such that
AH' = a'H'.
this relation becomes
In some later stage the restriction to real variables will be removed temporarily. 17
an'(a) = a'n'(a),
which implies that
n1(a) = 0
eA
Conversely, suppose that the domain H' # 0 value
whose representer a'.
Then
AH' = a'H'
eigen-vector with eigen-value spectrum; for, else would mean
contains a vector
for all values
for all
a
is an
belongs to the
Clearly, a'
a'.
in the spectrum
a
In other words, H'
holds.
n'(a) = 0
A
of
vanishes except for a particular
n'(a)
an'(a) = a'n'(a)
and hence relation
a # a'.
for
in the spectrum which
H' = 0.
It is quite convenient to use the term "eigen-value" for any value in the spectrum.
But then one must distinguish between the
"proper" eigen-values, i.e., eigen-values in the strict sense as described above, and the others, "improper" ones. are also called point eigen-values.
Proper eigen-values
The set of improper eigen-values
is also called the "continuous" part of the spectrum.
Naturally one may wonder whether or not there is an analogue of an eigen-vector for a continuous spectrum.
Such an analogue can
readily be introduced if it is assigned - not to a single, improper, eigen-value - but to an interval
Am
on the a-axis.
To every such
interval. we assign as "eigen-space" the space of all those vectors in Ch
A
whose representers
terval
Am.
n(a) vanish identically outside of'the in-
The vectors in this eigen-space - except the zero-vector
will then be called eigen-vectors asdociated with the interval For example, in our third problem all functions form
18
0(s)
Aa.
of the
f eisun(u)du
(3.2)
eu
are eigen-functions of the operator - id/ds terval
associated with the in-
eu.
The set of eigen-vectors associated with intervals includes those associated with points provided one regards a single point as an interval:
If the eigen-space associated with a point is empty
(except for the zero-vector) this point is not an eigen-value; otherwise it is.
Incidentally, intervals
differing only in the inclusion
em
of one or two end points must be counted as different in the present context; for, a non-empty eigen-space may belong to such an end point.
Still, one may wonder whether one could not introduce eigenvectors associated strictly with single improper eigen-values.
For
example, in connection with our third example, it might be suggested to introduce the functions
eisu'
since they satisfy the relation
as "'improper eigen-functions"
- i(d/ds)e
since these functions do not belong to
SAt'
lsu,
=
u 'eisu'.
Of course,
the domain of the
operator introduced, it would be necessary to enlarge this domain.
Such an enlargement would have awkward consequences, as was described when this example was discussed.
Therefore, we shall at present not
employ such an extension.
Improper eigen-functions will, however, retain a role in the explicit description of the spectral transformation.
The notion of simple spectral representation which we have described is too narrow since it does not coyer cases with a multiple spectrum.
For this reason it is necessary to introduce more general
notions of spectral representation.
It is expedient to do so also in
order to attain greater flexibility in handling copcrete spectral problems.
Appropriate generalizations may be carried out in two
19
directions.
A first generalization arises if the vectors are represented by functions variable
p
of the space
of an "auxiliary" spectral
in such a way that the vectors
A4), for
a(p)n(p); here
represented by the functions function of
n(p)
0
0
in
CcA, are
is an appropriate
a(p)
We express such a representation by the formulas
p.
$
(3.3)
{n(u)}.
C
A4 n) {a(p)TI (p)}.
(3.3) A
We then speak of an "indirect" spectral representation, while the representation will be called "direct" if
p = a.
The variable
a
will be restricted to an appropriate domain; the range of the function a(p), the "spectral function" is then the spectrum of the operator
A.
The spectral representation implied by the principal axis transformation given in the first example is evidently an indirect one. Every vector
is represented by a function
-
auxiliary spectral variable points
n(u) = np
whose domain consists of the
p
of an n
p = 1,...,n.
As another example we consider the operator
(3.4)
=-
M2
d2 ds2
applicable on functions
a(s)
defined for
-ir
< s <
it
which possess
continuous third derivatives and'satisfy the periodicity conditions 2 n
2 0, dO/ds, d p/ds 1-n = 0.
Suppose we represent these functions
by their Fourier coefficients the functions coefficients
M24(si
np
= -d24(s)/ds2
p2np; i.e.,
20
0(s)
according to (1.9), (1.9)*; then are represented by their Fourier
0(s)
f
: In u)
2
- ds2 0(s) E-
{u2nu}.
This representation is evidently an indirect spectral representation with the auxiliary variable
and the spectral function
u
a(u) = u2.
It is seen that the spectrum of this operator consists of the eigenvalues
0,1,4,9,..., those greater than zero occurring twice, i.e.,
having two linearly independent eigen-functions. The notion of eigen-space associated with an interval
Aa,
which was introduced in connection with the simple spectral representa tion, can immediately be carried over to the case of an indirect We assign to
representation. u
for which the values of a(u)
associated with the interval in
Aa
Ch
whose representers
Suppose to a single point n'(u) = 0
for
u
n(u) u'
yl u'; then
with the eigen-value
Aa
the set of lie in
of those values of
Then the eigen-space
Aa.
consists of all those vectors vanish when
a vector H'
Au
H'
u
is outside of
Ali.
is assigned, so that
is a proper eigen-vector of
A
a'
There are other generalizations of the notion of spectral representation.
We shall discuss them when we need them-.
We only
mention shortly the "multiple spectral representation" in which the vectors are represented - not by a single function - but by several functions.
For example in our second problem we may write our function 0(s)
in the form
21
I nl(U)cos is +
A(s) _
I n2(u)sin us u=1
u=0
and thus represent them by a pair of functions
This
{nl(u),n2(U)).
representation then affords a "multiple" and "indirect" representation of the operator M2A(s)
A = M2 = -d2/ds2
a(u) = U2.
For, the function
is-given by
M2A(s)
4.
with
_
u0
2
U2n2(U)Cos Us +
V n2(u)sin us. u
Functional Calculus
One of the major acnievements that may be attained with-the aid of a spectral representation of an operator consists in setting up; an "operational calculus".
Suppose the vector At
is such that the vector can be applied an
At.
will be denoted by
in the subspace
t
is also in
6'A: then the operator
The operator which transforms
0
into
A A(AA)
A2; i.e., A2@
Similarly, one can define powers r
6A of the space e
= A(A@) . Ar
of
A and polynomials
p(A) _
,c Ap.
P10 p
The question arises: trary functions
can one assign operators
f(A)
to arbi-
f(a) -- or at least to functions of a large class,
certainly comprising polynomials and possibly comprising continuous functions and even a variety of discontinuous functions.
One says
that such an assignment leads to an "operational calculus" if the following propositions hold,
(i)
f1(a) + f2 (a) = f (a) -* f1(A) + f2 (A) = f (A) 22
(ii)
fI(a)f2(a) = f(a)
fI(A)f2(A) = f(A)
(Iii)
f1(f2(a)) - f(a)
f1(f24A)) - f(A).
Here the provision is always to be added that the operations indicated be applicable.
Clearly, these propositions hold for the polynomials
as defined above and, conversely, whenever for an assignment of
p(A)
operators to functions these propositions hold, the operators assigned to polynomials are just those described above. Suppose now the operator
A
admits a spectral representation,
an indirect but simple one, say
0 (_> {n(u)) A@
Then we simply define vector
into that vector whose representer is
0
_y_ The tions
as the operator which transforms the
f(A)
f(A)Q
f(A)
{a(u)n(u)).
16
a vector
4,
0 {f(a(u))n(u))
question arises for which functions
defined in this manner? f(a)
f(a(u))n(u):'
is the function in
which functions
e. n(u)
f(a)
is the operator
In other words, we ask for which func nl(u) - f(a4p))n(u)
the representer of
This question could be answered if one knew are representers of vectors
0
in
6.
So far
in our discussion we have left open this point; but in a later chapter a precise answer to this question will be given.
At present we can state that the definition of the operator f(A)
derived from a spectral representation is such that the propo-
sitions i, ii, iii hold whenever the functions
f,fl,f2,..., of
involved are such that the corresponding operators are defined. 23
a
This is an immediate consequence of the fact that these propositions hold for the functions. fla(p)).
5.
Differential Equations
An elementary, but important, use of a spectral representation can be made in the treatment of linear differential equations.
We shall discuss three types of differential equations for vectors
t = 4)(t)
For every value linear space
(t)
(B.
as functions of a real variable, the "time" t. t > 0, the vector
0(t)
is to be an element of a t, the vector
In its dependence on the variable
is supposed to possess a continuous derivative
respect to
4)(t)
with
t.
What it means for an element of a space to depend continuously on a parameter and to be differentiable with respect to it will be explained in the next section.
At present -- where we are concerned
with only a preliminary discussion -- we may imagine the vectors
0
to bq represented by functions; continuity and differentiability of the vector may then be understood as continuity or differentiability of these functions in any ordinary sense of these terms. We furthermore assume that the vector .4)(t), for each
belongs to the domain
SA
of a (linear) operator
one of the following three differential equations: equation"
(5.1)
0 + A4) = 0,
or the "Schroedinger equation"
(5.2)
-i¢ + AID - 0,
24
A
t > 0,
and satisfies
either the "heat
or the "wave equation" R + A(D = 0.
(5.3)
The vector
0
The solution of the
t = 0.
should be prescribed for
wave equation is furthermore supposed to possess a continuous second derivative and its first derivative
'
should also be prescribed for
t = 0. Suppose now the operator For instance, let every vector
in
0
0
in
0(t)
of any of the three differential equations
is then represented by a function
n(p,t)
possesses a continuous first derivative n(p,t)
Ac,
a(p)n(p)
(B A, is represented by the function
The solution
tive
C be represented by a function
defined in an appropriate p-domain, such that the vector
ntp)
for
admits a spectral representation.
A
in the case (5.3) --
of
p
and
t
which
Mp,t) -- or a second deriva-
and which satisfies the equation
T(p,t) + a(p)n(p,t) = 0,
or
(5.5)
-in(p,t) + a(p)n(p,t) = 0,
or
(5.6)
respectively.
ri(p,t) + a(p)n(p,t) = 0,
Consequently, the representer
(5.7)
n(p,t) = e-ta(p)f(p,0),
(5.8)
n(p,t) =
n(p,t)
e-ita(p)n(p,0),
25
is given by
n(u,t) = cost
(5.9)
sin( t
)
-(11)) n(u,o) +
)n(u,0
respectively.
Thus the solutions of the differential equations in the
n-
representation are found.
If the transformation is known through which a vector given in terms of its representer
n(U), the solutions 0(t)
0
is
of the
three equations can be determined.
These solutions can be expressed with the aid of an operational calculus in the form
(5.7) 1
fi(t)
(5.8) 1
s(t) = e-itA4'(0)
=
e-tA't (0) ,
(t) = cos (t /A-) 0(0) + since ,rA-)
(5.9)
since the operators
f(A)
(0)
entering here are defined by virtue of the
assumptions made in the preceding paragraph. This elegant -- and helpful -- form of the solution of differential equations may serve to illustrate the striking effects that may be produced from spectral representations.
Projectors
At the end of this chapter we shall discuss a particular type of operators, the projectors, which play a dominant role in the spectral theory of operators.
Projectors are operators
P
for which the relation
P(D
26
holds for all vectors
Using the no-
on which it is applicable.
0
tation of'functional calculus the above relation can also be written in the form P2 = P
The manifold of vectors of the form linear space - we denote it by the "projection" of
@
J3:
into this space
P4'
evidently forms a
the vector q3,
P4'
is called
The relation
P2 = P
obviously expresses the condition that a vector in the space
¶3
is
projected into itself.
We should mention that in the literature operators
were
P
originally called projection operators or simply projectors; we prefer to call these operators "projectors" since we like to reserve the term "projection" for the result of applying the operator. To describe a projector in space .we,may consider a k-dimensional subspace
Ch ¶3
of a finite dimension and an ,(n-k)-dimensional
which has only the origin in common with
space
that then every vector vector in
10
0
and one in
vectors by P$
and
in
It is known
can be written as the sum of a
Cy
y3'
13.
Denoting these two
in a unique way.
we realize that by virtue of their
(1-P)O
uniqueness the assignmerfts of
and
P4'
(1-P)4+
to
f
constitute
projectors.
In a space of functions
with the-aid of
2k
functions
O(s)
one may define an operator
$l(s),101(s),...,$k(s),4ik(s)
through,
the formula k
P$(s)
(5)
j
(3)0(s)ds;
K=l
it is evidently a projector provided the functions the relations
27
P
satisfy
K,A = 1,...,k.
J OK(S)O1(s)ds = 6KA'
The space into which
P
projects consists of the linear combinations
it is thus finite dimensional.
of the
In connection with a spectral representation of an operator through functions'
n(u)
which projects a vector with an interval
to.
one should like to introduce a projector 0
into the eigen-space
for which
a(u)
J3
Aa
associated
We recall that this eigen-space consists of
all those vectors whose representers u
A
vanish for all values of
n(u)
lies outside the interval
to.
Such a "spectral
projector" can immediately be constructed with the aid of a functional calculus.
We need only introduce the characteristic function Act, given by
of the interval
If the operator
fAa(a)
fAa(a) = 1
for
a
in
= 0
for
a
not in
fta(A)
desired projector.
to to.
can be defined for such a function it is the
For, the function
fAa(a)
evidently satisfies
the relation
f2a(a) = fAa(a) ; hence the operator
fta(A)
satisfies the relation
f2a(A)
= fta(A)
and thus is seen to be a projector. lies in the eigen-space if
n(lp)
`AOL
Clearly, the vector
since its representer is
is the representer of '0.
28
If
0
fAa(A)41
fAa(a(U))n(p)
is already in this
eigen-space the relation
f0a(a(u))n(u) = n(u)
evidently holds and hence
fAa(A)'V = 9 . Thus it is seen that the eigen-space
$Aa
is exactly the space into
which the "spectral projector"
PAa = fa(A)
projects.
To exemplify the notions of spectral projector we consider our third example in which the functions tions
u(s)
are represented by func-
n(u),
y! F- {n (u) } in such a way that
MO (>
here
M = -id/ds.
From formula (1.16) we realize that the spectral projector transforms the function,
into the function'
0
Pauf(s) = I elsun(U)dµ Au
Substituting
n(u)
from formula (1.16)*
we find
ei(s-s ')uO(s')ds'd;
PAu0(s) = 1n 1
J
Au --
29
Thus we see that the spectral projector is given as an integral operator.
This is typical for cases of operators with a continuous
spectrum acting on functions of a continuous variable. For our second example, where the spectrum is discrete, we find from
(1.9)
and
(1.9)*, the formula IT
=
P
Au
1
(
2n
J
ei(s-s')'O(s')ds'.
The assignment of the spectral projectors and the eigenspaces to an operator of this operator.
A
is said to yield the "spectral resolution"
While in our presentation the spectral representa-
tion is adopted as the basic notion.
We have derived the functional
calculus from the spectral representation and the spectral resolution followed most frequently in the treatment of specific problems.
But
this procedure has disadvantages for the development of the general
The reason is that the spectral representation of an opera-
theory.
tor is not unique; there are many (equivalent) possibilities for it.
The spectral resolution, on the other hand, is unique inasmuch as the functions
f(A)
of an operator are uniquely assigned to it.
It is for this reason that in the development of the general theory the indicated procedure is completely reversed:
first the
Spectral resolution - or the functional calculus - is established; a spectral representation is then derived afterwards.
Thus, in the
general spectral theory of bounded operators which we shall present in Chapter IV we shall in fact first establish the functional calculus and then a spectral representation.
30
CHAPTER II
NORM AND INNER PRODUCT
6.
Normed Spaces
In order to be able to develop any specific theory such as a spectral theory in spaces of infinite dimension it is necessary to endow such a space with specific "structural" features.
The require-
ment of linearity does not give enough structure to a space for our purposes.
The central structure that we want most of our linear
spaces to carry is the "inner product".
Before introducing this con-
caPt we shall discuss the notion of "norm", a structural feature possessed by all linear spaces we shall deal with.
A norm associated with a linear space 11011
assigned to every vector
0
in
92
possessing the following
92
properties:
(6.1)
(6.2)
ICI
I1011;
note that this last property implies
(6.2) 0
11011 = 0.
Further properties are:
(6.3)
1k
=0
implies
0 = 0,
and
(6.4)
1101+011
<-
114)'11 + 114)11
31
is a real number
The latter relation is the "triangle inequality".
V - m
in place of V
or
Inserting in it
and using (6.2) for
-4`
c = -1, we ob-
tain the "second triangle inequality"
(6.5)
I
I
'
I
I
-
I
II
114, '-@II
I
If property (6.3) does not obtain we speak of a "semi-norm".
In a space of finite dimension whose vectors
-
are repre-
we may, for example, introduce the norm
sented by numbers
II
t = 1,...,n
II = max T
or the norm n
II = II = t=1
In the space of continuous
(s), defined in an interval
for example, assign to a function
3 we may,
the maximum of its absolute
q(s)
value as its norm
max 14(s)I,
s
in
s
or we may choose
Ilmll = f10(s)Ids.
All these assignments have the four properties (6.1) to (6.4) and hence qualify as norms; they would be only seminorms, if in its definition we had restricted the variable the variable
i
to a subset of the numbers
s
to a subset of
5 or
1,...,n.
With the aid of the notion of norm we may introduce the notion 32
a (linear) operator
of "bounded operator":
RA and producing vectors in number
defined in a subspace
A
is called bounded if there is a
't
such that
a
for all
aIIpII
Thus an integral operator
KO(s) =
(P
in
!?A.
K, given by
k(s,s')O(s')ds'
1
i1v
and acting on continuous functions
0(s)
in a finite interval 5 is max sin V
bounded with respect to the maximum norm case the "kernel"
in
is a continuous function there; for, then
k(s,s')
we have
IIK0II
with
max
I
Ik(s,s')Ids'.
<_ kiI0II
Differential operators are not bounded
s
however, as shown in Chapter VI.
We shall discuss bounded operators in great detail only in Chapter IV.
Still, we have mentioned this notion already here because
of its great importance and because we shall occasionally refer to it already before reaching Chapter IV.
Once a norm is established in a linear space the notion of "density" can be introduced and the notion of "dimension" can be given a more specific meaning.
A subset
9t'
it if to every vector such that the norm
of a normed space 4
in
Ik '-III
in other words, if every
(p
9t
a vector
92
is said to be dense in in
9t'
can be found
of the difference, is arbitrarily small; in
9t
33
can be approximated arbitrarily
closely by vectors of
92', closeness being measured with the aid of
the norm.
Thus, the space of polynomials of continuous functions
p(s)
is dense in the space
defined in the closed interval 9
4'(s)
is adopted as the norm. That Im(s)1 max sin 1 The set is the statement of the Weierstrass approximation theorem.
if the norm
1101 =
of trigonometric polynomials
cvelvs is not dense in
I
64
with
vl
respect to the maximum norm, but it is dense with respect to this norm qo, consisting of periodic functions, i,e., of
in the subspace
(n) = $(-n).
functions with
But the trigonometric polynomials are
dense in the space of continuous functions if the "absolute integral" norm
11011 = J
I0(s)Ids,
is adopted.
A subset the set of
92"
92'
4
92
is said to "span" the space densely if
consisting of all finite linear combinations of vectors
92' is dense in
spans
of
T.
Thus the space of powers
sn, n = 0,1,2,...
densely with respect to the maximum norm.
In Section 2 we said that a space is of infinite dimension if it does not consist of the linear combinations of a finite number of vectors.
We now say that a space is of "countable" or "denumerable
dimension" if there is a countable set, i.e., an infinite sequence of vectors which spans the space densely.
Thus the space of continuous
functions mentioned is of countable dimension with respect to the maximum norm.
Note that the notion of dimension, as introduced here,
depends on the choice of the norm.
In fact, as we shall show in a
later section, the same space can be made to have nondenumerable
34
dimension by imposing another norm on it.
Spaces of countable dimension are more customarily called "separable".
Note that the space of continuous functions
O(s) is of count-
able dimension, or separable, with respect to the maximum norm and the absolute integral norm, in spite of the fact that the vectors of this space possess a non-countable number of components - if we consider as these components the values of
(s) at all the points
s.
Inner Product
The inner product, the central structural feature of the linear spaces we shall investigate, is the natural analogue of the inner product of two vectors in Euclidean geometry.
The natural
analogue of the magnitude of an Euclidean vector is a norm which can be formed with the aid of an inner product.
Solely because of the analogy with Euclidean geometry, the geometry of spaces carrying an inner product would be worthwhile to investigate.
For us there is a more specific reason for doing so.
With refe;ence to an inner product the notion of "self-adjointness" of an operator can be defined. able property: are real.
Self-adjoint operators have a remark-
they admit a spectral representation and their spectra
Self-adjoint operators are the primary object of this
set of notes.
The inner product associated with a linear space complex number
(0',4)
assigned to any pair
It is linear in the second "factor": c10(1) + c241 (2))
= C1(0',0(1))
r It is symmetric in the Hermitean sense:
35
is a
@',¢ of vectors in
possessing the following propefties, (7.1) to (7.4):
(7.1)
YI
(2)
+ C2(0(D.
(4',4") _ (4',4);
(7.2)
the bar here indicates that the complex conjugate is to be taken. Property (7.2) implies that the associated quadratic form real.
(0,0)
is
With this fact in mind we can formulate the next property of
an inner product:
The quadratic form
(V,4)
is positive definite.
This property will be split into two parts:
(7.3)
(4, 4) > 0,
and
(4,4) = 0
(7.4)
implies
= 0.
A space in which an inner product is defined which has these four properties is called an inner-product-space; if it has the first three properties but not necessarily the last one, (7.4), it will be called a semi-inner-product-space. If the space
A4
is real, the inner product is linear also
in the first factor; if it is complex, the relation
(7.1)*
c2t(2),(r) =
c2(4(2)
holds, as follows from (7.1) combined with (7.2).
40)
One refers to this
relation by saying that the inner product is "antilinear" in the first factor.
(The term conjugate linear would seem more appropriate.)
The most important basic property of such an inner product space is embodied in "Schwarz's inequality"
(7.5)
36
For convenience we have written inner product
instead of
(m',t)
I(4',')j.
we first assume that the inner product arbitrary real numbers
for the absolute value of the
1k',')
To prove this inequality
(4',m)
is real.
we then derive from
a,a1
(7.1),
With two (7.l)* the
relation
0
<
(at + a1'', a; +
g1@')
a2(',4) + 2aa1('',t) + ai('',V),
which expresses the fact that the right member here is a non-negative quadratic form in real, we let
8
a,a1.
Therefore (7.5) holds.
be a number of absolute value
If 1
is not
(40,0)
such that
8($',m)
.is real and non-negative, i.e.
0(4'',(D) =
Then
W,8@) = 8('',41)
14) ',0, I
> 0.
is non-negative, and hence the statement
results from
Note that the property (7.4) was not used in this proof.
Consequently,
the Schwarz inequality holds also for semi-inner-product-spaces.
Furthermore, one readily verifies that equality holds in the Schwarz inequality if and only if there are two complex numbers and
cl, not both being zero, such that
CO + c1''I
c4 + c1'' = 0 - or
c
Ict+c101,
in case of a semi-inner-product-space.
The quadratic form
(4','') will be called the "unit form".
We
maintain that -- as in Euclidean geometry -- the square root of the unit form
37
II0I1.=
(7.6)
may serve as a norm.
is real follows from
The fact that
(7.2) as noted above.
Properties (6.1) and (6.2) are immediate con-
sequences of (7.3) and (7.1).
Property (6.3) is implied by (7.4); if is a semi-norm.
(7.3) does not obtain, 11011
It remains to prove,
property (6.4), the triangle inequality.
This inequality is an immediate consequence of the Schwarz inequality.
We first write this inequality in the form
(7.5)'
14.'.01-
II - 'II
114,11
and then proceed as follows:
1lmwll2
= (t+0',m+4')
(Q,0) + (0,$') + (0',0) + (0',0') II.P11114.'Il
11,p112
II.D'i!
+
+
II,DII
+ IIml12
111,11 + 114'11)2-
this is the statement.
It is clear from this proof and the remarks made at the end of the proof of Schwarz's inequality that equality holds in the triangle inequality if and only if there are two complex numbers such that
8.
c,cl
c$ + c V = 0. 1
Inner Products in Function Spaces
We proceed to discuss various specific expressions for inner products commonly adopted in specific linear spaces commonly considered. In doing this we shall frequently -- for convenience -- just describe
38
the unit form; the proper expression of the inner product can be derived from it in an obvious manner. in a real finite dimensional space of vectors sented by
1,...' n
components
n
-
repre-
the commonly adopted inner
product is the one associated with Euclidean geometry n a
l
e
In a complex finite-dimensional space one commonly adopts the analogous
Hermitean inner product
(_ _l
n_
The validity of the requirements (7.1) to (7.4) is immediately verified.
Of course, other bilinear forms associated with positive definite, quadratic forms couid be chosen.
As an example of an infinite-dimensional-space carrying an inner product we consider the space of continuous functions defined in an interval
$(s)
R of'the s-axis; for these functions we may
define as inner product the integral
(v.o) =
(8.2)
m'(s)+(s)ds.
J
Clearly, requirements (7.1) - (7.4) are satisfied. The associated norm in this inner product function space is evidently
(8.2)'
11411 =
[f
1/2 1 (s) 12ds-1
R
39
Instead of'a finite interval we may take for so <
s < -
<.s < -.
or the full s-axis,
at present that each function
*(s)
R a ray
In that case we require
be of bounded support, i.e.,
vanishes identically outside of an appropriate finite interval. such functions the inner product (8.2)
is defined.,
For
Later on we shall
enlarge this class of functions.
our aims in this chapter, as in Chapter I, are to present the formal aspects of spectral representation.
In Chapter III and sub-
sequent chapters we shall extend the classes of admitted functions so as to attain the desired completeness of the theory. Accordingly, we may suit ourselves in the choice of the class of functions been selected.
(s), once the region R and the inner product have On the other hand, we should already at the present
state attain a certain generality in the choice of the region R and the inner product.
Before describing such generalizations it is opportune to enlarge the class of continuous functions to the class of "piece-wise continuous" functions, and to employ a more general notion of integration, "integration with respect to a measure function". First, we introduce the notion of partition of the s-axis. As a partition
y° we consider an alternating sequence of open
intervals and points:
o: sa-1 < s < sa+l,
a
even,
Q:
a
odd,
s = sa
-W < a <m. We prefer this type of partition to other types of partitions which could, of course, also be used. We call a function-
f(s)
"piecewise constant" if, with 40
reference to an appropriate partition, f(s) = f
= constant for
s
in
3.
Clearly, such a function is also piecewise constant with reference to a subdivision of of the open intervals of
9, obtained by subdividing §ome or all
9 again into open intervals and points.
Since two partitions evidently have a common subdivision, it is clear that piecewise constant functions form a linear space.
Every continuous function can be approximated by piecewise constant functions, uniformly in every bounded interval. enlarge the space
4
We might
of all continuous functions to the space of all
functions which can be approximated by piecewise constant ones, uniformly in every bounded interval.
We are satisfied, however, with
considering a restricted extension of the space of
"piecewise continuous functions".
the space
t:
We call a function
at
f(s)
piecewise continuous if, in each open interval
of a sa-1 < s < so+1 agrees with a function which is continuous in the
partition, f(s)
closure, sa-1 < s < sa+1, of this interval.
(The values of this con-
tinuous function at the end points need not be related to the values of
there.)
f(s)
Clearly, each piecewise continuous function can be
approximated Ky piecewise constant ones, uniformly in every bounded interval.
-
Piecewise continuous, as well as continuous, functions evident
ly form linear spaces, and the product of two functions of any of these spaces again belongs to that space.
To attain a more general form of an inner product in a function space we employ the notion of integration with respect to a "measure function".
Such a function
r(s)
valued functioq defined for all values of The values, of
r(s)
at points
s
where
41
is a non-decreasing reals; i.e., for r(s)
-- < s <
is not continuous are
irrelevant in the following; therefore we introduce at each point the limit values
when
approaches
s'
and
r+(s) s
which the values
r -(s)
r(s')
from above and below respectively.
s
approach
We call
r - (s) a "measure function pair".
To each interval a of a partition
9 we now assign as its
measure the differ@nce
Aar = r (s(Y+l)
ya
i.e., if
-
for
r+(sa_l)
a
even,
is an open interval, and
A
a
r = r+(s a)
i.e., if a is a point.
- r (sa)
for
a
odd,
Clearly, if the open interval
9a is
subdivided
Q-
T
-)F r T
'
we have
AIr
Gar =
in obvious notation.
Let 9 be an open or closed interval; then for any piecewise constant function
f(s) - with reference to a partition which, contains
the end points of 2 - we define the integral of to
r
by
l
f(s)dr(s) _
a in.Q 42
f aAQ r
f(s)
with respect
is not changed by subdivision of
Clearly, this value If now
is a function in the spate.
f(s)
-60.
we may define
(A'
its integral simply as the limit
f(s)dr(s) = J
lim V -. =
f(V) (s)dr(s) J
M
of uniformly approximating piecewise constant functions
f(') (s).
From the inequality
dr(s),
f(s)dr(s) I < maic ,f(s) t 1
fm
it follows that this limit is independent of the choice of the sequence of approximating functions
f(V) (s).
On the other hand,
f(s)dr(s) > 0.
f(s) > 0, obviously
If
)
the relation
f(s)dr(s) = 0
together with
f(s) > 0
1
does not imply
f(s)
0
unless special provisions are made.
First of all the values of which
f(s)
in any open interval in
is constant do not contribute to the value of the
r(s)
Therefore, we may simply consider the function
integral.
defined only in the "carrier" of the measure function
J
f(s)dr(s)
which is
> 0
but
0.
point
at which
r(s)
s
not changed.
where
r(s)
r(s).
may be zero, for a function
For if the value of
f(s)
as
r(s), i.e., the
set obtained by removing all open constancy intervals of even the integral
f(s)
Still,
f(s)
is changed at a
is continuous, the value of the integral is
If, however, we require that
f(s)
is continuous, the statement that
43
should be continuous
1
f (s)dr (s) = 0 for f (s) > 0 implies f (s) = 0
becomes valid, as easily verified.
In the space of defined in the interval
41 functions _Q
44'
which are
we may adopt the expression
(s)1' (s)dr(s)
(0101) = J as semi-inner product.
in
$ (s)
The properties (7.1), (7.2), and (7.3) re-
qufred of an inner product are verified from the fact that they If we consider the function
obtain for piecewise constant functions.
as defined only in the carrier
4(s)
r(s)
stipglate that it be continuous where
r(s)
inside
.
and
is, the inner product is
a strict one, i.e., (7.4) holds, since then. (m(s){2dr(s) = 0
implies
4(s) = 0.
%
1
:ql
We may glad consider the space 4'
41' consisting of functions
41(s)
in
which vanish identically outside a bounded interval (depending on
0), and set. (41,4') = J 0(s)41' (s)dr(s) where the integration is to be extended over any interval.outside of which
4(s) = 0.
Such functions will be said to have a "bounded support", the support of a function being the closure of the set on which it does not vanish.
An extension to spaces of functions which are not re-
quired to be of bounded support will be given in Chapter III.
To illustrate the variety of cases in which the inner product is given by an integral of the kind described let us first take the function
44
r(s)
s<0
0 p + s
P+sl for
p > 0.
and
s > sl.
The functions
m(s)
0 < s < sl
,
s1 < s
,
may be defined as zero for
s < 0
with it-
The unit form, i.e., the inner product of
self, can be written as Si j
I$(s)I2ds + 0
note that the value
(0)
is an independent quantity not related by
a continuity requirement to the values of
(s)
for
s > 0.
Generalizing further, we may consider the unit form to be given by a
VIlPV1osV)
I2
+
j
I*$(s) I2ds,
0
where Pl,P2
sl,s2....
is an increasing sequfence of negative numbers, and
is a sequence of positive numbers with
PV <
We realize
V=1
easily that this expression could be written as an integral with respect to a measure function. Finally
we realize that a unit form given as a sum
E PVI$(sX2
V=1
may be written as such an integral if the
sV
are any sequence of
distinct numbers and the positive numbers
pV
are only required to
be such that their sum converges, numbers
£ PV < a. V=1
As a matter of fact, the
might form a dense set, e.g., they might consist of all
s V
-
rational numbers in a finite interval.
45
All such possibilities are important in spectral theory. Naturally, we may also consider functions variable
f(s)
of a multiple
defined in a region R of the k-dimensional
s = s1,...,sk
space and set up inner products with the unit form
where
ds = ds1 .. dsk.
The notion of integral with respect to a
measure could, of course, be extended to functions of several variables.
We may even consider functions variable
s = s1,s2,...
$(s)
of an "infinite"
In that case the notion of integral of
.
functions of infinitely many variables will have to be employed. may go even further and let for example by letting
s
s
We
stand for a continuum of variables,
stand for the values of a function
s(&)
of a continuous variable The latter possibilities are of importance in spectral.problems We shall discuss such problems in
of the quantum theory of fields.
detail at the end of
9.
these notes.
Formally Self-Adjoint Operators At the beginning of Section 7 it was said that the important
property of self-adjointness of an operator refers to an inner product
Z.
defined in space Suppose
%A and
3 t A
A
At
and
of
%, which have the property that the relation
(9.1)
holds for all vectors
are two operators, defined in subspaces
(40,At0)
0
in
= (AV, 0)
s t, A
46
@'
in
3A.
Then we say that
the operators
the operator
A A
and
are "formally adjoint" to each other.
At
is formally adjoint to itself, so that
At = A
If
and
the relation
('',A4) _
(9.2)
holds for all vectors
t,V
2ZA' the operator
in
A will be called
"formally self-adjoint".
The qualification "formal" is employed in order to distinguish "formal" adjointness from the more restrictive property of "strict" adjointness, which we shall discuss in later parts of these notes.
The notion of formal self-adjointness is the natural generalization of the notion of symmetry of a matrix; in fact, if the operator acts in a finite-dimensional real space, the formal self-adjointness just reduces to the symmetry of the representing matrix.
The formal
adjoint of an operator then corresponds to the transposed matrix. To see this consider a finite dimensional real space of
'vectors = given by sets of numbers inner product of two vectors
and suppose the
and
is given by
n
I qt,
CF=l
Let the operators
A
and
be represented by matrices
At
Ca
,} (To
{a.aso that
(-
'A
t_
t
_
,
-, _1I
n
I(
The identity of these expressions for all values of
47
{E0}
and
and
evidently leads to the relation
- as a
a o' {at}
which shows that the matrix
is indeed the transpose of
{a}.
Formal self-adjointness, At = A, is thus seen to be equivalent to the symmetry
as a,
= aa,a
of
{a}.
If the finite dimensional space is complex and carries the inner product n
a=1
the formal adjointness relation n
n
a a
between the operators
t =laaa'a'a = aIa'=i6a6a
A
and
as
a`
{at}
which shows that the matrix {a}.
At
leads to the relation
= aa'a is the "Hermitean transpose" of
Formal self-adjointness, At = A, then is equivalent with
"Hermitean" symmetry of
{a}
as
a' - aago
In view of these relationships, formally self-adjoint operators in any inner product
space are also called symmetric if the
space is real and Hermitean if it is complex.
