TRANSMUTATION AND OPERATOR DIFFERENTIAL EQUATIONS
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NORTH-HOLLAND MATHEMATICS STUDIES
37
Notas de Matematica (67) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester
Transmutation . and Operator Differential Equations R. W. Carroll Mathematics Department University of Illinois
1979
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM
NEW YORK
OXFORD
t
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ISBN: 0444853286
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Lihrary of Congress Calaloging in Publication Data
C a r r o l l , Robert Wayne, 1930Transmutation and o p e r a t o r d i f f e r e n t i a l equations. (Notas de matem&icas ; 67) (North-Holland mathematics s t u d i e s j 37) 1. D i f f e r e n t i a l equations, P a r t i a l . 2. Operator equations. I. T i t l e . 11. S e r i e s . QAl.N86 no. 67 [ Q A j 7 4 ] 5 l O l . 8 ~ [515'.353] 79-12341 ISBN 0-444 -85328 -6
PRINTED IN THE NETHERLANDS
PREFACE
With the advent of modern functional analysis and the theory of distributions it became possible in the early 1 9 5 0 ' s to systematically explore and develop the theory of partial differential equations (PDE) to a degree previously unimaginable. One important feature of this work was the determination of "natural" spaces of functions of distributions in which to pose various differential problems.
Then
linear differential operators were treated as abstract linear operators A
mapping
a domain of definition D(A) vector spaces F
and
G.
C
F
into G
for suitable locally convex topological
The abstract operational properties of
A
were studied
and existence-uniqueness theorems for example were then deduced as results in operator theory. This kind of approach was extensively developed also for certain types of nonlinear problems beginning in the early 1960's.
In theoretical and
applied work in other aspects of PDE one frequently has recourse to these now well established operational methods and distribution techniques.
It is no accident
that this material interacts naturally with various geometrical and variational points of view for example and enriches studies i n these contexts. Much of this material already appears in book form but research continues and we will report on some recent work as indicated below. One fascinating area of study which took form in these years involves abstract evolution equations of the form du/dt an
F valued function of
t
with
=
A(t)u
with
u(t) E D(A(t))
u ( 0 ) = uo.
TIere
in
gations of such problems played off operator properties of the A(t) d/dt
F.
Investi-
against
to produce a variety o f powerful results and in this spirit
one is led to study ordinary differential equations cients of the form
is
where we shall confine our
attention to families of densely defined linear operators A(t)
properties of
t+u(t)
(Dt = d/dt; u ' j )
= Diu)
(ODE)
with operator coeffi-
R. W. CARROLL
vi
where the A.(t) J
n
could be thought of as arising from differential expressions in
...,x
"space" variables
(xl,
) = x
but could of course also be more general
operators (e.g. pseudodifferential operators). type problems with Cauchy data incomplete Cauchy data usually take A (t) = I m (1.2)
d< = dt
A(t):
u(j'(0)
~ ( " ( 0 ) = uj
(0 <
uj
=
These could represent evolution
for some j
this is not necessary.
+ 7;
+
u(0)
+ can be studied where u is an
linear operator entries Aij(t).
=
1< m - 1) prescribed (or ( 0 , m-11); although we will
Similarly systems of the type
+ uo
Fm valued
m
= ((A
.(t))) with 13 To formulate guidelines under which problems of
the form (1.1) - (1.2) will be tractable let
vector and
us
A(t)
think of the A (t) j
as arising
a an where Dx = Dll...D
from differential expressions of the form Za. (x,t)DE la with Dk = a/axk. Now the Dk commute,in an obvious sense and generate groups
Tk(E) = exp ED,
in say' Co(Rn)
where
Tk(E)f(x)
= f(xl
surprisingly these features interact naturally with
d/dt
,...,xk + 5 ,...,x,).
Not
and corresponding to
the case where the a (x,t) = a (t) are independent of x one considers in ja ja general spaces F a family Ak of densely defined commuting operators generating in F with
one parameter groups Tk(C) = exp EAk
A.(t)
(t)Aa = ja Ca. (t)Ay '...Ann. Such situations as well as systems with A (t) = EaijaAa la ij to a variety of interesting solvable problems which will be discussed below.
a
J
Some siutations involving noncommutative families Ak related tractable situation occurs when may be employed. For example let F where the Riesz operators
\
=
F
= Ca
are also discussed.
L2 (Eln)
i a/axk = (-A)
and recall that
are continuous in F
and
A
=
1
+
is a
and a commuting
Ca^
(t,R)Ak with continuous jk Similar remarks gjk(t ,R) = xajka(t)Ra.
of bounded operators we can write A.(t) J
+
(-A)'
4Rk
a (x,t) = a (t) ja ja (t)RaAk where
again we have Ca (t)D: = Ca. (t)(-i)l"IR'(A - 1)la1 = 1, jka c1 ja Ja a R" = RI1.. .Rnn. Thus in terms of one nice unbounded operator A
polynomial valued operator functions t
Another
is a Hilbert space and spectral methods
densely defined positive selfadjoint operator. Assuming that
family Rk
lead
=
a apply to systems. We will present some recent results f o r ODE with operator coefficients of the types indicated and relate some of the results to formulas in
PREFACE
operational calculus.
vii
Some v a r i a b l e domain o p e r a t o r problems w i t h
Ak = \ ( t )
are
a l s o t r e a t e d as w e l l a s v a r i o u s o t h e r t e c h n i q u e s f o r h i g h e r o r d e r e q u a t i o n s .
We wish t o emphasize h e r e t h a t t h e u s e of o p e r a t o r methods t o s o l v e d i f f e r e n t i a l
problems r e p r e s e n t s a g r e a t achievement of mathematics i n t h e l a s t 25 y e a r s o r s o . I t i s n o t a n a r t i f i c a l c o n s t r u c t i o n of e f f e t e snobs t o avoid messy a n a l y s i s i n
Lp
(and g e n e r a t e p u b l i c a t i o n s ) b u t r a t h e r a s y n t h e s i s of such c o n t e x t s which i s o l a t e s t h e i m p o r t a n t n a t u r a l f e a t u r e s and produces e l e g a n t g e n e r a l theorems.
Anyone who
s t r u g g l e d w i t h d i f f e r e n t i a l problems i n t h e e a r l y 1950's s u r e l y must b e impressed and g r a t i f i e d by t h e power and economy of t h e s e methods.
Some of t h a t messy a n a l y s i s
was misguided, r e p e t i t i v e , o r unnecessary and came about because t h e wrong problem was b e i n g s t u d i e d o r t h e i m p o r t a n t f e a t u r e s were o b s c u r e .
One d i f f i c u l t y was t o
f i n d t h e r i g h t s p a c e s i n which t h e d i f f e r e n t i a l e x p r e s s i o n s would have good propert i e s when r e a l i z e d a s o p e r a t o r s i n t h e s e s p a c e s .
Today some messy a n a l y s i s i s s t i l l
u s u a l l y n e c e s s a r y (and h e a l t h y ) b u t i t can be b e t t e r f o c u s e d .
Another a s p e c t of t h e m a t e r i a l d i s c u s s e d i n t h i s book can b e i l l u s t r a t e d by t h e f o l l o w i n g examples of " r e l a t e d " d i f f e r e n t i a l e q u a t i o n s .
(1.3)
u
(1.4)
w
(1.5)
z
where
A
t
-
tt tt
2 A u = 0;
Consider t h e problems
u ( 0 ) = uo
- A'w
=
0;
w(0)
+ A2 z
=
0;
z(0) = u
=
uo;
wt(0) = 0
0
i s an a p p r o p r i a t e l i n e a r o p e r a t o r i n a s e p a r a t e d , complete, l o c a l l y
convex t o p o l o g i c a l v e c t o r s p a c e v a l u e s ) and
u
2
0
E D(A ) .
Schwartz t o p o l o g y ) , o r
F
(wherein
For example
D'
=
F
o'(TR) , w i t h
u ( t ) , w ( t ) , and
could be
E = E(R)
z(t)
take their
(= Cm(lR)
with the
A = d / d x , i n which c a s e we would be
d e a l i n g w i t h c e r t a i n Cauchy problems f o r t h e h e a t and wave e q u a t i o n s and a h a l f p l a n e D i r i c h l e t problem f o r t h e L a p l a c e e q u a t i o n .
A s w i l l be proved l a t e r , under
s u i t a b l e h y p o t h e s e s i t i s p o s s i b l e t o connect t h e s o l u t i o n s of ( 1 . 3 ) i n t e g r a l formulas of t h e form
-
(1.5) by
viii
R . W . CARROLL
(1.7)
z(t)
=
(1.8)
z(t)
=
],I
)1
where the kernels K
(1.11)
H(t.T)
=
I, and H are given by
-1 2 2tn (t
+
2 -1 T )
(and one checks easily that (1.8) follows from ( 1 . 6 ) and (1.7)).
On the other
hand in terms of operational calculus ,the solutions can be expressed formally as (1.12)
u(t)
=
2
exp(A t)uo;
w(t)
=
cosh(At)uo;
z(t)
=
exp(iAt)u
0
such that the formulas (1.6) - (1.8) are known integral transforms when replaced by a parameter
*A.
