K. 0. Friedrichs
Spectral Theory, of Operators in Hilbert Space
[S1 Springer-Verlag New York Heidelberg - Berlin 1973 ...
36 downloads
1057 Views
2MB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
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,
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
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
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
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
$.
0.
means Let
{0v}
Ov
+ 0
as
v + m
be the Cauchy
sequence consisting of the zero vectors, 0v - 0, then we have 110v - 0vll - 11011 {0v} + 0. Hence and {Ov} give the same ideal
71
element; but
was identified with
{0v} _ {0,0,0 ...}
0.
of
We have now come to the conclusion that the extension is a linear inner product space.
We still must show that this
space is complete. Let
Then let
iv
be a Cauchy sequence of ideal elements in
{0'v}
be an element in
such that
'j)'
Ilmv -
4Pv`I
p .
1/v
From the triangle inequality we have
Iliv - P II
so that
{mv}
k(a).
(a)
the vector obtained from by
0
-
and denote by
in the above inequality we find k = k(a)
Mil (P
B > a > a(e).
0,0.... }
and hence, as
E
verge to a limit vector components
Ka)
for
e
0.
The last term, therefore, tends to
2
{ 11
first term tends to zero as
m + m
since the difference quotient
1$(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)
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
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
$,V'
0, or,
b ± B
-b < B < b
(19.11)'
Theorem 19.2 can then be stated as saying that (19.11)' implies
(19.1)'
IIBII < b.
There are many ways of proving Theorem 19.2.
One concise
proof is based on the identity
2b(b2-B2) = (b-B) (b+B) (b-B) + (b+B) (b-B) (b+B)
,
and the resulting identity
2bfb2(0,0) - (B0,Bf))
_ ( (b-B) 41, (b+B) (b-B) O) + ((b+B) 9,
(b-B) (b+B) 4) .
The right hand side is non-negative since the operators non-negative.
Hence the left hand side is non-negative.
(19.1)' unless
b - 0.
any positive
If
bl; hence so does (19.12).
0
0.
for
a < T implies that
Since evidently
11Bt - Ball < 2b
we
may apply Theorem 19.1 which gives
(((Bt - Ba)Q((2 < 2b($,(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
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
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)
f-($11) for eigenvectors
of
4
9; for, these eigenvectors are also eigen-
vectors of the closure of 9 so that the statement for closed intervals can be employed.
Now we take any eigenvector
@
of 5
for which
.
and prove that
(4),f(B)4)) = f-(010)
0 = 0.
To this end we take any closed interval
of 9 and set
nJ (B) = Qa.
7a
in the interior
Clearly, the eigenvectors of
Qa
are
a
eigenvectors of P1 and, since eigenvectors
0
f(B) - f_
is nonnegative for these
we have
0 < (Qal, (f(B) - f_)(204))
f 0 = P. Similarly one proves
for all
< f+(0,0)
(4),f(B)0)
and thus the statement. can be proved.
The following converse of rule III
III
for
Suppose the function
:
in 5 and outside of
B
moreover that
0
_St
0
for
B
outside of
5.
Suppose
is a vector for which
(41,f(B)0)/(m,4') Then
has its values inrthe interval 5£'
f(B)
is an eigenvector of
lies in
SE'.
5.
To prove it one need only apply the second part of rule III) to the two open sets complementary to the closed set
1 to obtain
the converse of the first.part, and conversely apply the first part of the two closed sets complementary to the open set
9 to obtain the
converse of the second part.
We shall not give the details.
The first part of rule-III-,, when applied to an interval 5
130
B0, yields the statement that, for
that consists just of one point
all eigenvectors of this point interval,
- ($, (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
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
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
v£. KlAv j
v = ve
can be found such that
By (25.1)0, therefore,
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
0
we therefore can find a
chosen such that
110-kIll
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)
(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
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
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< B1
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
(0,0)
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
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-
$(x)
DO(x)
'a'.
which are
is square-integrable will be
Q)'; the space of functions in
port will be denoted by
Q i
Z'
with bounded sup-
With reference to the unit form
(0,0) = j I0(x)I2d:
the operators
D
in
Z'
and
-D
in
'3 )'
are formally adjoint to
each other since evidently
(Tf DO'(x) + D-TY O'(x))dx = 0
(0,Dm') + (D$,$') = J
186
for
m
')'
in
and
in' Z1.
Q'
Clearly; the domain the Hilbert space
and hence also
Sp', is dense in
since this space was defined by extension from
C)
the space of piecewise continuous functions with bounded support and 'ti
this space in turn obviously contains
We therefore know
densely.
from Theorem 28.3 (Chapter VI, Section 28) that the operators -D and D
Z'
and
inli '
and
by
possess closures in extended domains which we denote respectively; the closed extensibn of
domains will also be denoted by -D
Z and
in
D.
in these
Of course, the operators
D
and
i are formally adjoint.
We now maintain that these two spaces same.
D
Z
and
In other words, in the definition of the space
i are the SD
by exten-
sion Of a space of smooth functions it is no restriction to require that these functions have bounded support. Theorem 32.1. Proof.
Denote by
na(x)
na(x)
and
;a(x)
the functions defined by
1
fxj < a
=0
Ixj > a
5
Ca(X) , = a+l-lxl
;
a < (xj < a+l
,
a+l < jxj.
Let
(x)
be in
1i'; then a = ;a'
is in
D;a = 4aD + (D;a) . Now,
D;a < 1 - na' 1 - ;a < 1 - na.
187
Hence
'D'.
Note
IIDoa - Doll < IIVOa - CaDoll + II(1 - a)Doll
1/v.
x'
Con-
sequently, we may conclude that
1
D'jv(x-x')$(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