We recall that the matrix we dealt with in Example 1.
48
in Section 2 was required to be symmetric.
It is a well known conse-
quence of this symmetry that a principal axes transformation is possible and that the eigenvalues are real. In the present chapter we shall not yet be able to produce a
spectral representation -- the analogue of the principal axes transformation -- for operators acting in spaces of infinite dimension.
We are, however, able to establish the other basic property
mentioned, namely the property of formally self-adjoint operators that their eigenvalues are real.
This fact is a consequence of the important property of any formally self-adjoint operator
A, that the quadratic form
associated with it assumes only real values.
(4,,Al)
That this is so follows
from formulas (7.2) and (9.2) which yield the relation
(4',Ad)) = (All, 41) = (41,A(P) . To show that the point eigenvalues of a formally self-adjoint operator are real let eigenvector
a
be such an eigenvalue associated with the
H # 0, so that
AH = aH.
This eigenvalue can evidently,
be expressed as the quotient
(H,AH)
(9.3)
(H,H
of the forms
real, a
(H,AH)
and
(H,H).
Since the values of these forms are
is real.
We shall not attempt to prove an analogous statement for continuous spectra of formally self-adjoint operators.
For a strictly
"self-adjoint operator", the reality of its spectrum will be evident from its spectral representation, see Chapters IV and VI.
In these
chapters we shall also describe the form the inner product assumes if
49
the vectors of the space are given. in spectral representation.
10.
Adjoint Operators in Function Spaces In the present section we shall discuss the meaning of formal
adjointness for operators acting in "function spaces". Suppose that the space functions
m(s)
'Ii
consists of piecewise continuous
.'
defined in the carrier
of a measure function
in an interval 4 and suppose the inner product is given by
r(s)
($',c )
=
f 4' s $(s)dr(s) .
Suppose further that the operators A operators with kernels are in the carrier
and
a(s,s'), 0(s,s')
B
are given as integral
defined when
s
and
s'
(at present, we assume these functions to be
54"
continuous there):
A$(s) = J a(s,s')$(s')dr(s') '
(10.1)
B$(s) _ 10
Here the terms BO
and to
A
A'(s), Be(s)
at the point
(s).
stand for the values of the function Then the operator
B
AO
is formally adjoint
exactly if
(10.2)
.
6(s,s') = a(s',s).
.The integral operator is formally self-adjoint, or Hermitian, At = A, if its kernel
(10.3)
a(s,s')
satisfies the relation
a(s,s') = a(s',s).
50
This condition on the kernel of an integral operator is quite analogous to the condition of Hermitean symmetry of a matrix. For differential operators the analogy is less obvious. Let us specifically consider the operator D = d
ds acting on functions
¢(s)
defined in a region
', which may be a
finite closed interval of the s-axis, a ray, such as
0 < s < -, or
the total s-axis.
We require that the functions
4)(s)
are at least continuous
in . R; if the domain extends to infinity, we require the functions to be of bounded support; i.e., each function
0(s)
should be iden-
tically zero outside an appropriate finite interval (depending on
0).
Then certainly
0) _
10(s)12ds
f
is finite.
The operator space
2Z = CE
described.
D
will be applicable only in a subspace of the
of continuous functions with bounded support just
Let us denote by
the space of functions
X41
which have a continuous derivative defined in
ZA
4(s) = d4)(s)/ds.
Then
4)(s)
D
in 9 is
([1'
In case the interval is formally adjoint to
_Q
is the total s-axis the operator
D; for, we have
(4',D4)) + (D4',4)).=
_
0 '(s)4(s)ds + _M d ds
t;(S)O(s)ds o.
(0'(s)0(s))ds = 0
51
-D
since both
O(s)
and
were required to vanish identically out-
iP(s,')
The same statement is not true if
side appropriate finite intervals.
has one or two finite endpoints.
the domain
Still, for such an operator
D
we can find a formally adjoint 0
operator. 4(s)
in
of all functions a1 The (11, which vanish at the finite endpoints of -IV.
It will be defined in the domain
0
operator
D
will be called
when restricted to this domain
D.
0
Clearly, we may state that
is formally adjoint to
Dt = -D
0
D:
0
(O',DO) + (DO',O) = 0
for
in
0
C11,4'
in
(I 1.
since the left member here equals the expression
ds (q'(s)0(s))ds,
which vanishes since
4'(s)
vanishes at the endpoints of
If we restrict also the function
M.
to belong to the subspace
0
0
C1
the above relation remains true:
0 (0',D$) + (D4',4) = 0
so that
0
-D
for
in
$,O'
is also formally adjoint to
o
This fact is equivalent
D.
0
M = iD
with the statement that the operator
for
4,'''
al,
is formally self-adjoint:
in
0 91.
For, we have
0
0
(m',Mc) - (M4',l) = i(4',D4) + i(D4',O) = 0.
52
This self-adjointness was achieved by restricting the domain 0
641. One may wonder of applicability of d/ds to the domain 4 whether this restriction was not too severe. As will be shown in
0
Chapter VI, no spectral'representation of the operator, M
in
is
4il
possible and at the same time it will be shown that no extension of its
domain will help if the region R has only one finite endpoint. With the aid of the operators
D
ential operators of the second order.
(s)
space of those functions
.in
D
for which (Yl
0 L = -DD =-
(10.4)
we may form differ-
Let us denote by
in
Then we set
641.
and
D
64L
_
the $(s)
is
d2
ds2
Clearly, this operator and
0
0'
in
64L
L, defined in
64
L,
is Hermitean.
For, with
we have
-(m',DD4))
= (Dm',D4))
and just as well
-(DD04'.4))
(L(O'.O) _
D. and
since
0 D
= (DO',DO)
are formally adjoint.
Note that the boundary condition which restricts affects only one factor 0
the operator
d
ds
of
d (
ds)
2 .
D
to
0 D
It is for this reason that
-DD, (after its domain has been properly extended) can
be made into a strictly self-adjoint one; see Chapter VI. The condition that the function 2
-d /ds
2
4)(s)
on which the operator
acts should vanish at the endpoints is called the "first"
boundary condition.
The second boundary condition on the domain of
53
-d2/ds2, which requires the derivative of points, can be handled in a similar manner. the operator
-d2/ds2
0 -DD
as the operator
to vanish qt the end-
4
One need only interpret applicable on those func0
tions
0(s)
in
for which the derivative
D
- d /ds
is in
Ill
One immediately verifies that
and vanishes at the endpoints. this operator is also Hermitean.
11.
Orthogonality Two vectors
$,V
are said to be orthogonal or perpendicular
to each other if their inner product is zero:
(4,c') = 0.
(11.1)
Evidently, this terminology is chosen in analogy with that used in Euclidean geometry.
Just as in Euclidean geometry, the notion of
"orthogonal coordinate system" is of importance for inner product spaces.
n(2),...,
We say, a finite or infinite system of vectors
n(1)
is "orthonormal" if
(S2(K),n(K))
= 1,
K = 1,2,...
(11.2)
(D(K),Q(X)) = 0,
K # A = 1,2,...
A finite linear combination of such vectors
S2 (K)
.
is a vector
which can be written as a sum
(11.3)
c 0(K)
(D =
K=1,2,...
K
in which only a finite number of coefficients
cK
differ from zero.
These coefficients can be retrieved with the aid of the formula
54
cK = (c(K)
(11.4)
D)
which is immediately verified.
The linear space
[ft(K) ]
formed by all these finite combina-
tions will be said to be "spanned" by the system of vectors
52(K)
We observe that the spectral representations in our first two examples involved such an orthonormal system; the unit vectors were eigen-vectors accident.
H
of the operator considered.
2
That was no
Quite generally, the following theorem holds:
Eigen-vectors of a Hermitean operator which belong to
Theorem 11.1.
two distinct eigen-values are orthogonal. Let
A
be the operator, H(1) # 0, H(2) # 0, be the eigen-
vectors, al # a2
the eigen-values.
a2(H(1)H(2))
_
Then we have
(H(1),a2H(2)) (AH(1),H(2))
_ =
(H(1),AH(2))
1(H(1),H(2))
Here we have made use of the fact, proved in Section 9, that the eigen-values of a Hermitean operator are real. assumed, the relation
Since
al # a2
was
(H(1),H(2)) = 0, and thus the statement of the
theorem, follows.
12.
Orthogonal Projection Spectral representatidn of operators with a continuous (or
partly continuous) spectrum cannot be described simply with the aid of orthogonal systems of eigen-vectors in the proper sense; eigen-vectors assigned to intervals in the spectrum may be used.
In this connection
the notion of "projection" plays a leading part.
In Section 5 we defined a projector as a linear operator
55
P
for which
P2 = P; such a projector projects every vector on which it
Z
is applicable into the space
of vectors of the form
P4), in such
a way that every vector in this space is transformed into itself.
A projector acting in an inner-product-space property that the difference
of any vector
is orthogonal to the space
Pf
jection
f - P4)
ID
and its pro-
0
into which
Such a projector is called an "orthogonal projector".
may have the
2t
P
projects.
From the
property
(P4),(1-P)4)') = 0
(12.1)
for all
(D,4)'
(admitting
P)
we deduce the important fact that an orthogonal projector is formally self-adjoint.
For, relation (12.1)
together with
((1-P) 4),P4)') = 0
leads to the relation
(P(P,@') = (4,P(P')
(12:2)
admitting
for all
4),0'
P.
Conversely, every Hermitean projector is orthogonal. lation (12.1) follows from (12.2) by substituting Writing the vector
for
0.
in the form
4) = P(P + (1-P)0
(12.3)
and evaluating
@
P4)
For, re-
114,112
by using (7.1) and (7.1)* we obtain as imme-
diate consequence of relation (12.1) the identity
56
II(l-P)4,II2
(12.4)
II, I12 = IIPD112 +
which may be regarded as an extension of the Pythagorean theorem.
We denote by
the "complementary space" of
V
the space of all vectors orthogonal to
P.
In view of (12.1) we then
may interpret formula (12.3) by saying that every vector P
can be written as the sum of a vector in
i.e.,
j3
$
admitting
@
and one in
g31.
From identity (12.4) we infer the inequality
(12.4)'
<
IIPDII
which expresses the fact that an orthogonal projection of a vector is shorter than the vector itself.
In particular, this formula implies
that an orthogonal projector is bounded in the sense explained at the end of Section 6.
Projection on a one-dimensional space is always possible. this space
gal 'consist of vectors of the form
Then the projection
P14'
P
as is immediately verified.
1
(Q
(D) (Q,fl) a
4'
Projection into a space
dimension is also always possible. linear combinations of vectors
gtn
of finite
Let this space consist of the
S2(1),...,5Z(n)
Pn4 = C1Q(1) + ... +
0
f2 # 0.
on this space is given by
(12.5)
of a vector
cQ, where
Let
Then, the projection
(n)
on this space can be found by solving the
equations
57
n
linear
n (12.6)
E
(n (a) ,n(v)) &V = (n(a) ,'Z)
,
X = 1,...,n.
v=1
That these equations have a solution follows from the fact that every linear relation that holds between the left members also holds between the'right members.
identically in
Instead of verifying this fact -- which could easily be done we shall derive the statement from the fact that the vectors S1(n),
assumed to be linearly independent can be chosen to be
perpendicular to each other. If these vectors are mutually orthogonal and normal, (v)11
s 1, the projector into the space
11
Y)n
spanned by them is
simply given by n (12.7)
8=1
Suppose the statement holds for (n-1)
Q(n)
13.
by
Q(n)
(n-1); then we may assume
to be mutually orthogonal and normal. -
Pn-l,(n)
the statement follows for
Replacing n.
Remarks about the Role of Self-Rdjoint Operators in Physics In Section 5 at the end of Chapter I, we have described three
simple linear differential equations of physics in order to illustrate the use one can make of the spectral representation of the operator involved.
To explain the role of self-adjointness in problems of
physics let us consider more specifically the "wave equation"
0 + At - 0 for a vector. 0 - (t), and show that the self-adjointness of the
operator A implies the validity of the law of conservation of
59
It is mainly for this reason that the operators of physics
energy.
are self-adjoint.
The (real) vector
@
here may stand for the displacement of
a single mass particle, or of a system of displaced position:
4) _
k
particles, from an un-
We may assume that these
k}.
particles are connected by elastic springs or other mechanisms which try to bring them back to their original positions. may let the vector
0
stand for a function
$(x)
Instead, we also
which assigns to
each point in a continuous elastic medium the displacement of the particle which originally was located at the point In either case, the vector
-At
x.
stands for the forces per unit
mass exerted on the displaced particles by the "restoring" mechanism.
In the linear space of displacement vectors let us introduce an inner product by taking as unit form the expression k
KK or, in case of a continuous medium
!
Q, the expression
02(x)dx,
Op
where
dx
stands for a 3,2 or 1-dimensional differential according to
the dimension of
R.
Further,'let us introduce a "mass operator"
given by
M@ = {m141,...,mK0K}
where
m1,...,mK
are the masses of the
M(
u(x)4(x)
59
k* particles, or by
M,
where
µ(x)
is the mass density, i.e., the mass per unit volume, area,
or length at the point
Then the vector
x.
-MAO
gives the restoring 10
forces or force densities acting on the displaced particles. Since this force vector placement
-MAO
0, the potential energy,
mechanism in the position on
depends linearly on the disstored in the restoring
U
can simply be expressed in the form
U = . (O,MAf)
with the aid of the inner product. Suppose now the operator
MA
is formally self-adjoint, (i.e,
symmetric, since the present vector space is real):
(4',MAO) = (MAO', 0)
.
Then the time-rate of change of the potential energy is
U =.
($,MA@) + 2
Using the symmetry of the operator
M we find the time rate of change
of the total energy
E_
(t,Mi) + . (IP,MA4)
to be
E = 2 (O,M4+MAO) + 2
and hence zero by virtue of the differential equation. that the symmetry of the operator
MA
60
Thus, we see
implies the cpnstancy of the
total energy
E.
Also, for the solution
f(t)
of the Schrodinger equation
4 + iA4 - 0 an important constancy relation obtains provided the operator
A
is
formally self-adjoint, i.e., Hermitean, since we now assume the space to be complex.
In fact, under this assumption we have
+
dt
i{(41,A(D)
- (A4,4)} = 0,
so that
(0,0) = constant
for a solution of the Schrodinger equation.
This result is vital
since it is one of the stipulations of quantum theory that the vector i
representing a state should have the norm 1.
The Schrodinger
equation should be consistent with this stipulation; if the norm 1I4II of its solution equals I initially, it should equal 1 at all times.
That is the case if the operator
A, the Hamiltonian energy
operator, is Hermitean.
This fact is very important, but it is not the only important feature of quantum theory in which self-adjointness plays a vital role. We should like to discuss such other features.
When a quantity associated with a physical object is measured the outcome will depend on the "state" in which this object is.
The
states are identified with the elements of a vector space in which an inner product is defined.
Properly speaking, the states are supposed
61
with the norm 1; if the norm is not 1 one speaks of
to be vectors
a "state vector" belonging to the state
0/11011.
Any quantity, or
"observable", is associated with a self-adjoint linear operator acting on the state vectors
$.
We simply use the same letter, A
denote the observable and the operator.
say, to
The outcome of the measurement
of such a quantity is, not completely determined on the basis of the
laws of physics as formulated by the quantum theory; but, the expectation value of this outcome is determined.
It is given by the
expression
(',A$) . Since the operator
A
is assumed to be Hermitean, this expected value
This reality is one of the required features of
is a real number. quantum theory.
Another feature required of this theory is that to every -say piecewise continuous -- function an observable
t(A)
there should be assigned
in such a way that measurement of this observ-
able should give the value a.
f(a)
f(a)
if measurement of
A
gave the value
Such an assignment will be given by a "functional calculus" in the
sense described in Section 3. the operator
A
Such a calculus is indeed possible if
is self-adjoint in the strict sense; see Chapters IV
and VI.
Consider, in particular, the characteristic'functian of,the interval inside
Aa, and
nAa(a)
Am; that is, the function which equals L for a 0
outside.. The "spectral projector"
PAa
= nAa(A)
may also be regarded as an observable which can assume only'the values 0
and
1.
Measuring this value means to find out whether or not
measurement of the observable the interval PAa
Am.
A would yield a value inside or outside
The expectation value
($,PAa(b)
is then nothing but the probability that
62
of the observable
A would be found to
have a value in
Aa.
Thus, this probability is given by the expression
IIPAaotI2 = (,P$). The formulation of this basic contention involves the notion of spectral projector. It`is clear that this
formulation would not have been possible had it not been required that the operators which correspond to observables possess a spectral resolution.
Suppose the interval observation of the observable
Aa
is just a point
PAa = Pa
able
= Pam
a, the probability of finding/the value
evidently equals 1.
If, before
is to be nade.
measurement, the object was in an eigJnstate eigen-value
a, and suppose an
with the a
for
A
In such a case then, measurement of the observ-
A will yield with certainty the value
a.
This certainty of
finding ,a definite value could never be attained for the improper
eigen-values in a continuous spectrum.
63
CHAPTER III
HILBERT SPACE
14.
Completeness
The statements about spectral representations which we have made up to now were not quite satisfactory insofar as the linear spaces in which the operators were supposed to act were specified in a rather arbitrary manner.
For example, when we required continuous
differentiability of the functions for which the Fourier series expansion was formulated we knew that this requirement was stronger than necessary; the situation was similar with the other spectral representations considered.
The class of functions admitted could
have been enlarged without modifying the formulation of the problem. These classes could have been taken still larger if one had adopted a generalized notion of convergence of a series or an integral.
It is
an important fact that this process of enlargement of the linear spaces'has'a definite end.
There is a definite largest linear space
in all spectral. problems in which the spectral representation of an operator is possible.
The property characterizing such space is the
"completeness", which in turn involves the notions of "convergence" and "limit".
Let norm
11011
T
be a linear space of elements
t
provided with a
satisfying all the requirements postulated in Section 6.
Then the notion of convergence is defined as follows:
A sequence of vectors
10v
0v
converges to a vector
0
- 011 + 0 as v + -.
We note that the convergence of a sequence of vectors a limit vector
my
if
0v
to
implies that their norms converge to that of the
64
limit:
as
III(")II
(14.2)
This follows immediately from relation (14.1) if one uses the second triangle inequality (6.5) in the form
II -t (")II -
I
I V I
I
I
- II't (") - 4,II-
I
In a space of finite dimension convergence in the sense here defined, means just convergence of each component, but in spaces of infinite dimension that is not so.
In the space of continuous functions (s) interval
norm
defined in the
-.V: sl < s < s2, convergence with respect to the maximum
II4-II = max
is nothing but "uniform convergence",
9
while convergence with respect to the norm 1/2
IIPII =
[jISI2as1
is the same as convergence "in the mean":
I
Jr
(s) -$(s)I2ds+0
as
v -m.
Certainly mean convergence is weaker than "uniform" convergence since, clearly, there are sequences of functions which converge in the mean but not uniformly.
Convergence as introduced presupposes a given limit vector.
A notion of convergence can also be introduced without reference to a limit vector.
We say, a sequence of vectors
itself" or it is a "Cauchy sequence", if
65
O N)
converges "in
110(v)
(14.3)
i.e., if to every
(V)
(14.3)'
lit
c
- (fu)ll
> 0
- 0
as
v, u + -,
there is a
ve
such that
- m(u)ll
< e
whenever
u > v > ve.
From the second triangle inequality we conclude that the norms of the vectors of a Cauchy sequence themselves form a Cauchy sequence:-
(14.4)
110(v)ll
llt(u)ll + 0
-
as
v, u + -.
It is a consequence of this fact that the norms of the vectors of a Cauchy sequence are bounded.
It is clear that every sequence which converges to a limit vector also converges in itself. 11,(v)
that
- tll
< c/2
when
- 4 + 0 -
= 11@(v)
One need only determine v > v
then
e for u > v > v£
lit
(V)
ve
such
- O(")11
by the triangle in-
equality (6.4).
The converse need not be true, however.
There may exist.in
some spaces Cauchy sequences without limit vectors. For example, it is easy to construct a sequence of continuous
functions
(v) with (u) f if (v) (s) - $ (s) 12ds -
0,
as
to which there is no continuous limit function.
v, u - 0°. We may simply take
the sequence of functions
t (v) (s) - 0, s < 0,
= vs, 0 < s < 1/v, = 1, 1/v < s < 1
66
defined in the interval
obviously, this sequence
5: -1 < s < 1.
If it
converges in itself with respect to the square integral norm. 0(s), clearly, one would have
had a continuous limit function
1
0 0 J
-1
and
JIG(s)
- ll2ds 6, 0;
0
or
Cs) = 0, s < 0, (s) = 1, s > 0, and this function is not continuous. Of course, one could extend the space of functions by admitting piecewise continuous functions.
Then the sequence just
considered would have a limit function.
However, it would again be
possible to construct a Cauchy sequence of piecewise continuous functions without a piecewise continuous limit function.
We shall see
later on, in Section 15, that the function space can nevertheless be so extended that every Cauchy sequence has a limit. A space in which every Cauchy sequence of vectors has a limit vector is called "complete".
A complete normid space is called a "Banach space". For example, the space of continuous functions closed interval
_1
m(s)
in a
is complete with respect to the maximum norm
IIfII = max5 I0(s)I; it hence is a Banach space.
For, every sequence
of functions which is a Cauchy sequence with respect to this norm, i.e., which is a uniform Cauchy sequence, has a limit function which again is continuous.
With the aid of the notion of completeness we can formulate the notion of "Hilbert space": product space.
A Hilbert space is a complete inner
Here completeness is supposed to refer to the norm
67
associated with the inner product. According to this definition a Hilbert space may be of finite, countable,
space was r 14on.
non-countable dimension.
Originally, the term Hilbert
flied by von Neumann for the space of countable dimen-
The terminology here adopted is convenient, and rather commonly
used now.
The case considered by Hilbert himself was a special case of countable dimension, viz. the space of sequences 2
for which the series
f K
@ =
converges to a finite limit.
It is not
K
difficult to prove that this space is linear, and that the expression
(p0) =
KF'K
K
always converges for vectors
in this space and may serve as an
(P,@'
inner product so that the norm becomes
1/2
II4 II =
I
K12]
r'K1
We shall not give a proof of these statements here, since these statements will result as a special case of more general statements to be proved in Section 15.
What about function spaces?
Since the space of continuous, or
even the space of piecewise continuous, functions
p(s)
defined in
our interval 3 is not complete (with respect to the inner productnorm) we may wonder whether or not this space can be enlarged to a complete one.
This is indeed possible.
obtained in the manifold of all functions square
Such a complete extension is
in 9 whose absolute
(s)
is integrable in the sense of Lebesgue.
completeness of the resulting function space
68
-V2
The
is expressed by the
statement that to every sequence
Igv(s) - +'(s)l2ds
J
there is a function
1
1
(s)
for which
in
m(s)
v, u
0,
in the space
such that
Y.
Iov(s) - 4(s) 12ds - 0,
V -' M.
This statement is a part of the celebrated theorem of Fischer and F. Riesz.
We could rely on this statement if we wished; but it is not necessary to do so.
It is possible to attain the completion of function
spaces directly, without invoking the theory of Lebesgue integration. This will be shown in the following section.
15.
First Extension Theorem.
Ideal Functions
In this section we shall show that every inner product space
can be extended to a complete one, a Hilbert space it is dense.
ti, in which
An inner product space will, therefore, also be called
an "Pre-Hilbert Space".
We recall that the subset
23'
dense in it if to every vector in vector in
'Z3
of a space
a sequence of vectors in
V; see Section 6.
First Extension Theorem.
Let
there exists a Hilbert space
p-
%
is the limit of
be an inner product space. which contains
such a way that the inner product defined in originally defined in
was said to be
there is an arbitrarily close
$3'; in other words if every vector-in
£'
$3
1
£'
Then
densely in
agrees with that
V.
To establish this extension, let
69
4v
be a Cauchy sequence of
vectors in
To such a Cauchy sequence we assign an "ideal
1'.
element", or "ideal vector" denoted by element to two Cauchy sequences
Ilml - vll
as
0
We assign the same ideal
4'.
{(D v} provided
and
{@
we call two such sequences "equivalent".
v
In other words, each ideal vector corresponds to a class of equivalent Cauchy sequences.
Every vector in the space
$i'
garded as an ideal vector; for every such vector
itself may be re4>' generates the
and we simply identify the correspond-
Gauchy sequence ing ideal vector with
4>'.
ideal vectors contains
Having done so we may say that the set of as a subspace.
! '
'Note that the completion process described is the precise analogue to one .of the processes by which the set of rational numbers
can be extended to the set of rational and irrational real numbers. Of course, we must show that the set
t)
of ideal vectors
forms a linear space; furthermore, we must define an inner product in it and show that it has the desired properties. Let
0
and
m
be two (ideal) vectors in the extension
given by two Cauchy sequences Let
c,c
{4v)
and
{mv}
taken from
.10'.
be two complex numbers; then the sequence
{cm" + c4V}
is also a Cauchy sequence, as follows from the triangle
inequality.
We should like to denote the associated ideal element by
{co + c@}.
Furthetmore, the sequence of numbers
(4>v,mv) is a Cauchy
sequence since by virtue of the Schwarz inequality the estimate
n
I
I W,4") - (x",4")1 - I (m",4" - u) + (I". - P,0")1 IIVI I
holds and since
I14>"11
III" -
and
110`1
+ IIi" - P I1
110"11
are bounded as shown above.
approach a limit which we should
Consequently, the numbers like to denote by
4.11 11
(4',4>).
70
We must make sure that the assignments of (@,0)
to
0
and
Cauchy sequence
i
{0v}
CO + cR
and of
are independent of the choice of the defining and
{iv}.
To this end we make the following
obvious but useful remarks. 1)
Every subsequence of a Cauchy sequence is again a Cauchy
sequence, equivalent to the full sequence. 2)
The mixture of two equivalent Cauchy sequences, constructed
by taking, alternatingly, one term from the first and one from the second, is again a Cauchy sequence equivalent to'each of the components. Now we first observe that the limit of the numbers is unchanged if we restrict
and
@v
consider the mixture of the sequence {4P1v}.
iv
to subsequences.
(iv0v) Next, we
and an equivalent sequence
{@v}
Since the mixtures are Cauchy sequences the inner products for
them have limits; since the mixtures are equivalent with the components, the limit of the inner products for the mixture is the same as that found with the original and with the new components.
Hence the limit
of the inner product is independent of the choice of the defining Cauchy sequence.
The same argument, of course, applies to the linear combination.
Having assigned a linear combination
(cO+c$)
and an inner
product to the ideal elements we should verify that these assignments have the required properties.
This could be done easily.
We shall
carry out such a verification only for one of the properties, viz. the property (7.4) that
(0,0) - 0
To this end we note that where
{@v}
implies
0
(t,@) = 0
is the sequence defining
$.
0.
means Let
{0v}
Ov
+ 0
as
v + m
be the Cauchy
sequence consisting of the zero vectors, 0v - 0, then we have 110v - 0vll - 11011 {0v} + 0. Hence and {Ov} give the same ideal
71
element; but
was identified with
{0v} _ {0,0,0 ...}
0.
of
We have now come to the conclusion that the extension is a linear inner product space.
We still must show that this
space is complete. Let
Then let
iv
be a Cauchy sequence of ideal elements in
{0'v}
be an element in
such that
'j)'
Ilmv -
4Pv`I
p .
1/v
From the triangle inequality we have
Iliv - P II
so that
{mv}
<-
+ 2/v + 0
Il+v
is seen to be a Cauchy sequence, defining an element
One easily verifies
Il,v - 011 +
0.
0.
Finally, we note that the space
t)'
as is evident from the construction of
is dense in the extension Ti.
Thus we have established the first extension theorem.
We add
the obvious
Remark:
The completion of a space is the same as that of each dense
subspace of it.
This remark allows one considerable leeway in the
choice of the subspaces to attain a desired complete space. As a first application of the first extension theorem we let t)
be the space of all sequences 40 = {El'2' ...,0'0....}
of a finite number of components, the number of which is not restricted, and take the (finite) series
(15.1)
I
K=1
as unit form.
We maintain that the ideal elements which form the
72
quences {&
of
V
completion K
can be realized as (infinite or finite) se-
t)'
) of numbers
&
for which K=l E
K
IK
I2
In other
is finite.
words, we maintain that this completion is the original "special" Hilbert space, mentioned in Section 14.
It is clear that every sequence
(F1,c210,0....
forms a Cauchy sequence provided
{E1,&2,&3,0,...}
ICKI2 < -; all
Y
K
that we have to show-is that every Cauchy sequence of vectors in That is to say we should prove
is equivalent to such a special one.
of vectors in
that to every Cauchy sequence of vectors
{10 k}
a sequence
10'
can be assigned such that
@k = {Ell ...,Ek'0,0.... }
(a)
110
and appropriate
a +
for
- 0k(a)II + 0
k(a) + -. II,P (a)
To this end we note that the relation implies that to every
e
there is an
> 0
(CO
-
II'-
Set
ka)
_
ika),0,0.... }
replacing all components - 0(a)II
<
Il4,(ka)
B
tend to
=
We now choose 0(a)
of
K
0k
vanish for
such that
for
0
a
K > k.
Then, clearly, con-
Evidently, the
are independent of
k
for
K
II,t(a)
k so large that the components Then
ka)
the vectors
@k =
K > k(a).
(a)
the vector obtained from by
0
-
and denote by
in the above inequality we find k = k(a)
Mil (P
B > a > a(e).
0,0.... }
and hence, as
E
verge to a limit vector components
Ka)
for
e
<
(a),2a),...,9(a)....
_
O(a)
4) (6) 11
a(c)
-
0k(a) =
73
0(a)
< k.
- $ EK
and hence
Letting
k(«)II of
< e,
IIIt (a)
<
- tk(a)II
a > a(e).
for
e
Thus we have attained our goal. It is thus clear that the space of
'_nfinite sequences
with
(15.2)
is the extension
of the space
t)
hence a Hilbert space.
,'
of finite sequences and
As said before this was the original space of
infinitely many variables investigated by Hilbert.
A very important application of the first extension theorem is the completion of function spaces.
For example, we may complete
the space of piecewise continuous functions finite interval
.
(s)
defined in a
of the s-axis, and carrying the unit form
(15.3)
10(s)I2dr(s)
j
R involving a measure function
r(s).
The completion of this space
consists in adjoining "ideal elements" in the sense described above. In case the domain extension with the space bounded support.
-Q
extends to infinity we may begin our of piecewise continuous functions with
G'
We then may carry out our extension in two stages
and first extend the space
f,4'
piecewise continuous functions
f
to the space
(s)
fl
c1
of those
for which
I0(s)I2dr(s) < -.
Clearly, any such function may be identified with the ideal element which is given by the Cauchy sequence formed by the functions
74
v(s) = $(s) j'.
The space
for
Isl
< sv , =0 for
co, belonging to
? sv, sv
Isi
is therefore an inner product space which
CA' n C)
now may be completed to the space
R).
The "ideal elements" entering the completion of a function space will be called "ideal functions".
As a matter of fact,.we
shall refer to these elements simply as "functions", and use the notion 'or even
for them, although most of these "functions" do
$(s)
not assign definite values to the values of
in interval
s
it
is true, some of these "functions" may be materialized by functions that have definite values everywhere in
R.
In fact, by virtue of
the Lebesgue theory, realization of ideal functions is always possible. But, we do not make use of such realizations; at least not for most of those parts of spectral theory we shall deal with.
In case we need
such a realization, we shall make special, provisions.
We say that an ideal function
is zero,
= 0, if
0.
j
This does not mean that the values of any proper function affords a realization of the ideal function of
f(s)
which
vanish for every value
In fact, as we had seen earlier, in Chapter II, Section 8, the
s.
w
values of a piecewise continuous function outside of the carrier of r(s)
do
not contribute to the integral
vanishes in the carrier of at some values of the situation that
where
s 0
= 0
r.
r(s)
so that
Also, 0 = 0 is continuous.
does not imply
$(s)
0 = 0
$(s) = 0
if
if
except
In other words, = 0
arose already
with proper functions.
While we do not want to make use of the possibility to ascribe values to the ideal functions at each point
75.
s, we do want to be able
to say when an ideal function vanishes identically in a subinterval
5 of the interval Let i.e., let
n(s)
.
be the characteristic function of the interval
n(s) = 1
for
5, = 0
in
s
otherwise.
sequence of piecewise continuous functions sequence if the sequence
tion of
{4v(s)}.
forms a Cauchy
when
nEs)m(s)
Then the
The limit function of this
does.
sequence will be denoted by
Jr,
(s)
is the limit func-
Evidently, multiplication by
n(s)
induces a
projector, P9 , which thus is defined in the whole complete space of ideal functions.
Having defined multiplication by
n(s), we may say that the
9 if
ideal function vanishes in the interval
n$ = 0; i.e., if
P = 0. Suppose the interval
consists just of the'point
..P'
.which is not a jump point of the measure function
Then, we maintain, P
= Ps
s = so
r(s).
= 0. 0
Clearly, for any piecewise continuous function approximating an ideal function
j
In(s)4
(s)
¢
0v(s)
we have
1 2dr(s) =
9
IOv(s)I2dr(s) = 0
J
so that P5¢v = 0
and hence P9
= 0.
In this context the following question arises. ideal function
ing the point
0
Suppose the
is identically zero in every interval not contain-
so, is it identically zero?
is indeed the case if
s0
We want to show that this
is not a jump point of
r(s).
In other
words, we want to prove the
Remark:
Suppose the ideal function
containing a point
s0
$
vanishes in every interval not
assumed ndt to be a jump point of
= 0. 76
r(s).
Then
Proof.
Let
be a sequence of piecewise continuous functions
v (s)
which approximates 4(s), so that to every
e
>
there is a
0
v
such
that
IIm,,
- 0II2 =
I4V(s)
- 4(s)I2dr(s)
<
e2.
J
Furthermore, since
is piecewise continuous and
0v(s)
r(s), a numbek
jump point of
J 0 Is-sI
a > 0
so
is not a
can be chosen such that
v(s)I2dr(s) <
e2
By assumption we have
14(s)I2dr(s) = 0
j
Is-soI>a
and, therefore
I
4(s) 4v(s)dr(s) = 0.
Is-soI>Q Consequently, we have
I I,II2 =
J
0(s)
(4(s) - 4v(s))dr(s) +
R <-
whence
J
(s)4v(s)dr(s)
I s-soI
IImIIc + II0Ik
IIsII < C.
The statement of the remark then follows.
Ideal functions are particularly appropriate entities to employ in those branches of physics which involve linear operators, in particular in quantum theory.
77
In their mathematical operations
physicists have always been guided by the conviction that formal operations are justified somehow or other; to a large extent such formal operations can be justified by interpreting the functions involved as ideal ones.
To be sure, in the end one wants to work with proper functions which assign definite values to values of the independent variables; and in fact, the solutions of most concrete problems can in the end be shown to be given by proper functions.
But, it is of greatest
advantage, not to insist too early on showing that the ideal functions one works with are proper.