A
is
One objective in this monograph will be to report on
some relationships of this sort and show how they arise via transmutation. I n fact we develop in Chapter 2 a formalism based on the transmutation concept which connects eigenfunction transform kernels of ODE and displays the transmutation operator accordingly; numerous examples are given. As a byproduct one obtains a new derivation of inversion formulas such as the Hankel transform. Using the idea of a generalized convolution this also yields elegant expressions for the solutions of certain PDE. An abstract treatment of generalized translation operators is also given and the application of transmutation methods to problems in inverse scattering theory is indicated.
An abstract transform theory is also developed.
In order to make the abstract presentation of differential problems more accessible we have included an appendix wherein certain basic facts from functional analysis, distribution theory, etc. are outlined. Such material is of course standard knowledge for specialists but it should be helpful for graduate students or others
PREFACE
t o have i t r e a d i l y a v a i l a b l e h e r e .
ix
We s u g g e s t t h a t t h e r e a d e r g l a n c e a t t h e
appendix b e f o r e r e a d i n g t h e t e x t s i n c e s p e c i f i c r e f e r e n c e s t o t h e appendix w i l l n o t be made.
Standard n o t a t i o n s w i l l be used throughout and i n t h i s r e g a r d i f a
r e f e r e n c e t o formula chapter.
(x.y)
If a reference
while a reference
(1.y.z)
= Introduction (resp.
labeled consecutively.
a p p e a r s i n a g i v e n c h a p t e r i t means
(x.y.z)
a p p e a r s i t means formula
( r e s p . (A.y.2)) means formula
Appendix 1).
(y.2) (y.2)
(x.y)
of t h a t
of Chapter
x
of t h i s P r e f a c e
Theorems, L e m m a s , Remarks, e t c . w i l l be
We have a l s o found i t u s e f u l t o s t a t e v a r i o u s "formal"
theorems wherein n o t h i n g i s r e a l l y proved o r assumed beyond t h e formalism b u t p r e c i s e v e r s i o n s of t h e r e s u l E a p p e a r l a t e r a s theorems o r examples; such theorems a r e denoted by t h e symbol Theorem x . y ( F ) .
The a u t h o r would l i k e t o e x p r e s s h i s g r a t i t u d e t o Leopoldo Nachbin f o r h i s encouragement t o w r i t e a monograph f o r t h e Notas de Matematica s e r i e s . r e s u l t i n g c o n t r i b u t i o n i s d e d i c a t e d t o my c h i l d r e n David and Malcolm.
This
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TABLE OF CONTENTS
PREFACE
V
CHAPTER 1.
GENERAL TECHNIQUES
1.1. Preliminary results
1
1.2. General systems
12
1.3. Examples
25
1.4. Other techniques
43
CHAPTER 2.
TRANSMUTATION 2.1. Preliminary remarks and examples
69
2.2. Further examples
83
2.3. Separation of variables and transforms
98
2.4. More on kernels and transforms
117
2.5. Generalized translations
131
2.6. Miscellaneous topics
154
APPENDIX 1
181
APPENDIX 2
205
REFERENCES
209
INDEX
2 39
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CHAPTER 1
GENERAL TECHNIQUES
1.1
Preliminary results.
In order
to
deal with ODE having operator coefficients
there are many techniques available when Remark 1.9 for some references).
F
is a Banach or Hilbert space (see
However for various reasons, some of which are
connected with properties of solutions, it is essential to work in "larger" spaces
F, which for convenience will always be assumed to be complete, separated, locally convex topological vector spaces. It will be instructive to begin our discussion with some constructions having immediate application to problems of obvious interest and to proceed from such particulars to more general theorems.
For background
material on functional analysis let u s refer to Bourbaki [l; 2 1 , Carroll [ 4 ] , HorvLth 111, K&the [ l ] , Reed-Simon [l], Schaeffer [l], and Treves [l]. N
11 E C ( W ; F)
(i.e.
x E R, h
is N
h(x) E F
for x
R , and as an F valued function of
E
times continuously differentiable in the topology of
N topological vector space C (R;F)
and
6c
W.
It is known that
N C (IR; F)
=
N
C (W)
CN(R) 8 F
denotes the completion of
5N
As a
E
CN(IR)'
on compact N C (lR)
F where
in the
convergence on products of equicontinuous sets in [ 2 ; 31 and Treves [l]).
F).
is assumed t o have the topology of uniform
convergence of the functions and their derivatives of order subsets of
First let
=
N C ( a ;C)
topology of uniform X
F'
(cf. Schwartz
N
N
f" E F
(finite sum in
K C IR is compact then C (K) = C ( K ; E) has the N N natural Banach topology with C (K; F) = C (K) 6E F a s above. Let now h = 8 :f
Lh:
+
h
in CN(R)
If
gE F
where :h
Then we are able t o define <S,h>
S
E
CN(R)'
since taking
E ' C F'
<
,f'>= L CY a this asserts that V
+
=
CN(lR)
E
and
= lim a
=
V
lim Cf U
C
equicontinuous, one knows that
converges in
=
<Sxf',h>
uniformly for
F, to what we define a s
is a closed convex disced neighborhood
(nbh)
of
0 in
V).
for
Y
=
f' t B'.
But
< h , S > , since if
I: then
V
=
V
00
with
R. W. CARROLL
2
Vo = B'
(Vo
equicontinuous
-
denotes F
F
polarity) and the topology of
F'
coincides with the topology of uniform convergence on equicontinuous subsets of
F'. This leads to N Definition 1.1 If h E C (R;F)
and
S E CN(R)'
then
<S,h>
E F
=
is
well defined, Remark 1.2 where
It is of particular interest to apply this construction to the case
El
S E
=
€'(w)
Set K = {x : 1x1 CN(K)I
5u
5
is of order
€1
<S,h> E F
=
5
<
ci
m}.
Then by Schwartz [l] S E
and let h E CN(K; F).
i
and we can define
supp S C L = i x : 1x1
with
N
as above.
This will be of use when
working with hyperbolic problems (cf. Carroll 15; 6 ; 12; 13; 141 and Carroll-
[a]).
Showalter
tinuous for any h
Proof:
C
N
(L)
E
N
C (L; F)
fixed.
is a Banach space, hence barreled, and bounded sets in
Sn
consequently equicontinuous. Let :
I S E CN(L)'
ChV Q fl
v a
-+
h
I f' E
B'.
C (L; F)
of
0 in F
,h> E 2 V
for
n
=
v
(S,h)
-+
u n B'
one has the quantity n > n
( s o that
n
a
F
<S,h>
for
=
Sn E A ' )
while
can be put in any closed convex = Vo,
n
2 no,
and
a 2 uo(V).
ll arbitrarily small for n u
are
I -
for
n
2
n (u,V).
1
S E CN(Lf'
But for
sufficiently
Consequently
sufficiently large.
<S,h> : CN(L)'
space CN(L)'
[ll)
+
U)
and
C fv E V
Since our construction of : CLY(L; F)
a
by taking
fixed one can make each u n
-
This means that
also equicontinuous. Then if ha
(finite sum in
2
CN(L)'
and set A ' =
0 in CN(L)'
+
B' C F'
with
N
uniformly small for u
large so that n'<
11 SI1 2 11 in
disced nbh V c1
* F is con-
compact the composition S * <S,h> : CN(L)'
Lemma 1.3 For L C €3
QED fixed shows that
h
+
<S,h> =
is continuous one has established now that the bilinear map N
x C (L; F)
is barreled
so
+
F
is separately continuous.
But the Banach
we know for example (cf. Bourbaki [2] and Horvith
3
PRELIMINARY RESULTS
Lemma 1.4 The bilinear map
(S,h)
hn
h
+
in
Sn
<Sn,hn>
then
N
F
-+
is hypocontinuous
N
In particular this means that if hn
N
< S , h > : C (L)' X C (L; F)
C (L; F).
relative to bounded subsets of
N C (L; F)
+
-+
+
in
S
<S,h>
and
CN(L)'
is a sequence with
hn
F. We note in this respect that
in
will be a Cauchy sequence automatically and hence bounded (cf. Bourbaki [l; 21,
Carroll 1 4 1 , HorvLth [l], etc.). d/dx : CN(L)
QE
C d/dx h: Q :f
V
CN-l
(L; F)
F
QE F
CN-'(L)
-f
One remarks also that the continuous map d/dx ZhV 8 :f
defined by
v a
(dldx Q l)Chv @ :f v(y.
=
CN(L; F)
extends by continuity to be a continuous linear map
for N
2
=
-+
1. a s indicated
When working with abstract differential problems in general spaces F
it is extremely useful to utilize the concept of locally equicontinuous group or semigroup as developed by T. Komura [l] and Dembart [l] (cf. a l s o Bablola 111, Komatsu [l], Lions [l], Lb'fstrGm [ 2 ] , Miyadera [l], Oucii [I], Schwartz [ 4 1 , Waelbroeck [l], Yosida [l], etc. for various other general semigroups
-
for the
standard strongly continuous semigroups in Banach spaces see Hille-Phillips 111). We recall that a closed densely defined linear operator A generate a group T(x)
I, and
lim(T(x)
-
E
L(F)
I)f/x = Af
primarily with groups T(x) way except that
T(x)
(continuous) seminorm p
on
5
+
0 for
is only defined for x x
p(T(x)f)
x
as
T(x)T(y)
q(f)
=
T(x+y)
f E D(A).
is said to
for x,y E IR, T(0) =
We will be concerned
and simply mention that semigroups arise in the same
locally equicontinuous if
such that
provided
in F
+
T(x)f
F
E
Co(F)
2 0.
The group
for any
fE F
is said to be
while given any
there exists a (continuous) seminorm q
for
1x1
5 xo
<
m
(any € k i t e x
group is equicontinuous if this last condition hqlds for all called quasi-equicontinuous if
T(x)
p(T(x)f)
5
q(f)exp
cllxl
will do). x
E
for some
IR il
= T(x)Af
Example 1.5 Let subsets of
IR.
when
F
=
F
The
and will be (x E IR).
I n general with any of these groups or semigroups i t is well known that
AT(x)f
on
d/dx T ( x ) f =
f E D(A). Co(lR)
with the topology of uniform convergence on compact
The operator A
=
d/dx generates a locally equicontinuous group
4
R. W. CARROLL
T(x)f(y)
=
f(x+y)
which is not equicontinuous.
We will now sketch formally an important technique due to Hersh [l; 21 for treating differential equations with operator coefficients (precise theorems are spelled out when needed).
The method was first developed by Hersh for Banach spaces F
and later extended to general F
by Carroll [ 5 ; 61 using the construction <S,h>
of Definition 1.1; it has proved very useful in the study of various abstract
differential problems (cf. Bobisud-Hersh [I], Bragg 111, Carroll [ S ; 6 ; 12; 13: 1 4 ;
15; 16; 171, Carroll-Showalter [ E l , Donaldson 11; 2; 3; 4 ; 5; 7; 81, DonaldsonHersh [ 6 ] . Hersh [l; 2; 3 1 ) .
The idea goes as follows, assuming everything makes
sense. In (1.1.1) let A.(t)
=
a.
3
operator in
F
1 ajk (t)Ak
where A
is a densely defined linear
T(x) = exp Ax.
generating a suitable one parameter group
We set
m
so P(t,Dt,A)
=
1 A.(t)D:
O
and consider the abstract Cauchy problem for
t
2- 0,
J
u E Cm(F),
M. where
u
j
E
D(A ')
for suitable M
for distributions Gn(t,x),
where
0
5
6nk is the Kronecker symbol and 0 < k < m-1
+ 111, Gelfand-Silov [l; 2; 31). Gn(t,x)
Now replace A by j' n 5 m - 1 , satisfying
-D
D't,x
able distribution pairing or bracket
h(x) = T(x)u,
Z)'(R)
for large enough
but we will employ this notation anyway
E F
and appropriate h ( * ) Q
be illustrated by examples below). written as
and look
(cf. Carroll [ 4 ] , A. Friedman
since no confusion is likely to arise. Assume now such G n
E
-d/dx
There i s an obvious abuse of notation in writing
for a distribution G E n
corresponding Gn(t,*)
=
E
exist and that a suitcan be defined for t h e of the form
C4(E; F)
(this may involve growth at
m,
etc., and will
Then formally a solution of (1.2) can be
PRELIMINARY RESULTS
5
m- 1
(1.4)
u(t
since we have then formally
m- 1
1 <6nk6(*),
=
T(*)un>
=
uk
n=O for
0
5
k
k
5 m - 1 while, since
k
A T(*)un = DxT(*)un,
the differential equation in
(1.2) takes the form m
(1.6)
P(t,Dt,A)u
=
1 A.(t)D'u
j=o
=
J
m- 1
1
The key fact of course is that <(-Dx)Gn(t,.).
h(.)>
Theorem 1 . 6 ( F ) . and let
u(t),
Example 1.7
F.
>
=
AT(*)un
=
DxT(-)un
while
We summarize this in
Under the types of hypotheses indicated let the given by ( 1 . 4 ) . be well defined.
=
Gn
satisfy (1.3)
u satisfies (1.2) formally.
Then
A s a simple but important application of this technique let us apply A
generates a locally equicontinuous group
T(x)
We are led to consider
Gtt - Gxx
(1.7)
= 0;
G ( O , x ) = 6(~);
Gt(O.x)
The (unique) solution of (1.7) is evidently u
0
for a reasonable distribution pairing.
it to the problem (1.1.4) when in
T(*)u
2
E D(A )
=
0
1
G ( t , x ) = -[6(x+t) 2
we write the solution of ( 1 . 1 . 4 ) as
w(t)
=
+
6(x-t)]
T(*)uo>
and for where
6
R. W. CARROLL
upon r e s t r i c t i n g
t o be suitably f i n i t e (e.g.
t
d e f i n e d r i g o r o u s l y a s i n Remark 1 . 2 .
T < x
+
x
have s u p p o r t at
-
t = 0
and t a k e s a y
L = { x : 1x1
2
TI
and
5
T <
so
with
1x1
2
T
< G ( t , * ) , T(.)uo>
i n (1.1.12).
we o b t a i n
K = { x : 1x1
and l o o k a t
1
C (K)'
in
F.
is 9
write
xol
Dt
lim
-
where
T(x).
Dt < G ( t , . ) ,
a s indicated
Since
AG/At
w
tt
-
A2w = 0
f o r t h e s o l u t i o n of
2 C (F)
in
-
Gtt
with
Gxx = 0 ,
in this
u E D ( A 2) .
for
-
AG = G ( t , * ) -
Gt(to,')
T ( * ) u > =
c o n s t r u c t e d above i s t h e unique s o l u t i o n of (1.1.4)
w(t)
is
< G ( t , * ) , T(*)uo>
f o r e x a m p l e we w r i t e
Gt(to,*), T(-)uo>.
x
Upon e v a l u a t i o n
T(-t)]uo = cosh(At)uo
A < G ( t , * ) , T(*)uo7 = < G ( t , * ) , A T(*)uo>
s o l u t i o n of
G(t,s,x)
5
2
i t f o l l o w s by Lemma 1.3 t h a t
To show t h a t t h e w
+
2
Finally i n order t o obtain
in
w(t) = T[T(t)
Note h e r e t h a t from t h e s p e c i f i c n a t u r e of
c a s e it i s c l e a r t h a t
G(to,*)
1
G
w h e r e w e w i l l assume
t h e number a s s o c i a t e d w i t h t h e l o c a l l y e q u i c o n t i n u o u s g r o u p of
t h e b r a c k e t can b e
m)
Indeed i n t h i s event t h e &-functions i n
x
t = 0
t
w(0) = w t ( 0 ) = 0.
+
0
T(*)u >
suppose Let u s
G ( s , s , x ) = 6 ( x ) , and
1
+
G(t,s,x) = y[6(x + t - s) x+t-s S(C)dC ( i n t e g r a l i n E ' ( l R ) 6(x - t s)]. Similarly l e t H(t.s,x) = x-t+s s e e e . g . B o u r b a k i 13; 4 ; 51 o r C a r r o l l [ 4 ] f o r v e c t o r v a l u e d i n t e g r a t i o n ) b e t h e
Gt(s,s,x) = 0
(0
5
s
5
t)
so t h a t e v i d e n t l y
I
+
s o l u t i o n of
Htt
-
Hxx = 0 ,
I t i s e a s i l y checked t h a t
H ( s , s , x ) = 0 , and Hs = -G
and
G
H ( s , s , x ) = 6(x) t
= -H
tt
= -H xx.
(0
5
s
2
t).
We u s e now a t e c h n i q u e
f o r u n i q u e n e s s i n t h i s s i m p l e example which i s a s u b s t a n t i a l v a r i a t i o n of a p r o c e d u r e o f F a t t o r i n i [ l ] and H i l l e [ Z ] and was d e v e l o p e d by C a r r o l l [ 5 ; 6 ; 1 2 ; 1 3 ; 1 4 1 and C a r r o l l - S h o w a l t e r
[ 8 ] t o a p p l y t o much more g e n e r a l s e c o n d o r d e r s i n g u l a r and
d e g e n e r a t e h y p e r b o l i c p r o b l e m s ( c f . a l s o C a r r o l l [ 1 6 ; 1 7 1 , Donaldson [ 4 ; 5: 7 1 , and l a t e r s e c t i o n s of t h i s c h a p t e r ) . b2(t s) (1.8
where
We c o n s i d e r t h e f u n c t i o n
@ ( t , s )= @ , ( t , s )+
7
PRELIMINARY RESULTS
The brackets make sense as before and for completeness we will check the continuity
Q1
and differentiability of recall that
w
ss
2 A w
-
0 with
=
Q2 as functions of s
and w
E
2
C (F)
that one need only consider sequences sn
Ab2
example
-
T(*)W(S,)>
T(*)w(so)>.