It should be mentioned that the "distributions" of L. Schwartz represent a class of "ideal functions" less restricted than the class of ideal functions considered here, inasmuch as they are not related to a norm; but rather to a particular seminorm.
To be sure, there are
questions in spectral theory in which distributions of certain kinds are the appropriate entities to employ; in fact, we shall employ distributions in later chapters.
For most of our purposes, however,
it is exactly the restriction to a Hilbert space which makes the ideal functions as we have introduced them an effective tool.
16.
Fourier Transformation The role of ideal functions may be illustrated in connection
with the Fourier series expansion discussed in our second example in Section 1.
There we had considered a space of vectors defined for
functions
m(s)
sequences
nµ, v - 0,±l,±2,...
inverse (1.9)* functions
4(s)
-w < s < .
0
given by
and also represented by
The transformation (1.9)
and its
connecting these two representations are valid if the are sufficiently smooth, for example, if they have
continuous derivatives
d$(s)/ds
and are periodic
Furthermore, the identity
78
$(n)
InuI2
(16.1)
2n
?n
u=
holds for such functions. of these func-
We now know that we can extend the space
(s)
tions
consisting of ideal functions
to a Hilbert space
which may be realized by more or less smooth proper functions. {nu}
The sequence of components tions in {n
t'
for which
corresponding to the func-
form a subspace.of the complete space of all sequences ul2
Clearly, the complete space
is finite.
In
corresponds to a complete space of such sequences, which we shall call 'admitted for the moment. InVI2 I u
We maintain that all sequences with finite
are admitted, i.e. that
actually corresponds to the
t)
space of all such sequences, that is to the special Hilbert space of sequences.
To prove this we need only take an arbitrary set of numbers nv
such that
nv = 0
for
IvI
and assign it the function
> k
nveivs
mk = k(s) _
L
IvI
These functions are in the original function space follows that all finite sequences are admitted.
rP'; hence it
Since the space of
all sequences is the closure of the space of the finite ones, our contention is proved.
Thus we see that the complete space of functions
O(s)
corre-
sponds to the complete space of all square summable sequences.
The statement expressing this correspondence is essentially that part of the Fischer-Riesz theorem which is concerned with Fourier series.
We should note that the inverse transformation
79
(1.9)*
is
-meaningful as it stands even in case the function
m(s)
it is any ideal function of the complete spaces
entering in
for, the integral
;
involved in this formula may be regarded as the inner product of the two functions
eius, both being in
and
¢.(a)
hand, the direct transformation formula
On the other
'0.
(1.9). cannot be interpreted
as an inner product; it may, however, be interpreted as the symbolic expression of the statement that the ideal function
can be
approximated in the sense of the norm by the function
nuelvs
k(s) _ when
nu' is given by (1.9)*.
The situation is similar, but not quite as simple, in the third example, concerned with'the Fourier integral transformation.
Before discussing it, we like to fgrmulate a general lemma,.which describes the structure of the argument to be used.t
Lemma.
£
Let
and
subspaces in them.
respectively
rP'
® be two Hilbert spaces, Let
T
and
and
into
), and having the following
properties:
(16.2)
(0,S9') = (TO,Y')
for
0
in
t',
''
i
(16.3) T
(16.3)S
dense
be linear transformations mapping
S
W
into
16'
IITIPII = 114-11
for
0
in
i 1SY'11 = JIT11
for
Y'
in
®'.
This argument follows a suggestion by P. Rejto.
80
in
(b',
Then the transformations
tors to in
4'
and
Q)
,
can be extended as linear opera-
T
such that the same three relations hold for all
t)
in
W
and
S
Furthermore, the relations
0.
(16.4) T
ST4' = 0
for
0
in
rp,
(16.4) S
TS9' = `Y
for
'Y
in
0.
hold, which express the fact that
and vice
T
is the inverse of
S
versa.
Since
&)'
0v
a sequence
is dense in
we may find to every
Sp
IIOv - II - 0
such that
in
(16.3) T it then follows that the vectors in
which we denote by
Q0
ator
T
operator
now defined in S
fp
0
to all of
4
in
f IF
0v; it then follows that the operis linear.
Similarly, we may extend the
so as to be linear there.
in
From
converge to a vector
relations (16.2) and (16.3) hold for the operators
all vectors
-.
-
Clearly, this vector is independent
TO.
of the choice of the sequence
TOv
as- v
in
f
Evidently, the
T and
S
with
d.
Using (16.3) S and (16.2) with
`Y - TO
and further (16.3) T
we derive the relation
IIST4-4112 = IISTO112 - (ST4',4)
- (4,ST@) + 110112
= IIT@112 - 2(Tm,Ta) + 110112 = 110112 - IIT4II2 = 0,
which shows that indeed (16.4) T holds.
Similarly (16.4) S follows.
These two relations, incidentally, imply that the range of the operators
T
and
are the complete spaces
acting in the complete spaces
S
®
can be written in the form
and
V)
0 = S(TO)
81
0
and
respectively, since every or every
IF
in the form
4
W = T (S4') . The lemma can, of course, also be used for the Fourier series We may take as
transformation.
(J'
the space of all sequences with
only a finite number of non-vanishing components.
As space
!D'
we
may take the, space of periodic functions with periodic continuous first derivatives.
The transformation
may be taken as
and
(16.1).
S
T.
and its inverse (1.9)*
(1.9)
The unit'forms are the two sides of
Properties (16.2) and (16.3) S are then immediately verified
and property (16.3) T may be taken from the theory of Fourier series,
or derived by arguments similar to those employed below for the Fourier integral transformation. and
0
The extension to the complete spaces
is thus seen to be possible.
In case of the Fourier integral transformation, we may take as space
.}/'
the space of all functions
having continuous first derivatives. n(s}
is defined correspondingly.
+(s)
The space
(n,n) = J
0'
of functions
The unit forms will be taken as m
cc
(16.5)
with bounded support
In(u)I2du,
(0.4) = 1n j I0(s)12ds.
The transformations
(16.6)
Sn(s)
Je)SMfl(u)dM,
J e_"s+(s)ds
T+(u)
then produce continuous functions of
s
finity at least of order
Jul-'
Js{-1 -and
these functions are respectively in
and
y
respectively.
(16.3), which assume the form
fits 2ds
fn(u)12du,
82
Clearly,
and,
Property (16.2) is immediately verified.
(16.7)
which decay at in-
To verify identities
we could rely on the theory of the Fourier transformation. we shall give a brief derivation of them.
m m
m
In the relation
m ei(s._s)u$(s)$(s')ds'dsdu
r
r
4n2
Still,
1
I
-m
we may interchange the order of integration since the function 0(s) = 0
Assuming
has bounded support.
_f or
+(s)
> a we obtain for
I s I
the left member the expression a 2
a
s'-s)
sinsm ('
Jf
( s ' ) = ds' ds
-a -a a
a
0s)ds'ds
sin m(s'-s)
2 1
1
-a -a a-s
ra 1
1
sin "
-a
Note that the integral over m
tends to infinity for
do"
ds.
-a-s
11
s"
tends to the value
-a < s < a, since then
-a -s < 0
as
n
and
to
a -s >
0.
The last term, therefore, tends to
2
{ 11
first term tends to zero as
m + m
since the difference quotient
1$(s)I2do.
-
The
a
by the Riemann-Lebesgue Lemma
(4(s') - $(s))/(s'-s)
virtue of the assumed continuous differentiability of
is continuous by Q(s).
Hence
the desired relation a
2n
I2dp = f -a
IIT
(S)12
do
ensues; i.e. we have established relation (16.3) T for Relation (16.3) S for
Y
in
tl
in
t1'.
is established at the same time.
83
It is possible to avoid use of the Riemann=Lebesgue Lemma in Instead of letting the spaces
the following way.
and
(S'
consist of differentiable functions we simply let these spaces consist of all piecewise constant functions of bounded support.
These
functions are finite linear combinations of the unit step functions
3.
associated with intervals
In order to prove identity (16.3) S
for both functions it is then evidently sufficient to establish the relation
27 where and
is the length of the intersection of the intervals
a22
associated with
12
ffl1(ii)n2(ii)di = alt
Snl(s7sn2(s)ds
1
n1(u)
_ e-isu )
Sn(s) _ (is)-1(eisu
n2(u).
and
where
11
Now, since,
are the end points of the
u
%
1, we need only prove the relation
interval
isu+
Wf
2n
j_W It
1
- e-
isu
isu+
1] Le
2
- e isu
s-gds
12 '
which is easily done vy complex integration. In both cases it is seen that the Lemma is applicable.
Conse-
quently, the Fouriei transformation can be extended to the complete function spaces
and
(16.3) and (16.4) hold.
d, in such a way that relations (16.2), Relations (16.6) may also be adopted in these
complete spaces and regArded as a symbolic expression of the transformations
S
and
T.
The result is related to the theorem of Plancherel, which describes the Fourier trapsformation in these complete spaces in a more specific way.
In the presaent course we are satisfied with
establishing basic relations (for the present as well as other cases)
in subspaces of sufficiently smooth functions and their extension to
84
complete spaces.
A more specific description of the nature of such
relations in the complete spaces will be given only if there are special reasons for doing so.
17.
The Projection Theorem The wider a space is, the easier should it be to find in it
an entity with desired properties.
in Section 12, we dis-
Earlier ,
cussed the operation of orthogonal projection of a vector into a subspace in an inner product space and asked whether or not there al-. ways is such a projection.
We shall show that indeed there is always Here then we shall be
such projection if the subspace is "complete".
rewarded for our efforts in making spaces complete.
The statement is
embodied in the "Projection Theorem", the basic theorem of the geometry of the Hilbert space.
Projection Theorem. Every vector in an inner product space possesses a unique orthogonal projection on any complete subspace.
In general, the inner product space
IS
will itself be
complete, i.e. a Hilbert space.
We recall that the projection is a vector in
space
vectors
4'
such that
0 - P4D
0
on a sub-
is orthogonal to all
We maintain that the distance of the vector 0
in
from any vector
distance from
$
of a vector
P4
'Y'
in
other than
P'
is greater than its
PO; for,
II, -'''I12 = II(
- P(b) +(Pb
-'x')1i2
= 110 - P0112 + IIPO - T-112,
since
$ - P4
particular to
is orthogonal to all vectors in
P@ - V.
$, and hence, in
This minimum property of the projection is
85
the'starting point for the proof of the projection theorem. Consider the set of numbers over
5.
110 - I'll
where
runs all
'Y'
Certainly, this set of numbers has a greatest lower bound.
d, and there is a sequence of vectors
TV
in
approaches this lower bound if
Ilt - T"11
(A)
for which
v +
d
I14- - Pull
and
I10- Y' II -d as
(B)
We call
a "minimizing" sequence.
4'v
v-0. M.
We want to prove that this
sequence has a limit and that this limit is the desired projection. To this end we use the following identity which holds for any
triple of vectors ,'Y' ,'Y"
I Ii.-'r'i12+I II.-'r"112= IIt_I (I, +I")1f2 +
I
IZ
and is immediately verified by working out the squares formally. This identity may be related to the fact that the norm 11* - III
is a convex function of
I.
we also mention incidentally
that this identity has a simple geometric interpretation:
The sums
of the squares of lengths of the two diagonals of a parallelogram is the sum of the squares of the lengths of the sides. In using this identity we note that
so that
1
1 $ - y (Y" + if"'))
I
'V' + 'V"
is a vector in
> d by A, and hence
86
IIIV ' -T"I1 12111-'V'II +2110-Y'"II - 4d. u
Taking
and
'V' = 4'v
and letting
W" = %F
v,µ
tend to in-
B, the relation
finity.we find, by
IIIv
-'Yu II
+0, v.u+°°
{TV}
This relation just says that
is a Cauchy sequence.
Now we make use of the assumed completeness of the space and conclude that there exists in as
TQII i 0 1I(t - '3v)
t
Ij$ - 'rvll
(C)
T o
such that
0, so that, using (14.2), we may conclude
-
'f'
.I10-'3'vII Now since
a vector
v i . This relation may also be written as
- (t - 'o)II,
that the norm of
$
SD
tends to that of
-
II0-'YoI1
tends to
d, by
as
o t - f:
V - W.
B, we have
lit - X011 - d.
In other words, the greatest lower bound of
lit - 3II
is assumed;
it is a minimum.
In a standard way we derive from this minimum property of that the."first variation" of the functional 'Y - 'Y0.
vanishes fo;
By this we mean that the first derivative of the function
lit - T(t)II
function
lit - "II
T0
of
'P(t)y
t
vanishes for
t = 0
for every differentiable
Actually, it is sufficient to take
linear; then
87
''(t) - To + t'1'1
III,
-'(t)112' 114, -Y'O-tY'11I2= I1 -% 112 -
A
Since by have
and
Re(Y'1,' -
C V
2tRe('11 (P
-
+ t21I'y1112
Y'O)
this function attains a minimum for
)
for all
0
the imaginary part of
iTl
since
(4l,$ - TO)
we
The same*is t?e for
in
Y 1
t = 0
is also in
Consequently we may conclude that
(Y',0 - Y' O) = 0
for all
Y'
in
a.
Since this relation expresses the orthogonality of we realize that
of
0
'
- T0
to all of
is the orthogonal projection
Y'o
Thus the projection theorem is proved.
on
.s
The uniqueness of the orthogonal projection was already established in Section 12.
Another fact, also mentioned in Section 12, should be recalled:
which possesses a projection on a subspace
Every vector
can be written as the sum of a vector
TL the space of all vectors in
to the complete space
in it and one, (1-P)'P,
We may amplify this statement now.
orthogonal to this space. denote by
P4
$
We
which are perpendicular
i.e., the "orthogonal complement" of
9
$.
As an immediate consequence of the projection theorem we then have the
Corollary:
Every vector
t
in
3
88
can be written as the sum
0 = Pf + (1-P) 4' !
of a vector in a complete subspace complement
$
and one in its orthogonal.
.
We may express this fact also by saying that the linear combination of the vectors in the complete sub-space
t
and those in
This fact is symbolically expressed in
span the whole space the form
We said before that we shall in general deal with cases in which the space
itself is complete; we then call it
93
the orthogonal complement
a
Hilbert space is also complete. a limit in
of a complete subspace
ro.
a
of a
5, and hence in
Frequently, we shall deal with incomplete subspaces
the space
a'
$t. $'
of
Before the projection theorem can be applied
fp.
must first be "closed".
of a normed space
$
A subspace
belong to
91
is "closed" if every
Here we-mean by limit a
limit of elements in
$
limit element in
The closure of an incomplete subspace
91
has
For, any Cauchy sequence in
V, which is also orthogonal to
a Hilbert space
We note:
fl.
is obtained by joining to If the space
t)
a'
$'
of
all limit elements.
is complete, every closed subspace of ii is
complete, as easily verified.
The process of closure is then the same
as the process of completion.
But this process of closure is much
simpler than the process of completion described in Section 14 since the elements to be added in a closure process are already available, and the linear combination and the inner product are already defined. As an application of these considerations we make the following
89
Let
Remark:
{Y}
be an orthonormal system in a Hilbert space
and suppose that no vector in , except unit vectors
nV.
Then the system
t)
0, is orthogonal to all spans the space
{S2v}
t)
densely.
In other words, the space
spanned by the vectors
a'
cv,
the space of their finite linear combinations, is dense in
i.e.
To prove this statement we may consider the closure ,
and its orthogonal complement But, by hypothesis,
hence
Ci
= a
1
a
1
of
Then
.
contains only the zero vector:
Thus, it follows that
.
.
a'
is dense in
t.
If in the formulation of this remark one drops the requirement that the space be true.
t
be complete, the statement would not necessarily
There are counter-examples. This fact, played a considerable role in the earlier theory of
integral equations in which one did not require the underlying function space to be complete.
A system
{S2'}
as described in the
Remark was then called "complete"; it was called "closed" if it spanned the function space densely. alent.
These two notions were not equiv-
But they are equivalent if the underlying space is complete
and then the discrepancy disappears. It may be felt desirable to have examples presented in which concrete closed subspaces and the projections on them are exhibited; but such examples will not be given here.
One may just as well be
satisfied with the assertion that the projection theorem will be used over and over again in the course of our presentation of spectral
theory. The subject matter treated in the next section will give an indication of this fact.
90
18.
Bounded Forms A "linear form"
such that
X0
(18.1)
IX(4.)1
for every
9t; it is bounded if there is a
of a normed space
every vector number
is an assignment of a complex number to
x(f)
0
x0114.11
W.
in
A simple example of a linear form of space is the inner product
(A,$)
of
0
in an inner product
$
with a fixed vector
A; this
form is bounded by virtue of the Schwarz inequality
(18.2)
IA,01 I IIAij il.ll
If the form is defined in a complete inner product space the converse is true:
Theorem 18.1.
Let
a Hilbert space
X(f)
be a bounded ]{inear functional defined in
0; then there is a -'vector
(18.3)
in
A
such that
*
x(f) _ (A,O).
The proof follows immediately from the projection theorem. Let
be the subspace of all those vectors
0.
X (T)
This space is closed; for, if
a T
1'
in
$
- T0 with
for which 'Ya
j , we have Ix(0) 1 = 1(a - V0) 1 < X011ta - '1011 - 0, hence X(T0) = 0. Therefore, 'Yo is in 1
and
TO
in
If the space A = 0.
in
a
is the full Hilbert space
ff.
!D, we may set
Otherwise, by virtue of the Corollary to the Projection
Theorem, there is a vector
X0 + 0
in the orthogonal complement
91
a
1
Xo
a.
of
for such a vector; for otherwise
Clearly, x(X0) + 0
would be in
as well as in
5
a
and would hence be the zero
vector.
1
We now maintain that the space
other words, we maintain that every vector of the vector
is one-dimensional.
a X
1 in
a
In
is a multiple
X0; specifically, we claim that
X = [X(X0))-1X(X)X0.
Clearly, the difference of these two vectors annihilates the form hence, being in space
a
1
i
a
,
this difference is zero.
x;
Thus we see that the
is indeed one-dimensional.
' By the corollary to the projection theorem every vector in ' can be written in the form
4=
where
tt
is in
$t
and
(P
t
+ cXo
c _ (X0,$)/,(XO,Xo). 0
have
X(O) = cx(X0); setting
(18.4)
A =
(Xo,Xo)-1X(X0)X0
we find the desired relation
xCf = (A,0) .
92
Since
X((Dt) = 0
we
Theorem 18.1 is thus proved.
As an immediate consequence of Theorem 18.1 we shall prove a corollary concerning bounded (bilinear) forms. 4'B0
A "bilinear form"
is an assignment of a complex number to two vectors
space
93
which is linear in
and anti-linear in
m
bounded if there is a number
b
(18.5)
<- boll,P'II
k 'B411
for all
0,4'
in
fined in
t).
in a
0'; it is
such that
0
114,11
23.
Corollary to Theorem 18.1. Hilbert space
0,0'
Let
'B4'
be a bounded form defined in a
Then there is a bounded linear operator
B
de-
such that the relation
VB0 =
(18.6)
holds for all
4,0'
in
!D.
ness relation (18.5) for all
Furthermore, the validity of the bounded0,0'
in
implies the validity of
the relation
(18.7)
1184'II
for all
< bo11(P II
(V
To prove this corollary we keep
T BT 18.1
is a bounded linear form in ,
there exists a vector
O'B@ = (t'',A)
Obviously, A
A
for all
rp .
fixed and observe that then
Hence, by virtue of Theorem
4'.
in
'
in
such that
k)
0'
in
V'B
T).
is uniquely determined since the relation
93
= (A,0')
or
for all
(9',A1) _ (0',A2)
The relation implies the relation
0'
,'Bm(1)
c1 A(1) + c1A(2) = A
A
to
Thus, the vector m
for all
@'
1
= A2. (2)
depends linearly on
constitutes a linear operator
(vA(2)
)
and hence
by virtue of the uniqueness of A
_
c1@'Bm(1) + c2$'B@(2)
0
in
A
@' BO
and
_ (4)',A(1))
@'B(c1'(1) + c20(2)) =
_ (s', (c1A(1) + c2A(2)))
rived.
implies
in
A
just de-
@; the assignment
B.
From the relation (18.6) thus established, combined with (18.5) we derive the relation
b0110'11
Setting
m' - BO
we find
11011.
11B$112 < b011B$I1
11011
and thus (18.7).
Because of the close relationship between bilinear forms and operators we shall not discuss specific bilinear forms now.
We shall
do so in connection with the discussion of specific operators in the next chapter and again in Chapter VI.
94
CHAPTER IV BOUNDED OPERATORS
19.
Operator Inequalities, Operator Norm, Operator Convergence An operator
B
acting in a Hilbert space
bounded in Section 6 if there is a number
The bilinear form
(W,Bc)
defined for
bounded if there is a number
(19.2)
I1Y,B'I
< b'II'II
b'
>
0
in
'
4,T
! .
in
T)
was called
such that
for all
IIBII
such that
b > 0
for all
(19.1)
was called
4D
in
4,''
Every bound for the bilinear form is one for the operator, and vice versa.
b'IIB@II
For setting II411
W = BO
and hence
in (19.2) we obtain
I1R4I
2
<
IIB011 < b'11011, and by Schwarz's in-
equality together with (19.1) we obtain
I1f,B01
The least upper bound for the operator
< IIBII b11011 B
is denoted by
IIBII; i.e.
(19.3)
IIBII = l.u.b. IIB'11/ IIBII
for
0 # 0.
This notation anticipates the fact, discussed later on, that the least bound may serve as a "norfi".
The statement made above shows that the bound the same time the least bound for the form
(19.4)
IIBII = l.u.b. IV,BctI/ 114,11-11Y11
95
IIBII
(',Bl)); i.e.
for
4),T # 0.
is at
An operator
acting in the space
B*
"formal adjoint" of the bounded operator
(19.5)
(B@,Y') = (,D,B*Y')
B
holds for all
$
was called the
if the relation
4,Y'
in
.10.
Later on, in Chapter VI, we shall introduce a "strict" adjointness property and sh.,w that every formal adjoint of a bounded operator is the strict adjoint.
For this reason we shall drop the qualification
"formal" in this chapter.
It is an important fact that to every bounded operator
B
acting in a Hilbert space there is just one such adjoint operator. Since
(BO,Y)
is a bounded bilinear form the existence of
8*
fol-
lows immediately from the corollary to Theorem 18.1; its uniqueness is obvious since for the difference
of two adjoint operators the reti m,Y', whence BY = 0 Y,
for all
lation ($,BY) = 0 holds for all ti i.e.
B = 0.
Since the least bound
IIBIJ
same time the least bound of the form it is clear that
118IJ
of the operator
('Y,,Bf)
B
is at the
(@,B*Y') = T;BTj,
and
is also the bound of the operator
B*,
I IB*11 = JIBH .
(19.6)
The bounded operator
B
will be called "self-adjoint" (with-
out the qualification "formal", see Section 9) if it is equal to its adjoint, B = B*.
The statement that the least bound of an operator is the same as the least bound of the associated bilinear form can be strengthened for self-adjoint operators. "quadratic form"
To such an operator
B
we assign its
(41,B$).
Note that the bilinear form
96
(Y',B$)
of an operator
B
is
(P,B0
determined by the quadratic form
is self-adjoint:
B
if
this is seen from the identity
2('Y,B4,) + 2('P,B'Y) = (('Y+t, B(4'+4)) - (('Y'-4, B(T-P)). The left member equals
4('Y,B'')
if the space is real.
If
the space is complex we need only add to this identity the identity obtained from it by substituting The quadratic form
for T.
i'Y
of the self-adjoint operator
(@,B4')
B
will be called "non-negative" if the inequality
(19.7)
holds for
> 0
(iV,BP)
0
in
I'D .
(Note that the value of this form is real for Hermitean B.)
We use
the notation B > 0
to express this property.
The bilinear form
(4',B(D)
of such a non-negative self-adjoint
operator may be considered a semi-inner-product since it satisfies the requirements (7.1), (7.2), and (7.3).
Therefore, the Schwarz
inequality
(19.8)
Y',B TI2
holds with any
$,V'
<
(`',B41)
(41,B41)
if
B>0
in
Using it we may derive the general
Theorem 19.1.
Suppose the quadratic form
and has the bound
b, i.e.
0
< B < b, or
97
(1',B4')
is non-negative
bII4.Ii2.
0 < (t,B0 <
(19.9)
Then the inequality
(19.10)
IIBPII2 < b(,D,B4,)
in fact, we need only set BO
(19.9) to
holds.
' = B1
0, and divide by
instead of
in (19.8), and apply IIB@II2.
If
IIB'II - 0
relation (19.10) holds anyway.
Next we state the important
Theorem 19.2.
operator
B
b
has the bound
b,
Ia,BhI K b!I0II2;
(19.11)
then
Suppose the quadratic form of the bounded Hermitean
is a bound for the operator
(19.12)
IIB,II
_
B
bIItIl.
In case the inequality
It,B0I
holds for all
< bI10112
# 0, also the inequality
IIBsII
< bII.II
holds.
98
for '0 # 0
Condition (19.11) may be expressed by saying that the forms are non-negative, b ± B > 0, or,
b ± B
-b < B < b
(19.11)'
Theorem 19.2 can then be stated as saying that (19.11)' implies
(19.1)'
IIBII < b.
There are many ways of proving Theorem 19.2.
One concise
proof is based on the identity
2b(b2-B2) = (b-B) (b+B) (b-B) + (b+B) (b-B) (b+B)
,
and the resulting identity
2bfb2(0,0) - (B0,Bf))
_ ( (b-B) 41, (b+B) (b-B) O) + ((b+B) 9,
(b-B) (b+B) 4) .
The right hand side is non-negative since the operators non-negative.
Hence the left hand side is non-negative.
(19.1)' unless
b - 0.
any positive
If
bl; hence so does (19.12).
<
are
This implies
b - 0, inequality (19.11) holds for
in this case, (19.1)' holds for
in place of
b i B
b = 0.
Since
bl > 0
is arbitrary
The statement involving
<
is proved in the same way.
Incidentally, the last result:
B =-0
if -(O,B$) = 0
for all
0, holds even if the operator is not Hermitean, provided the space S1
is complex.
The least bound
IIBIJ
of a bounded operator may serve as a
norm in the linear space of all bounded operators.
For, postulates
(6.1) to (6.3) are evidently satisfied and (6.4), i.e. the triangle
99
inequality (19.13)
IIB111 + IIB211
IIB1 + 8211
follows immediately from the relation
II(BI + B2)0II'
< {IIB1I1 + IIB211}114'111 which states that
upper bound for
B1 jE B2.
IIBIII + IIB211
For this reason the least bound
also called the "minimal" norm of
is an IIB11
is
B, sometimes simply called the
For restricted classes of operators we shall on occasion, see
"norm".
Section 20, use other - non-minimal - norms, given by other than least bounds.
Naturally, a notion of convergence of operators can be introduced with the aid of the norm operators
By
II
II:
We say, a sequence of bounded
tends to a bounded limit operator "in minimal norm" or
"uniformly" if
IIB"-Bt1 -0 as
(19.14)
Convergence in any other (non-minimal) norm may also be introduced;
convergence with respect to a non-minimal norm is stronger than uniform convergence.
We shall frequently use the notion of convergence in norm; but for many purposes a weaker kind of convergence, called "strong" convergence will be more suitable since a sequence may converge strongly even if it does not converge in norm and important conclusions can often be deduced from strong convergence.
We say, a sequence of bounded operators to a bounded operator (1)
B
the vectors
By converges strongly
if
BV4
converge to
100
Bt
for every vector
- B4,{ - 0 as v -, -
I IB'
(2)*
a number
b
0
the operators
are uniformly bounded; i.e. there is
B
such that
< b
IIB"II
Clearly, the norm
0
IIBII
for all
v.
of the limit operator
by any common bound of the sequence
IIBVII
B
is bounded
or a sub-sequence thereof.
This sub-sequence may be so chosen as to approach the inferior limit inf
IIBvI I
B"II From this one derives the
of the sequence
v
inequality (19.15)
IIB11
< inf IIBv!! V
for the strong limit
B
of the sequence
Bv.
We also mention the important, though immediately verified fact that the products
of the members of two strongly conver-
B)B2
gent sequences form again a strongly convergent sequence. A sequence of bounded operators
B
V
is a strong Cauchy se-
quence if (1)
the vectors
Bvt
form a Cauchy sequence for each
vector in (2)
the operators
By
are uniformly bounded, IIB"I1 < bo.
*We mention incidentally that condition (2) could be omitted since the existence of a uniform bound b0 could be deduced from condition (1) by virtue of an extension of the theorem mentioned in the footnote on
We shall not have occasion to use this remarkable fact since the procedure by which we shall introduce our sequences of bounded operators will always automatically yield a uniform bound for page
145.
them.
101
Such a sequence evidently possesses a limit operator bo
which it strongly converges and for which
B
to
is a bound;
IIBII < bo.
Clearly, the product of two strongly convergent Cauchy sequences is again a strongly convergent Cauchy sequence.
Finally, we mention.the notion of "weak" convergence of a BV; by this it is simply
sequence of uniformly bounded operators
meant that the bilinear forms
converge for all
(T,B"O)
,1.
The
main weakness of this type of convergence is that the product of two weakly convergent operators need not converge weakly.
There is oBe particular case of weak convergence of operators which automatically implies strong convergence, as seen from Let
Theorem 19.3 on monotone convergence.
B1,B2,...
be a sequence
of bounded Hermitean operators which increases monotonically,
B1 < B2 < B3 < ... .and is uniformly bounded, ((Bt(( < b, t quence
Bt
1,2,...
.
Then the se-
is a strong Cauchy sequence.
The assumption
B. < Bt
for all
B0)*4 > 0
0.
for
a < T implies that
Since evidently
11Bt - Ball < 2b
we
may apply Theorem 19.1 which gives
(((Bt - Ba)Q((2 < 2b($,(Bt - B0)5).
Since the sequence the sequence
(5,B15) increases monotonically and is bounded,
($, (Bt - Ba)4) tends to zero as
a,t + m.
Hence the
statement follows.
Inasmuch as a strong Cauchy sequence of bounded operators leads to a Limit operator, this limit process may be used,to define
102
specific operators with the aid of operators already defined before. we shall use this procedure extensively.
Before doing so, however,
we shall describe another procedure of defining an operator, the extension of a bounded operator defined in a dense subspace of Accordingly, we formulate the rather obvious
Second Extension Theorem. dense subspace
Suppose the operator produces vectors in
of
V'
IIB$IJ < b11shI
for
$
in
Then there exists an operator defined in all of there, and agreeing in
b
Every
in
0
choice, of the vectors BO
if
t), having the bound
- $'JJ
the vectors
Bov
The limit is evidently independent of the $'; we may denote it by
0
is in
to
B$
since it evidently
The linearity of the operator
so defined is obvious and also the relation
20.
f,'.
B.
b0"
BO" -
form a Cauchy sequence.
agrees with
!D, and is bounded
can be approximated by a sequence v from
By virtue of
'.
with
ID'
is defined in a
B
B
1IBSII < b11sH.
Integral Operators
Specific cases of bounded operators are naturally given by integral operators.
These are operators which act 'on the functions
of some function space and produce functions in this space. For example, let the functions be the continuous functions $(s)
defined in a closed interval 3 of the s-axis.
tegral operator
(20.1)
K
may be given in the form
K m(s) = f
k(s,s')4(s')ds'
1 103
Then an in-
with the aid of a function over the interval
9.
of two variables, both running
k(s,s')
This function
"kernel" of the integral operator.
is called the
k(s,s')
For the present let us assume
that this kernel is continuous over the square 9 x -0. function
K$(s)
Then the
Moreover, we have
is also continuous.
max IK4'(s)l< k max 1o(s)j
s
S
with
k = I max
(20.2)
..-
-where
I
8,8'
fk(s,s')I
is the length of the interval
(20.3)
IiK4II
3.
Consequently we have
11.11
no matter whether we take the maximum norm or the square integral norm.' In any case the operator
is bounded.
K
Still we cannot immediately apply to the operator
K
the
general theory of bounded operators which we have begun to develop. 1
For in this theory it is assumed that the bounded operators are defined in the whole complete Hilbert space; but the space of continuous functions is not complete with respect to the square integral norm.
The obstacle we thus have met can be easily overcome.
We
need only employ the second extension theorem described at the end of Section 19.
This is possible since the space of continuous func-
tions is dense in its completion. does
According to this theorem there
bounded operator acting on all (ideal) functions
in the Hilbert space with the unit form
104
0(s)
(m,$) =
(20.4)
j
Is)12ds,
which agrees with the given integral operator.
We denote this ex-
K; in fact we shall use the'formula (20.1) to
tended operator by
describe it symbolically. It is necessary for us to introduce a more general class of On the one hand we must consider a more general
integral operators.
function space in which the operator acts, and on the ether hand we must consider a more general class of kernels.
In defining integral
operators we shall employ both, the second extension theorem, and approximation by a Cauchy sequence of operators already defined.
in
this connection we shall have to use bounds for the operators which are less crude than the bound
k
given by (20.2).
It seems advisable to discuss such less crude bounds already for the simple integral operator (20.1) with a continuous kernel acting on functions with the unit form (20.4), although for this simple integral operator as such we do not need these bounds, Using a number
between zero and
a
(20.5)
1
0 < a < 1,
we use Schwarz's inequality to estimate 2
f
IKm(s) I2 <
Ik(s,s') Ialk(s,s')
3
Ik(s,s')12ads'
<_
I
l aIm(s9 Ids'
Ik(s,s-)12-2oI,(s")I2ds"
1
1
Jr
whence, after interchanging the order of integration,
105
IKO(s)I2ds <
f
-
1
g2a(s")IO(s")I2ds"
with
Ik(s,s')I2alk(s,s")I2-2ads
g2a(s")
(20.6)
f
Using the abbreviation
k2a =
(20.7)
g2a(s")
s
we evidently obtain the estimate
kI(s)2ds,
IK(s)2ds
(20.8)
1
L'r or
(20.8)'
For
k2a 110 11.
IIK,II
a
the bound
1
becomes the "Hilbert-Schmidt" bound
k2a
1/2
(20.7) 2
for
k2 =
a = 1/2
j Ik(s,s')I 2dsds'
J9 J
we obtain the bound 1/2
k1 =
(20.7) 1
l.u.b. 1 S.
Ik(s,s')I Ik(s,s")Idsds'
J 9 1
related to the "Holmgren bound" to be discussed below.
We now consider a general measure function space
1
of functions
(s)
r(s)
and the
complete with respect to the unit form
106
(20.10)
Ik(s) 12dr(s).
J
We recall that this space was the completion of piecewise constant functions with bounded support.
We enlarged this subspace first by
adjoining the general piecewise continuous functions with bounded support, then by adjoining those piecewise continuous functions defined for
for which the unit form is finite, and
- < s < W
finally, by adjoining the ideal functions needed to complete the space.
In order to define integral operators we first introduce piecewise constant kernels.
P x P
We introduce the product
of a
partition of the s-axis and the same partition of the s'-axis.
The
cells of this partition are open rectangles, open segments in the sor s'-direction, and points.