~(T(x)w(s~)) <
m
2
q(w(sn))
for suitable x
G(t,so,*) Aw
=
-+
0
in
@2(t,*)
and that of
we consider first
1'
as
above).
q(w(sn))
But
K
5
q(Aw)
+
0 for
@l(t,*)
-f
T(x)Aw(s
)
will be bounded in
is clearly sufficient to verify that
F)
Thus
AG
=
+
(i.e. T(x)Aw
Then with
since
5
xo
T(x)w(sn)
G(t,sn,') 0.
is
-
Similarly for +
0 in
For differentiability
+
As
=
(a/as)@,(t,s)
T(x)w(sn).
is continuous when
=
sn - so we l o o k at
Gs(t,so,*)
(d/dx)T(x)w(sn)
as well as
Aw(-)
1x1
0. This proves the continuity of
T(*)ws(s)>.
Co(K;
(here
is a Cauchy
which formally can be written
will be bounded in C1(K; F)
)
w(sn)
follows by a similar argument.
F we have
F
on
as above and
xE K
-
T(')w(sn)>
will be bounded.
Now, with a view to applying Lemma 1.4 again, AG/As T(x)w(s
=
hence by Lemma 1.4
+
and consider for
so
+
+
for suitable compact
(a/as)@,(t,s)
sn
T(')w(so)>
< x
when T't
Co(K)';
and hence
to determine continuity, etc. of
T(*)w(s
Here we
~ ~ ( =0 0. ) First, recall
for some (continuous) seminorm q
w(sn) - w(so),p(T(x)Aw)
Co(K; F))
=
5 t).
s
is a (continuous) seminorm on
sequence in F, hence bounded, so bounded in Co(K; F)
-
)>
=
so
w(0)
(cf. Carroll [ 4 ] ) , so let
iR
functions defined on
+
and
5
(0
in
C1(K)'
= AT(x)w(sn)
and =
To see this it w(*)
and
2
A w(*)
are continuous. This "obvious" fact actually needs a little proof which we supply
for completeness. Thus (cf. T. Komura [l]) when (T(x)
-
I)w(sn)
T(n)Aw(sn)dn
= ' 0
and
x
5x
( T ( q ) - I)Aw(sn)
for example one can write ri 2 = T ( 5 ) A w(sn)dE so that
I
0
8
R. W . CARROLL
(1.10)
-
(T(x)
I)w(s ) =
T(E)ALw(so)dSdq. that
Aw(sn)
+
Aw(so)
Thus
xAw(sn)
since
A
and hence
Aw(sn)
Aw/As
-t
ws(so)
Co(K; F)
in
t o t h e desired expression.
+
< H ( t , s , * ) , T(*)wss(s)>.
Next we l o o k a t
2
Dx
( a / a s ) @ , ( t , s ) =
S i n c e now
( t , s ) = < H S ( t , s , - ) , T(*)ws(s)>
Here w e o b s e r v e t h a t f o r s u i t a b l e transport
K
H
<
= -G
G
= -H
,>
supp H ( t , s , . )
xx
we o b t a i n
C
L
C
K
we can
i n a d i s t r i b u t i o n s e n s e a s was done
$ ( t , t ) = w ( t ) = $(t,O) = 0
Therefore
and
+
a s above w i t h
around under t h e b r a c k e t
formally i n (1.6).
and hence t h e b r a c k e t t e n d s
I n view of p r e v i o u s arguments t h e r e i s o b v i o u s l y no
problem i n j u s t i f y i n g t h i s formula.
(1.11)
so
F o r t h e second b r a c k e t i n (1.9)
2 w E C (F)
since
w(sn)
Aw(*) E Co(K; F ) .
is c l o s e d and c o n s e q u e n t l y
T h e r e f o r e Lemma 1.4 a p p l i e s t o t h e f i r s t b r a c k e t . we know
converges w i t h
and
w ( t ) :0.
This
concludes o u r d i s c u s s i o n o f Example 1 . 7 and we s t a t e t h e r e s u l t a s
Theorem 1 . 8
Let
F
be a complete s e p a r a t e d l o c a l l y convex space and l e t
g e n e r a t e a l o c a l l y e q u i c o n t i n u o u s group i n
in
C
2
(F)
i s given by
w(t)
=
F.
A
Then t h e unique s o l u t i o n of ( 1 . 1 . 4 )
cnsh(At)uo.
T h i s theorem i s a s p e c i a l c a s e of more g e n e r a l r e s u l t s t o f o l l o w b u t i t h a s been proved as an i l l u s t r a t i v e example t o i n t r o d u c e v a r i o u s t e c h n i q u e s .
We w i l l g i v e
a n o t h e r such r e s u l t f o r ( 1 . 1 . 3 ) b e f o r e proceeding t o g e n e r a l e q u a t i o n s and systems
9
PRELIMINARY RESULTS
in Section 2 (see Example 1.10). Remark 1.9
Differential equations with operator coefficients and operational
differential equations in various spaces have been studied extensively in the last 25 years or so and a complete bibliography would be virtually impossible to assemble.
In addition to the references already cited in this respect, let u s
mention however a few others which contain valuable background material, provide other points of view, or have in some way influenced our thinking.
For material in
book or lecture note form, information involving both ordinary and partial differential operators in Hilbert space (linear and nonlinear) can be found in Akhiezer-Glazman 111, Berezanskij [l] , Bre>is [l],
Caldera% [l] , Coddington-Levinson
[l], Ekeland-Teman [l], Foias-St. Nagy [2], Garding [l], Garnir [l], Gohgerg-Krein [l], Lax-Phillips [l], Naimark [l], Showalter [l], and Stone [I]; developments of the theory in Banach and other spaces appear in Aubin [l], Barbu [l], Berger [l], Browder [l], Colojoara-Foias [l], Courant-Hilbert [l], Daleckij-Krein [l], DunfordSchwartz [l], Duvaut-Lions [l], Ehrenpreis [l], A. Friedman [2; 31, B. Friedman [I], Goldstein [ 4 ] , Hille [ 3 ] , Hb'rmander [l], Kato [4], Krein [l], Ladas-Lakshmikantham [ 11,
Ladygenskaya-Uraltzeva [ 11, Ladygenskaya-Solonnikov-Uraltzeva
[ 21, Lakshmikan-
them-Leela [l], Lions [4; 5 ; 6; 71, Lions-Magenes [81, Martin [l], Massera-Schaeffer [l], Maurin [l; 21, Palamodov [l], Pascali [l], Schechter [l], Sobolev [l], Strauss [l], Treves [2; 3 ; 4; 51, and Zaidman [l].
Some further papers dealing with differ-
ential problems in general spaces or in distribution spaces are Carroll [l; 2; 31, Dettman [ 2 ; 41, DuChateau [l], Fattorini [2], FoiaS-Gussi-Poenaru [l], MillionEFikov [l], Ovsyannikov [l], Schwartz [ 5 ] , Sebastiao-Silva [l; 2 1 , Steinberg-Treves [l], and Treves [5; 6; 71. We will omit any discussion of hyperfunctions, ultradistributions, microfunctions, etc. beyond mention of Komatsu [2] and Shapira [l].
For
problems in abstract operator equations and abstract boundary value problems see Aubin [2], Browder 12; 5 ; 6 71, Calkin 111, Carroll 17; 27; 28; 301, Cordes [l], DaPrato 11; 2 ; 41, Dezin [l , Milman [l], Vigik [2], and Wyler [l].
For linear
abstract evolution problems and semi-groups see for example Balakrishnan [l; 2 1 , Beals [l; 2 1 , Emami-Rad [I] Feller [l], Goldstein [I; 2; 3 ; 51, Goldstein-Sandefur
R. W . CARROLL
10
[ 7 ; 8 1 , G r i s v a r d [ l ] , Hackman [l], Kato-Tanabe [l], Kato [2; 31, K i s y n s k i [l], Ladyzenskaya [ 3 ; 4 1 , Lyubi? [ l ] , P h i l l i p s [l], P o u l s e n
111,
Raskin-Sobolevskij
S o b o l e v s k i j [ l ] , Tanabe [ l ] , Vis'ik [l] , and Vizik-LadyFenskaya
[l],
Some a b s t r a c t
[3].
Cauchy problems a r e t r e a t e d by s p e c t r a l methods i n C a r r o l l [18; 1 9 ; 20; 2 1 ; 221, Carroll-Neuwirth
[ 2 3 ] , Carroll-Wang [291, Sandefur [l], and Walker
[I]. V a r i a b l e
domain a b s t r a c t e v o l u t i o n problems a r e s t u d i e d i n Baiocchi [l; 2 1 , B e r n a r d i [ l ] , Bernardi-Brezzi
[ Z ] , C a r r o l l [ 3 1 ] , Carroll-Cooper [ 2 4 ] , C a r r o l l - S t a t e
Mazumdar [ 3 2 ] , Cooper [ l ] , Kato [6; 71, and Mazumdar 11; 2; 3; 41.
[25], Carroll-
Some fundamental
p a p e r s on n o n l i n e a r e v o l u t i o n problems, n o n l i n e a r semigroups, v a r i a t i o n a l i n e q u a l i -
[I], B r & i s
t i e s , e t c . a r e Bardos-Brffzis
Crandall-Pazy [ 2 ] , C r a n d a l l - L i g g e t t Lagnese 11; 2 1 , Leray-Lions
131, Browder [ 3 ; 4 1 , C r a n d a l l [I],
[ 3 ] , Dorroh [l], Kato [ 5 1 , Y. Komura [l],
[l], and S e p a l [l].
For Sobolev e q u a t i o n s s e e t h e book
of C a r r o l l - S h o w a l t e r [8] and f o r example Galpern [l], Lagnese [ 3 ; 4 ; 51, Levine
[ 4 ] . Rao [l], Showalter [6; 7 ; 8 ; 9 ; l o ] , Sobolev [ 2 ] , Ting [l]. c a l c u l u s l e t u s mention Berg [ l ] . Boginov [l], Bozinov-Dirnovski
For o p e r a t i o n a l
[2], C h u r c h i l l [I],
Ditkin-Prudnikov [ I ] , Doetsch [l], T. Donaldson [l], E r d e l y i [ 2 ] , B. Friedman [ 2 ] , Groetsch [l], Hirschman-Widder [l], J u r y [l], Krabbe [l], Laptev [l], L i g h t h i l l [l], Maslov [ Z ] , Mikusinski [ l ] , van d e r Pol-Bremmer
[l], Rubel [l], Sneddon [l],
S t r u b l e [l; 2 ; 31, Ungar [l; 2 ; 31, Zemanian [11.
Example 1.10
We w i l l now s o l v e ( 1 . 1 . 3 ) and produce t h e c o n n e c t i o n formula (1.1.6)
r e l a t i n g s o l u t i o n s of (1.1.3)
and ( 1 . 1 . 4 ) .
Such f o r m u l a s have l e d t o t h e concept
of r e l a t e d d i f f e r e n t i a l e q u a t i o n s s t u d i e d mainly i n a c o n c r e t e a n a l y t i c a l s e t t i n g
by Bragg [l; 2 ; 3 ; 4 ; 51, Bragg-Dettman [ 6 ; 7 1 , and Dettman [l; 2 ; 3 ; 4 1 .
Such
problems a r e a l s o connected w i t h t h e i d e a of t r a n s m u t a t i o n o f o p e r a t o r s developed by D e l s a r t e [l], D e l s a r t e - L i o n s [ 2 ;
3 1 , Hutson-Pym [l; 21, L e v i t a n [l; 3 1 , Lions
[ 2 ; 3; 4 1 , MarFenko 11; 31, anctPovzner and f u r t h e r r e f e r e n c e s g i v e n . t e c h n i q u e a s i n Example 1 . 7 .
111;
t h i s w i l l be d i s c u s s e d in d e t a i l l a t e r
F i r s t l e t u s look a t (1.1.3) Thus one c o n s i d e r s
and a p p l y t h e Hersh
11
PRELIMINARY RESULTS
A solution to this is easily seen to be
x
form
1 K(t,x) 7
where
K
is given by
This can be derived for example by taking Fourier transforms in (1.12)
(1.1.9). in
G(t,x) =
to obtain
F @ ( s )=
$(s) =
2
exp(-s t)
6+ +
2-
t(0,s)
s G = 0 with
L
@(x)exp
where we use the Fourier trans-
1
=
( f @ ' ()s ) = - i s $ ( s > .
isxdx s o that
1 2 Fx K(t,x) = exp(-s t) 2
and it is known that
(cf. Titchmarsh [ l ] ) .
generate a quasiequicontinuous or equicontinuous group
Let now A
complete separated space F
(recall
5 q(f)exp
p(T(x)f)
Thus c(t,s) =
T(x)
in a
alxl in this event).
We
consider formally then as a solution of (1.1.3) (1.13)
u(t)
=
T(*)u >
=
O
I
-1
(m K(t,x)T(x)uodx
2 I-,
m
=
m
K(t,x)cosh(Ax)uodx
K(t,x)w(x)dx
= 0
0
where
is the unique solution of (2.1.4) obtained in Theorem 1.8. Because
w(x)
of the nature of
K(t,x)
=
2
(nt)-+exp(-x
/4t)
differentiate under the integral sign for Theorem 1.11
+ m
and the growth estimates
t > 0. This leads u s to state
F be a complete separated locally convex space and let A
Let
generate a quasiequicontinuous group in in C (F)
x
it is easily seen that the integral in (1.13) makes sense and one can
for T(x)u
1
as
F.
Then the unique solution of (1.1.3)
is given by (1.13).
Proof: It remains to prove uniqueness s o we look a t the solution of t
2
s
with G(s,s,x)
Then let
u E C1(F)
=
6(x)
given by
G(t,s,x)
@(t,s)
=
m
G( t ,s , x) T (x) u (s)dx. It is clear that
T[n(t
be any solution of (1.1.3) with
t,s,o (1.14)
=
1
G s = -Gt
so
formally
=
- s)]-'exp(-x
2
Gt
=
Gxx
/4(t - s ) ) .
u ( 0 ) = 0 and consider for
for
12
R . W . CARROLL
since
2 T(.)u(s)> = < G ( t , s , * ) , DxT(*)u(s)>.
make s e n s e f o r q u a s i e q u i c o n t i n u o u s
T(x)
and now from
The c a l c u l a t i o n s e v i d e n t l y $(t,s)
c o n s t a n t we h a v e
=
= 0.
$ ( t , t ) = U ( t ) = $(t,O)
QED
There is no need t o d i s c u s s a fancy b r a c k e t
<, >
h e r e i n t h e manner o f Example
1 . 7 s i n c e w e p u t e v e r y t h i n g i m m e d i a t e l y i n t h e form o f i n t e g r a l s .
This could a l s o
h a v e b e e n d o n e i n Example 1 . 7 o f c o u r s e b e c a u s e o f t h e n a t u r e of t h e d i s t r i b u t i o n s
H.
and
G
However i n g e n e r a l v a r i o u s b r a c k e t s c l e a r l y must be d e f i n e d and we
c o n t i n u e i n t h i s s p i r i t i n C h a p t e r 2.
1.2.
General systems.
F o r n o n h y p e r b o l i c problems t h e d i s t r i b u t i o n s
a r i s e i n ( 1 . 3 ) w i l l n o t h a v e compact s u p p o r t i n pairings in
x
and d i s t r i b u t i o n s i n
F
valued functions
d e l i m i t e d by g r o w t h c o n d i t i o n s a t
x
example t h e c o n s t r u c t i o n i n Example 1 . 1 0 ) .
which
and one i s o b l i g e d t o d e f i n e
x
r e l a t i v e t o s u i t a b l e s p a c e s of
< G n ( t , * ) , T(*)un>
Gn
m
(cf. f o r
I n t h i s s e c t i o n we u t i l i z e p a i r i n g s of
t h e t y p e employed by Donaldson [ 4 ; 5 ; 7 1 when
F
was a Banach s p a c e .
c o n t i n u e t o work i n g e n e r a l c o m p l e t e s e p a r a t e d l o c a l l y convex
We w i l l
F, thus generalzing
D o n a l d s o n ' s c o n s t r u c t i o n s , and w i l l s p e l l o u t t h e d e t a i l s more c o m p l e t e l y a s i n C a r r o l l 116; 171.
F o r h e u r i s t i c p u r p o s e s i t seems more a p p r o p r i a t e t o d e a l w i t h
s i n g l e e q u a t i o n s s e p a r a t e l y and t h e n e x t e n d t h e t r e a t m e n t t o s y s t e m s .
L (F)
b e t h e s p a c e o f c o n t i n u o u s l i n e a r maps
o p e r a t o r topology ( i . e . f E F). a
jk
We c o n s i d e r ( 1 . 2 )
independent of
ajk(-)
Q,
t
-t
in
Q
(with
with
Ls(F)
F
means
T(x)
in
F
Q,f
provided w i t h t h e s t r o n g +
Qf
f o r each fixed
m u E C (F)) and f o r c o n v e n i e n c e t a k e
ajk(t) =
A ( t ) Z 1; e x t e n s i o n s of t h e t h e o r y f o r c o n t i n u o u s
m
are i m m e d i a t e ( c f . Theorems 2 . 6 and 2 . 8 ) .
continuous group
+
Thus l e t
F
Let
A
generate a quasiequi-
f o r example s o t h a t i n p a r t i c u l a r
T(*) E Co(Ls(F)).
13
GENERAL SYSTEMS
b e a n y ( c o n t i n u o u s ) seminorm i n
q(u(t)
-
u(t)
+
u(to)
for
x
near
u ( t o ) ) exp a l x l
values i n
for
u E Co(F)
x
F
DtT(x)u(t)
L D(A )
F
valued function of
+
(xo
for
t
and o b s e r v e t h a t
and h e n c e
0.
-t
0
T h i s shows t h a t
and
whereas f o r
x
finite.
fixed
t
f o r a l l terms i n (1.2)
-
q(u(t)
q
in
*
x
*
0
+
V(t,x)
t o make s e n s e .
u(to)) t
+
5
t
0'
uniformly
Now o p e r a t e on
the
is continuous with
T(x)Dtu(t) =
and we want
D:T(x)u(t)
=
Let
T ( x ) E Co(Ls(F))
Next i t i s e v i d e n t t h a t
T(x)AJu(t)
and l e t
As
F.
u ( t o ) ) e x p ctlxl
(t,x)
-
p(T(x)(u(t)
On t h e o t h e r hand s i n c e
finite).