A function
will be called
k(s,s')
piecewise-constant if it is constant on the cells of such a product partition; the function will be called piecewise continuous if on each .product cell it agrees with a function which is continuous on the With the aid of such kernels
closure of this cell.
define the integral operator
Kf(s)
(20.11)
First we assume
$
K
k(s,s')$(s')dr(s').
are piecewise constant, then so is
not
I
1!
number
JK$(s)I2dr(s)
K$(s).
K$(s); if
from
0
$
i4.
and
If k
$
and
are piece-
<
to
1, introduce the number
107
k
So far we leave open whether br
We now introduce bounds for the kernel a
we
through
to be of bounded support; $ c
wise continuous, then so is
k(s,s')
k(s,s').
We fix a
1/2 (
k2a =
(20.12)
1.u "I s
i
1
jk(s,s')12ajk(s,s")I2-2adr(s)dr(s')
f
I
i
1
that for it
k(s,s')
and require of the kernel
k
(20.13)
2a
A literal repetition of the arguments that led to formula (20.8) now leads to the inequality
J IK4(s)12dr(s) < k2a J 10(s)12 dr(s)
(20.14)
for piecewise continuous functions of bounded support. shows that the function
This result
is in the Hilbert space
K$(s)
We may
write (20.14) in the form
(20.14)'
IIK,II <_ k2a11011
Since the space
q
of piecewise continuous functions of bounded
we may apply our second
support is dense in the Hilbert space extension theorem.
The operator
is then defined in all of
K
and inequality (20.14)' holds there.
We shall use relation (20.11)
as the symbolic description of this operator. For later purposes it is desirable to introduce instead of k2a
a different bound which (except for
bound
k2a
a = 2)
is weaker than the
but has the advantage that it may serve as a norm in a
space of kernels.
First we observe that
k2
is evidently a square integral norm
in the space of these piecewise continuous functions for which it is finite.
We call this space
b2
and adopt
108
(20.15)
IIKII2 = k2
as norm -in it.
2a = 1
For
(20.16)
We call
IIKII2
the Hilbert-Schmidt norm.
we introduce the quantity
IIKII1
= max I1.u.b.1 lk(s*,s')Idr(s'), l.u.b.J lk(s,s")Idr(s) s"
s*
As easily verified, we have
(20.17)
IIKIII > kl;
hence the operator
K
is defined whenever
IJKII1 < C..
Also
IIKIII
satisfies the triangle inequality since evidently
J
Ikl(s*,s') + k2(s*,s')Idr(s'), <_'1 K1II1 + IIK21I1
J
Ik1(s,s") + k2(s,s")Idr(s)
;
at the same time it follows that the set tinuous kernels
k(s,s')
with
IIKIII < -
may serve as a norm.
Therefore, IIKIII
gji
of piecewise con-
forms a linear space.
We call
IIKIII
the
"Holmgren norm".
Of course, other norms could be introduced instead of the ones described.
For example, employing 1/2
q(a) (s") _ 1
in place of
J
Ik(s,s')Ia(s')Ik(s,s")l'a-'(s")dr(s)
dr(s')
ql, one easily verifies the observation made by Carleman
109
that instead of
(20.18)
the expression
ItKII1
max l.u.b. b-1(s*)
IIKJ ll,a,b
jk(s*,s'ja(s')dr(s'), 1
s*
11
jk(s,s")Jb(s)dr(s)
l.u.b. a-1 (s") s"
involving two positive functions
1
may be taken as norm.
a(s), b(s)
Setting
An application may be mentioned incidentally. a(s") s 1 ki
Ik(s,s")(dr(s)/ki
of the adjoint 4operator
and K'
one obtains the k1-bound
b(s) - 1
as bound for, K.
Obviously there are kernels for which 11K111
, 11
infinite or the other way around.
is finite and
11K112
Examples could be furnished
by product kernels
k(s,s') a g(s)h(s'). Evidently jh(s')I2dr(s')11/2
IEKI12 =
!
J (g(s)12dr(s) f
and
max
11.u.b. jg(s*) j s*
l
Ih(s') Idr.(s4) ,
1/
l. u.b. Ih(s") I s" '
1
lg(s)ldr(s)
. I
Any one of the quantities entering here could be made infinite while the others are finite.
Important kernels for which the Holmgren norm is appropriate are kernels of the form 110
k(s,s') = g(s-s').
Here we assume
r(s) = s,
IIKII 2 = M
Evidently, always
< s <
unless g(t) a 0; but g(t)Idt
IIKII1 =
(20.19)
may well be finite.
An example for which the Holmgren norm is not sufficient but Carleman's modification is sufficient is given by the integral operator (related to the "Hilbert matrix")
1
which acts in the space of functions Is(s)I2ds. = Ills-
$(s)
for
s > 1
Here the Carleman norm with
is finite; in fact
with
a(s) = b(s)
IIKIIl,a,a = 7/2, as a simple computation
shows.
Finally we mention that the class of integral operators we have introduced incltudes infinite matrices acting on sequences of numbers
_ {Ca)
for which the infinite series
is finite, i.e. on vectors in the special Hilbert space. tor
The opera-
K, given by
K=I
of
kao'
. o,
.
is defined and bounded if any of the norms k2a is finite, sucH as ill
1/2 2
I ikoa'l QUO '
or
JJKIll = max1.uu.b. J,Ik,, 6,I,
The kernels
k(s,s')
I Ikaa"1
so far considered were assumed to be
piecewise continuous in the specific sense explained above.
It is
necessary to extend the class of these kernels and the class of
operators generated by them. To this end one may adopt a norm in an appropriate space of-kernels acid then complete this space with respect to this norm. the norms
IIKI12a
tinuous kernels.
Clearly, we can adopt for this purpose any of defined in the space
'2a
of piecewise con-
We then can close off these spaces by adjoining
"ideal kernels" thus obtaining a complete space of such kernels. When a sequence of kernels
R2a
kv
in
R2a approaches a kernel
in the sense of the norm, the corresponding operator
approaches a limit operator function
k
K, i.e.
11Kv - KII2a - 0.
k
in
KV
The ideal
may be regarded as the kernel of this operator, which
therefore may symbolically be written as an integral operator. This method of defining integral operators may frequently be used in cases in which a kernel is given which is piecewise continuous except at the diagonal
s' = s, where it may be singular.
The
"Volterra operator", whose kernel is continuous up to the diagonal on one side, and vanishes on the other side, is such a case. We exclude the case in which points on the diagonal
s' = s
where
112
k(s,s') r(s)
differs from zero at has a jump.
We then
introduce an approximate kernel
which vanishes in a neighbor-
kE
hood of the diagonal and agrees with
outside of this neighborhood.
k
Specifically, we assume this neighborhood
and containing the strip
< 2e
we first adopt the norm
to be an open set
9 x 9, contained in the strip
composed of cells of a.partition Is-s'I
nE
Is-s'I
IIKII2
E.
and accordingly require
that the integral
IIKII2
= ,J f
over the complement A as
c - 0.
Ik(s,s') I2dr(s)dr(s') rE
of the neighborhood
remains bounded
nE
The limit of this integral will then be denoted by
Ik(s,s')I2dr(s)dr(s').
I
J
The approximate kernels
IIK6-KEI12
kE
evidently form a Cauchy sequence
Ik(s,s')I2dr(s)dr(s')
= Jj
-o-
as
0
d
> 2e - 0.
nd- nE Therefore, the operators by the kernel
k(s,s').
KE
tend to a limit operator
K, represented
Also we may set 1/2
J J Ik(s,s')I2dr(s)dr(s')
Similar argifnents may also be used to define operators that
have singularities at places other than the diagonal. arguments can be used to show that the operator
ka(s,s') = k(s,s') = 0
for
for other
113
Isi
< a,
s,s',
Ka
Is'I
Also these
with the kernel
< a
tends to the operator
K, in the Hilbert-Schmidt norm,
as
IIKa-KII2 + 0
a -
For the.Holmgren norm a corresponding procedure to extend integral operators might not succeed since the norm IIKEIII does not necessarily increase monotonically as
IIKaII1) (or
a + °°).
e
(or 0
Still, the Holmgren norm can be used to extend the
class of integral operators, if one is satisfied with using strong convergence of operators, rather than convergence in norm.
What may happen in such a case will be illustrated in a special case which is of considerable importance. We consider the Hilbert space fined for
-°° < s < -
t
of functions
$(s)
de-
and carrying the unit form
(0,0) = 1 I*(s)I2ds.
We then consider the integral operators
t
is any positive number and
function of the real variable
(20.21)
with the kernel
jT(s,s') = tj(t(s-s'))
(20.20)
where
Jt
j(x)
2)
j(x) = 0 r1
is a real continuous
with the following properties
> 0,
1)
3)
x
j(x)
for
IxI
> 1,
j (x)dx - 1.
1
-1
The Holmgren norm of these operators is obviously
114
(20.22)
IIJTII, = 1.
for
jT(s,s') = 0
Clearly, then
Is-s'I > 1/T
and
T(s,s')ds' = 1.
Now we maintain that the operators identity as
T - m, i.e.,
(20.23)
IIJT*-OII
-
as
0
tend strongly to the
JT
for every
T
For continuous functions
$
in
of bounded support the state-
ment follows from the estimate
IJT$(s) - $(s) )
=
jT(s,s') [$(s') - (s) ]ds' I
I J
<
max
TIS'-SI<1
I$(s') - 4(s) I.
We need only note that the last term here can be made arbitrarily small for sufficiently large
T
independently of
(s).
the uniformity of the continuity of
To an arbitrary continuous also
0E
and any
in
e
with bounded support such that
IIJT+E-JT+II
IIJTa-mII <
$
IIJTII
IIJTO-JT$EII
s, by virtue of
1100-0II = 11$e-III
> 0
we may find a
II0E-ail
Then
Hence
110E-oil < 3e.
+
< c.
Thus the
statement is proved.
Since the identity is thus recognized as the strong limit of a sequence of integral operators it may be suggested to write it symbolically as an integral operator.
.Dirac who introduced the notation
115
In fact, this was done by
6(s-s')
for this kernel.
(The
symbolic kernel is regarded as a function of
s - s'
since the
approximating kernels have this property.)
We should note that this delta kernel is not obtained as an ideal function of
s
and
by a closure process with respect to
s'
a norm in the space of functions
k(s,s').
It was said above that the extension of an integral operator in norm corresponds to a "strong" extension of the corresponding %kernels to ideal elements in a normed space of kernel functions.
we now realize that the extension of kernel functions corresponding to strong extension of an integral operator does not correspond to a strong extension of the kernel; it is of a different character.
In
fact, it could be described with the aid of the notion of distribution. Later on we shall come back to this question. 21.
Functions of Bounded Operators
We now return to the general theory of bounded operators. shall establish a functional calculus of such operators.
bounded piecewise continuous function operator
f(B)
we shall assign an
f(B)
We assume the self-adjoint operator
act in a Hilbert space
and we let
may be taken as the norm
IIBIJ
B
To every
which obeys the rules of functional calculus,
described in Chapter I.
Since
We
< b,
acts in all of
I1BOII
be a bound for it, which
< b11011
the operators
t)
to
in any case
JIBIJ.
or
b
B
for
0
in
B2,B3,...
t).
are defined
n
and so is every polynomial
p(B) =
p I aBP of degree
At present
n.
p=O
we assume the coefficients verify that the operator
ap p(B)
to be real.
Then we immediately
is self-adjoint just as
aim is to extend the definition of the operator nomials to functions of a more general class.
116
f(B)
B
is.
from poly-
The main tool in
Our
doing this is the
be a self-adjoint operator in
Main Lemma.
Let
bound
< b < -.
IIBIH
variable
a
B
Let
J1
with the
be a real polynomial of the real
p(B)
for which
p(B)
for
> 0
JBI,< b.
Then
p(B) > 0. The statement involves the terminology introduced in Section 19; accordingly it means
(cD,p(B)(P)
for
> 0
0
in
Many proofs of this Main Lemma have been given. a proof which goes back to
We present
F. Riesz, but uses a modification of
Riesz's argument suggested by K. Brokate. In addition to the class p(B)
> 0
for
1B1
< b
(p)
of real polynomials
we introduce the class
which are sums of polynomials of the form where
g(B)
is any real polynomial.
product of polynomials in
(q)
(q)
of polynomials
q(B) = g2(B), (b+B)g2(a)
We note that the sum and
belong to
(q)
again.
product of two squares is a square and the product of (b-s)
can be written in the form
+ 1b (b-B)(b+B)2.
Evidently, (q)
converse is also true:
117
with
p
For, the (b+B)
with
(b+6)(b-B) = 2b (b+B)(b-B)2 is contained in
(p), but the
(q) = (p).
Lemma on Polynomials
We need only prove that every polynomial of class longs to
Let
(q).
< n
of degree
in
and
(p)
be the classes of polynomials
and '(q) n
(p) n
Then we prove the lemma by in-
(q).
duction, assuming that the statement Let
where in 8o
< b.
101
Evidently
longs to
p
we may write (p)n-1, in
p2
which vanishes somevanishes at a point
p(B)
< b, we may write it in the form p2(B)
is a polynomial of degree
belongs to the class
belongs to (q) n. If
p(B) = (b+B)p1(B).
hence to
(q)n-1.
(p) n-2
p(B)
since
151
p
be-
belongs to
vanishes at Bo = ±b
Evidently, p1
Therefore
does not vanish in
(p) n
is true.
By induction assumption, therefore, p2
(p)n:
(q) n-2, hence
I
where
p(B) _ (6-B0)?p2(6)
(p)n
If the polynomial
in the interior, I$
< n-2.
(q)n-1 = (p)n-1
be a polynomial in
p(8)
be-
(p)
belongs to'
p belongs to
(q)n.
p
If
< b, it has a positive minimum,
there, and hence can be written in the form p(8) = P min P min + po(B) where now po(8) is in (p)n and vanishes somewhere in Again we conclude that
101 < b. _ (p).n
p(B)
is in
(q)n.
Thus
(q)n
follows and the lemma on polynomials is proved. Setting
g(B)O _ T, we can write each form
(@,q(B)o)
as the
sum of terms of the form
('Y,fl
,
('Y, (b-B)'') , ('Y, (b2-B2)'Y) = b2('Y,'Y) - (B4',B'Y).
Each of these terms is non-negative, hence we have
(m,q(B) @) > 0. since every polynomial in (O,p(B)O)
> 0
(p)
is of the form
q, the inequality
follows and the Main Lemma is proved.
118
As a consequence of the Main Lemma we may state the
Corollary to the Main Lemma.
Let the real polynomial
be such
p(B)
that
Ip(B)I r po
for
jai
< b.
Then
HP(B)II ? po.
Since the polynomials
p
0
+ p(B)
Main Lemma is applicable to them. < po
or
< p0.
-pa < p(B)
are non-negative in
101
It yields the relation
< b, the I(0,p(B)flj
The statement of the corollary then
follows from Theorem 19.2.
We now can prove the
Main Theorem.
Let
B
be a Hermitean operator with the bound
every continuous function
f(B)
assigned an operator, denoted by
defined for
fBJ
< b
To
b.
there can be
f(B), which obeys the rules I, II,
III, IV of operational calculus:
I. fl(B) + f2(B) = f(B). implies f1(B) + f2(B) = f(B), II. f1(B)f2(B) = f(B) implies f1(B)f2(B) = f(B). III.
Whenever the values of.the real-valued function
in an interval
lie
3L', also the values of the ratio
(P,f(B) P)/(ID,41)
lie in
f(B)
for any vector 0 # 0 in
.SY.
This ratio may be interpreted as a mean value of the operator
119
f(B)
generated by the vector
$; thus this mean value lies in the
same interval to which the function IV.
Whenever
for
f(B) = 0
is restricted.
f(B)
IBI
< b, the oper4tor
f(B) = 0.
An immediate'corollary to rule III, via Theorem'19.2, is the rule III'.
Let
f
0
be the center
of the interval
and
it'
its
26
diameter; then
1)
II(f(B) - f0)0lI < 6II$II
i.e. if the interval 2)
is given by
it
II(f(B) - f0)$II < 611411
i.e. if the interval
is given by
SL'
for any
$
If - f0I
< d;
for any
4
If - f0I
< d.
if i is closed,
0
if
is open,
5W
To prove the main theorem we rely on the Weierstrass approximation theorem.
Accordingly,-to a given continuous function
is a polynomial
pE(0)
Hence IpE(B) - p6(8)I < c + d IIPE(B) - p6(B)II < e + 6. PE(B)
If(B) - pEMI < c
such that
for
181 < b.
and the corollary yields the relation
Clearly, then the sequence of Operators
is a "uniform" Cauchy sequence as
tends to zero.
a
operators, therefore, approach a limit operator, denoted by such that
there
f(B)
IIpE(B) - f(B)II - 0
These f(B),
c - 0.
as
Note that the validity of Rule IV is implied by this construction.
Since the sums and products of approximating polynomials respectively tend to the sums and products of the approximated functions and since the product of two strongly converging operators it
converges strongly we conclude that also the rules I and II of the operational calculus are obeyed by the assignment of Furthermore, if
f(B)
> 0
for, all
a
we have
f(B)
f(B).
p (B) + e > 0, e
120
to
p£B + e
hence
>
0, whence in the limit
f(B)
>
0.
- fl
f(s)
Applying this result to the functions
and
f1 < f(B)
f2 - f(s), rule III follows for the closed interval
< f2.
To derive rule III for an open interval we need only consider the case in which an
f(s)
such that
a > 0
IV we may change f(s)
> a
'
for all
0
> a
f(B)
s.
for all IB-1
f(B) outside of
for all
B.
Then
Clearly, in that case there is
f(B)
By virtue of rule
b.
in such a way that
< b
> a > 0
It is thus seen
follows.
-that rule III holds also for an open interval
SL'.
It is not sufficient for our purposes to have operators assigned to continuous functions; we must set up such an assignment for piecewise continuous functions.
Having done this we shall be able to
-establish "spectral resolution" of an Hermitean operator.
We first take step functions, i.e. characteristic functions of
intervals 1 defined by
n3 (s) = 1 for
s
in
9, = 0 for
8
outside
1.
We here regard also a single point as an interval, a closed one, of course. If
n 1(6)
1 is an open interval, s_ < B < B+ , we approximate
from the inside by the piecewise linear functions
n3 (B) = f(B - B)/6 1 0
for 0 < ±(B+ - B)
<6
for a- - 6 < B < S+ + outside 1.
Evidently, these functions increase monotonically as
If 1 is closed, B_ < B <
121
B+ , we take
6 - 0.
=1 n6 (B)
in 9,
1 t (B+ - B) /6 for = 0 for t (B-B,t) > 6.
< 6,
These functions decrease monotonically as 6 ; 0. Since these functions
are continuous we may conclude
n-,(s)
from rule III of the main theorem that the operators in the-same way; i.e. the forms
1
(6,nS
V (B)
)
behave
increase monotonely if
ii an open interval and decreases if 9 is closed. We-now may apply the theorem
.19.3
of monotone convergence
formulated in Section 19, and conclude that the operators
n6
converge strongly to limit operators, which we shall denote by To formulate a counterpart-of rule I for the operators
(B)
nJ(B). n1(B)
we assume that the interval 3 is the union of three other adjacent intervals,
1= 1 U where either
.1+
1- are closed and
U1+
9 are open and
and
.1 open. 0
n, (B) + n,
I n:
0
closed, or
o In either case we have
0(B} +
9+ and
n5( B) .
To verify that this is so we need only observe that the piecewise linear functions that approximate the three step functions can be so chosen that in their sutra the linear sections interior to 1 cancel
away, so that this sum is an approximating function for the total interval
.9.
On occasion we shall express a closed interval as the union of the open interior and the two endpoints, regarded as closed intervals. Formula
In
then allows us to express the operator 122
n '(B)
for such
a closed interval as the sum of the operators these three parts of it.
Also we observe that if an endpoint of an
interval lies outside of > b
B
or
IBI
< b
it does not matter where it lies in
We, therefore, may just as
B < -b, by virtue of rule IV.
a = -
well allow an interval to have an endpoint at
we shall rely on a lemma concerning the operator with a continuous function
f(B)
of
converges strongly to
n 1 (B)
Lemma 1.
if
Suppose the continuous function
an interval -.1 and has the absolute bound
0 < f(8)n(8) < K then for all vectors
0
f(B)n1 (B)
formed
which exists since
f(B)n; (B)
n,,(B)
B = -m.
Clearly, this operator is
B.
defined as the limit of the operators
or at
and related facts
n(B)
In deriving rule II for the operators
6
associated with
n(B)
6 - 0.
f(B) K
for all
is non-negative in there, so that
B;
the inequality
(O,f(B)n,,(B)a) < K(O,m) holds.
. To prove this statement we choose, for a given
approximating function n1 (B)
c > 0, the
in such a way that
-E
which is possible by virtue of the continuity of
f.
we then conclude that the inequality
(K+E) (0,@)
123
From rule III
holds.
Since
n6
converges strongly to
(B)
n3(B)
and
a
is
arbitrary, the statement follows. Taking
we are immediately led to the
K = 0
Suppose the continuous function
Corollary to Lemma 1.
f(B)
vanishes,
in the interval 5; then
f(B)n5(B) = 0. From Lemma 1 we can derive an important second lemma which refers to a monotone sequence of intervals
9a C
i.e. a < T implies the operators
and
1
qa(B) = n 1a(B)
9a
fla 1o = 0.
with empty intersection;
The statement is that
tend to zero for such a sequence.
is sufficient for our purpose (and actually no
It
restriction) to con-
sider a special case.
Lemma 2. na(B)
Proof.
be the open interval
3a
Let
If 'a
tends to
which we denote by linear function for T
Then the operators
tend to zero.
0
monotonically the functions
monotone sequence; hence the operators
a <
0 < B < a.
Q.
Let
nT(B).
na(B)
tend strongly to a limit
be an approximating piecewise
nT(B)
Then we can, for every
such that
na(B)nT(B) < C. By the lemma we have, for every
na(B) form a
0,
(O,na(B)nT(B)0) < E(0,m).
124
c > 0, find a
a, d, t, and
Letting successively
tend to zero we find
a
successively,
(4, ,Qn(B)')
< E(,4), <
Clearly, Q
on (B)in < c(V 0.
E(
is self-adjoint, since the
IIoII2 = ($,Q20) = 0
and therefore
na(B)
Q = 0.
are; hence
Thus the statement of
the theorem is proved.
An immediate consequence of this theorem is the
Let 3 be an open (or closed) interval and
Corollary to Lemma 2.
5a
be a monotonely increasing (or decreasing) sequence of intervals
covering all of
-', so that Ua 1a = 3.
Then
J
(B) - n(B)
n -1110
Without restriction one may assume
9a
to be closed if 3
is open; then one needs only apply the theorem to the two shrinking
sequences of open intervals left over from 3 after removed.
The statement for the case that
by going over to the complement of
9' is
has been
o closed then
follows
9.
We are now ready to prove rule II for step functions.
This
rule assumes a special form since the product of two step functions is again a step function. two intervals
91,
Specifically, if 5 is the intersection of
12,
yl n 32 = 1,
125
its step function is the product of those of the two intervals:
n9
(a) n
1
92 (B) - n,(a) .
Accordingly, we expect'rule II for step functions to be
implies
n 9 (B) n, (B) = n3 (B) . 2
1
In proving that this rule is valid we shall employ piecewise linear approximating functions
and
n 1 (S)
n 2
(6)
and use the
2
1
relation 6 T1
if
61 -
0
and
6
1(B)
1
62 ± 0
n
92 (B)
n
31
(B) n
in any manner.
-r2
(B)
Actually, it will turn out to
be sufficient to prove rule IIn in cases in which the intersection is empty,
Jl n 12 = P.
61 n
-Wl
In such cases we need only show that
92 (B) - 0
as
(B) n62
There are four such cases.
In case both
61 +
91
0,
and
62 - 0. 12
are open with
empty intersection, the product-of the approximating functions is zero; hence the rule holds with
n}B) = 0.
The same argument applies
when the two intervals are at a positive distance from each other. If one of these intervals is closed one must, of course, choose the linear parts of the associated approximating functions. sufficiently steep.
126
In case one interval,
boundary point
of
R2
J'1
9l, is open and-the other,
32, a
we apply the corollary of Lemma-1 to the
with f(s) = n6(0) being an approximation to nl(s). j2 Since evidently f(s2) = 0 we have n6(B)n6(B) - 0 and hence interval
nl(B)n2(B)
0.
To handle the general case one should obser-Ye that any two in-.
tervals can bg referred to a common'subdivision and be written as the sum
of open intervals and vertices of this subdivision.
The products
of the two step functions are then sums of products, either of the four special types discussed first or of the products of the form n2
with 1 being open or a point..
To handle these two cases one
need only write
2 n1=n-n,nj,* ,
when
1*
is the complement of
5, which evidently can be written
as'the union of intervals either away from but no% intersecting
1.
Jr or adjacent to
That is to say, the product
9,
n 5 (B)n,*(B)
consists of the sum of products of the four types considered first Thus the relation
and hence is zero.
nj (B) results.
= n-V (B)
Rule IIn has thus been established.
Relation
2
(B) = n ),(B)
implied by rule 'In
Pr
shows that the
operators
P' = n, (B) are projectprs.
Moreover, rule IIn
127
shows that the projectors of
foreign intervals annihilate each other,
P
The projector
P
_JV1 2 P16I
if
=0
-1 fl 2 = fd.
assigned to the interval
< b
j8
is the
identity,
PtBk
as follows from rule IV and the corollary to Lemma 1.
The assignment of the projectors P, = n 3(B) vals
to the inter-
Y constitutes the spectral resolution of the operator
will be called the eigenspace.of the
The range of the projector P5 operator
3 and the vectors
for the interval
B
B.
range will be called eigenvectors of
B
P_ Y@
for the interval
in this This
_V.
terminology is justified by the fact that these vectors are eigenvectors in the original sense in case the interval consists just of
That this is so will be implied by the subsequent con-
one point. siderations.
For eigenvectors of an interval we can formulate a rule analogous to rule III.
III7 :
For all eigenvectors
t = P9 t # 0
of the interval 9 the
mean value
lies in if the continuous function interval
'f
for
B
in
f(8)
S['
takes its values in the
5.
We here assume, without serious restriction, that
128
SL' is
closed if 9 is closed and that
In case 3 and
St'
lemma
consequence of the
;
.$'
is open if 3 is open.
are closed, the statement is an immediate (P5 $,f(B)P
one need only replace
by
9 ')
(P@,f (B) n5(B) 0) . In case the interval
SY
is open, given by
f_ < f < f+ , we
first observe that
(,f(B)4)) > f-($11) for eigenvectors
of
4
9; for, these eigenvectors are also eigen-
vectors of the closure of 9 so that the statement for closed intervals can be employed.
Now we take any eigenvector
@
of 5
for which
.
and prove that
(4),f(B)4)) = f-(010)
0 = 0.
To this end we take any closed interval
of 9 and set
nJ (B) = Qa.
7a
in the interior
Clearly, the eigenvectors of
Qa
are
a
eigenvectors of P1 and, since eigenvectors
0
f(B) - f_
is nonnegative for these
we have
0 < (Qal, (f(B) - f_)(204))
<
(@, (f(B) - f_M.
Hence
(Qa4),(f(B) - f_)Qa(P) = 0.
Since the values of positive lower bound
f(B)
as
- f_
are positive by assumption there is a
for this function in the interval
therefore 129
(Q04,(f(B) - f_)Qa4)) a a.(QQ@,QQ$).
Q.0
It follows that
is zero.
increase so as to cover all of
Now, if one lets the intervals -7, the operators
Q.
by virtue of theScorollary to Lemma 2 proved earlier.
Pam = 0.
But, Pam = 0, and hence
0 = 0.
Jpa
tend to P5 It follows that
Thus we have proved that
(4,f (B) 0) > f 0 = P. Similarly one proves
for all
< f+(0,0)
(4),f(B)0)
and thus the statement. can be proved.
The following converse of rule III
III
for
Suppose the function
:
in 5 and outside of
B
moreover that
0
_St
0
for
B
outside of
5.
Suppose
is a vector for which
(41,f(B)0)/(m,4') Then
has its values inrthe interval 5£'
f(B)
is an eigenvector of
lies in
SE'.
5.
To prove it one need only apply the second part of rule III) to the two open sets complementary to the closed set
1 to obtain
the converse of the first.part, and conversely apply the first part of the two closed sets complementary to the open set
9 to obtain the
converse of the second part.
We shall not give the details.
The first part of rule-III-,, when applied to an interval 5
130
B0, yields the statement that, for
that consists just of one point
all eigenvectors of this point interval,
- ($, (f (B) - f(B0)) 0) _, 0.
It follows that the operator say, all eigenvectors
(f(B)
- f(B0))p1
is zero.
That is to
of this point satisfy the-relation
0 - Pf 0
f (B) O = f (B0) 0;
in particular,
BO = 800.
Thede eigenvectors are thus seen to be eigenvectors in the usual sense. The converse is also true as seen by applying rule III function
f(B)
to the
B.
Finally we introduce the class of "piecewise continuous" functions as the functions which can be written in the form
f(B) = E f_JV (0)n,(0) Jr
F0:
where each function
f)r(B)
is continuous in
IBI
< b.
We can now
assign to such a function an operator
FB
f (B) _
1
f, (B) n jr (B)
From the corollary to Lemma 1 we infer that this operator is in-
dependent of the choice of the function f, (B) 'employed to express it.
From rule 11 we infer that the operator is independent of the.
131
choice of the partition to which the intervals 5 belong.
From
then clearly holds for our piecewise continuous functions. rule i19 and III.
we conclude that also rules II and III hold for That is to say the assignment of
these functions.
Rule I
f(B)
to
f(S)
obeys the first three rules of operational calculus. Accordingly we may formulate the
C
rQ
ollary to the Main Theorem.
function
f(S)
an operator
of the form
f(B)
FS
To every ordinary piecewise continuous
there can be assigned, throug
FB1
such that the four rules I, II, III, IV of opera-
tional calculus are observed.
22.
Spectral Representation Having established a functional calculus for a bounded Hermitean
operator by operators f(S)
f(B)
to ordinary piecewise continuous functions
we have established the spectral resolution of the operator
which is given by assigning projectors a partition.
t)
to intervals 9 of
rl 1(B)
We now proceed to establish a spectral representation.
To this end we first select any vector space
B,
and form the subspace of
0
from the }(filbert
which consists of all vectors
of the form
0 = h(B)S2 = f(B)S2 + ig(S)Q. where
h(S) = f(S) + ig(S) runs over all piecewise continuous functions. This subspace will be denoted by
132
@'(S2); its closure will be
and called the space "generated" by
0(1)
denoted by
S).
With this space we shall associate a measure function To this end we consider the open interval point
function of no(B').
with the upper end
9B
while the lower end point is any number
a
The step
of the open interval from
B'
will be denoted by
< b
< -b.
associated with this interval will be denoted by
B'
The step function of
any point
r(s).
to
a
1 - n+ W). In other words we
have
nB(B') = 1
for
B'
< B, = 0
for
B' > B,
n+B(') = 1
for
B'
< 8, = 0
for
B'
> B.
Now we introduce the functions
(n,ns(a)n).
Evidently, these functions are monotonically increasing. that both
and
r+(B)
from above, and to
r (B) .,amend to
r (B0) when
B
tends to
follows from the corollary to Lemma 2. of the open interval
B0 < B'
< B
+
na(B)
when
r+(B0)
is
_
B0
0 1
finds
- r+(B0)
r+(B)
>
B
instead of -
0.
tends to
B
from below.
B0
This
For, the step function of no(B') - no (B'). + 0 0
- nB(B) - 0 and hence r M - r (B0)
this to any
We remark
0
B, and observing
as
Therefore,
8 300.
r+(B)
B'
Applying
< r-($
I
)
one
Similarly, one derives the statement for
6180. The remark made shows that the pair
r
r -(B)
could indeed be
used as a "measure function pair" as introduced in Section 8, Chapter II. We now consider a partition open intervals and points.
9 of the interval
IBI
< b
In agreement with the procedure in.
Section 8 we set
133
into
A-: Ba-1 < B < Ba+1'
a
even, open, interval,
Q: B - Ba,
a
odd, point.
Aar = r (Ba+1)
A
a
- r+(Ba-1)
r = r+(B0) - r-(Ba)
for
for
a
even,
a
odd.
Clearly,
Aar -
In
-j6
(B)$112 = (n,n
With complex constants
(B) n)
6
ha
we form the'piecewise constant
function
k(B) _ I hn,a(B) and the operator
h(B) _
han 3a(B).
By virtue of
n
-X(B)n9 (B) = n3 (B) 0
for
t=a
for
t + a
we have
lhaI2Aar.
IIh(B)2I 12 = a
134
Now, the sum here can be interpreted as the integral of respect to the measure pair
r-(s).
with
Ih(B)I2
In other words,
I Ih(B)SI 1 2 = 1 Ih(B) I2dr(B) . Let now
V, i.e. a piecewise con-
be a function in
h(B)
tinuous function; since it can be approximated uniformly by piecewise h'(B), we clearly have
constant functions
-
(h (13)
as
A
hA(B))RI12 - 0
as well as
I
h(B)
-
hA(B)I2dr(s)
+ 0
evidently then
-+ -.
Ih(B)I2dr(B)
IIh(B)II2 = J
and
(h(B)R,h' (B)S2)
for any function in this class
=' h(B)h' (B)dr(B) . as.
We can go one step further. of the space 1
-
(ir
of functions with respect to the unit form
la'
by adjoining ideal functions
Ih(a)I2dr(B)
Ih(B)
We may introduce the closure
hA(B)I2dr(B)
-
0.
Now, the sequence
h(B).
hX(B)
Approximating forms a Cauchy
J
sequence and hence the vectors converge to a vector the ideal functions generated by
Q.
-
h(B)
hA(B)S:
such that
form a Cauchy sequence; they
II(P-ha(B)SiII - 0.
Clearly, then
correspond to vectors in the space
%(S2)
The converse, of course, holds true just as well.
Thus we have established a one-to-one correspondence of the
135
vectors in
Q(S2)
and the functions in the closed function space
ir,
h(B)
0
such that
(P,(P)
Of course, we must set
= fIh(8)(2dr(B).
h(B) = 0
also evident that the vector
B(
whenever
corresponds to the function
Ci', the relation
since for functions in relation. Bh(B)R = h1(B)12
It is
1 Ih(S)I2dr(B) = 0.
Oh(O) = h1(B)
Oh(B)
leads to the In
by rule II of the functional calculus.
other words
B k 441 WS)
Thus we have achieved a spectral representation of the operator
the subspace
of the Hilbert space
(S(S2)
B
in
h.
Symbolically, we may express this relationship by the formula
4) = h(B)0 for all ideal functions in
(fir.
From here it is only one step to the full spectral representa-
tion of
.
For simplicity we assume the space countable dimension.
Ql'n2"" every
(n)
f
in
and
1D
of al, ... ,SJn
to be of (at most)
Then there is certainly a sequence of vectors
which spans the space
e > 0
b
there is an
n = n£
such that 136
V densely.