2
F
(t,x)
T(x)u(to) - T(xo)u(to).
f o r some ( c o n t i n u o u s ) seminorm
(T(x) - T(xo))u(to)
term
i s an
U(t,x)
U ( t , x ) - U( t ,xo) = T ( x ) u ( t ) - T ( x ) u ( t o )
us write p
so
U(t,x) = T(x)u(t)
Set
u(t) E
(1.2) with
T(x)
to
obtain
(2.1)
0
for
5k5
m-1
be working with
k
D U(0,x) = T(x)uk
= 0;
P(Dt,Dx)U(t,x)
t
w h e r e we h a v e w r i t t e n P(D , D ) t x
i n s t e a d of
Note h e r e we w i l l
P(Dt,A) = P ( t , D t , A ) .
as i n t h e H e r s h t e c h n i q u e i n
P(Dt,-Dx)
S e c t i o n 1.1.
Now i f ( 2 . 1 ) c a n b e s o l v e d t h e n
u ( t ) = T(-x)U(t,x)
( 1 . 2 ) s o l e t u s c o n c e n t r a t e f i r s t on ( 2 . 1 ) . regard
U ( t , . ) E Ls(O; F )
tions.
Here
Q,
in
+
Q
Ls(O: Ls(O; F)
w i l l b e d e n o t e d by
For f i x e d
means
L ( @ ; F)
Q,($)
-t
$ ( a ) >
E F
formally a
s o l u t i o n of
i t is u s e f u l h e r e t o
t
0 2 D(IR)
f o r some s u i t a b l e s p a c e
denotes
F)
w i l l be
of
Cm
test func-
with t h e s t r o n g operator topology ( i . e .
Q(@)
in
F
and s i n c e
f o r each
@ E 0). The p a i r i n g is a continuous
U(t,.)
F
03
v a l u e d f u n c t i o n w e can w r i t e integral in
F)
$(*)>
=
I,
U(t,x)@(x)dx
and t h i s c a n a l s o h e w r i t t e n f o r f i x e d
t
as
(Riemann t v p e (rT(x)@(n)dx)u(t) -m
with t h e i n t e g r a l i n n a t u r e of t h e g r o u p ;,(t,s)
and
5
Ls(F). T(x)
(t,T,s)
The s p a c e
O must b e c h o s e n i n a c c o r d a n c e w i t h t h e
and w i t h t h e growth p r o p e r t i e s of c e r t a i n m u l t i p l i e r s d e f i n e d below - examples w i l l be gi ven.
t a k e F o u r i e r t r a n s f o r m s s o l e t u s assume
C
S'
with
F
: O
+
We w i l l w a n t t o
i
i s o m o r p h i s m and i t w i l l b e u s e f u l t o make a b l a n k e t h y p o t h e s i s t h a t c o m p l e t e and h a r r e l e d .
Let
IJS
a topological
0 and
are
r e c a l l n e x t t h e P a r s e v a l f o r m u l a ( c f . S c h w a r t z [l])
R. W. CARROLL
14
=
Q,
If
spaces.
1
s
Q, E
for
= F($)
and
1
$(x)exp i s x dx =
and one h a s
given t h a t
Definition 2 . 1 * *
Q E Ls(O;
6
0
isomorphism
+
FQ
&
where
-1-
F Q, + , F-'$
L s (O ; -
Q
F)
=
8E
=
+
Q, i n
=
I$
and hence
+
0.
Then
d e f i n e d and
Q
-+
6 : Ls(O;
F)
so t h a t f o r f i x e d
Ls(O; F)
follows t h a t continuous.
[l]).
l i n e a r operator
FQ, and
by t h e r u l e
F is a topological
+
F
a Banach s p a c e and w e w i l l
4, 6
+
+
in
8
in F
so t h a t
since
Q E
Zn i s a ( u n i q u e l y d e f i n e d ) c o n t i n u o u s l i n e a r map
t h e a c t i o n of which w e d e n o t e by
F
Ls(i; F)
First l e t
F.
n
: 0
( c f . Gelfand-Silov
6.
show now t h a t i t makes s e n s e f o r g e n e r a l =
Hence v
f o r m a l l y as an F-valued
FQ
and d e f i n e
@ E 0
for
=
= Zn
T h i s d e f i n i t i o n was used by Donaldson 14; 5 ; 7 1 f o r
@,
and we o b s e r v e t h a t
F).
Q E Ls(@; F)
Let
Zn E F
=
=
$ ( - < ) e x p ( - i s < ) d c = 2n(F-l$(-x)) ( s ) .
I t i s i n t h i s way t h a t one d e f i n e s
6
a r e t h e s t a n d a r d Schwartz
-m
(FF$)(x) = 2n$(-x)
F0 =
s'
and
m
-m
on
(S
now one o b t a i n s
m
(F$) ( s ) =
S'
Q E
+
@ E 0
in
F in
+
and t h e map Ls(6;
F)
8
Consequently
is c l e a r l y onto.
one h a s
6, 6
Similarly i f
F)
Ls($;
<e,&. Q
then
+
+
Now l e t
Q,
+
Q
in
F)
in
Q, + Q
6 : Ls(O;
is w e l l
in -+
F; i t F)
Ls($;
L s ( O ; F)
is
s o we
can a s s e r t
P r o p o s i t i o n 2.2
The map
Q
-+
6 : L s ( @ ; F)
+
Ls($; F)
of D e f i n i t i o n 2 . 1 i s a topo-
l o g i c a l isomorphism f o r any complete s e p a r a t e d l o c a l l y convex
We observe i n p a s s i n g t h a t t h e s t a n d a r d formulas s e r v e d s i n c e one h a s f o r example
<(DxQ)*,$> A
=
F.
(DxQ)" = -is6, e t c . , a r e pre-
2n = -2n =
h
2n = <;,(Dx@)^> = -is. Now a p p l y t h i s F o u r i e r t r a n s f o r m to
for
U(t,.>
*
(2.2)
for
f i x e d i n (2.1) s o t h a t w r i t i n g
t
P(Dt,-is)U(t,s)
0
5
k
5
m-1.
= 0;
kA
$(t,s)
for
Fxu(t,*)
Q
-f
;
we have
n
D U(0,s) = T ( * ) u ( s ) t k
In c o n j u n c t i o n w i t h t h i s problem w e c o n s i d e r t h e o r d i n a r y
d i f f e r e n t i a l equations ( c f . (1.3))
15
GENERAL SYSTEMS (2.3)
(t,s) = 0 ;
P(Dt,-is)G
h
DtGn(O,s) kA
=
6 nk
(0
5
n,k < m - 1).
Assume that the equations ( 2 . 3 ) can be solved for
6
are multipliers in
for each
t
and suitably differentiable in
t
6 (t,s)
which
with values
A
L (4)
in
as multipliers (examples will be given later).
Then the solution of
( 2 . 2 ) can be written formally as
and taking inverse Fourier transforms we obtain the formula
(Gn(t,*)
=
F-16n(t.s))
m
mi1J n-0
Gn(t .S)T(x - 5)undS
-m
There is some conscious abuse of notation in writing integrals for distribution action but this should cause no confusion. We will continue in a formal manner to prove the following theorem; a more precise statement and a rigorous proof are contained in Theorem 2 . 6 to follow but this result seems worth stating separately. Theorem 2 . 3 ( F ) .
F
: 4
*
i
Let
0
T(x)
and
T(*)un
E
functions (i.e. the continuously on U(t,x)
t
and
D
C
C
S'
with
be a suitable group generated by a closed densely
Ls(@; F ) .
Suppose
kDtGn(t,s)
are multipliers i n
in L s ( $ ) ) .
0
$ are also assumed to be barreled
in the complete separated locally convex space F
defined operator A
II
E L(F)
test functions where
Cm
a topological isomorphism; @
and complete. Let
D(A )
be a space of
Then
t
+
G(t,s)
(t,s) E Cm(Ls(i)) for 0 5 k
with
un E
as multiplier
5
m
depending
is a solution of ( 2 . 2 ) in L
(8;
F),
satisfies (2.1) in Ls(@; F ) , and a solution of (1.2) is given formally by
the formula m- 1
=
provided
n=O
Gn(t ,S)T(-S)undS -co
T(-x)[Gn(t,*)*T(-)unl
= [Gn(t,*)*T(-)un]x=O
makes suitable sense
R. W. CARROLL
16
(1
Proof:
(A
C a l c u l a t i n g f o r m a l l y one h a s f r o m ( 2 . 6 ) m
(2.7)
P(Dt,A)u(t)
m-1
=
1
i=o
AiDt
i
1
=
A.(t) =
1
a
Ak = P j ( A ) )
jk
=
m
1 < 1 D:Gn(t,
=
n=O i = O
m- 1
s i n c e from ( 2 . 3 ) ( 2 . 3 ) we o b t a i n
P(Dt.Dx)Gn(t,x) k D u ( 0 ) = uk t
0.
=
for
0
Further because
5
k
5 m-
To o b t a i n u n i q u e n e s s t h e o r e m s f o r t h e s o l u t i o n
k DtGn(0,x)
= 6nk6(x)
1.
hy
QED
u(t)
o f ( 2 . 6 ) f o r ( 1 . 2 ) we l o o k a t
u n i q u e n e s s f o r ( 2 . 2 ) and employ a v a r i a t i o n of a t e c h n i q u e d e v e l o p e d i n C a r r o l l [ 1 2 ; 1 3 ; 141 and C a r r o l l - S h o w a l t e r [8] which was i n d i c a t e d h e r e a l r e a d y f o r t h e The u n i q u e n e s s argument i n Donaldson [ 4 ; 5 ; 7 1 f o r
wave e q u a t i o n i n Example 1 . 7 .
Banach s p a c e s i s i n c o m p l e t e ( a l t h o u g h c o r r e c t l y d i s p l a y e d i n Donaldson
[a])
and
t h e p r e s e n t e x p o s i t i o n f o l l o w i n g C a r r o l l [ 1 6 ; 1 7 1 , i n t h e r i g o r o u s form g i v e n b e l o w f o r s y s t e m s , s e r v e s t o g e n e r a l i z e and c o m p l e t e t h i s a r g u m e n t , s i n c e i n t h e p r e s e n t c o n t e x t t h e two a p p r o a c h e s a r e shown t o b e e q u i v a l e n t .
The u n i q u e n e s s q u e s t i o n h a d
b e e n open f o r a number o f y e a r s and e a r l i e r p r o o f s by Hersh [ l ; 2 1 and C a r r o l l [5: 61 r e q u i r e d ( u n n e c e s s a r y ) h y p o t h e s e s on
F'
and
We p r o c e e d f o r m a l l y a t f i r s t
A*.
h
and w i l l make e v e r y t h i n g r i g o r o u s l a t e r .
(0
5
( t , i - , s ) = 0;
P(Dt,-is)E
(2.8)
n
2
m-1;
0
5
T
< t < T < -
a s a r i s i n g from a system
(0
2
a;
Thus Let
Dkb (T,T,s) t n 0
5k5
k,n < m - 1)
=
m-1).
Gn(t,T,s)
satisfy
Ank One c a n t h i n k of t h e s e e q u a t i o n s
GENERAL SYSTEMS
+ (6
where
17
h
nk
)
i s t h e column v e c t o r w i t h e n t r i e s
kA
We are assuming
(DtGn(t,T,s)).
= A (t) Z 1
A
m
m
Ank
+
and
i s t h e column v e c t o r
G
so t h e matrix
P(-is)
has the
form
(2.10)
P(-is)
f o
1
0
...
0
0
0
1
...
0
=
'
. . . 0
-i
-A,
-A,
1 6 ( t , ~ , s ) = (G ( t , T , s ) )
kDtGn(t,7,s)
(0
column index.
with
0
5
h
-*m- 1,
k,n
5
m-1)
with
k
d e n o t i n g t h e row i n d e x and
I
a l s o depends on
t
Such a formula g e n e r a l i z e s e a s i l y t o t h e c a s e
(cf below).
I n p a r t i c u l a r , a p p l i e d t o t h e columns
h
t ( t , T [ , s ) , we w i l l have v a r i o u s formulas f o r t h e terms
of
o f which t h e most r e l e v a n t =re
(2.13)
where t i o n s of
(0
5
-6
D ~ G ~ ( ~ , T=, S ) n- 1( t , T . s )
E 0. '1.
the
and i t i s w e l l known t h a t
-+
Gn(t,r,s)
n
Then one h a s
~(T,T,s) =
lP
...
b e t h e fundamental m a t r i x w i t h e n t r i e s
( c f . C a r r o l l [ 4 ] , Schwartz [ 5 ] ) . when
1
-+
h
Now l e t
0
Let
Suppose now t h a t $(T,s)
T
kDTDtGn(t,T,s)
n < m - 1)
+
P ( - i s ) C m- 1( t , T , s )
+
C
(t,T.s) E
1
c (L~($))
be 9 s o l u t i o n of ( 2 . 2 ) w i t h
t
a s m u l t i p l i e r funcr e p l a c e d by
T
and
18
R. W. CARROLL
for 0 < k
DY(0,s) = 0
Clearly $(t,t)
=
5 m - 1.
Eo(t,t,s)6(t,s)
m-1, E G (t,T,s)D;+%(T,s) n=O n m-1, k, k~lGk-l(t,T,s)DTU(T,s) But
For
fixed consider the expression
t
while
$(t,O)
=
0. However, formally, using ( 2 . 1 3 )
( t ,T ,s ) D
can be written in the form so
F ( T ,s )
+
that (2.14) becomes mrl
(2.15)
$T(t,T)
=
6m-1 (t,T,s)[Dy + 1 Pn(-is)D:]6(T,s) n=O
which vanishes identically. Consequently G(t,s)
=
G(t,O)
=
0 and fi(t,s) E 0
Thus formally we can state (cf. also Theorem 2.8 below) Theorem 2.4(F). E
C1(Ls(o))
Assume the hypotheses of Theorem 2.3(F) and that
as multipliers. Then the u(t)
(1.2) in Cm(F)
with
T
+
En(t,T,s)
of (2.6) is the unique solution of
T(*)un E Ls(@; F).
We consider now the case of general first order systems and will be more rigorous in our treatment. Thus consider
* -t
(2.16)
where
dt
+
u(t)
=
IP(t,A):;
+
G(0) = uo
is a column vector with entries u.(t) J
i s an
mxm
matrix with entries
(0 5 j
5 m-1)
and
IP(t,A)
GENERAL SYSTEMS
k
where
5 Mij 5
and t h e
are c o n t i n u o u s .
cijk(*)
g e n e r a t e a s u i t a b l e group
A
defined
M
f o r t h e v e c t o r w i t h components
T(x)
T(x)u.(t). J -+
a s s u r e d by a p p r o p r i a t e h y p o t h e s e s on T(x)
19
u
Let t h e closed densely -t
as b e f o r e and w r i t e
We a s s u m e
(cf.
E D(A ) , w h i c h c a n b e
ui(t)
(2.22))
U(t,x) = T(x)z(t)
M
on ( 2 . 1 6 ) w i t h
and o p e r a t e
t o obtain d - t -t U(t,x) = T(x)P(t,A)u = P(t,Dx)G(t,x)
(2.18)
;
+
-t
U(0,x) = T ( x ) u o m L (Q; F)
D e n o t e by
t h e s p a c e of m - v e c t o r s
F o u r i e r transform (2.18) i n
x
w i t h components in
Ls(O; F )
and
t o get
A
d - t U(t,s
(2.19)
h
G(0,S)
=
h
for f ( t , * ) E
where
~"'(6
E(t,s)
i s an
multipliers in of
t
F).
In c o n j u n c t i o n w i t h ( 2 . 1 9 ) w e c o n s i d e r
mxm
$ f o r each
matrix.
If the entries
and b e l o n g t o
t
(which w i l l b e a s s u r e d i f
t
-+
C1(Ls($))
;..(t;)
0)
Ls(O; F)
can be w r i t t e n as
,S)U(O,S)
Oh
(2.22)
and p o l y n o m i a l s are
h
E(t
-+ U(0,s) = T ( * ) u (s) E Lz(8; F ) .
g i v e n by
are
-t
=
0
h
where
i(t,s)
m h
t h e n a s o l u t i o n of (2.19) i n
h
+ U(t ,s)
(2.21)
of
as m u l t i p l i e r f u n c t i o n s
E Co(Ls(8))
1 1 1
h
multipliers i n
iij(t,s)
A s o l u t i o n of ( 2 . 1 8 )
-t
U(t,x) = F
-1-t
U ( t , s ) = G(t,*)*u(O,.)
-t
-+
u ( t ) = T(-x)U(t,x)
in
L:(Q;
F)
is then
-+
=
and f o r m a l l y
T(-x)[G(t,*)*T(.)u
'I
i s a s o l u t i o n of (2.16)
L e t u s examine t h e c o n s t r u c t i o n s and see what c a n b e a s s e r t e d p r e c i s e l y .
First
R. 14. CARROLL
20
+ M note that ( 2 . 2 2 ) formally displays u ( f ) E F and shows that ui(t) E D(A ) + whenever the components of u belong to D(AM). Now under our assumptions + i(t,s) E Ls($m) for fixed t and we have defined a map (t(t,s), Q ( s ) ) + h
h
A
: Ls($m)
t(t,s)$(s)
; $ ( : L
X
F)
h
<$(.),iT(t,.);(*)>
$E
for
defining amap from Ls(Qm) (6(t,.)
* $(.I
+
L : ( $ ;
(OT
?? X
F)
*
6(*)) =
since F(E(t,.)
@ transpose).
denotes
L:(O;
to
Lm(O; F) s * &(t,s)d(s).
in Ls(Qm).
But under our definitions it is then clear that
cation (resp.
(resp. Ls(Lz(O;
means
$&$)
+
A
for + h
h
in
F
for
acting by multipli-
in Ls(L:(8;
F))
6E
Lm($; F); but this is equivalent to
+ Hence we can say that Et(t,*) * T(.)uo= + T(*)uo in Lz(O; F). Let us explicitly
8".