That is, to
and a linear combination
ll-a(n)II
01,...,On
The space of linear combinations of
Z(521,...' n)
in = so that we may say that Now set
and let
= S21
St(l)
O(l)
jects into the space
is in
0,(n)
will be denoted by
In.
be the projector which pro-
PI
O(St(l)) genc=ated by
=
R1
and let
successively
St (2)
= (1-P1)02, and P2 be the projector into
0(n)
= (1-P1 ... -Pn-1)Qn,
(fin = 0 6, (n) )
Then, we maintain, or, in other words, P P
n 1
and
Pn
be the projector into
.
O m'...' 0(n-1)
is perpendicular to
0(n)
... = PnPn-1 = 0, for all
= P P n 2
To show this assume the statement proved up to (P1 +
+ Pn-1)V'(r) =
On = hn(B)0(n) hn(8)
and
be in
and
4'(r)
is also in
®(r)
(4)n,41 r) _ =
(hn(B)St(n),
Then
n - 1, so that r
in
4) r = hr(B)St(r)
are continuous functions.
hr(s)
fi(B)h? }St(r)
for every for
Y(r)
(St(n), ,(r))
Pn-1) if (r1 ) = 0.
(Stn, (1-P1
137
Now, let
be in
and
hr(B)S0(r))
n.
d(r), where =
Hence the statement is true for
n.
Next we maintain that the space X1(1)
q) d(n)
e)
is contained in
do
To show this we set
.
Stn = Plnn + ... + Pn-1S2n + (1-P1 - ... - Pn-1)Sln. The first d (1)
® ...
is in
vectors on the right hand side are in
n - 1
0(n),
d(n-1)
Now let
0
(p1 + ... + Pn)4
that
4)(n)
P1,...,Pn-1; the last vector Assuming the statement has
by definition of this space.
been proved up to
vector
by definition of
in
n - 1
it follows for
be any vector in
tends to . 11n
4(n), being in
V.
n.
We maintain that
In fact, for a given e > 0
for which
II$-- (n)II
< E.
Since we now know
d(1) ® .. H, d(n)
Xn, is in
P1 + ... + Pn
minimum property of the projeotion
take the
we can use the
into this space.
We find
I'D -(P1 + ... + Pn)O11 . II0-4,
(n)jI
<
E
Then the statement follows.
This statement implies that every vector
can be written in
the form
= P1( + P24) + ...
as a series of vectors in
0(1),
0(2)....
so that
IIP11,112
110112 =
Now each vector in
O(n)
+
can be written in the form
138
hn(B)n(n)
with
an appropriate (ideal) function
hn(B).
In other words, the vector
can be represented by a sequence of functions
41
in the spaces
4-0 {h1(B), h2(a).... I
such that
Qrl , Qr2
Ih1(B)12dr1(B) + f Ih2(8)12dr2(B) + ...
At the same time
is represented as
BO
B$ a*D (Bhl(B) , Bh2(B) ,...
).
Thus we have established one of our major aims:
we have established a
spectral representation of any bounded Hermitean operator.
Of course, we do not claim that this representation, which is of the direct multiple type, is the only one - or the most suitable one.
In particular, it could happen that some of the measure func-
tions are identically zero so that the corresponding terms drop out.
We also should mention - as Hellinger in 1909 has shown - that a "minimal" spectral representation can be established, i.e. one in Al rm = 0
which A
rf = 0
for
k
for some interval 9 always implies > m.
We do not intend to discuss the proof of this
fact.
Finally, we should mention'that we could eliminate the assumption that the Hilbert space
should have a countable dimension.
To
handle spectral representation in a non-countable Hilbert space one may employ a well ordered set
{i2m}
of vectors which span
densely.
All arguments given can then be carried over; we do not want to give details.
139
Normal and Unitary Operators
23.
Suppose
B
and
are two bounded Hermitean operators which
C
commute:
BC = CB.
Then powers, polynomials, and hence piecewise continuous functions of B
commute with such functions of
and
of
Y
B
and
C
in a
We may plot the eigenvalues
C.
(3,Y)-plane and introduce as common
spectral resolution of the pair
the projectors
B,C
which correspond to the step function n cell
(B),n ma(y)
riip, (B),nV,(C)
of the product
It is also clear that to any bounded piecewise con-
X Y.
_3d
R
tinuous function
an operator
f(B,y)
can be assigned obeying
f(B,C)
the rules of the operational calculus.
With the aid of two such commuting operators we may form the operator
B + iC which, unless "normal".
C = 0, is not Hermitean.
The eigenvalues
s + iy
The common spectral resolution of the spectral resolutions of
of B
B + iC.
Any such operator is called B + iC
and
C
are complex numbers.
may then be regarded as
Also, it is clear that to every
(complex-valued) bounded piecewise continuous function B + iy
an operator
f(B+iC)
f(B+iy)
of
may be assigned obeying the rules of
operational calculus.
in particular, if
0
is an eigen-function of the product cell
5 X/ the value of the ratio'
(41,
(B+iC)0/(2,4)
lies in this cell, as follows from Rule IIIy in Section 21.
140
If the normal operator U* = B - iC
U = B + iC
together with the adjoint
satisfies the condition
U*U = 1 it is called "unitary".
we have also
Since
and
U
U*
U* = U-1.
UU* = 1, so that
commute, i.e.
UU* = U*U,
Clearly, a unitary operator
is "norm preserving"
IIUtII = II'DII
3d x/ be a cell in which
Let IYI
Y2
with either
Bi + yl >
an eigenvector of this cell. B2
and
C2
0 < B1 1
or
IBI
< s2
and
2 + Y2 > 1.
0
Let
< Y1
a # 0
be
From Rule III5 of Section 21 applied to
we conclude that the ratio
(0,(B2 + C2)41)/(ID,9)
is either greater or less than 1, which contradicts the condition B2 + C2 = 1.
Hence there is no such eigenvector
f + 0.
follows that the eigenspaces of any product cell vanish lies outside of the unit circle.
It then if this cell
We express this fact by saying that
the spectrum of a unitary operator lies on the unit circle.
As a consequence of this fact we note down thg following
Lemma.
Suppose the piecewise continuous function
on the unit circle; then For, such
f(B,y)
f(B,C) = 0.
a function may be written in the form
f(B,Y) _
5'/
fy, j(0,Y)n 1(a)n,(Y)
141
vanishes
with continuous functions
£
5 1(B,y).
Bylsubdivision one can
in those cells that intersect the
achieve that
If 1,1-(B,Y)I <
unit circle.
The contributions from the product cells
E
5 xJ
lie outside of the unit circle may evidently be omitted.
we have
HHf(B,C)II
We now-let function of
B
< c, hence
B2 + Y2 = 1.
c
is arbitrary.
for which
y
=B+iy
ei8(8,Y)
on
since
Consequently
be any real-valued piecewise continuous
6(8,y)
and
f(B,C) = 0
that
To be sure such a function can be constructed; one
need only take a function whose values on the unit circle agree with the polar angle
8
restricted by
-n"< 0 < n.
From the lemma we then conclude that the relation
e
holds.
°
Now, the operator
is real-valued.
' C) = B + iC = U 8 = 8(B,C)
is self-adjoint since
8(B,Y)
Thus, we have established the important fact that
every unitary operator
U
can be written in the form
U=ei8 with the aid of a bounded self-adjoint operator 8 with spectrum in
-n < 8 < n.
142
CHAPTER V
OPERATORS WITH DISCRETE\SPECTRA
24.
Operators with Partly Discrete Spectra There are various classes of operators whose spectra have
significant special properties.
In this chapter we shall discuss oper-
ators whose spectra are "discrete" or "partly discrete".
A discrete spectrum is a pure point spectrum; but the term "discrete" is to imply more, namely, that each point eigenvalue has
a finite multiplicity and that there is only a finite number of eigenvalues in each interval.
This requirement is rather severe.
We shall
require somewhat less by allowing the eigenvalues to accumulate at zero, and by allowing zero itself to be an eigenvalue of infinite mul-
We then say the spectrum is "discrete away from zero", or
tiplicity.
simply "essentially discrete".
We may describe the property of discreteness in an interval as a property of the eigenspace associated with this interval, without mentioning point eigenvalues explicitly.
In doing this we
assume the operator - denoted by K - to be bounded and Hermitian, so that we can refer to its spectral resolution, i.e. to the projectors
n(K) and the eigenspaces 11
associated with intervals J' of the
K-axis.
Using these notions, we say that the spectrum of the operator
K is discrete in an interval / if the eigenspace dimension.
intervals /A: (or -IIKII
has a finite
2 In particular we shall consider'for any-positive
<
eigenspaces.
discrete above
<
A
<
K
K
< -
(or
A
<
<
K
A) and denote by
11
A
JJKJJ) and and
finite dimension.
> 0 (or below
-A <
0)
K < -A
<
12-, the associated
Then we shall say: the spectrum of the operator A
the
A
if the eigenspace
4
K
is
has a a
Finally we say that the spectrum of the operator
143
K
is discrete away from zero or simply essentially discrete if the eigenspace of each closed interval that does not contain zero has a finite dimension, i.e. if it is discrete above every positive and below every negative value of
K.
Since the operator
Qr
into the vector
transforms every vector
K
(K)m of
0 _
which also lies in
K@ = Knj(K)(P =
it may be regarded as a Hermitean operator acting in the finite-dimensional space
This space, unless it is empty, is
Q,.
therefore spanned by a finite number of mutually orthogonal normed eigenvectors
Q.
We apply this remark to the intervals
and the corresponding eigenspaces
IIKII
Q N
0 <
A
< K <
and conclude that every
vector O(a) of this space can be written as a linear combination n, (24.1)
a=1
of eigenvectors
Si(1),... of the operator
K
such that
n
(24.1)'
=
Kc
C
K
C==1
Here
a
Kn
runs from 1 to
(a)
a a
n,, the dimension of the space
are the eigenvalues of
lying above
K
QA, and
K1,
A.
We further conclude that the unit form is given by
E
a=1
and the associated quadratic form by n, E
CF=1
Every vector
4)
in
t)
can evidently be written as the sum
(24.2)
of its projections (and L
41
into the eigenspace
complementary space Q , so that the relations
144
Qx
and the
n
L41 (24.3)
012 + a=1 n,
(24.3) '
EaIY +
K
ail
hold.
are given by the formula
Note that the coefficients
Q =
(24.4)
and are hence independent of the choice of the number Finally we state that for the vectors
0x
X.
A
< a(OL,(
(v,,K
To show this we may introduce the unit step function of the interval
1 - rlA(K)
Q
orthogonal to
K
<
so that
A
4x = nt(K)(b A
.
nx (K)
_
Now, since
1 evidently
(a-K)nA(K) > 0
,
we have
1
1
.1
0LX-K)b a) _ (Ox,(A-K)n,(K)4x)> 0 by rule III of the functional calculus, whence relation (24.5) follows. Similar statements, of course, hold for the eigenspace and the spectrum of
below
K
-A.
Suppose now that the spectrum of zero.
vals
Then we introduce the eigenspaces 0 < K < -
K = 0.
Q+
is discrete away from and
Q_ Ci0
of the inter-
of the value
By virtue of the corollary to Lemma 2 of section 21 in Chapter
the projectors jections
0(±,l)
K
and -- < K < 0 and the eigenspace
IV the projectors
d)(;)
Q _X
of
r1
'(-A) 0
into
n(K) (K)
of the eigenspace
of the spaces
of a vector Q ±.
are independent of
t
into
Q-.
Q±
tehd (strongly) to In other words, the protend to the projection,
Since the coefficients A
of the projections
we obtain the following statement con-
cerning the spectral representation of an operator with an essentially 145
discrete spectrum.
The vector n+
s(a)
_ F
0
0=1 where
PO
(P
may be written as an infinite series n_ +
-0
0=L
is the pro)ection of
(o) +
D
0
into
'
and
10
sequences of orthonormal eigenvectors of
f2 (0),
n(-0)
with positive and nega-
K
tive eigenvalues respectively; the (finite or infinite) numbers are the dimensions of the spaces
n K(P _
`
I
K(P 0 = 0.
0=1
n
At the same time the expansion
.
n_ +
0 0
0==1
holds since
0(0)
K
q
are
7(-a)
K
-a -o
For the unit form and the form of the operator
we have n_
n+
IF-012 + II@0112
i
f
0=1
0=1 n_
n+
KaIFaI2 +
I
-0IF-0
£
0=1
0=1
It is not implied here that there actually are positive or negative eigenvalues, i.e. that none, i.e. if
n+ > 0
and
n_ > 0.
K = 0; if there are
n+ = n_ = 0, we have, of course,
such eigenvalues
If there are
n- > 0 they can be arranged in decreasing and in-
creasing order respectively, K1 > K2 > ...
> 0;
K-1 < K-2 - ...
since their number above every positive -a
A
<
0
,
and below every negative
is finite.
It is also not implied that there are infinitely many positive or negative eigenvalues
n+ = m
;
n+
if
W
or
n- = W
these
eigenvalues tend to, zero:
K1 > K2 > ...
+0 ;
K-1 < K-2 <
.
0
,
as follows from the fact that away from zero there is only a finite 146
number of them.
These statements and formulas give the spectral representation of operators with an essentially discrete spectrum.
25.
Completely Continuous Operators
The question naturally arises whether or not one can tell beforehand
from the nature of the operator
K
- that this operator
has an Issentially discrete spectrum.
In fact it is possible to do so,
as was discovered by Hilbert in 1906.
Hilbert found that operators
with an essentially discrete spectrum can be characterized by a simple property, which he called "complete continuity",.of its form.
Very
frequently, it is easy to test whether or not a concretely given operator has this property.
Instead of describing Hilbert's property of complete contin-
uity, we shall at first describe a different - but equivalent - property which, in general, is still more easily verified in concrete cases
and from which.the essential discreteness of the spectrum can frequentWe shall call this the property of
ly be inferred immediately.
At first, however, we shall describe
"almost finite-dimensionality". a less restricted property.
We shall say that the form of an operator g
above a number
A >0
if there exist vectors
K
has dimension
Z(1),...,Z(g) in
C)
such that the inequality
(25.1) +
((D,K4)
IZ(Y) 4I2
I
<
+
Y=1
holds for all vectors have dimension Z(-g)
(25.1)_
g
4
below
in
t); similarly, the form will be said to
-A < 0
if there are vectors
Z(-1),...,
such that (4,K(P)
> -
I
IZ(-Y).0I2 - A($,4)
Y=1
Finally, we say the form of
K
is "almost finite dimensional" if for 147
every
41,MPI
0
of vectors
g = g(E)
Z(1),
E) such that the inequality
(also depending on
...,2(g)
(25.1)
there is a finite number
e >0
I
<
Y=1
holds for every vector
{
in
Of course, the latter property coul,
also have been described
by saying that the form is almost finite-dimensional if it is finitedimensional above every positive and below every negative number.
As seen from the two theorems that we shall prove, discreteness above
A
and having finite dimension above
discrete above
> 0 (below
A
>
0 (below
-A <
0)
K
is
it. is finite-dimensional above
-X < Or.
M
To prove this statement we write tion 24, and set with
are equivalent.
If the spectrum of an (Hermitean bounded) operator
Theorem 1.
X
A
g, = nA
+
as in Sec-
In view of (24.5), formula (24.3)'
Z(Y)
then assumes the form (25.1)+.
Similarly, one estab-
lishes (25.1)
Theorem 2, the converse of Theorem 1, will be proved together with a corollary. Theorem 2.
dimension sion Proof.
g
If the form of an (Hermitean bounded) operator has g
above
above
A
>
A
> 0 (below
0 (below
-A < 0)
its spectrum has dimen-
-A < 0).
Suppose the first statement of the corollary were not true;
dim qX > g.
Then there would exist (at least)
dent vectors
(1),...,m(g+1) in
in
11
a vector
(P
¢ 0
(FX.
g+l
line..rly indepen-
Consequently, ther: would exist
perpendicular to the
g
vectors
Z(1),...,
2 (g) . For, the condition that the linear combination cl1. (1) + ... + cg+l 0(g+l)
is orthogonal to 148
Z(1),...,Z(g)
(
_
represents
g
linear conditions for
efficients
(cl,...,cg+l)
The resulting vector
g+l ,
unknowns; hence there is a set of co-
not all zero, satisfying these equations.
is not zero because of the linear independence
4,
Inserting this particular vector
of
(P
in (25.1)+
we find
On the other hand, since the vector
((P,K4)
D
X 0 is in
F x
A (41, f)
5
holds by virtue of Rule III of Section 21, Chapter IV. dim
0, this is a contradiction; i.e., proves
dim
the inequality
(Ix
< gx.
Since
(I,4) ¢
Similarly, one
Q_ < g_x.
The main statement to be made in this section is Theorem 3.
If the form of an (Hermitean bounded) operator is almost
finite-dimensional its spectrum is essentially discrete.
It is an
immediate consequence of Theorem 2.
As we shall see, it is in many cases easily verified that the form of an operator is almost finite-dimensional; therefore the essential discreteness of the spectrum of an operatof is also easily established in many cases.
Moreover, an estimate of the manner in which
the eigenvalues of such an operator approach zero is given at the same time.
For the corollary to Theorem 2 shows that the nth positive
eigenvalue is less.than or equal to g,.
X
if
n
is chosen greater than
In fact, one of the methods of estimating the behavior of the
sequence of eigenvalues is based on just this situation. In order to describe Hilbert's property of complete contin-
uity, which we shall do now, we must introduce the notion of "weak convergence" of a sequence of vectors, which differs from the ordinary or "strong" convergence introduced in Section 14 of Chapter III.' We
149
,(v)
say that the sequence of vectors
if for every vector
tends to zero "weakly",
in a dense subspace
W
(25.2)
(T, 1P')
-
as v
0
holds while at the same time a number
the relation
of
-
C exists such that
11x"11
(25.3)
TO
v.
Clearly, it follows from these two properties that (1) holds for every* vector in
0 a vector
for every
e >
and a
such that
vE
lb.
For, if in
4"
11V1,4,v1
is any vector in
'Y
< E/2
14',4v1 < c
such that
!D'
', there is < e/2C
11T" -T11
v > vE , whence
for
for v > ve
.
It follows from this remark, for example, that every sequence @(v)
= {
i,E2.... }
of vectors in the special Hilbert space converges
weakly to zero if every component v converges to zero as provided
11$v11
< C.
v
For, we need only take the space of vectors with
a finite number of components as subspace
b'.
Now we may give the definition of complete continuity. form of an Hermitean bounded operator
The
is said to be completely con-
K
tinuous if it converges to zero whenever the sequence
my
tends to
zero weakly, i.e. if
implies
(t;'K@v) 1 0
.
In fact, this stronger version of property imp ies property (2)., by virtue of the principle of uniform boundedness (due to Hellinger and Toeplitz); but we also do not need this fact. 150
We then formulate If the form of the (Hermitean bounded) operator K is
Theorem 4.
almost finite-dimensional, it is completely continuous. Proof.
Let
41 v , 0.
To any
> 0 take vectors Z(1),...,Z(g) such
c
Then a
that inequality (25.1) 0 holds. IZ(Y),Ov12
<
E
for
v > v£. KlAv j
v = ve
can be found such that
By (25.1)0, therefore,
<
(g+C) E.
;
hence the statement is proved.
The converse of Theorem 4 also holds, as could be shown by deriving the statement of Theorem 3 directly from the complete continuity of the form of
K.
This converse could also be proved directly.
We find it preferable to take the property of being almost finitedimensional as the basic one.
A related property was already used by
Hellinger and Toeplitz, although they did not give this property a special name.
The notion of complete continuity is, however, very '-seful if one wants to e$hibit counterexamples. Remark.
If the form of the (Hermitean bounded) operator
K
is not
completely continuous its spectrum is not essentially discrete. This statement follows immediately from Theorem 4 by combining it with the converse of Theorem 3, which is implied by Theorem 1. We shall use this remark in Section 25 to show that certain operators do not have an essentially discrete spectrum. We should mention that there are several different formulacontinuity
tions of complete to a Hilbert space.
which are all equivalent when they refer
F. Riesz has given the following two striking
such formulations. 1.
The operator
K
is completly continuous if it transforms any
151
weakly convergent sequence into a strongly convergent one. 2.
The operator
K
is completely continuous if it transforms any
bounded subset of the space into a set whose closure is compact. (It is now customary to call an operator "compact" if it has the last property.)
Note that in the last formulation no reference to an inner product is made (this reference could also be eliminated from the first one by replacing the inner product by a bounded linear functional in It is thus understandable that
the definition of weak convergence).
these formulations are important in the theory of operators in spaces However, in the work presented
more general than the Hilbert space.
in these notes we shall have no occasion to employ these formulations.
26.
Completely Continuous Integral Operators Before showing that integral operators of a wide class are
completely continuous, we make a general remark which will be helpful in this connection.
We say a Hermitean operator is a number of vectors
is "of finite rank" if there
K
and numbers
Z1,...,Z'
kXX, = kA,X
,
for
a,1' = 1,...,g, such that (26.1)
kAA,(Z
Km =
,0)
Z
X,A'
Evidently, the range of such an operator is finite-dimensional. we say a Hermitean operator e >0 there is an operator (26.2)
is "almost of finite rank" if for every
K
of finite rank such that
K
jjKe-Kjj
Next
<
C
.
Here we have employed the "operator norm" of an operator.
If such an
estimate holds for any other norm, it certainly holds for the minimal norm.
Remark.
Using these definitions we make the If a Hermitean operator
K
152
is almost of finite rank its form
is almost finite-dimensional.
bp of the form (26.1);
KE
To prove this statement we let then we have
I0,KEPI
Iza',01
ikaa,l
2
< maxi
Ika A' a'
Inserting this into
I41,K(DI
17X,4I
<
a
we obtain the statement
I4,,KE4I + c10,01
(25.1) 0 (except for an irrelevau't factor).
We recall from Section 20 in Chapter IV that an integral operator
K
with the kernel
assigns the function
k(s,s')
KG(s) =
k(s,s')0(s')dr(s') 1
(s).
to the function
Here
is a non-decreasing (real) measure
r(s)
function.
Let us first assume the kernel
9 x 9
tinuous with reference to a partition which is the product of a partition I.e., let 9 ;
./,a
to be piecewise conof the
(s,s') -plane
9 of the s-axis with itself.
stand for the open and closed cells of the partition
then the partition 9 x 9 of the
product
k(s,s')
la X Q,.
The kernel
k(s,s')
(s,s') -plane is given by the
is piecewise constant if in
each such product cell it agrees with a function which is continuous in the closure of this cell.
For any piecewise continuous function port the function
K$(s)
4(s)
with bounded sup-
is defined and piecewise continuous.
Section 20, Chapter IVa these functions
were extended to a Hil-
4(s)
bert space with the unit form
11112
=
J
(s)12dr(s)
153
In
.
Suppose now the kernel
k(s,s')
has a finite bound, such as
the Hilbert-Schmidt bound or the Holmgren bound.
Then the operator
K
is bounded and can be extended to the whole Hilbert space of functions m(s).
Certainly the kernel
k
,
and hence the operator
K
,
is bounded
if this kernel has bounded support.
We now state An integral operator
Theorem 26.1.
K
having a piecewise continuous
kernel with boundedsupport is almost finite-dimensional and hence completely continuous.
Let us first assume that the kernel
is piecewise con-
k(s,s')
It can then be written in the form
stant and of bounded support.
knA,(s')nx(s)
k(s,s') _
which shows that the operator
K
is of finite rank.
the number of terms is finite since
k(s,s')
Note that here
was assumed of bounded
support.
Next we let the kernel boundedsupport.
be piecewise continuous and of
k(s,s')
Clearly, such a kernel can be approximated by kernels
which are piecewise constant with reference to appropriate subdivisions
of the partition 9x piecewise constant kernel - k(s,s')I
For every e
.
k6(s,s')
is so small that
see Section 20.
>
0
we therefore can find a
chosen such that
110-kIll <
e
and hence
maxs's,Ike(s,s') IIKe-KII
< e;
The remark made at the beginning of Section 26 then
gives the statement of the theorem.
This theorem immediately leads to the following Corollary.
An integral operator which can be uniformly approximated by
an integral operator whose kernel is piecewise continuous with bounded support is
almost
finite-dimensional and hence completely contin-
uous.
154
For, since
K
can be approximated by an operator
almost of finite rank such that
which is
K'
is arbitrarily small, it is
JIM-KII
itself almost of finite rank and hence the "remark" is again applicable.
To be sure, the class of integral operators thus covered is very large.
It does not only comprise integral operators wich contin-
uous kernels defined in a finite region; it comprises also certain integral operators in infinite regions whose kernels may have certain singularities; the criterion is whether or not these kernels can be approximated in norm by non-singular kernels in finite regions.
Also
infinite matrices are covered, since the measure function
is
r(s)
permitted to be constant except for a sequence of jumps.
For all the operators thus covered a spectral representation is furnished by the results of Section 24. For all such operators there exist (infinite, finite, or absent) sequences of orthonormal eigenvectors
S1 -1),S2(t2),...
with eigenvalues
K+1,K+2,...
which
In addition,
tend to zero if there are infinitely many of them. 0
0
K =
may be an eigenvalue of finite or infinite multiplicity. Of course, not all integral operators are covered by the class
described. There are integral operators by operators
K
K
which can be approximated
of finite rank in the strong sense and still are not
completely continuous.
It then follows that these operators cannot be
approximated uniformly by operators of finite rank.
A typical example is any integral operator h(s,
')
is generated by a single function
h(s)
H
whose kernel
in the form
h(s,s') = h(s-s') while the measure function is simply be piecewise continuous and such that
f Ih(s)Ids < -
155
r(s) = s.
We assume
h(s) to
then, evidently, the Holmgren norm is finite and hence the operator
K
is bounded. Let
be the unit step function of the interval
na(s)
with the kernel
Ha
Then, we maintain, the operator
Isl <.a.
ha(s,s') = na(s)h(s-s')na(s')
approximates the operator K
$.
Ham
Ha
The convergence bounded support since
vided
a
for a dense subspace of functions alll is clear since Ith < Ilhll1.
tends to
The boundedness of
is bounded independently of
Ha
ient to show (1) that the operator .a, and (2) that
To prove this, it is suffic-
strongly.
Ha. -
HO
HO
is evident for functions
Ha,(s) = na(s)H@(s)
is chosen sufficiently large.
continuous.
H
To show that Hilbert's property is violated we
need only exhibit a sequence of functions
weakly to zero while
(0v,HOv)
my
in
and for which
fJshs_s's'ds'ds Then we set
0v(s) = Cs-v) For these functions, clearly
IlmviI
t)
which converges
does not converge to zero.
To this end we take one particular function < a
is not completely
At this place it is advantageous to use Hilbert's defini-
tion of this notion.
Isl
of
for these functions pro-
Now, we can easily show that our operator
in
0
= 11.11
and
156
0
.
m(s)
with support
1P b(s)0v(5)d5 = f 1,b(s)4)(s-v)ds = 0 1
if b.
provided the function
v > a+b
$b
in
b
Since the space of these functions
lows that
converges weakly to zero.
0v
fQ
has support in
is dense in
Isl
rp, it fol-
On the other hand, setting
HO(s) = f h(s-s')O(s')ds' we find
Hev(s) =
1
h(s-s'-v)0(s')ds'
and
(10 v,H40v)
=
1
f
independently of
v.
f
0(s-v)h(s-v-s')0(s')ds'ds
J
m(s)h(s-s')0(s')ds'ds ¢ 0
Therefore
(0V,H¢v)
Thus it is shown that the operator
H
does not converge to zero.
is not completely continuous;
its spectrum is not essentially discrete.
27.
Maximum - Minimum Properties of Eigenvalues if the spectrum of an Hermitean operator
> 0
K
above a value
A
is discrete and not empty, so that it possesses a largest eigen-
value, this eigenvalue can be characterized as the maximum of the quadratic form
($,K0)
taken for all vectors
This fact is evident from the formulas n
(@,Ko) =
I
KcinaI2 + (Ix,KcX)
a=1
157
m
with
II0II - 1.
(p.@) =
In12 +
I
o=1
L
in conjunction with (41
)
1 1 < A(C ,(P I
given in Section 24.
For,
K1 taken as the largest eigenvalue we deduce from them the rela-
with tion
n
I (K1-Ko) In012 - (KA) I I4 i 12 < 0,
Kl(4r4)
a=2
Hence the statement
0 = n(1).
the inequality being assumed for follows.
The mth eigenvalue
Theorem 27.1. > K
n
Km in the sequence
K1.> K2 > ...
can be characterized as the maximum of the quadratic form
for all vectors
with norm 1 which are orthogonal to
0
R(1)
,...,
(m-1) R For, with such a vector.
the relation
@
n
Km(@,0)
< -
(Km-Ka)InQ12 - (Km_A)II4, AII2
o
o=m+ = 9(m)
holds, the equality being assumed for
It is an important fact that the mth eigenvalue can also be characterized as a minimum without reference to the m-1 first eigenvectors.
This fact is expressed by the
Theorem 27.2.
Suppose the eigenspace
bounded), operator
value of
K
has the dimension
K
>
n > m.
on the (Hermitean
0
Then the mth eigen-
is the minimum with respect to the choice of
X(1),...,X(m-1)
for vectors
%A' A
0
of the "maximum" of the quadratic form
m-1 vectors (t,K4) taken
X(1),...,X(m-1).
with the norm 1, orthogonal to
(By
"maximum" here we mean the "least upper bound" since we do not intend to prove that an actual maximum is assumed.)
Evidently, one can choose a vector if 0
spanned by.the
gonal to any chosen
m m-1
eigenvectors vectors
t = n1 n(1) + ... + nmR(m)
0(1) ,...,n
(m)
X(1},...,X(m-1) 158
,
which is orthosince the deter-
m-l
mination of such a vector involves the solution of equations for
unknowns.
m
for this vector is evi-
The value of the ratio > Km, the smallest of the eigenvalues
dently
homogeneous
K1,...,Km.
hence true of the "maximum" of this ratio for vectors Since this "maximum" equals Km for X(m-1) _ Q(m-1), as observed above, the value X(m-'1).
01X(1),...,
= Q(1),...,
X(1)
Km
The same i$
is indeed seen to
be the minimum of this "maximum".
The fact stated in Theorem 27.1 enables one to study the effect which a change of the operator
has on its eigenvalues.
K
or differ-
ential operators the corresponding fact was derived and widely employed by Courant.
Another, complementar' way of characterizing the mth eigenvalue should be mentioned. Theorem 27.3.
Suppose the eigenspace
bounded) operator value of
K
(0,K4')
has the dimension
K
A > 0, of the (Hermitean
n > m.
Then the mth eigen-
is the maximum with respect to the choice of
independent vectors form
Q'A,
=(l)
=(m)
taken for vectors
combinations of
H(1)
m
linearly
of the "minimum" of the quadratic with the norm 1, which are linear
0
. Z(m).
To establish this fact we observe that there is at.least one such combination this vector t
0 jO 0
we have
which is orthogonal to (4',K4')/(4',@)
true of the "minimum" of the ratio.
now, this minimum evidently equals
For
< Km.
For
The same is, therefore, E(1) _ (1) gy(m) _ (m)
Consequently,
Km.
Km is the max-
imum of the "minimum".
The maximum (minimum) property of positive (negative) eigenvalues can be used to derive the spectral resolution of almost finite-dimensional operators without relying on the general spectral theory of bounded operators developed in Chapter IV.
159
.
We first prove
the existence of a largest eigenvalue for an operator which is finitedimensional above a number Theorem 27.4.
A > 0
under a simple condition.
Suppose the form of the selfadjoint bounded operator
is (I) finite-dimensional above a number (II) that a vector (27.1)
K
> X(00,00)
.
possesses a largest eigenvalue
given as the maximum of the ratio To prove it, let and let
(27.2)
V
(
Furthermore, assume
(P 0 ¢ 0 exists such that
(00,K(b 0)
Then the operator
A > 0.
Kl
for
K1 > A.
be the least upper bound of this ratio
be a sequence with (4)v,K4,v) + K1
11$"11 = 1 for which as
V + OD .-
each of the g inner products (Z1,@V) z
0v
- such that
tends to a limit, where Z(1),...,
are the vectors figuring in the inequality (25.1)+ which ex-
presses the hypothesis (I) that A.
It is
m ' 0.
,From it we may select a subsequence - also denoted by
(g)
K
(4,KO)
be finite-dimensional above
Introducing the difference VV =
v
V
we may express this requirement by the relation (ZY,Ivu) + 0
as v,V + m
We maintain that the subsequence
0V
for y = 1,...,g .
so chosen is a Cauchy Sequence.
To prove this we first note that the quadratic form
= K1(0,$) - (,,K0 K1.
(0,(K1-K),P)
is non-negative, by virtue of the definition of
Hence we may employ the same identity which we have employed in
proving the projection theorem:
160
(4v-(P u),(K1-K) (4v-4P)
+ ((4v+4u),(K1-K)(4v+4u))
= 2(4v,(K1-K)4v) + 2(u,(K1-K)4 4v-4u = 4vu
With
we derive from it
(4vu,(K1-K)4vv)
< 2(4.(K1-K)(Dv) + 2((P 11,(K1-K)4U)
.
To this inequality we add the inequality (25.1)+ for
4vli,
which we may write in the form (4vu,(K-A)(D VL)
<
I 1Z(Y)vu12 Y=1
in obvious notation.
Thus we obtain
(K1-a)(4vµvp) <2(4v,(K1-K)4v) + 2(4u,(K1-K)(Pµ) + I IZ(Y),4vv12 Y=1
We maintain that all terms on the right-hand side tend to zero as -.
For the first two terms this follows from relation
v,v
(27.1), for
the last term from relation (27.2).
From this fact we may conclude II4v"II
since
K1 >
0
,
because of hypothesis II and the definition of
Thus we have proved that
K1.
is a Cauchy sequence.
4v
By virtue of the completeness of the Hilbert space there is a vector
R(1) such that 4v + a(1)
,
furthermore, I (n(lI I
=1
and
(Q(1),(K1-K)9(1)) = 0
.
161
From the last relation we may conclude (K1-K)
S2(1)
= 0
follows that
K1
is an eigenvalue and
0(1)
orthogonal to
only apply Theorem 27.3 to the space
K@
is in this space if
(Q(1) ,KO)
an eigenvector of
K.
existence of the second eigenvalue we need
To establish tht
serving that
Thus it
is non-negative.
K1-K
by Theorem 19.1, since the form of
=
R(1), ob-
is, as follows from
4)
(KS2(1) ,4') = K1(a(1) ,4))
Of course, this theorem is only applicable if hypothesis II holds also
in
C 1. Continuing in this fashion we may establish the existence of a*
non-increasing sequence of eigenvalues, as long as hypothesis.II remains valid.
This hypothesis ceases to be valid before the
For, if there were
step.
would be
>
A
g+1
eigenvalues
for any linear combination
>
(g+l)st
A the ratio
f # 0
of the first
g+1
eigenvectors, while such a combination exits which is orthogonal to the
ZY
and hence contradicts this statement by virtue of inequality
(25.1)_. If the form of the operator
K.
is almost finite-dimen-
sional one obtains in the manner described a sequence of non-increasing positive and non-decreasing negative eigenvalues which tend to zero if there are infinitely many of them.
The spectral representation des-
cribed in Section 23 can then be established.
We shall not describe
the rather obvious steps needed to complete this argument. I
162
CHAPTER VI
NON-BOUNDED OPERATORS
28.
Closure and Adjointness
Operators which are not bounded will not be defined in the whole Hilbert space
h, but only in a subspace of
"domain" of the operator and denoted by in a subspace of
ti
OA-
occur quite frequently.
ID, called the
Operators defined only Of course, integral
operators are naturally defined only in such subspaces of t.
But if
they are bounded - and those that we have considered are bounded they can be extended to the whole space
ff
.
Differential operators
also are defined only in subspaces - as was already indicated in Chapter I; but they are always strictly non-bounded, as will be shown in Chapter VII.
Of course one will naturally try to extend the domain of
a non-bounded operator as far as possible.
In this section we shall discuss procedures for extending an operator defined in a subspace of to be defined.
f)
as far as possible - in a sense
If the operator is actually non-bounded these extended
domains are actually subspaces of the Hilbert space.*
In these proce-
dures we shall use the notions of closure and of "formal" and "strict" adjointness.
These notions are importaAt aside from their role in
extension procedures.
In particular, we mention that "strict self-
adjointness" of an operator is the property which makes its spectral representation possible, as we shall show in Section 30. In the discussions of the present and some later sections it is advisable to extend the notion of "operator" slightly by allowing
It should be mentioned that a Hermitean operator which is applicable on every vector in a Hilbert space is automatically bounded, provided it is closed (see below). This remarkable fact represents one of the deep results in the theory of operators. Still we shall not have occasion to use it; for, in order to exhibit specific operators that are applicable on-all of Hilbert space, we shall always presuppose the existence of a bounds see Section 20. 163
the range of an operator
A
Cr, which need
to lie in a Hilbert space
not be the same as the original space
43.
First we introduce the notion of "closedness" and "closure".
to C
Then we form the space
S2.
t) A
defined in a domain
A
To this end we start out with an operator
of all vectors
in
(Do
that can be approximated (strongly) by a sequence of vectors rp
A
such that the sequence
tV
in
converges (strongly) to a vector
Atv
t1
in A = N, i.e. if each such vector
If 4)
1
= Atg, we say that the space
to
is in
'A' and if
is'closed.
t)A
$A is not closed we may try to extend the operator
If
At0 That can be done if
01
$1
depends uniquely and linearly on the vector
t0, independently of the approximating sequence. the operator
A
A
the closure of
then the domain of
A.
to
by setting
K defined in -.:VA
an operator
A
in
K
Clearly,
i
In that case we call
The space
tA'
closed in
A =
A
is
tA.
Below (in Theorem 28.3) we shall give a condition under which a closure exists.
Before doing so we must describe the notions of
adjointness.
In accordance with the definition given in Section 9, we say that an operator to
A
in
$A
At
defined in a domain
1W
if the relation
(At'V,$) _ (Y',AO)
(t)
holds for all
t
in
0A,
in
W
r t. A We furthermore say that the operator
domain
is "formally adjoint"
A
Q A*
is the "strict" adjoint of
firstly formally adjoint to
A
in
164
A*
defined in the .Sp
A
in
if it is
(*)1
holds for
(A*'Y,4) = (W,AI')
in
4'
Y'
and in addition has the following "second property": vectors
C 'l
in
`Y
in
('Y1,d))
holds for all
in
@
in
QA*,
whenever two
are such that the relation
t
= ('Y,A4')
I'D A, the vector
a A*
is in
W
and
A*'P =
Ti. This second property may be expressed by saying that whenever a vector
together with a vector
'V
would hold if 'Y
is in
were in
'Y'
and
41 A*
and
(1A*
satisfies the relation that
4'1
'Y1
were equal to A*P, then
Note that the domain of
'Y1 = A*'Y.
A*
must be
taken sufficiently large for this second property to obtain, while it might be necessary to narrow down this domain to insure that the first property, (*)l, obtains.
Without giving the obvious proof, we note the fact that the strict adjoint
of an operator
A*
A
is closed in its domain
rA*.
This fact illustrates the "wideness" of this domain. If the domain of both full spaces
and
*
A
and its formal adjoint
91, the operator
At
in
t
9)
whence A*'Y = 'P1.
(Tilt) = (A*'Y,4')
for
This remark shows in particular that
bounded formally adjoint operators defined in joint.
are the
is strictly adjoint to* A.
For, relation (*)2 then leads to the relation all
At
are strictly ad-
The qualification "formal" or "strict" can therefore be omitted
for bounded operators.
The following theorem gives a simple condition under which the operator
A
has a strict adjoint.
Theorem 28.1. in
A.
Then
Suppose the domain A
4)
A
of the operator
possesses a strict adjoint.
165
A
is dense
Let
Proof.
be a vector in
''
to which there is a
ry
such that relation (*)2 holds for all space formed by all these vectors 0 for all
we conclude
Suppose
4'1.
Call
Consequently, the assignment of
'Y1 = 0.
CA*.
the
G4A*
Y' = 0, then
(4'1,f)
to
T1
is
4'
A*, with the
unique and hence constitutes a linear operator, called domain
C)
5)A; and since this space was assumed to be dense
in
$
C)A.
in
$
in
4'1
By its very definition this operator has the two prop-
erties characterizing it as the desired strict adjoint. We mention that the converse also holds: A
a strict adjoint,
'
is dense in
.
if
A
in
IA has
We omit the easy proof.
We can use Theorem 28.1 for an extension of the operator To this end we apply this theorem to a formal adjoint of Theorem 28.2.
adjoint
At*
A
in
in a domain
A.
has a formal adjoint
Then there exists to
with a dense domain
At
A
Suppose the operator
At
which is an extension of
t*
A.
a strict
A
in
f
in
A 'PA.
A*
In other words,
A
C
i%
A
t*
The existence of the operator
and
At*$ = At
for
At*, the "strict adjoint of the
formal adjoint", follows from Theorem 29.1.
The second property, ex-
pressing this strict 4djointness, now assumes the following form. Relatign
(*)2
(AtY',$) - ('Y1m
Nt* and
for all
Y'
in
a
implies
0 is in
Q1 = At
does satisfy 4*)2 because of relation M. Hence
OA * and
At*$ = At.
At*.t = @1.
Now, any
0
in
$A with 0
is in
Thus the theorem is proved.
Note that the operator
At*
is a closed extension of
A
(in
view of the remark made above that any strict adjoint is closed), but need not be its closure.
It is remarkable that the closure exists
under the same circumstances under which the operator
166
At*
exists.
Suppose the operator
Theorem 28.3.
A
)A
in
has a formal adjoint
At; then it possesses a closure,
At with a dense domain
A
in
'D X.
We define
as the space of all
A
approximated by vectors tor
These vectors
pl.
by (t)
4
A
in
t)A
i _ 1v
this relation holds for
C
t 1i
such that
in
h which can be
Acv tends to a vec-
certainly satisfy condition (*)2 since
@, @1
from Theorem 28.2 that words,
v
?
admits
At,
and
0l = A@".
and that
It now follows
Therefore, the assignment of
hAt*.
constitutes an operator,
A
Clearly this operator
In other
At*O _ @l.
A, the restriction of
to
At*
0
in
to
is the desired closure.
Under the hypotheses of Theorems 28.2,3 we thus can form extenA
sions of the operator
in
to closed operators in two ways:
9A
A, also called "minimal" or "strong" extension (since
the extension
strong convergence was used in its formation), and the extension
At*,
also called "'maximal" or "weak" extension (since the statement that a At*
vector admits
is weaker-than the statement that it admits
A).
The relationship between the corresponding domains is indicated by
0A C
A
C
f)At* C
Naturally the question arises whether or not the strong and the
weak extensions agree. That this need not be the case will be shown in connection with differential operators. Note that the weak extension formal adjoint
At.
domain of
A t*
is possibly reduced.
domain of
At
it exists).
At'
depends on the choice of the
If the domain of this adjoint is enlarged, the The maximal enlargement of the
is reached by taking the strict adjoint itself (assuming The adjoint
weak extension of strong extension of
A**
of
A*
furnishes the strongest possible
A; it is a remarkable fact that it agrees with the A.
167
Suppose the operator
Theorem 28.4. joint
A*
joint
A**
with a domain
strongest weak extension
A**
A
in
dense.in
Q A*
A
is the closure
A*
of
A
of
A
Then the strict ad-
in
That is, the
g) A.
K
is the strong extension
A
of
Ci.
has a strict ad-
of
A.
The formulation of this theorem and its proof (as all notions developed in this and many other sections) are due to von Neumann. The proof is based on the notion of "graph" of an operator. The space of pairs forms a Hilbert space
of vectors in
ro + (
9
graph the subspace of in
A.
A = A
If
an operator
A'
A inA one may assign as its
is closed, then so is obviously the graph.
acting in a subspace
The statement that
of
is a formal adjoint to
At
At
in
is the strict adjoint of of those vectors in
t) A.
G A'.
A
in
with
A
in
T
can then
is perpendicular
A
Moreover, the operator A
To
we assign as its
(2
{-A'4,T}
simply be described by saying that the graph of to the graph of
with
formed by the pairs
graph the subspace of all pairs of the pairs QRA,.
again
Q1
with respect to the unit form
To every operator
1101I2 +
and
A*
in
rOA*
if its graph consists exactly
which are perpendicular to the graph of
A
By virtue of the projection theorem the orthogonal space
of the orthogonal space
of a subspace is exactly this subspace if
it was closed to begin with.
In other words, the contention
A**
is proved.
An operator All be called "strictly self adjoint" in a dense domain if it is equal to its own strict adjoint: FO
At
A* = A.
12
A*
The question then will arise how one can tell whether or
not a formally self-adjoint operator is strictly self-adjoint, or canbe extended to a strictly self-adjoint operator.
Instead of trying to
answer this question directly, we shall show that strictly self-adjoint
operators can be built up with the aid of a pair of strictly adjoint 168
In doing this it is convenient to make use of some theorems
operators.
on non-negative forms.
Closed Forms
Z9.
A bilinear form space
of
YP1
'
defined for vectors
(4',4)1
is said to be "Hermitean" if it has the properties 4, anti-linear in
(7.1,2) of an inner product (linear in
said to be non-negative if it has property (7.3),
D
in
!Dl.
the space
Then
4'); it is
(4,4)1 > 0
it is said to be a "sub-inner-product" if
holds in
in a sub-
4,4'
for
(4,4)1 >
(4, 4)
will be called a "sub-norm".
[(4,4)111/2
4
If
is complete with respect to this sub-norm the sub-
norm, as well as the sub-inner-product, will be called closed. In the present section we shall assume the sub-norm to be closed.
In Section 31 we shall give various conditions under which the
sub-norm can be extended to a larger domain so that is closed there. At present we prove the fundamental Theorem 29.1.
111W1
Let
*1 which is dense in
be a closed sub-norm defined in a space Then there exists a sub-space
t .
dense in
P1 with respect to defined such that
29.1)
in
4I
Moreover,
F
rp l,
42
t2.
( 29.2)
The range of F is all of
C) 2.
4
in
B
with bound
1
defined in
The relations
FB(P = 4
hold for all
in
has a bounded inverse
with the range
in
in which an operator F can be
(4i1F42) = (41'42)1
for all
F
$2
$,
,
42
BF42 = 42
in
9) 2.
t), are strictly self-adjoint.
169
The operators
B
in
t)
and
This theorem will be one of our major tools in establishing To prove this theorem we first note
strictly self-adjoint operators.
that the bilinear form (4Pis bounded in
with respect to
t)l
I III; in fact,
I4'i.tll
_ II0iII
110111 _ 114-i111 11 1mlll
.
Consequently, according to the corollary of Theorem 18.1, there is an operator
B
defined in
with the
S?l
for all
(pl'B(P 1)1
('D 1'01)
Ill -bound
II
01'41
1, such that
!0
`n
1
Moreover,
IIB@1112
whence
1IB(P 1112 = (B4>1,41)
IIB4III
<
That is,
1I4>111.
respect to the original norm. dense in
C), the operator
operator with the bound 11B4>l11
completeness of
for any
0
-V l
01 was assumed to be as an and
IIB4l112 < IIB$111 114111
IIB.1111 < 110111
with respect to
II
111
and conclude from the that
BO
lies in
(1)i0) _ (01'BO) 1
therefore holds for all
in
4
@i
in
$1.
Moreover, for all
t
in
(29 .4)
IIB4111
Suppose all
with
$. The forn0ala
in
(29.3)
4
1
can be extended to all of
From
we deduce
Ilmlll
<
has even the bound
B
Since the space B
1.
IIB4111 II$lII
<
Oi
in
tl
< IIBmlil 1 114111
B m = 0.
Then formula
and hence
Consequently, the operator
0 = 0 B
since
gives
(4i, 4>)
=0
is dense in
for )
..
has a linear inverse, which we denote by 170
F, defined in the range of B, which we denote by
all
in
0'
for all
0
in
40 1
such that
!D 1
($1,B@')1 = 0
That is to say, by (29.3) we would have
C .
in
Q'
For, else
$l with respect to
maintain that it is dense in there would be a vector
we now
is contained in
$2
We know that this space
$2.
and hence
t)
by definition every
in
4
2
(4,1,@') _
(P 1 = 0.
follows from the definition of
FBO = @
The relation
for
is of the-form
C2
B@
ml
F; since
we have
BFm2 = BFB' = BO = 02.
The formal self-adjointness of the operator from the relation Vl:
Since
$ 1
adjoint in $ .
(B$
1)
(B41'B41)1
Since
by interchanging
it follows that
$
is dense in
B
1 follows
in
B
01
and
,
is formally self-
is bounded, it is strictly self -adjoint
B
there, according to a remark made in Section 28. Relation
for all
(0'FO2) _ (F0Z,42) self-adjoint. Suppose
O2
in
in
00
t,$'
$2.
in
4V2,@2
and
02
00
for all
for all
(B40,@) = (',B4))
C)
$
in
Hence
implies
is formally
F
are such that
41 i "2)
_
t2; then
(BV,4 0) _ (0'4) 2) for all that
0'
in
From the strict self-adjointness of B we conclude
$ .
BO0 = 02; hence
self-adjointness of
F
02
is in
C)2
and
F@2 = (D0.
Thus the strict
is proved.
In employing this theorem we shall use Remark 1. 11
Let the operator
be closed in its domain.
A
with domain
$ A C $
Then the sub-inner-product
(40 ',0)1 = (M',A(P) + (4',(P) 171
and range in
is closed in
A form
* A.,
is a sub-inner-product for a suitable
c(AM',A$)
if the form
c > 0; i.e., if
cIIA0II
itself is closed
(A4',AO)
A
In that case the operator
1101 1.
>
is
called inverse-bounded. Clearly, if norm
111,, it,is a Cauchy sequence with respect to the norm
11
of the space in
that
3'.
A
is in
0
Consequently, there are vectors tv + 0, Atv
such that
12
is a Cauchy sequence with respect to the sub-
0v
completeness of the space
The closedness of
01.
At
and
0
in
$1,
A
11
11
III
now implies
01; and this fact just expresses the with respect to the subnorm.
tA
We may now formulate Let the operator
Theorem 29.2.
be dense in
A
let
adjoint of
A
for which
At
in
4>
be closed in its domain A*
Q A*C
in
Then the subspace
.
lies in
is the full space
The operator
.
A'
$
The operator
in
C)A
A*A
id
Moreover, the range of the operator
C)A*A.
*
0
and
* A
be the strict
Wr
!DA*A of all
is dense in
QA*
has a bounded in-
A*A+l
B = (A*A+1)-1, and the range of B is
verse (with bound 1)., say $A*A,
Let
t .
strictly self-adjoint in A*A+l
A
i.e.:
(A*A+1)B = 1
B(A*A+l) = 1
,
in
A*A
In view of Remark 1 made above it is clear that Theorem 29.1 applies. in
A bounded selfadjoint operator
FPF
'A such that relation
(0',0) = (A$',AB(D) + ($',B$)
(*)
holds for all
in
$
adjoint inverse
that
exists with range
B
F
!0,
of
B
$'
$
in
A'
a strict adjoint
A*
in
6
is defined with range
' F !DA*A' Here we make use of the assumption that r
A*'
a strict self-
F
In
* .
A
in
We must show
C)A
possesses
From the second property of strict 172
adjointness with
'A*
'' is in
if
is bounded with bound
C
I1B4112
Suppose now the domain
to
A*
*
is in
B
A
More-
1; for we have
=
so that
the operator
IIABpI12
11c4.112
11
whence
-1
plays a role; it is defined since the range of over,
is
A*A we
is in
@'
8 = (A*A+1)-1
C = A (A*A+1)
B4
The statement then follows.
*F .
In addition to the operator
we conclude
01
(d'-B(A*A+1)0',$) = 0
may deduce from (*) the relation B(A*A+1)4)' _ @'; hence
and
m
A*AB$ _ (1-B)O; i.e.,
and that
(A*A+1)B@ _ 0. Conversely,
and
IVA*A
instead of
O-BO
and
AB4 is in
from (*) that in
ABA
A*- and an operator
A**A*
of the operator
QA*
has an adjoint
11x112
Then Theorem 29.2 can be applied
A**.
can be established.
V. Neumann's Theorem 28.4 we have
is dense in
A*
A** = A, hence
Now by virtue of A**A* = AA*.
With
this in mind we formulate
each other in domains operator
(AA*+1)
£'A,
Q1
dense in
A*
A defined in
are strictly adjoint to
A, A*
Suppose the operators
Theorem 29.3.
equals
.0 A
(AA*+1)-lA - A(A*A+1)-1 = C
To prove it let A*At
t-401.
A4-A01, or by
m
be in
It follows that
is in
and
(; .
Then the
C:
A
in
and set
t A
A*AG1
£
t1 = Bt
$ A
and
so that
AA*A01 =
whence the statement after division
(AA*+l)A(A*A+1)-10 = A$
(A*A+1), i.e. after multiplication by
B = (A*A+1)-l.
Theorems 29..2 and 29.3 will be used in the next sections.
Theorem 29 .2 and other applications of Theorem 29.1 will be used in our treatment of differential operators.
173
Statements analogous to Theorems 29.2 and 29..3 could be
proved - by essentially the same arguments - in the case in which
eIIAsH > 11011.
We do not carry this out.
The case in which the closed operator
this case the form a A2
Q =
At - A, and
adjoint,
(At,A$)
of the domain
a A.
29.2 yields the inverse of
is formally self-
A
*, is of particular importance. can be written as
in a subspace
(41,A20)
One may wonder whether or not Theorem A2+1
(or of
A2)
in this case.
need not be so; for it may happen that the formal adjoint not the strict adjoint
A*
In
of
That
At = A
In fact, the strict adjoint
A..
is A*,
Examples of
need not even be formally self-adjoint in such a case.
this occurrence will be given in connection with differential operators in a later section.
The question naturally arises whether or not a closed formally self-adjoint operator can be extended to a strictly self-adjoint one. The domain of this extension would then be a proper subspace of the A*, unless
domain of
A*
itself is strictly self-adjoint.
Such an extension can be constructed if the operator inverse-bounded.
the operator 'A
E
but for the operator E
has been established.
E = A2. as
The extension of
A = E1/2
We shall formulate and prove
It should be mentioned that, once an inverse and hence the inverse
A
BA
of
A
once the functional
the pertinent extension theorem at the end of Section
tion of
is
This extension will at first be carried out not for
can then be obtained from that of calculus for
A
3 1.
of
B
A2,
A, has been found, the spectral resolu-
can be derived directly from those of
B
employing the theory of Section 30 to be developed.
and
BA, without
We do not carry
this out.
30.
Spectral Resolution of Self-Adjoint Operators
The operator A
defined in.a dense subspace
174
$ A
of
is
strictly self-adjoint, or "self-adjoint" for short, if firstly relation
holds for all
($',AO) = (AM',Q)
(1)
in
and if second a vector
T)
in
(P,('
with M = 01
A
belongs to
g) A
whenever the relation
(2)
(01 ,4 1) = (A01,4')
holds for all
in
4)
t A
.
N We shall establish a functional calculus for such operators We observe that
using a variant of the original method of Neumann.
A with
the hypothesis of Theorem 29.2 is satisfied for this operator A* = A
Q =
and
adjoint operator sists of all
@
B = (A2+1)-1 in
t)A
C = A(A2+1)-1
operator
Consequently there exists a bounded self-
S .
with bound
for which
A$
whose range
1
is in
FP A2
con-
Also, the
A.
and bounded with the bound
is defined in
1.
We now maintain that these operators
B
and
C
commute:
BC = CB This follows immediately from Theorem 29.3, according to which
(A2+1) -1A (A2+1)- 1 = A (A2+1) -1(A2+1)- 1
.
Consequently (as was already mentioned in the last section, 23, of Chapter IV), all piecewise continuous functions of With the aid of such functions of of
B
and
C
B
and
C
commute.
we shall define functions
A.
First we restrict ourselves to piecewise continuous bounded functions suitable
f(a)
of
a
which are continuous for
a > 0 and approach definite limits as
tat
> a
a
with a Every such
function can be written as the sum of an even and an odd function of a.
Every evep function of a may be regarded as a function 175
g(9)
of
6= while each odd function of
a
is of the form
Y =
0(Y)h(y), where
a
a2+1 and
0(y) = 1 for
note that
82(1)
= 1 for all
We therefore can write
y > 0,
y < 0
= -1 for
;
Y.
f(a)
uniquely in the form
f(a) = g(R) + 8(y)h(R) where
g(B)
± h(6) = f(± 4-1 - 1) for
0< B<1
for for
B>1
f(0) 0
By virtue of the assumptions made on
f(A)
Since
8(y)
since
is real the operator
are self-adjoint. f(a)
g
and
h
by
real, the same is true of f(A)
.
We can therefore define the
f(A) = g(B) + 0(C)h(B)
(30.1) A
<0
f(a), the functions
are piecewise continuous and bounded. operator
a
g
and
is self-adjoint; if
8(C) h.
.
Hence
g(B), h(B) and
Note that the operator
is.
f(a)
f(A)
is bounded
_
In order to show that the rules of operational calculus are obeyed, let f1(a)f2(a) = f(a)
176
;
is
then, evidently,
g(B) = gl(B)g2(B) + hl(B)h2(B) h(B) = gl(B)h2(B) + hl(B)g2(B)
From the fact that functions of
and
B
commute we conclude
C
g(B) = g1(B)g2(B) + h1(B)h2(B)
h(B) = gl(B)h2(B) + hl(B)g2(B) and hence
f1(A)f2(A) = f(A)
.
we need not prove the rather obvious addition rule.
Next we must extend the definition of
To any such function
tinuous not necessarily bounded functions. we introduce the
for
Suppose now that the vector a
Jul
(D
for
a'
> a.
IIfa(A)(DII
Cauchy sequence. is linear.
Clearly,
f(A)'.
Hence
.
Ilfa(A)411I
Ifa'(a)-fa(a)J2
Ifa,(A)-fa(A)12 = fa,(A)-fa(A)
a
f(A)
Thus,
-.
-
Obviously, the operator
and
of all vectors
f(A)
$ F0.
is self-adjoint in
The formal self-adjointness of
177
f(A)
fa(A)(D
is a
defined by the limit
f(A)
defined in this manner is evidently dense in shall show, the operator
> a
1Ifa,(A)4, I12 - IIfa(A)1P 112 _ 0
is monotonic as
The space
Jul
faWO converges to a
. Then, we maintain,
II(fa,(A)-fa(A))IDI12 = since
= 0 for
< a,
is such that the norms
limit vector, which we denote by fa,(a)-fa(a)
f(a)
function
fa(a) = f(a)
remain bounded as
to piecewise con-
f(A)
for which
f(A)
Moreover, as we uf(A)'
follows from the fact
is
that the approximating operators have this property. t
strict self-adjointness, let
and
(v ,@1) = (f(A)4
holds for all Pa = na(A)O'
where
na(a)
Now, by definition of
f(A)
be such that
,0)
is the unit step function of
Iai
< a.
f(A)na(A) = fa(A); hence the
we have
0a' instead of
above relation for
10
Then this relation holds for all
'f(A).
in
0'
in
1l
To prove the
(na(A)0',0l) _
V, i.e.
(fa(A)'',01)' yields
OV'na(A)ml) = ((',fa(A)41) for all V
in
FPf(A), hence for all
fa(A) " = na(A) ml By definition, then,
is in
$
Note that the operator of
A
-+
01
0'
Therefore
C).
as and
V P(A)
A
in
f(A)(P _ 01.
itself is one of those functions
now defined; one readil)r verifies that the new operator
its domain agree with the old one.
A
and
Thus we have established all
essential features of the functional calculus for the operator particular, the spectral projectors
niA)
A.
In
and the eigenspaces into
which they project are defined.
We add a few remarks about the spectral representation of the self-adjoint operator
A.
We confine ourselves to considering a
space CO0) which is generated from a vector bounded piecewise continuous functions
in
0o
,
by all
f(A); i.e., which is the clo-
sure of the space of all" f(A)(P o; see Section 22.
Let / be any bounded (a).
c*-interval with the unit step function
Then we may define a measure function
IIn(A)moII2 178
r(a)
such that
Let
in obvious notation.
valued function and
be any piecewise continuous complex-
f(a)
fa(a)
its cut-off function.
From the spectral
representation of bounded operators, Section 22, we deduce the relations
I f (a) 12dr (a)
fa (A) 4) U 112 Ia
and
I
I fa. (A) 0o - fa(A)
f
I2 =
I f (a) 12dr (a)
a<jaI
fp
is in the domain of
f(A)
then is exactly
given by the condition
JIf(a)I2dr(a)
(*)
The vectors
t
<
of the space Q(0 ) are now represented by the (ideal)
functions
f(a)
obtained by closure from the piecewise continuous
functions
f(a)
satisfying (*)
The functions
f(a)
which together with the function
satisfy condition (*)
which also the vector
clearly represent those vectors t1 = f1(A)md = Af(a)00
f(a)
A
f(a)
0 = f(A)0
belongs to *(it
other words, these functions represent the domain of that the domain of the operator
f1(a)
A.
0
).
for In
Thus we see
is represented by those-functions
for which
f a2If(a)j2dr(a)
< -
.
This result implies the fact that the domain of an unbounded 179
self-adjoint operator cannot be the whole Hilbert space. a > a0
is constant for
hand, the carrier of functions
a0.
If, on the other
is not bounded there evidently exist
r(a)
with
f(a)
J
r(a)
for sufficiently large a
is bounded with the bound
A
the operator
a < -a0
and for
For, if
If(a) I2dr(a) < - but J a21f(a)12 dr(a)
_
Clearly, i.e., there are vectors in the Hilbert space which are not in the domain of the operator
31
A.
Closeable Forms
,
In Section 29 we introduced the notion of closedness of a bilinear form.
We shall now slightly extend this notion and call a
(Heriitean) bilinear non-negative form 1
if the relations
obvious notation - and is in
!01
and
closed in its domain
I10v- Oj j' - 0
IIi -sIt - 0
for
Ov,0u
for a
in
'
If the form
II41v-cP11' - 0.
$l - in
in
imply that
0
is closed in
this space is obviously complete with respect to the subnorm given by
(4',4)) 1 = (O,4')' + (O,$) in case there is a constant
c-> 0
>
we may take
c(',4)'
.
such that
(4,,4)
itself as subnorm
(4,4)1.
If the domain is not complete with respect to the subnorm it may be possible to extend it to one that is complete. call a form
(,)'
defined in a domain
i' "closeable" if the rela-
tions 0
and
0'*'
0
for
4v, 011 V
180
We
in
imply + 0
If a form in space PV
to which there is a sequence
in
P
and
0
0
11
11
.
(P1,0V)'
By virtue of the first relation, the inner product
We must show that this limit depends only on the
tends to a limit. limits
Pi11 ,
of the approximating sequences
Pl 12
P2
and not on
To prove this we introduce differ-
the choice of these sequences. _V
_P
ences
Pi, P2
110
+ 0
0
of two such pairs of sequences and observe that
and
+ 0
P2
would imply
IIPiII' + 0
and
(PP2
by virtue of the closeability assumption, so that also
depends only on
P1
and
closedness of the form quence.
to the
FP'
such that
t'
in
is closeable we may extend
!t)'
of all those
1
as
(V 2
and moreover is bilinear.
(P,P)'
in
Iv1
Clearly, the
is now an immediate conse-
We formulate this result simply in A closeable form can be extended to a closed form.
Theorem 31.1.
t'
One might have extended the space
to a space
%)
1
complete with respect to the associated subnorm by simply adjoining ideal elements; but then it would have been necessary to identify these ideal elements with vectors in
possible if there existed a sequence IIPv-Pull, - 0
and
114V11 + 0
This would not have been
C).
PV
but not
in
for which
110vIll - 0.
ideal element would correspond to the vector being 0 in
C)'
P - 0
in
For then the $>
without
To exclude this occurrence, the "closeability"
t 1.
requirement was introduced.
Clearly, the form if the operator
A
in
(P',P)' = (A@',AP) LI
A
is closeable in
possesses a closure. 181
The following
0 A
theorem describes, another important class of closeable operators. Theorem 31.2.
Let the non-negative Hermitean quadratic form
(t,f)'
be the least upper bound of a set of bounded forms in a domain Then the form
To prove this statement find for every c > 0 11ty-$P1H' < e
that
where the superscript
Now, let
inition of
11
11'
µ
ve
such
jj@v-tullb < E
tend to infinity. 11
and the boundedness of
0
II4VHH
,I0v-4.u(`b + 11,vllb
a
refers to any of the bounded forms entering
b
the definition of hypothesis
Then also
p > v > vE.
for
'
is closeable.
t'
in
f
as
Hence
u
we then infer
11
11b
vl`b < E.
11
V11' < E.
From the
we conclude From the def-
Thd% Theorem 31.2
is proved.
In employing Theorem 31.2 we shall make use of the obvious Lemma.
If t3* non-negative forms
a common domain then so is
and
(,)"
are closeable in
(p) s + ( ,) ".
It follows from this lemma that the form (A@',Ad) + (4',f)'
is closeable in a domain there and if the operator
,'
if the form
A defined in
(1',O)' Sp'
is closeable
possesses a closure.
It furthermore follows that the sub-inner-product
(@',0)1 - W ,$) _+ (At',A@) + (0',0)' in
0
can be extended to a closed sub-unit-form.
By virtue of
Theorem 29.1 this sub-unit-form is associated with a strictly selfadjoint operator.
This procedure of defining self-adjoint operators will prove to be very helpful in establishing the strictly self-adjoint character of various differential operators.
182
Finally, we shall prove the extension theorem mentioned at the end of Section 29.
Extension Theorem* for inverse bounded self-adjoint operators. formally self-adjoint operator
defined in a domain
and having a lower bound
dense in
domain
E
J
T)
= 2E
'1)
c, admits an extension in a
in which it has the same lower bound as
'I)
Any
and is
E
strictly self-adjoint.
To prove the statement we assume the operator
E
satisfies
the inequality'
c(m,E$)
>
(0,0)
and show that the form
for all
in
0E
with a constant
c > 0
is closeable.
(0,Em)
To this end we consiaer a sequence
{,v)
of vectors in
for which
((QU-4v)
0
as
µ,v -
while at the same time
11011 -0 as After writing
(@v,Ew)
+ (Emv,mu
and then estimating V (0 ,Emu)
< (v,E,v)1/2(("-0),E(O"-0 u))1/2 + IIE."II 1I-`'I1
we can choose
v
such that
Proved by the author in 1
,
.
183
zE
E(4v-4u))
and then choose
for
E2
<
u > v
such that
u > b
E2
We are then led to the inequality
< (4v,E(Pv)1"2E + E2
(4v,E(Pv)
,
which implies < 3E2
(0v,E4v)
That is to say,
(4v,E$v)
0
-
v -
as
Thus we have proved the closeability of the form
c(4,E4),
and it follows from Theorem 31.1 that this form can be extended to a closed form (4,4)
D ZE
z1
in a domain
(4,4)1
for which
(4,4)1
>
holds.
Denoting the extension of
c(4',E4)
(4',4)1, we can apply
by
It yields the existence of a self-adjoint operator
Theorem 31.2.
in a domain
2
D Z 1
F
such that
(41,F42) = (41'42)1
holds for all
4i
in
zl'
42
z 2.
in
Hence
(4i,F42) = c(E$i,42)
41 in
holds for all adjointness of and
cE4i.= F4i.
extension of
in
F
E
Z E
Z 2
and all
z F.
in
Z2.
it then follows that c-1F
That is to say, in
42
Clearly
184
in
2
From the self41
is in
Z2
is a self-adjoint
holds for all
c(',c-1F(D)
= (P, 4)) 1 > (t,@)
in
Thus our statement has been proved.
$? Z.
185
CHAPTER VII
DIFFERENTIAL OPERATORS
32.
Regular Differential Operators D
In Section 10 we introduced the differential operator to
acting on functions of a real variable running from which have a piecewise continuous derivative.
D
trans-
In this section we shall show that this operator
D
can be
$ (x)
into
Dm (x) =
4(x)
J-X
OW.
extended into a dense subspace
of
SD
where it is closed and
g)
where it is strictly adjoint to the operator iD
The independent var-
so that the operator
iable will now be denoted by
forms
+w
is strictly self-adjoint.
-D, so that the operator
In later sections we shall consider
various modifications: we shall define various operators of first and second order, defirid in finite or infinite domains of one or more variables and shall prove that these operators are strictly selfadjoint.
The space of functions ivative - for which the operator denoted by
D
was defined so far - will be
611; the space of those functions in
square-integrable and for which denoted by
with a piecewise continuous der-
$(x)
DO(x)
'a'.
which are
is square-integrable will be
Q)'; the space of functions in
port will be denoted by
Q i
Z'
with bounded sup-
With reference to the unit form
(0,0) = j I0(x)I2d:
the operators
D
in
Z'
and
-D
in
'3 )'
are formally adjoint to
each other since evidently
(Tf DO'(x) + D-TY O'(x))dx = 0
(0,Dm') + (D$,$') = J
186
for
m
')'
in
and
in' Z1.
Q'
Clearly; the domain the Hilbert space
and hence also
Sp', is dense in
since this space was defined by extension from
C)
the space of piecewise continuous functions with bounded support and 'ti
this space in turn obviously contains
We therefore know
densely.
from Theorem 28.3 (Chapter VI, Section 28) that the operators -D and D
Z'
and
inli '
and
by
possess closures in extended domains which we denote respectively; the closed extensibn of
domains will also be denoted by -D
Z and
in
D.
in these
Of course, the operators
D
and
i are formally adjoint.
We now maintain that these two spaces same.
D
Z
and
In other words, in the definition of the space
i are the SD
by exten-
sion Of a space of smooth functions it is no restriction to require that these functions have bounded support. Theorem 32.1. Proof.
Denote by
na(x)
na(x)
and
;a(x)
the functions defined by
1
fxj < a
=0
Ixj > a
5
Ca(X) , = a+l-lxl
;
a < (xj < a+l
,
a+l < jxj.
Let
(x)
be in
1i'; then a = ;a'
is in
D;a = 4aD + (D;a) . Now,
D;a < 1 - na' 1 - ;a < 1 - na.
187
Hence
'D'.
Note
IIDoa - Doll < IIVOa - CaDoll + II(1 - a)Doll
<
I
I (1-na) o I I+
and at the same time
I
I (1-r1a) D¢
11(l-ca)4II
I
II(l-T) a)oll
<
From the definition of the space closure of
21 C
D
as
0
I
-
0.
as the domain of the
ID
it then follows that
in
a
¢
is in
ID .
Hence
is proved.
ID
Since the inclusion
'D
SD C
is obvious, the statement of
Theorem 32.1 is proved.
From Theorem 28.1 (Chapter VI, Section 28) we know that the operator
in I
D
possesses a strict adjoint:
extension of an extension of as well of the closure
Z*
the space
to
i =
in
Z* =
in
1)
;
iD
or
prove this theorem, let tion in
-T)
.
i'
Z*, an
in
and hence just
We now maintain that even SD.
Z.
In other words, the operator -D
Sp
is the same as the space
Theorem 32.2. I
-D
in the space
-D
-D
in
D
ID
is strictly adjoint
is strictly self-adjoint there.
in
be a function in
4(x)
to which there is a function
41(x)
To
ID*; i.e., a funcin
such that
the relation
f *(x)o1(x)dx = holds for all j).
J
in
ID ,
To show that this
O(x)
*(x)
Day (x)O(x)dx
exists a sequence of functions IIOv-$l!
0
,
hence in particular fof all (x) is in
ID
V(s)
IID4v-41111
-
0
in
we must show that there
in
ID'
such that
as v
Such a sequence of approximating functions can be formed in 188
We shall present a method which can easily be exten-
different ways.
ded to a rather large class of ordinary and partial differential operators although for ordinary differential operators one could proceed in a somewhat simpler manner.
We shall construct the functions V (x) appropriate smoothing operators on the function
by-applying certain
(x).
These valid
smoothing operators are given as integral operators
Jvf(x) = I jv(x-x')Cx')dx'
whose kernels are functions of
j(x-x')dx' = 1
1
(32.1)
x-x'
so chosen that
.
Specifically, we choose a non-negative function ported by the interval
ICI
<
j(&)
in
a
1
sup-
and for which
1
and then set jv(x-x')
= vj(v(x-x'))
Clearly, this kernel is supported by dition (32.1). in
Ixl
If
$
bounded support, namely in Jv
!x-x'1
is a function in
< a, say, the function
of the operator
.
lxl
JV$(x)
< 1/v
and satisfies con-
Gi with bounded support is defined and has also
< a + 1/v.
Since the Holiilqren norm
evidently equals 1 (see Chapter IV, Section 20),
the inequality
(32.2)
IlJv,ll_ Hell
holds for such functions.
This inequality shows that the operator
189
$
Jv can be extended to all of
Operators of this type were treated at the end of Section 20
identity strongly. tions
Jv4(x)
3V
There we saw that the operators
in Chapter IV.
I.e., for every function
tend to IIJv4l-'PII
- 0
as
v
which approximate the
Jv
Integral operators of the form
The reason is that
identity strongly have been called "mollifiers".
they transform the function $(x)
into a function
4(x)
this kernel to have continuous derivatives.
jv.
II, but are
The degree of
is required to be.
$(x)
smoothing depends on the choice of the kernel
J"$(x)
'Pv(x) = Jvo(x)
in the sense of the norm' II
smoother than the function
functions
the func-
in
$(x)
$(x); i.e.,
(32.3)
which is close to
approximate the
We have required
As a consequence the
have continuous derivatives, as we proceed to show.
For any finite interval
the inequality
< x0
IxI
max IJv4(x)I _ CV1loll IxI<x0
holds with an appropriate constant holds for any m
in
Q .
From this inequality we conclude that
is approximated uniformly by
Jf$(x)
Jv$(x)
is approximated with respect to
fp
quently, tion i4
= $
Jvq(x)
Jva(x) .
II
is continuous for
m
II
in
by
IxI
< x0
if
4
in
Conse-
in
$
Moreover, the func-
in
has a continuous derivative
DJ"4(x)
(x)
if
is in
Since for such functions the inequality max IDJ"4'(x)I Ixl
< CvII'PII
holds with an appropriate constant Jvo(x)
For this relation evidently
Cv.
C,, it follows that the function
has a continuous derivative for any function
We are now ready to prove the statement
190
'P
in
SD* _ 'D
made
Let
above.
in
I1$v-$11
Q', and that
Moreover,
+ 0.
DJv$(x) = 1 Djv(x-x')$(x')dx' = -
D' = d/dx'.
where
Z *
for all
point
$
x
I
D'j'>(x-x')$(x')dx'
We now make use of the assumption that
so that there is a
J
chosen the kernel
is such a function in t
is in
D'$(x')$(x')dx'
Z, in particular for all
in
$
such that
C
in
$1
$(x')$1(x')dx' _ -
J
$V(x) - J'$(x) is
't *; then we know that
be in
$(x)
in
$
¶'.
Now, for each
considered as a function of
jv(x-x')
since it vanishes for
jx'-xI
> 1/v.
x'
Con-
sequently, we may conclude that
1
D'jv(x-x')$(x')dx' _ - I jV(x-x')$1(x')dx
= -Jv$1(x)
.
In other words, we have DJ"$(x)
From
IIJv$1-$11
- 0
= Jv$1(x)
we therefore may conclude IIDJv$-0111 - 0
Thus we have shown that. $
in
t'; i.e.,
.
.
is in the domain
)
of the closure of
SD* C Z.
Since the opposite relation 3's.2 is proved.
191
'
C
is obvious, Theorem
D
33.
ordinary Differential Operators in a Semi-Bounded Domain In this section we shall deal with the differential operator
d/dx
defined on the half-axis
+(x)
acting on functions
0 < x <
We shall carry over the method of Section 32; but in doing this we shall get noticeably different results.
(x)
The functions
defined for
0
< x< - which have piece-
wise continuous derivatives and for which
4,0) =fo 14(x) 12 dx
(DO,DO) = f 1D4() 12dx
,
are finite are said to form the space of the functions in
and
troduce the spaces Z ;
and
iD'
with bounded support.
'.a'
The operator
-D
in
in
SD'
and
'aU
i)0
noted by
Z6
in
-D
'
as well as
'
in
D
can be closed.
'a ,
i
z0'
,
D0
in
Zd and
Since the spaces
are obviously dense in $ it is and as well as D0 in
Z0
and
or
One immediately verifies that the oper-
D0.
are formally adjoint to each other.
SD' , Z' , i0 that the operator and
SD'
.
D = d/dx, when restricted to functions in
Z ', will be denoted by D0
4(x) in
for which ( (0) = 0
ator
In addition, we in-
of those functions
i)6
consists
is
'i'; the space
i'
i'
clear ZQ
The domain of these closures will be de-
lb
0'
In the same manner in which Theorem 1 was proved in Section 32 one may prove
Theorem 33.1.
_
`
so that the superscript
'a0
z0
may be omitted.
The question naturally
arises whether or not the subscript can also be omitted. show that this is not so.
192
We shall
Z
Z
Theorem 33.1°.
0
-
In other words, the boundary condition 0(0) = 0 is essential. The proof of this statement is based on various lemmas. For every function
Lemma 1.
aIIDOII+
I0(x)I
and every
a > 0
holds for every
Z'
in
$(x)
the inequality
a II0II x
0 < x < a.
in
To prove it we need only observe that by the Schwarz inequality the relation x I $ ( x )
-
I
=
1 D@(x")dx" + 0 ( x ' )
<
I
I
Yra-
I
I D O I
+
I
1 0W )
I
x'
holds for
0
together with division by
Z C Qi.
Lemma 2.
from
x
< x < a; integration with respect to
0
to
a
then yields the statement.
a
Z
That is, every function in
more precisely, every (ideal) function in
is continuous;
Z "equals" a continuous
function.
Let '
4
such that. I1pv-011 0,
¶ and
be a function in
IIDOv-DO1I ; 0
-+ 0,
IIDOv-D4u11
-
v,p - =.
as
0
Lemma 1 we may then conclude that the sequence uniformly to zero in every interval interval
0'(x)
as
v - 0.
Clearly
Then
From the inequality of 0'
x)-011(x) converges
< x < a, so that in each such
converges uniformly to a limit function
of course is continuous. 0v
0
be a sequence in
4v(x)
II4v-OII - 0
q(x) whia
and the fact that a
tends uniformly to
0
in
0
< x < a
implies
j
0,
0
so that Lemma 3.
0 = 0.
SDO
is the space of all 'o(x)
in Z
for which
Note that this last condition makes sense since 193
q(x)
0(0) = 0.
is
4(x)
function in
vanishes at
z 0
x = 0.
Z which vanishes at
function in
of functions in IIDmv-DOII
the same is true for (x) by Lemma 1; thus any
x - 0
vanish at
0.
-
$(x) be a
Conversely, let
Then there is a sequence
x = 0.
0, and
0v(x), such that
I DI, say
According to Lemma 1 this implies pointwise convergNow the elements of the sequence
"(0) - 0.
ence; hence
approximating
Clearly, if the functions 0V(x)
continuous by Lemma 2.
2
0VI
'ti.
are in
Taking
(x)
_ 0v(x) -
t
tv
0v(0) e-t x
tvl$V(0)I
such that
0, we clearly
-
have
II,"'-,,II - 0
which shows that if
is in Z
0
IIDm"'-DOII
and
consists exactly of the functions
(x)
(x) with
00(x)
0(0) = 0 't
01(x) be any function in
Let
Corollary.
and
in
-
0
,
then
not in
is in
0
Z0.
0.
Then
of the form 00(x)
z 0.
it follows from Lemma 3 that Theorem 33.10 is proved as soon as a single function
(x)
in
Z with
0(0) ¢ 0
Such a
is exhibited.
function evidently exists.
Clearly, then, we must consider both spaces ' is obvious that the operators -D adjoint. D0
by
D0
Since these spaces are dense in
have strict adjoints 'fl*
and
and
Z*0.
D*
and
and
't 0.
in these spaces are formally the operators
D
and
D**; their domains will be denoted
Evidently, we have the inclusions
194
It
and
z0
C
Z**
,
9*0
C
'D
.
Actually, there are only two different ones among these spaces, as shown by the analogue of Theorem 32.2:
Theorem 33.2.
Z
Z0=
'a 0,
D.
To prove this theorem we proceed as in proving Theorem 32.2. Let
be (1) in
0
*,
quence of functions 0v y 0, DO' - D$.
(2) in
'D 0*.
Then we should exhibit a se-
(1) in
0v(x)
(2)
in
q)'
such that
We use "shifted" rnollifiers by setting
V(x) = J" c(x) = 1 jv(x-x' + 1)0(x')dx'
Note that in the first case then the argument
-x' -
consequently
v(0) = 0
case we have
jv = 0
tion
when
iv = 0
of the kernel so that
when
x' = 0
iv
and
x'
> 0
since
is outside its support;
is in
4v(x)
Q. In the second
and x > 0; consequently, the func-
considered as a function of
jv
x = 0
x'
belongs to
2)6.
Keeping
these facts in mind one may literally carry over the arguments of the proof of Theorem 32.2 to proving the present Theorem 33.2.
Thus we have to deal with only two operators, -D0
D
Z 0, which are strictly adjoint to each other.
in
in
Z and
Using these
operators we can form the two strictly self-adjoint operators
-D2 = }
-DD0
and
-D0 = -D0D
defined in appropriate dense spaces.
The func-
tions on which the first of these operators is applicable satisfy the first boundary condition
0(0) = 0; those on which the second one is
applicable satisfy the second boundary condition
DO(0) = 0.
The spectral representations of these operators are given respectively by
195
0(x) =
CO) =
(
sin ux E(u)du
0
r sin ux $(x)dx
IF
1
0
and
$(x) - f cos ux C(u)du 0
(u) = n 1 cos ux O(x)dx 0
The eigenvalues of these operators are given by The operator
iD
in
is evidently formally self-adjoint
z 0
but not strictly so since its domain
D ator ator.
of its strict adjoint in
in
differs from the domain
We may wonder whether or not the oper-
z 0
would imply
Z* - D in
z 0
can be extended to a strictly self-adjoint operA If that were possible the domain D of this extension would
iD0
be contained in the space
D
iD.
in both cases.
u2
D *
C
by Theorem 33.2.
D .
D
For certainly the relation
D*; but
0
C
by assumption and ID
Now, let
not
be be a function in
D 0; then the corollary to Lemma 3 implies that every function.
D is'in
extension of operator
iD
D .
z 0 in
That is to say,
contained in
S
n D =
a
D .
other than
There is hence no
D itself; but the
D is evidently not self-adjoint.
We formulate this result, due to von Neumann, as Theorem 33.3.
The operator
iD
in
D
is formally self-adjoint but
cannot be extended to a strictly self-adjoint operator.
This result shows that the requirement of strict self-adjointness is essential; it is not simply a matter of mathematical completeness which could always be attained once the operator is Hermitean.
196
34.
Partial Differential Operators Our treatment of ordinary differential operators can be carried
over to partial differential operators nearly literally. functions
of
defined in the whole
x = xl,...,x
variables
n
< x1.<
space ri'
0
We consider
We introduce the spaces
< xn <
4.
of piece-
of continuous and piecewise continuous functions,
wise continuous functions with bounded support,
i
of functions with
finite support having a piecewise continuous first derivative. functions in
i'
and
For the
the integral
f I0(x)12dx
dx = dxI ... dxn
is defined where
over the whole space.
and the integration is extended
Using the same extension procedure that we have
used for functions of a single variable, we may extend the space to a complete space of ideal functions, the Hilbert space
ai
Clearly,
As dense in
t).
t .
We introduce the space
of all functions in
Z '
have a piecewise continuous derivative which in turn is in then define the operator
D
which
rp
which transforms the function
t)
$
.
We
in
T
into the system of functions DO(x) _
n0(x)} _ {ax (x)......ax 0(x)} n
which are piecewise continuous and in convenient to consider a set of
n
'.
In the following it is
functions as a single entity.
set
1, = {p1,...,yin) and denote the spaces of functions by
,
The unit form in
*(x)
of course -T
is
197
whose components are in rP
is a Hilbert space.
(f
12dx
J
where
1*12 =
T' we define the operator D which trans-
In the space forms each function
D' into the function
in
J
D-0 1
ax *1+...+ax *n n
1
V and
which belongs to
The operators operators
and D in
t'
in
and
D'
in
-D
-D
9).
t'
in
D.
Z' as well as the
are formally adjoint to
each other; for, the identity
dx
dx J
holds whenever and
in
J
is in
0
Z'
in
and
Z1-
or
0
t'
is 'in
Note that no boundary terms appear in the integra-
,
tion by parts since one of the two functions vanishes identically outside of a finite region.
Since thus the operator in
t'.
and
i
,
t.,
,
I
in
Z '
is
and.
as well as
D-
have formal adjoints defined in dense domains,
't '
The domains of these closures willbe denoted by
they admit closures.
t,
D
Clearly we have
.
=-(D$,*)
for
0
in
D,
,
1. and for
in
m
in
,
4,
in
t
.
Note that by this closure process we have extended the partial differential operator
D
{al,...,an)
to a class of functions
q(x)
which do not all possess partial derivatives in the strict sense. would be possible, though, to characterize the extended operator the terms of the Lebesgue theory.
It D
in
It does not seem possible, however,
198
to characterize the extended divergence D as producing the sum of anon, each being obtained by applying an exten-
the functions
The approach to extending the operators'
ded differential operator.
gradient and divergence as described here avoids this difficulty. In analogy to Theorem 32.1 we formulate 0
Z?0
Theorem 34.1.
This statement is proved in nearly literally the same way as Theorem 32.1 was proved.
M- #
The counterpart,
We need not give details.
of this statement is also true.
We shall not attempt to
prove this statement directly; its validity will eventually be ascertained without a special effort.
Since the operators D
in
1),
possess formal
and
adjoints in dense domains they possess strict adjoints in domains *
and
We have oslittedIthe dot as multiplication symbol
t *.
because these domains are spaces of single functions; in fact, these 1
D
domains are extensions of o
*
D( 'a
Z*-
Z' we have
C
The adjoint operator (D-)* defined in
) *.
be denoted by
Since
.
since it is an extension of
-D
-D
o
defined in
will z .
We now formulate 0
Theorem 34.2.
Because of
D
'fl .
* _
t * _
'D * j
Z *
D ti .this statement is implied by
To prove the latter statement we use mollifiers as for We define the mollifiers
the proof of Theorem 32.2.
J1
as the inte-
gral operator with the kernel jV(x-x') = v nj(v(xl-xi))...j(v(xn xn))
This n-dimensional mollifier has the same properties as the one-dimensional one.
For
0
I I-J"OI
I_
in
11011
C)
,
I IJv,-,l -. 0 as I
199
V - 0
and
is in
J'O(x)
Furthermore, for each point
4 j.
is in
p x') _ {j'(x-x'),0,...,0}
Z'
the function
x
and
aijv(x-x') 0
Hence, for
S*,
in
0
jV(x-x')ajc(x')dx' _ -
f
=
I
1
aij'(x-x')O(x')dx
aljv(x-x')O(x')dx' = al 1 7v(x-x'),(x')dx'
Since a corresponding relation holds for each component we have proved the identity
for
0
From
in
the domain of the closure of domain; i.e.,
0
and from the definition of
J'Do - Dp D
in
t '
we infer that
0
Z
as
is in this
M
is in
In the same manner one could prove the identity but it is not necessary to do so.
1.* _
z
For this identity follows immediately
0
from the identity states that 0
Z_
Z
'
A** = A.
and
D=
=
t*, using a theorem of von Neumann which The identity
Z =
'I *
in turn follows from
tj *.
In the same manner von Neumann's theorem, applied to the iden0
tity
¶ *, yields the identity,
Z
=
D . This is the
counterpart of Theorem 34.1 which we have claimed to be valid. Having proved the fact in
'
at the operators
D
in
Z and
-D-
are strictly adjoint to each other we can assert that the
operators -D-D
are strictly self-adjoint.
and
-DD,
The first of these operators is the negative
200
Laplacean, and the second one is on occasion used in the theories of elasticity or electro-magnetism.
It follows from the general theory
that these operators admit spectral representations.
Of course the spectral representations of these operators can be given explicitly with the aid of the Fourier transformation.
But
the aim of our theory is not just to show that these particular operators admit of such a representation.
Our primary aim is to show this
to be the case for more general though related operators whose spectral representation cannot be given explicitly as simply as that of
35.
-D'D.
Partial Differential Operators with Boundary Conditions In Section 33 we imposed on our functions of one variable the
-condition that they should vanish at one point and found that the domain of the closure of the operator
D
changed by imposing this condition.
We shall find that the situation
acting on such functions is
is quite different if we impose the condition of vanishing at a point on functions of more than one variable.
We denote by entiable functions
the space of piecewise continuously differ-
D U
(x)
in
which
-e (identically) zero in a
neighborhood of the origin fi(x)
(If we required only
= 0
$(0) = 0
for 1xl
<
p2
as in the case
n = 1, we would obtain
a somewhat weaker result.)
Since the operator the domain a closure:
D
Z;
in
t ', which is dense in D
Theorem 35 1 .
in
D0.
Z0=
is formally adjoint to D
', it follows that it possesses
We now formulate
Z
For
n > 1.
In other words, the imposition of the condition not make any difference on the closure of the operator
201
in
m(0) = 0 did D
in
Z '.
We introduce the function
eP log P
{P(x) =
EP = 1/log p-1.
defined by
,(p)
for
p2
=0
for
p <
IxI
,= 1
for
IxI
< P2
<
IxI
(x)
Then we have for any function
IIDSP0I < IIkPDfll +
P
in
Z '
IID IIP + IID;PII M P 0
II
with ID4I2dx
IID,II2 =
J
Now we have n
IID;PII2 < nn2 Pn-2 tp
= E P
where
if
n >
if
n=2
is the surface of the n-sphere.
Qn
IIDCPal I
-
as
0
p
0
x = 0
In any case
which is zero in the neigh-
approximates the function
IIDOP-D$II - 0
,
,
.
Consequently, the function P = (1-;P)o borhood of
2
II0P-4'II - 0
0
in such a way that
as p - 0
Thus Theorem 35.1 is proved.
The situation is quite different if we impose a boundary 202
(n-l)-dimensional part of the
condition not just at a point but at an
I
boundary of a region R in the x-space.
Let us take a rectangular
cell
R : 0 < xv < av
v = 1,...,n
,
Q and denote by
as such a region
the space of those functions
Z 0
R which vanish on the
with piecewise continuous first derivatives in part
-40 0: xv = 0,
v = 1,,...,n 0
of the boundary
.g
of 9 .
we denote the space of those
By
O(x) = (q,1n(x)) which have a piecewise continuous
vectors
derivative and vanish
on the remaining part of .
-4 ;: xv = av
Clearly the operators
D
adjoint to each other. domains
'
and
0
in
,
v = 1,...,n
and -D
Z 6
.
in
z .'
are formally
Hence these operators possess closures in .
The analogue of Theorem 33.10 is the statement that the space 'D 0
is not the same as the space
boundary condition.
[J
D
defined without imposing a
It can easily be proved by using the inequality
...
1/2
f
x =0 n
[a
J
xn `a
x
in place of the inequality for 33.10.
IO(x),2dxJl/2
IDO(x)I2dxjl/2 + [a-1 J
I*(x)I
n
used for the proof of Theorem
We shall not give details.
We denote by
Z 0
and
i 203
the domains of the closures of
the operators
D
in
D U
and D in
Z '.
Theorem 33.2 we then state that the operator are strictly adjoint to each other.
Z
D
As the analogue of in
and
z 0
D.
in
Again this can be proved in
the same way as Theorem 33.2 was proved, by using a mollifier kernel with shifted arguments: jv(x-x') = v-nj(v(xl-xi) - 1)
Considered as a function of 0
time the function
x
is in the region Q .
vanishes on the part
Jv4(x)
.
this function vanishes on, the part
x'
of the boundary provided
... j(v(xr-xr) - 1)
At the same
g0 of Q .
The
arguments used in proving Theorem 32.2 then yield the statement.
We have chosen the rectangular cell as region Q and imposed a boundary condition oily on the part
,Q 0
of its boundary because
then the arguments used for the one-dimensional case carry over with nearly no modification.
Actually, the corresponding statements'hold
for any region with a sufficiently smooth boundary when the boundary -(n-1)-dimensional part of the boundary or
condition is imposed on any on the whole boundary.
To prove the analogue of Theorem 33.2 one may
employ mollifier kernels with arguments which are shifted by the addition of a function of
x+ rather than by just a constant.
We shall not
In any case it follows that the operator
carry out the details here.
-D-D, the negative Laplacean, is strictly self-adjoint in an appropriate dense space of functions and hence possesses a spectral representation.
Various modifications of the Laplacean can be shown to be strictly self-adjoint.
For example, we may consider the operator
where the operator uous function operator
v(x).
V
consists in multiplication by a contin-
Clearly, if the function
Iv(x)I
is bounded the
is strictly self-adjoint in the same domain in which
is.
Suppose
v(x)
is not bounded but non-negative.
204
Then we may
invoke Theorem 31.1 proved in Section 31 to ascertain that in an approis strictly self-adjoint.
priate dense domain the operator
For clearly, the function
v(x)
may be regarded as the least upper
bound of bounded continuous functions. If the function
v(x)
is not bounded below the self-adjointness
is not so easily established; it might not even be true without imposing additional conditions on the type of boundary condition.
In some
such cases, however, the problem can be handled by writing the operator in the form -(D + g)-(D + g)
g(x) = g1(x),...,gn(x)
where the vector tion of
If
x.
g(x)
is a given real-valued func-
is differentiable the operator in question when
applied to twice-differentiable functions can be written in the form
(-D + g) (D + g) =
V
with V =
If the function operators
D +.g
jg(x)j
is bounded the domains of the closed
and -D + g are the same as those of
Moreover, the operators
D + g
strictly adjoint to each other.
and
(-D + g)
(-D + g)-(D + g)
and
in these domains are
This statement is proved in the same
way as Theorem 32.2, without noticeable modifications. the operator
D
Consequently,.
is strictly self-adjoint in an appro-
priate dense domain; in general, this domain is not the same as that of
-D'D.
An interesting special case arises if we take
with any constant
a.
Although this function is discontinuous at
205
x =
0 it is bounded; the statements made above are therefore valid.
Thus
the operator
is strictly self-adjoint in an appropriate dense domain. course, this operator is non-negative.
Also, of
When applied to continuously
twice-differentiable functions which vanish at the origin this operator may be written in the form
a n
L=
+ a2
T7X
it is thus recognized as Schr8dinger's energy,operator for the hydrogen atom (in case
n = 3) except for the addition of the constant
a2.
In
this manner the strictly,self-adjoint manner of the Schr5dinger operator
L-a2
is established.
Incidentally, the function
(x) - e is an eigenfunction of
L
ajxj
with the eigenvalue
A = 0
since it satis-
fies the equation
(D+ax TXT Since the operator eigenvalue.
is non-negative the value
A = 0
is its lowest
Thus it is seen that the SchrBdinger operator has no eigen-
value less than
36.
L
-a2.
Partial Differential Operators with Discrete Spectra In our discussion of operators with discrete spectra in Chapter
V we assumed the operator to be bounded.
Since differential operators
are not bounded it is necessary to modify the theory of Chapter V so as to cover differential operators.
This is easy in case the differential
operator is non-negative (except for an additive constant).
206
Let F
be a strictly self-adjoint operator in a dense domain
F
for which F > C
with an appropriate constant
C.
Then we say that
xIt$112 <
,(36.1)
such that
Z(1),...,Z(g)
A" if there are vectors
rank below
is "of finite
F
IZY,4'I2 + (4,F@)
L
Y=1
for all
in
4'
F.
We maintain that the spectrum of such an oper-
ator is discrete below
X.
in proving this statement we may assume restriction.
Let
B
be the inverse of
F > 1,
A > 1, without
which exists by virtue of
F
the operational calculus; then the square root / is defined and sat-
isfies the relation vF B = 1, when applied to vectors
/
is in
t
F
in the inequality (36.1) above, we
t = fB
Setting
Y'. for which
find the inequality
Y=1
with
yY
117UR ZY
to be valid for all
hence, by closure, for all
in
'Y
`Y
with
rB'Y
in
* F and
$ .
From the theory of Chapter V we may now conclude. that the
spectrum of the operator
B'
the spectrum of the operator
is discrete above F
is discrete below
We shall say the spectrum of
F
Consequently,
A-1. A.
is "discrete" if the eigen-
space of every finite interval is finite-dimensional.
If the sequence
of eigenvalues with multiplicity is infinite these eigenvalues tend to infinity.
From the statement just proved we may then immediately con-
clude the validity of the
207
Suppose inequality (0) holds for every value of
Corollary:
appropriate vectors
Z
depending on
a).
x
(with
Then the spectrum of
F
is discrete.
We shall first consider a partial differential operator in a bounded domain and prove that its spectrum is discrete.
For simplicity we assume the operator to be simply the nega-D-D, acting on functions
tive Laplacean, tangular cell
condition
V = 1,...,n, and satisfying the boundary
0 < xv < a,,,
= 0
xv = 0,
on
defined in a rec-
m(x)
v = 1,...,n.
The following arguments, which go back to F. Rellich, can be
carried over to more general operators, more general regions and more general boundary conditions provided that the coefficients of the differential operator are not singular and that the region is bounded and has a smooth boundary.
A major tool in the proof of discreteness is Poincare's inequality: 2
11112 < R-1 11,12 + i IIDf112 where
R = al...an
is the volume
of the cell .y'
and
d = (ai+...+
To prove this inequality we connect any two points
in R
by a zig-zag path, going from
x(2) = (xi,x2,...,xn), then to
to
x(n+1)- (xi,...,xn) - x'.
x = x(1)
x (v+l) I0(x(v+l))-4(x(v))
=
_ (xl,...,xn)
x(3) = (x i,xZ,x31...,xn)
Evidently
(aVO(X))dxl
IJ
x(v)
x(v+l) laV012dxv]1/2
[a v 1
x(v) 208
x
and
x'
to
and finally
hence
x(v+l) n d2
a-l
E
v-1
x(v)
x(v+l)
n d2
<
r
a-1
i
V=1
J
x (v)
Integrating this inequality with respect to
x
and
x'
over R
we obtain
2RIIIPI12 - 21,,t 12 < d2RIIDOI12
as claimed.
We apply this inequality to any of the sub-cells obtained by dividing each side into
= 1 in
equal parts.
R
RY
Letting
of
R
nY(x)
R Y, = 0 outside MY we obtain
i
R-1knIf1,,12 + d2
II4,112
IIDmHIY
2k
in obvious notation.
112
11
Addition.over all such cells gives the relation
R-lkn
which is of the form
d 2
in Y=l
2k
(36.1) with
A = 2k2/d2
and
Z(Y)
= aR 112kn/2
since ((P,F9)
Since
k
=
(D4,D(P)
is arbitrary it is thus proved that the operator
finite-dimensional below every value is discrete.
Since the space
x
and hence that its spectrum
is not finite-dimensional the
sequence of eigenvalues is infinite; it follows that these eigenvalues tend to infinity. 209
is
Although the result thus obtained pertains only to finite regions, the argument that led to it can also be used for infinite regions.
We shall prove,the
Theorem.
Suppose the function
to infinity. funct..uns
,
v(x) tends to infinity as
tends
(acting on
Then the spectrum of the operator
(x)
lxl
defined in the whole x-space) is discrete; its eigen-
values tend to infinity.
w
be an arbitrary positive number and let
v(x)
> w
when
(36.2)
x
is outside
wIIOIIR
throughout and let
v > 0
To prove this statement we assume
.Q
be a region such that
To the inequality
5e'.
lZy,t12
Y.1 I
+ IfD$112
obtained from Poincar6's inequality we add
wllm$12
where
R *
is the complement of
wll0l12
1
< (O,VO)
<
R .
Thus we obtain
1Zg,m12 + (DO,Da) + ($,VO)
Y-1
which is the desired inequality (36.1) since
(DO, DO) + ($,V$) _
V) 0)
.
Thus the statement follows. Finally, we shall consider a differential operator whose spectrum can be shown to be partially discrete on the basis of our criteria, namely the Schr8dinger operator F-a 2 = (-D+g).(D+g) - a 2
,
g(x) - ax/Ixi
acting on functions defined in the whole x-space. can be
(f,F@)
Here we make use of the fact that the form expressed in the form
(c,F(D) = ((D+g)', (D+g) (D)
.
;(x)
We introduce a continuously differentiable function which vanishes for which
0
<
;
<
1
Ixi
< p/2
throughout.
and equals 1 for
Ixi
> p
and for
Then we h.ve
II(D+;g)0II < {II(D+g)4-II + II(1-;)g'ii}2
(l+E-1)a21I,PIi2R0
<
.
Further,
II(D+;g)IDII2 = IIDOII2 -
as verified by integration by parts. choose
p
-(n-l)aixl-l > -a6/2
so large that
(aid/2, so that D-;g > -6a2.
(l+c)II(D+g)4 II2 > IID'II2
-
(2+E-1)a2I14.112
RP
Next we apply the inequality (36.2) to the region w >
and
we
ID;I
Then we obtain
9p
chosen
d > 0
Now, to a given
+ (1-6)a2II-PII2
Ixi <.2p
having
(2+E-1)a2 so that we obtain
(1-6)a21,0112
lzy,.Dl2 E
Y.1
Since
(1+e =1)(1-d)a2
can be made arbitrarily close to
found that the spectrum of the operator negative n!lmber.
F-a2
a2
we have
is discrete below any
This is a well-known property of the Schr(dinner
211
operator; but our derivation of this property evidently does not very specifically depend on the special form of the function
g(x);
it would yield the same result for a wide class of such functions for which the spectral representation of the operator cannot be given explicitly.
212
CHAPTER VIII
PERTURBATION OF SPECTRA
37.
Perturbation of Discrete Spectra The method of perturbation is one of the most effective meth-
ods of determining approximately the spectral representation of a given operator.
The method is applicable if the operator
tion is sufficiently near another operator
L
which depends on a parameter
£
whose spectral repre-
in such a way that it reduces to' to expand the
e =
for
spectral representers of senters of
L£
in powers of
Le
The approximate repre-
c = 1.
If the undisturbed operator
has a point eigenvalue
L0
one may assume that the disturbed operator ae
E.
are obtained by breaking off the series after a few
terms and then setting
eigenvalue
in ques-
One then introduces an operator
sentation is known - or partly known. L£
L0
L
L.
X0,
possesses also a point
which with an appropriately chosen eigenfunction
possesses F_.i expansion with respect to powers of
e.
4e
If this attempt
succeeds, at least partial description of the spectral representation In the present section we
of the disturbed operator is attained.
shall be concerned with this problem of the perturbation of a point eigenvalue and closely related questions. Let us first-describe the perturbation procedure in a formal manner.
Weesuppose that the operator
L£
is analytic in
e, and thus
admits an expansion
L£ '= L0 + eL1 + .. . (with bounded
L1,L2,...).
A = A0 + eal +
Then we try to determine expansions
0e = 213
4)
0
+ £ l + ...
of an eigenvalue anL an eigenvector of eigenvalue
XO
which reduce to a given
LC
and a given eigenvector
00
of
L0.
Assuming the ex-
istence of such expansions we may write down a sequence of equations satisfied by the terms in it:
(LO-A0)t0 = 0
(LO-x0)m1 = -(L1-X1)tot (LO-X0)42 = -(L1-X1)o1 - (L2-A2)40
The question arises whether or not these equations allow us to determine the terms
A1OX2,...
and the vectors
01,02,...
.
If
this is possible it remains to be shown that the series formed with these terms converge.
First we note that for every vector
m
(,0,(L0-X0)41) = 0
holds if, as we assume, the operator
L0
is Hermitean.
This fact
implies that the right-hand sides of the sequence of equations above are orthogonal to
0O1
(tot (L1-X11 )00) - 0
,
(tO,(L1-X1)@l) + (001(L 2-X2)00) = 0
.
Clearly, then, we can express successively the expansion coefficients .An
of the eigenvalues in terms of the vectors
Suppose
the right-hand side of th' equation
(LO-XO) On - `fin has thus been made orthogonal to from this equation? possible if
X0
00; is it possible to determine
to
This is not at all always possible, but it is
is a simple isolated eigenvalue of 214
L0; i.e., if the
eigenspace of containing
1'0
A0
associated with a sufficiently small open interval
consists just of the multiples of
For, in this
00
case, as is seen from the functional calculus, the restriction of to the orthocomplement of
L0-A0
has a bounded inverse.
t0
Therefore
the,above equation always has a solution, unique except for the addition of,a multiple of enough'
This multiple can be made unique (for small
00.
at least) by adding the condition
c
(toItC
1
together with (10,@0) = 1
so that n = 1,2,...
(to,on) = 0,
.
Because of its linearity this condition is preferable to the condition
E1, which of course could ke achieved afterwards. Instead of trying to.pxovp the convergence of the resulting series for
A.
and . directly, one can proceed more effectively as
follows: one allows the parameter then the Hermitean character of unique existence of solutions (Le-AE)@E = 0, (00,OE) = 1
is lost.
L. @E
to be complex, never mind that
c
and
aE
Next one establishes the
of the equations
for sufficiently small
establishes the unique existence of solutions
Moreover, one
jc$.
and
0
of of the equa-
tion (LE-AE )VV = -(L'-A')OE
with
,
(00, OF'
0
LC - dLE/de, which would hold for the derivatives
aE - dae/de
if they existed.
Next one proves that the difference
quotients of e. and
aE
do approach-these solutions
that indeed
aE
have derivatives.
@E
and
mE = dOE/dc,
to
and
XE, so
Evidently, these deriva-
tives are independent of the direction in which the difference quotients
215
Hence
are taken.
and
P E
are analytic in
ae
e
and, consequently,
That these expansions are the same as
possess power series expansions.
those obtained before is clear from the uniqueness of their construction.
Thus convergence of these series is established.
We shall carry out the main step in this argument not just for the problem of the disturbance of a single point eigenvalue as described.
Rather, we shall carry out this step for a more general prob-
lem.
L. is
For simplicity we shall assume in the following that of the form
L C = L0 + 6V where
is bounded (not necessarily Hermitean).
V
In the following we shall either omit the subscript
a
or
replace it by 1.
We assume that of
L0.
/0
is an isolated segment of the spectrum
By this we mean that the eigenspace of the operator
ass-
L0
ociated with a slightly larger interval / is the same as that associated with intervals.
so that the spectrum is empty in the intervening Then we maintain that there exists an operator
transforms the eigenspace an eigenspace
of
CEO
possesses an inverse
associated with
which transforms
U+
11
back into
which
/0 into
L; moreover, we assume that
of the operator
11
L0
U
U
140.
This
being so, the operator L = U+LU a
transforms the space
eigenvector 01
of
L
of
L
0
into
(S0, but in such a way that any
with the eigenvalue
with the same eigenvalue A.
.
216
a
leads to an eigenvector
More generally, if, with reference
to a spectral representation of the operator U+0
in
L
4. 44b,
' ()L)
affords a spectral representation of the vector 11
of
L.
L
sentation of
in
L
in
44
(40.
same is true of the eigenspace
9
0
of
that in a finite-dimensional space. Q O
in the eigenspace
is reduced to that of the spectral repre-
If in particular the space
in
0
In other words, the problem of the spectral representation
of the operator
L0
the vector
9 0
'f(A), the representation
has the spectral representer
110
in
is finite-dimensional, the
L; the problem is reduced to
If, for example, the spectrum of
consists of just one eigenvalue
r, the spectrum of
in
L
G
each point with multiplicity.
with the multiplicity
A
consists of-exactly
r
points counting
The problem of finding the
r
eigen-
values is thus reduced to a corresponding problem in a r-dimensional space.
It should be said that the problem of the splitting up of a multiple eigenvalue in a finite-dimensional space is far from trivial; it was first completely solved by Rellich in 1937.
But the present
approach shows that the problem of such a split-up in infinite-dimensional space can be reduced to a corresponding problem in finitedimensional space.
How can one show the existence of such operators
Uf
as
described? Of
if the eigenspace 0
is one-dimensional, con-
sisting of the multiplesof an eigenvector 00, the space 6
is also
one-dimensional consisting of the multiples of a vector b 1, which may be assumed to satisfy the relation
(Q0,01) = 0.
U± given by ±
U
= 00 ± (00, -P) (00 - -V 1) 217
Then the operators
U-'
have the desired property, for then
0
U+01 =
and
= 01
00'
Thus the present problem covers the previous one. Q
Instead of the eigenspaces projectors
P
P.
and
P0
assume that about
0
O
which project into these spaces.
P
We shall
is Hermitean, but we shall not make that assumption
Instead we shall require that PP
(37.1)
we shall work with
Q
and
0O
POP = P
=
Q
Note that in the case the spaces
satisfy the,two conditions
P
0
0
Q
and
are one-dimensional,
these conditions are satisfied if the projectors
Pop P
are defined
by PO
= (OO,@)*0
and
PO = (00,§)@1
.
Also note that the two relations (37.1) imply the relation P 2 = (PP0)P = P(P0P) = PP0 - P
in other words, they imply that
if
lies 4in
and consequently
$
(1-P)LPO = 0.
should hold. Conversely, projects into a space
sert
does, so that
implies that
L
LPO, always lies in
In other words, the relation
(1-P)LP = 0
(37.2)
P
is a projector.
is aneigenspace of
The condition that L$
P
f
L = L0+V
the validity of this equation implies that Q
transformed into itself by
L.
in this equation and observe that
PL0P = PP0L0P = PL0P0P = PL0P0 = PP0L0 = PLO
so that (1-P)L0P = L0P - PL0 = L0(1'-P0) - (P-P0)LO
Accordingly, we may write equation
(1-P)LP - 0 218
.
in the form
We in-
L0(P-P0) _ (P-P0)L0 - (1-P)VP
This equation, essentially due to C. Bloch (1958), will be solved to determine
P.
We employ the function A-1
for
A
outside
for
A
in
/0
C(Z)
i=O
/0
and introduce the operator
S(L0) = ZO' Since
PO = rl(L0)
with
= 1 for
A
in
/0
n(A) = 0 otherwise
we clearly have
Z0LO = L0Z0 = 1 - P
.
0
Multiplying our equation by
Z0
and observing
(1-P0)(P-P0) = P-P0
we find P-P0 = ZO(P-P0)L0 - Z0(I-P0-Q)V(P0+Q)-
Suppose we have found an operator together with
P2 = P,
PP0 = P,
P
which satisfies this equation
P0P = P0.
(1-P0)((P-PO)L0 - (l-P)VP} _ (P-P0)L0-(1-P)VP
Setting
219
Then
L0(P-P0)
and hence
(1-P)LP - 0.
f(Q) = ZOQPOLO - Z0(1-P0-Q)V(PO+Q)
we can write our equation in the form
Q = f (Q) It is assumed, without loss of generality, that denote by and
,( ,
a8-1
and
the maximum and minimum of
9-1
respectively, where / is the complement of
to the spectrum of
1'I
E
0.
in
/0
/ 0
We
relative
It then follows that
L0.
a6-1
<
IIPOLOII
0
< 9
I1z01I
.
and
aq + 9(1+q)21Iv1I
Ilf(Q)II
IIf(Q)-f(Q')11
<
if IIQ11 < q
[a + 29(1+q)1IVII)IIQ-Q'II, if IIQII,IIQ'II _ q
.
We now require that the disturbing operator is sufficiently
sms,1: 2911+q
I
This bpund is positive because spectrum.
Then, with
/0
is an isolated segment of the
8 = aq+29(l+q)IIVII < 1,
IIf(Q)II < aq + 12a(l+q)
and
11f(Q)-f(Q')II < 81IQ-Q'II
We now perform iterations
Qn+l = f(Qn)
beginning with we require IIQn+111 < q
Q0 = 0, Q1 = -(l-P 0)VPO.
To make sure that
q(l+q) > (1-0/28; to make sure that we require
IIQn+l-QnII < e11Qn-Qn-11I
q > 1.
IIQn1I < q
I1Q11I < q implies
Having done this we find
and it is then clear that the sequence 220
On
satisfies the condition tor
Q P0 n
Qn
Suppose the operator
Q.
converges in norm to a limit operator
POQn = 0; then so does the opera-
Qn'
For then we have
Qn+l'
f(Qn)PO = Z0QnL0P0 - ZO(1-PO-Qn)V(PO+Qn)P0 = f(Qn)
POf (Qn) =
0
since
Hence the limit operator P = PO+Q
operator
P0Z0 = 0
satisfies
PP0 = P, P0P = P0'
and P0Q = 0
Q2 = 0. is equivalent with
relation
Thus
P0.
P
Q2 = QP0Q = 0.
imply P2 = P
as verified from
is seen to be a projector.
defined as. the range of
Relation
QP0 = Q, P0Q = 0, i.e. the
satisfies
Q
Relations Q = QP0
PP0P
.
The space
rA
The
PPO = is
P.
(1-P)LP = 0, now established, which is equivalent to
LP = PLP, shows that the operator
L
transforms
into itself.
.Finally, we set
U -
Clearly, a 0
+
U PO = P, U P = PO, and
onto
Q , while
U+
U+U
transforms
= 1.
a
Hence
onto
U (10.
transforms
Thus we have
established the main step of the theory outlined.
We do not intend to carry out the details of the remaining steps which consist in showing that' P = P£ PE
with respect to
E.
has a unique derivative
This can be done by arguments quite similar
to those commonly employed to prove the differentiability of a solution of an ordinary differential equation with respect to a parameter. In many respects the existence of the solution that we have proved is more important than the possibility of expanding it in a power series.
We have seen that the Hermitean character of the perturbation V
was not important; but it was important that this operator is
221
The bound depends on the width of the gap
bounded.
the spectral interval spectrum of
L0.
/
(1-a)B-1
between
considered and the remaining part of the
If the minimal bound of
V
gets too large this
remaining part of the spectrum may interfere with the part of the spectrum associated with the space
%
There is a great variety in
.
what the spectrum may suffer when this happens.
If the remaining part
Qf the spectrum is continuous, for example, it may happen that when
/
the point eigenvalues coming from
reach the continuous spectrum
they are absorbed and disappear.
38
Perturbation of Continuous Speptxa In the preceding section we have seen that the eigenspace be-
longing to a disconnected segment of the spectrum may move under perturbation, but keeps its dimension.
If this dimension is infinite
the nature of this segment of spectrum may change considerably; a point eigenvalue of infinite multiplicity may become a continuous spectrum; point eigenvalues may disappear when they come in contact with a continuous spectrum.
In general one may say that the spectrum
of an operator is very sensitive to disturbances.
There are oases,
however, where the spectrum does not change'at all under certain perturbations.
Such is the case, for example, if the undisturbed opera-
tor has a purely continuous spectrum and if the disturbing operator is sufficiently smooth in a sense to be explained.
We assume that the undisturbed operator spectrum running from
to +m.
L0
has a continuous
(Later on, we shall show that the
theory we shall develop automatically covers cases in which this spectrum covers only a finite or semi-infinite segment.)
The vectors
of the Hilbert space may, therefore, be represented by functions
@ 4 !$ (A ) such that
L0$
is represented by
X*(A):
222
.1
(A),
L00<= > A*(A) The "values" of the function
may be complex numbers; but
'
+y
may
also more generally be a complex-valued function of some accessory variables.
may itself
In fact, we may simply say that the value of
be a "vector" in an "accessory" Hilbert space.
The expressioji
is then understood as an inner product in this accessory space.
Keep-
ing this in mind, we assume for the unit form the expression
(s,
)
= 1 I*(X) I2dA;
here, and in the following, integration always extends from +m
to
unless otherwise stated.
The "smoothness" requirement we shall impose on the disturbing operator
V will be expressed by saying that V
should be represented
by an integral operator,
Vi <=.=->
1 v0,A')*(A')dA'
having a sufficiently smooth kernel.
If the values of
$
are vectors
in a Hilbert space, the values of V are bounded operators acting on vectors in this accessory space.
The smoothness condition we shall
impose on the kernel will be forhulated later on.
At present we discuss
our aim in a formal manner and assume that the disturbed operator
L=L0 tV has the same spectrum as
of the Hilbert space functions
L0.
That is, we assume that the vectors
' admit representations by square-integrable
(A), 223
0
4(A)
@ I.
such that
is represented by
Lm
a¢(A),
L@
The two representations will be referred to as the representation.
L0- 'into the
L0-
and the
L-
Our aim then is to find a transformation from the L-
representation and vice versa.
In cases in which the representing functions of two representations belong to the same class (the class of square-integrable functions in our case) it is convenient to effect the transformations with 4.
the aid of a pair of operators of
t
is the
L0- representer of
is the
of
so chosen that the
U-
U+
,
and the
L-
representer
L0- representer of
That is,
U
and
Up
'P(a) .
4, (a)
From the first formula of the first line we may conclude
since
U+L(P 4 applying
L0
on this first formula, on the other hand, we obtain L U 0
+
0 4 a0(A)
Thus,
(38.1)
U+L = L0U+
Similarly, we derive (38.2)
L4
LU
= U L0
224
.
'Y
The equivalence of these two lines yields the
from the second line. formulas
U+U
(38.3)
= 1
and
U U+ = 1
(38.4)
the fact that
which express
is the inverse of
UT
U
and vice
versa.
Our aim now is to find a pair of operators
U -
which satisfy
Clearly, once such a pair has been found,
relations (38.1,2,3). the transformation of "the
L0-
into the
L-
representation (or vice
versa) is established. Setting
L = L0+V
we write equations (38.1),
L0U+
- UL0 = UV
L0U
- U L0 = -VU
(38.2) in the form
This form of the equation suggests that we should first investigate the solution of the equation (LO,Z) = L0Z - ZL0 = R
where
R
is a given operator.
The theory of this equation is the
basis for the treatment of the perturbation of continuous spectra. Somewhat later on we shall describe a class of operators for which this equation has a solution. this is so. of L0.
L0
At present we shall assume that
Evidently, this solution is not unique since any function
may be added to it,, since any function of
L0
commutes with
Later on we shall select a particular such solution which has
particular desirable properties.
depends linearly on
We shall denote this solution, which
R, by
225
Z = FR Our operators
Ur
.
will be solutions of this basic equation with R
= U+V,
= -VU
R
but they will not be of the form
but rather of the form
FR±
U1+rR +
V = 0
This is natural since for as it should be.
The operators
we will have
so that
R- = 0
U
= 1
are now to be found by solving the
R±
equations
R+ = V + (rR+)V
,
-R
= V + V(rR )
We shall naturally try to solve these equations by iterations beginning with
R+ = 0,
R1 = ±V.
If we want to be sure that these iterations
converge we must introduce a class R(1)FR
and
(rR(1))R
(R)
of opdrators
belong tb this class if
such that
R
R
and
R(1)
do;
furthermore, this class should be complete with respect to a norm IIRII
for which the inequalities
IIR(1)rRIJ, II(rR(1))RIi hold.
IIR(1)II
Then the iterations will converge provided
IIRII
IIVII
<
We proceed to discuss possible classes$of operators
1.
R
having
the desired properties as described.
First of all we assume that the operators by integral operators with kernels
RY
R
r(A;A'):
when
1
or simply
R 4 r(A;A') 226
W
are represented
Evidently, the operator
is also an integral operator
L02 - ZL0
(LO.ZJ = L0Z - ZL0 if
(A-A')z(A;A')
Z is an integral operator
Z 4 ' z(A;A') The relation
(LO,ZJ = R
thus implies
(A-a')z(a;a') = r(A;a') or
z(A;A') =
Thus the kernel
z(A;A')
1r(X;a')
.
would be singular unless
r(A;A) = 0.
In
fact it is possible to introduce such singular integral operators and show that they have the desired properties provided the kernels have an appropriate smoothness property.
A smoothness property suit-
able for the purpose is H6lder continuity with respect to 1/A, and
1/A'.
A, A',
The details of justifying this statement involve con-
siderable technicalities which we do not want to describe here. denote by
!'0R
r(X;X')
the operator
Z
with the kernel
Let us
1r(x;A').
Actually, we shall not work with this solution of the equation (L0,ZJ = R; we shall rather work with the solution PR
which transforms the vector
0
into the vector
FRO
represented by
(A-a')-lr(A;A')rU(a')da'
+ iurr(A,A)I)(A)
I'R0 1
and which may be regarded as an integral operator with the sym-olic kernel 1
+ in6(X-X')1r(X;a') 227
.
rR will become
The reasons why we shall work with this operator apparent later on.
We prefer to describe in detail a different class of operators The properties characterizing this
which have the desired properties.
class will be described in terms of their Fourier transforms c
P(a:J') = 2n
r(A;A')dadA'
f
f
The relationship between
and
p
will be indicated by
R
p(a;c') We now require that the functions
f
1
p
are absolutely integrable
jp(a;c')jdada' < M
and observe that this class J of functions is complete with respect
to the norm HP11 =
2n
J
J 1P(o;')jdada'.
Let us first proceed in a formal manner, without asking under which conditions on the functions involved the operations are applicable.
If the kernel
z(a;X')
corresponds to
z(A;a') E-- L(a;a') we have, with
as = a/aa,
(a-X') z(a;A') 4--.
Hence the equation
(LQ,z] = R 228
C(c;a'), i.e. if
goes over into the equation
-i(aa= P(a;a') A solution of this equation is given by the kernel 0
C(a;a') = yp((Y;a') = i f p(a+T;a'+T)dT;
for,
may be replaced by
as+aa,
aT
and then integration with res-
pect to
T
z(A;A')
corresponding to this particular kernel
may be carried out.
[(A-A') +
(It could be shown that the kernel is exactly
yp(a;a')
and thus improper.)
The kernel
yp
belongs to a class of kernels
b
for which
the H81mgren norm
max {maax J Ic(a;a')Ida',
ma ax
J Ic(a;a')Ida}
is finite, for evidently IIYPIIl < IIPII
.
Moreover, the kernel
P(2)(a;a') = J p(1)(a;a")YP(a";a')da"
which corresponds to the operator p
and
p1
do.
IIP(2)II
R(1)rR, belongs to the class
For
J
J
f J IP(1)(c;a")IIP(a"+T;a'+T)IdTda"dada'
J J IP(1)(0;a")Ida"da JJhP(cY;G')dodcl'
= HP (1)H IIPII 229
gr
if
Assigning the norm
to the operator
iioli
IIRII = IIPi!
,
R:
'
we may write the last result in the form !JR(1)rRII
<
IIR(1)II
IIRII
I
In a similar way we derive < IIR(1)II
II(rR(1)) RII
IIRI I
.
It is then clear that the iterations set up to solve the equations
R
the norm
= V+(rR+)V
and
provided
IIRII
= -V-V(rR )
R
converge in the sense of
hull < 1.
From the fact that the kernel
YP
of the operators
has
rR
a finite H8lmgren norm we deduce that the integral operator with this kernel transforms square-integrable functions of
a
Since the square integral is invariant under
integrable functions.
Fourier transformation, we may say that the operator without referring to a kernel
Of the operator
V
addition to being in class = VU
0
are bounded.
we shall require that it be bounded in -' .
Then the operators R+ = U+V
and
holds when applied to vectors which admit
is represented by the Fourier transform
Suppose now
iaa.
need not be differentiable; still the
(a), application of
it$ representer
maintain
YP(o;o')
[L0,rR) = R
If the vector
tion of
It thus
are also bounded.
The kernel relation
= 1 + rR
U
is bounded,
rR
of this operator.
z(a;a')
follows that also the operators
R
into square-
fyp(c;(7') m((3')da'
only show that
iaa
0(o)
L0
is applicable weakly.
ia0.
,rR(P)
-
(rR)L041) _ 230
Then, we
To show this we need
We show a little more,
namely that the relation
(1)
of
is represented by applica-
admits this operator.
also admits
¢(o)
L0.
41(1)
Rf)
holds whenever
0
and
admit
0(1)
and
L0
R
is bounded.
p(CW)
end we need only approximate the kernel
in the
11
To this 11-norm by
For such a kernel the identity
a continuously differentiable one.
above can be proved in the same way as we had done it above in a formal The desired relation then results in the limit.
way.
Thus we have proved: if the vector admit
U
L0
0
admits
L0
the vectors
and relations
L0U+4) =
U+(L0+V)4)
U L0IP = (L0+V)U (D
,
hold.
Having constructed the operators they satisfy the relation
U+U
= 1.
U-
we proceed to prove that
In terms of the operators
Rr
this relation takes the form r(R+ + R ) + rR+rR
= 0
.
To prove this relation we first write down the identity
R+U
= U+VU
= U+R
or
R+ + R
+ R+rR
+ (rR+)R- = 0
.
Secondly, we make use of the identity r(R(1)rR(2) + (rR(1))R(2)) =
which we shall prove presently. R )
U+U
rR(1)rR(2)
Combining it (for
R(1)
= R+,
R(2)
_
with the previous relation we indeed obtain the desired result = 1.
To prove the above identity we observe that in terms of transformed kernels it takes the form
231
0
0
-1
11
0
0
-
f
f
P(1) (a+T';a(2))P(2) (a(2)+T(2);0'+T'+T(2)dT(2)do(2)dT,
f J
00r JJ
P(1) (a+T';a"+T')P(2) (o"+T";a'+T")dT"dT'dT" 1
which is immediately verified by taking
a" = a(2)_T', T" = T(2)+T'
and
a" = a(1)_I", T' = T(1)+T"
as new variables in the integrals on the left side.
While it is thus seen that the identity quence of the equations for
R±,
identity
= 1
U+U
U U+ = 1
is not such a
consequence unless additional requirements are imposed on requirement
IIvII
may serve for this purpose. 11 ell < IIvII
< 2
For this relation implies
{1 + IIR+II) < 2 {l + IIR+II}
hence
IIR+II
< 1
and consequently IIrR+II1
< 1
so that U+0 = 0
or
@ = rR+@
232
is a conse-
V.
The
implies
Now
4, = 0.
U(U U1) _ (U+U -1) U+ = 0 so that U+(U U+-1)4 = (U+U -1)U+l) = 0
so that (U U+-1) 4) = 0 follows for any
Having thus established the remaining relation
0.
U U+ - 1, we have attained our goal.
We just add that for practical purposes one will preferably use the expansions.
R+ = v + (rv)v + (r(rv))v + R to determine
R-
...
,
= -v + vrv - vr(vrv) +
rather than iterations.
First,'we note that the
Two additional remarks should be made.
theory developed covers the case in which the vectors
t
of the Hil-
bert space - now called
t)+ - are represented by functions
defined for
The kernel
only for
A
A
> 0 only.
> 0, A'
v(A;A')
(A)
is then also defined
> 0.
We now define
t)+
represented by functions
$
as the sub-space of
defined for all
ye(a)
A
whose vectors are from
-
to
+m
and for which
for
-ye(a) = 0 We extend the kernel
v(A;A')
A
>0
by setting
v(a;a') = 0
for
A < 0, A' < 0
Then, clearly, the theory developed is applicable provided that the 233
kernel
for
v(A;A')
> 0,
A'
> 0
is such that the extenged kernel,
a,A', is the Fourier transform of an absolutely in-
for all
defined
A
v(A,A').
tegrable kernel
We must make sure that the operators in this theory also transform vectors of
3+
which are Fourier transforms of kernels
p(a;a')
and for
A < 0
A'
21T max A,A'
lr(A;A')l
_V+
We must further show that
+.
of those kernels
which vanish
r(A;A')
< HHPII
is closed with respect to
$+
transforms
TR
HP1I.
into
if
t +
Y+.
belongs to
To this end we consider a vector
t
in the space
whose
4p(A), is of finite support and possesses a coptinuous
representer,
Then the Fourier transform
second derivative.
out at least like
o-2
as
the same propel ies as borhood of
constructed
Since
< 0.
it is clear that this class
R
into
$i+
To this end we introduce the classes
for
and rR
Rt
0
Using a vector
jol
whose representer
t(A)
A - 0, we consider the inner product
(X)
of
qt(a)
dies
with
in
is zero in a neigh-
(0-,PRO)
and show
We have
that it is zero.
0r
If m_(a)
1
-t 0
If ;_(a-T)p(a;a')4(a-T)d0da'dT I
where the interchange of the order of integration was permitted since $(o-T)
and
-(a-T)
at least like T2
decay
Fourier transform
e-iTA4,(A)
tor
while the transform
4) (T)
in
sents a vector
lb
t(T)
in
of
t _.
$(o-T)
T - -°.'
Now, the
is the representer of a veca- 1T A0-(A)
Consequently
234
as
of
o-T)
repre-
(a-T)dodo' _ (0-(T),R4)(T)) = 0
if 0_0-0p(o;o since
RO(T)
Thus, our statement
_L
Since the spaces of the vectors
, and
dense in
rR
i.e.,
'T
,rR4)) = 0
is proved.
of the kind considered are
4), P_
rR4 1
Ziit follows that
transforms
(4)
-
for all
41
in
into
This result implies that all the operators which are formed in
the process of iterations as approximations to R belong to We conclude that the operators
R and
rR-
_W +'
themselves belong toHencfe
R
into
transform vectors of
9)+.
We should like to elaborate on the condition, imposed above, that the extended kernel
v(A;A')
be the Fourier transform of an ab-
solutely integrable kernel
Since the Fourier transform of
an absolutely integrable function is continuous, this condition implies that the original kernel boundary
A = 0
and
v(A;A')
A' = 0
is continuous and vanishes along the
of its domain of definition.
Although it is not necessary
tinuity condition is quite significant. for the existence of transformations
This con-
U-
(of a wider class than here
considered), some such condition is necessary.
To show this we shall consider an example of a kernel which violates our condition, namely the kernel v(X;a') = eb(A)b(A')
where the function
b(A), defined for
and bounded, but does not vanish for
A > 0, is absolutely integrable A = 0.
Moreover, we assume
W
fIbA)I2dA = 1
We maintain that for
e > 0
the operator
L = L0+V
possesses a point
eigenvalue, so that clearly its spectrum is not simply the same as
235
A > 0.
L0, namely the semi-axis
that of
Such a point eigenvalue is
easily exhibited.
Suppose there were such a point eigenvalue
with the eigen-
AE
t'; then we would show
vector
(LO-\E) (
= -VI
or
(A-AE) SE(A) = Ecb(A) with
c=
b(A') mE(A)dX'
J
0
A < 0, we would have
Assuming
SE(A) = CE b(A) A_X
E
and hence cc
b(A') 2 A,-A
c =
Then
A = AE
dA'
E
0
satisfies the equation W 12
b(),A)
F(A) =
J
dA' = e
0
A simple discussion, using tion as
tion
F(A) A
if
to E
0.
the funA 0
to
Obviously, there is exactly one solu-
> 0.
Suppose the function
b(A)
vanishes at
A = 0
Then our theory is applicable.
a ppwer of
A, say.
tion
remains finite and the equation
F(A)
A < 0
on the left-hand side is positive and grows from
varies from
A = AE
b(0) # 0, shows that for
236
after all, like
Indeed, the func-
F(A) = l/E
has no solution
is sufficiently small.
if, c
again a solution.
If
c
is sufficiently large there is
This does not contradict our theory since we had
restricted the norm
of
JIVI,
V
to be sufficiently small, but it
shows that some such restriction is essential.
Scattering
39.
One of the important features of the theory of perturbation of continuous spectra is its role in the description of scattering.
The
process of scattering is described with the aid of the operator e
(where
may stand for the Hamiltonian energy operator of quantum
L
theory).
The aim is to relate the limit that this operator approaches
(the time) tends to
as
t
to
+W.
-itL
-w
with the limit approached as
Note that the operator
(t) =
a-itL
t
tends
gives the solution
e-itLIV (0)
of the Schr8dinger equation
i !Lit t t= L4, (t) Actually, the operator
eitL
does not approach a limit as
t -* ±=, but the operator
eitL0e-itL
does.
If a spectral representation of +i tL
of e e +ita
0
L0
is employed, application
is represented by multiplication by the "phase factor"
, which for the present problem is insignificant. +itLO e- itL The operator e can easily be described with the
aid of the operators
U
Since
L = U L0U+ 237
we have
f(L) = U f(LO)U+ and hence
e
itL
Oe-itL = e
itL
0
-itL
U- e
0 U+ = S(t)U+
with
S(t) = e
itLO
-
-itL
U
e
0
rR
= 1 + F tR
0
where -
rtR
itL
= e
itL
_
e-
0..
It is easy to give the Fourier transform of the kernel of the operator rtR .
Since the Fourier transform of the kernel of any operator
ff eiaa-ia'Q'r(A;a')dada' = p(c;a')
1
2n
eitL0Re-itL the transform of the kernel of
2n
is
ff ei(aa-a'a')eitar(a;a')e-itx'dada' = P(a+t;cr'+t)
The transform of
r
t
R
is therefore
t
0
Ytp(o;o')
f p(6+t+T;o'tt+T)dT =
Now this kernel tends to zero as
t -
f
p(a+T;o'+T)dT
and, as
t -. M, to
Y-P(o;a') = f P(a+T;o'+T)dT This is true even with respect to the Holmgren norm: 238
R
is
t - --
as
as t - -
0
,
as is easily verified (by first'verifying this for kernels of finite Evidently,
support).
Yap(o;o') = 11(G-o')
with
u(6) = 1 p(o+T;T)dT
.
-=
kernel
y_p(o;o')
multiplication of
on a vector cf(A)
corresponding to the
r.R
Consequently, application of the operator
represented by
0
4(1)
consists in
by the factor
m(a) = f e-iavp(o)do. 1-w
The operator' S(t) = 1+rtR
,
in particular, approaches the
operator
S = 1 + r.R which consists in multiplying just by the factor
y0,(X) = 1 + m(X) for the
L0- representation.
operator".
This operator
It transforms the limit
into the limit
S.U+
U+
of
of this operator as
S. e
is the "scattering
i tL 0e-itL
as t'-
t - +-.
We recall that at the beginning of Section 38 we said that the values of the representing functions
4(A)
may themselves be functions
of some accessory variables and that then the values of the kernels v(A;A')
or
r(A;A')
are matrices or integral operators acting on 239
In this sense, then, the scat-
functions of these accessory variables.
tering operator.is also a matrix or an intergral operator acting on Aside from this, however, the
functions of these accessory variables.
scattering operator is a-function of the operator mutes with
L0
and hence com-
L0.
Frequently, in place of the operator
S.
the operator
U SoU+
is designated as the scattering operator. the disturbed operator
L
in place of
240
LSO.
It evidently commutes with
REFERENCES
Akhieser, N.I., and Glasmann, I.M. - Theorie der Linearen Operatoren im Hilbert-Raum, Akademie Verlag Berlin, 1954. Dunford, N., and Schwartz, J.T. - Linear Operators, Irlterscience Publishers, Inc., New York, 1958.
Riesz, F., and Nagy, B.S. - Functional Analysis (translation), F. Ungar Publishing Company, New York, 1955. Stone, M.H. - Linear Transformations in Hilbert Space, American Mathematical Society Colloquium Publications, New York, 1932. Taylor, A.E. - Introduction to Functional Analysis, J. Wiley & Sons, New York, 1958.
241
INDEX
adjoint, formal, 46,50ff,96,164 strict, 164,165
partial, 197ff regular, 186 in a semibounded domain, 192
Banach space, 67 bilinear form, 3,93
dimension, 13,34 domain, 163
bounded forms, 91ff,93 bounded operator, 33,95ff
eigenspace, 18 associated with an interval, 21, 128
Carleman norm, 110 Cauchy sequence,65 strong, 101 characteristic function, 28 closed subset of a normed space, 89 closure of an operator, 164 compact operator, 152 complete space, 67 completely continuous operator, 147ff,150
eigenvalue, 18 of bounded operator, 128 of differential operator, 8 improper, 19 max-min property of, 157 of a quadratic form, 5 eigenvector of bounded operator, 128 of differential operator, 8 of quadratic form, 5
extension theorem first, 69
convergence, 64ff
second, 103
in the mean,, 65 monotonic, 102 strong, 100 uniform (or in minimal norm), 100 weak, 102,150
Fischer-Riesz theorem, 79 form
almost finite-dimensional, 147,148 bilinear, 93 bounded, 91
dense subset, 33
closeable, 180
differential equations, 24ff heat equation, 24
closed, 169,180 Hermitean, 169 linear, 91
SchrBdinger equation, 24,61 wave equation, 25,58 differential operators
with discrete spectra, 206
unit
Fourier integral, 9ff transformation, 83,84
242
Fourier series, 7ff,82
measure function pair, 42
function spacc,14
minimal extension, 167
completion of, 74 functional calculus, 22ff
moklifiers, 190 monotonic convergence, 102
of bounded operators, 116 functional representation, 14
Neumann, F. von, theorem of, 168
non-bounded operator, 163ff graph of an operator, 168 Hermitean, 48,50
norm, 31ff Carlemann, 109 Hilbert-Schmidt, 109 H8lmgren, 109
Hilbert space, 67 pre-, 69
normal operator, 140 Hilbert-Schmidt bound, 106 norm, 109
H8lmgren bound, 106 'norm, 109
normed space, 31ff operator, 15ff of almost finite rank, 152 bounded, 33,95ff
ideal elements, 69ff,75
closed, 164
inner product, 35ff,39 norm of, 39
closure of, 164 compact, 152
completely continuous, 147ff,150
integral operator, 103ff,152
convergence of, 100
differential - see differential operator
kernel, Dirac, 115 ideal, 112
with discrete spectrum, 143ff
of integral operator, 104 Kronecker symbol, 2
domain of, 163 of finite rank, 152,207
Laplacean, 201
formally self-adjoint, 46ff graph of, 168 integral,. 103ff,152
linear form, 91
inverse bounded, 172 linear space, 12
maximal or weak extension of, 167 minimal or strong extension of,
matrix, Hilbert, 111 infinite, 11
167
non-bounded, 163ff normal, 140
max-min property of eigenvalues, 157
maximal extension, 167
polynomial, 116ff associated with a quadratic form,
measure function, 41
scattering, 239
5
strictly self-adjoint, 168
carrier of, 43 243
unitary, 141,142
complete, 67
Volterra, 112
eigen-, 18,21 function, 14
operational calculus, 22ff
Hilbert, 67
orthogonal, 54,55 complement, 88,90
_f2, 68 linear, 12 normed, 31ff
projector, 56 perturbations of continuous spectra, 223ff
of discrete spectra, 213ff
special Hilbert space, 68 spectral projector, 28 spectral representation, 16ff of a bounded Hermidean operator,
piecewise continuous function, 41
132ff, 139
Poincare's inequality, 208 principal axis, 4
of an operator with essentially discrete spectrum, 146 minimal, 139
projection theorem, 85
of a quadratic form, 5
projectors, 26ff
Pythagorean theorem, 67
spectral resolution of Hermitean operator, 121,128 of an operator, 30,128 of a self-adjoint operator, 174ff
quadratic form, lff,36
spectral variable, 17 Rejto, P., 80
spectrum Riemann-Lebesgue lemma, 83
continuous. 12,222 discrete, 12,143,206
Riesz, F., 117
essentially discrete, 144 perturbation of, 213ff of a unitary operator, 141
scattering, 237ff scattering operator, 239
Schwarz inequality, 36 self-adjoint operator (formally), definition; 46ff
sub-inner product, 169 sub-norm, 169 support, 44
role in physics, 58ff
spectral resolution of, 174ff
triangle inequality, 32
(strictly), definition, 168
semi-norm, 32
unit form, 37
separable space, 35
unitary operator, 141,142
space
Weierstrass approximation theorem,
Banach, 67
120
closed, 89
244