@ E
*
*T(O):~ =P(t,A)E(t,-)
P(t,Dx)E(t,*)
Ga-+6
Note here that
F))).
in Lm(i; F)
$($)
h
6,
h
<s,C$),$> + <$(a),$> A
satisfies
satisfies
A
A
+
acting by convolution) satisfies ( 2 . 2 0 ) (resp. ( 2 . 2 3 ) ) in
(6,
F))
Ls(L:(6;
(E(t,x),d(x))
We know & ( t , s )
(8")
C(t,x)
=
This corresponds to
by the action
(2.20) in L
so that
h
by the rule
F)
observe here that one can pass the verification of continuity and differentiability for t(t,s)
in the above spaces to transpose action. Thus if ?(t,s)
is a typical multiplier element and
L~($; F )
since
<;(t,s)G(s),$(s)>
we have
A h
plier implies
AVW +
o
=
in
G(t,*) E C'(L~(~)) Remark 2 . 5
Setting A;
<w,Av@>
so
~ ~ ( F8) ; as
that t
t to
+
G(t,*)
+
or
t
+
<(Aj/At)G,$>
= <;,(&/At)$>
+
€
=
Co(Ls($))
.G(t,*) E h
h
<w,vt $>
G(t,s)
Let u s note here that if
f
E
Lz(O; F)
;6
and
and applying the Parseval formula one obtains h
h
+-+
< IP(-is)T,@>
= 271<
-
CO(L~(L~(&; h
=
as a multi-
h
++ lP(Dx)T,$(-x)>
2
=
T
2
am
F))).
h
for
as a multiplier.
distribution pairings yield
(2.25)
1J
A-
h
Similarly one looks at
g..(t,s)
then G(t,s)c(s)
Ls(&; F)
E
.
= h
G(to,s)
c ( s ) = T(')uok(s)
=
then natural
t
+
E
GENERAL SYSTEMS
21
is any matrix of the form IP(t,D )
(here P ( D x )
notation i(t,s)
=
E(t,-is) A
=
FE(t,Dx)
=
above).
FE(t,x)
Similarly using the
we have
h
++ <E(t,-is)T,p = 2n<E(t,D )?,$(-x)>
(2.26)
h
2n
=
);](-x)>
=
A
+ T +
. i *(t,s)
These formulas show for example that using the multiplier "adjoints" G * (t,-is)
= 6
"adjoints"
T
m h in L (Q; F)
(t,-is)
G * (t,x)
=
L * (t,Dx)
=
=
is consistent with using the operator
T
L (t,-D
in LZ(0; F).
)
+ + Now we have shown that Et(t,.) *T(')uo =P(t,A)E(t,*) *T(*)uo in L:(Q; F ) with + + E(O,.) *T(.)uo = uo and it remains to particularize to x = 0 in order to obtain
(2.22).
It is to be noted that in this process the space
Q virtually disappears
from consideration in (2.22) but it really occurs implicitly in two ways in that T(*):(t)
L:(Q;
E
F)
and
E(t,*) E Ls(Qm)
under convolution (satisfying (2.23)).
In general there may be many suitable 0 , or perhaps none, €or which this is the The requirement T(.):(t)
case, and this question will be examined later.
E
is obviously not too much of a restriction in any event. In order to + -f + set x = 0 in U(t,x) = L(t,*) *T(.)u we want U(t,*) to be a function of x, + + say continuous, and to satisfy (2.16) we must have DtU(t,O) = [D,U(~,X)]~,~=
LZ(0; F)
[Et(t,.) E(t,*)
+ M + M m Now for uoED(A 1, T(-)u E C (F ) by assumptions and thus + + Et(t,-) * T(-)uo = IP(t,A)E(t,.) * T(*)uo involves the convolu-
* T(*):olx=O.
* T(*):o
and
tion of elements evaluation
E
L(t,-)
maps
p(h(x))
5
g..(t,*)
at
1J
x
Ci(Fm)
kp exp alxl
in
E(t,*)
5
m)
for
f E F.
Hence
=
0 will be possible for example when, under convolution,
+
Co(Fm)
where
h(.)
for any seminorm
for quasiequicontinuous groups). (N
with terms T(-)f
means
E CZ(F)
p
in
F
h(.)
E
Co(F)
(recall p(T(x)f)
Further requirements oE the type
played off against the order o f the distributions g . . ( t ; )
5 u
and q(f) expalxi
ok
in
D(AN) E(t,.)
1.l
will give other situations where evaluation would be possible (cf. A. Friedman [l], Gelfand-gilov [ 3 ] ) .
Under the present hypotheses one expects a suitable evaluation
to exist for parabolic and hyperbolic systems for example and this will be illustrated later.
For now we will simply assume
u
ok
E
D(AN)
for some N
such
R. W. CARROLL
22
that evaluation E T(*):o].
+
E[E(t,*) *T(-)uo] = DtE[E(t,.)*
Then we can state
Theorem 2.6
F
x = 0 is possible and
at
: @
8
-t
Let
be a space of
@
C"
test functions where
a topological isomorphism; 0
and complete in general.
Let T(x)
and
F with the components uok
Assume E(t,*)
evaluation E -t
at
x
= 0
maps
in the complete locally convex space
C:(Fm)
N
for suitable N
)
c1 ( L ~ ( $ ~ ) )as +
a
l*multiplierl'satisfy-
-t
*T(*)uo]
=
DtE[C(t,*) * T ( * ) c o ] .
Then u(t)
given by (2.22) is a solution of (2.16).
Remark 2.7
It was mentioned earlier that if the entries iij(t,s)
are continuous multiplier functions of then t
* iij(t,s)
The integrands values in
8
T
E
-t
since
C (Ls(8))
t
and polynomials are multipliers in
,.
h
* iLk(~,s) E Co(Ls(8))
as
and
EC1(Ls(&)) jk
multipliers and
p
T
with
(T,-is) =
in.
Hence by standard facts about
vector valued integration (cf. Bourbaki [ 3 ; 4 ; 51)
8
6
as a multiplier. To see this write from (2.20)
Z C ~ ~ ~ ( T ) ( -is ~ ~a )polynomial ~ (cf. (2.17)).
in
of t(t,s)
~~~(',-is)g~~(~,s)~(s)are continuous functions of T
and
under convolution and that
Co(Fm)
satisfies E[Et(t,.)
1
S' with
are also assumed to be barreled
belonging to D(A
E ~~ ~~ ( F) 0. ; Suppose t * i(t,s) E ~ ( * ) u
ing (2.20).
C @ C
be a quasiequicontinuous group generated by
a closed densely defined linear operator A
* of u
8
D
(A;.
Jk
/At)$
+
D
2
t jk
(t,s)$
as a multiplier.
Now for uniqueness we will give two proofs and show them t o be equivalent in the present context. The first is based on the technique of Carroll [12; 1 3 ; 141 whereas the second is the author's version of the approach in Donaldson [ 4 ; 5; 71 modeled on Carroll [16; 171.
First we consider the ordinary differential equations
( c f . (2.20))
and as with (2.12) one knows that
GENERAL SYSTEMS
Theorem 2 . 8
23
Assume the hypotheses of Theorem 2 . 6 and suppose further that as a multiplier while polynomials are multipliers in
i(t,T,s) E Co(Ls(fm)) -+
solutions satisfying T(*)t(t)
E L : ( @ ;
that
T
h
-+
be any solution of (2.19)
will show that
im
E
-+
E 0. First consider
U(t,s)
5
0
fixed and
T
5
t
so
that
+ with U(0,s)
in L z f f ; F)
n
E
* &(t,T,S)
1 C (Ls(Cm)).
A
Let now U ( t , s )
h
f.
F).
Proof: Following Remark 2 . 7 we know from ( 2 . 2 9 )
t
*
of ( 2 . 1 6 ) given by ( 2 . 2 2 ) is unique in the class of
Then the solution u(t)
for
T
$(t,T,s)
0 and we
-+ =
m h
= i(t,T,s)lj(T,s)
n
+
$(t,t,s)
=
n
-+
U(t,s)
E Ls(@; F)
-+
while
+(t,O,s)
= 0.
Let
n
and look at
T
+
q ( t , ~ )= <;(t,T,s),
$(s)>
E
To check continuity and
F.
differentiability it suffices to consider sequential AT
= T
-
T
(cf. Carroll [ 4 1 )
+
F
is separately
and in an obvious notation we write
We recall that the bilinear map continuous and since
f
(6,i)
+
:
Ls($; F)
X
f
is barreled it follows that this map is hypocontinuous (cf Horvith [l]).
relative to bounded sets in LS'\$; F)
*
Now the sequence
h
h
U(T + A T , s ) n T -+ GT(t,T,s)$(s)
is Cauchy in
im
in
Lm($; F).: and hence bounded while
ZT(t,T,s);(s)>
+
fo the first bracket in ( 2 . 3 0 ) converges to n
n
h
<;(T,S),
(AiT/AT)J
in F.
I
Similarly the sequence &/AT
is Cauchy in
Lm(f; F) and hence bounded so the second bracket in ( 2 . 3 0 ) converges to s, + in F. Hence we obtain ,
h
+T(t,T)
(2.31)
This shows that $JT(t,T)
$(t,t)
=
9
h
9
$(s)>
is continuously differentiable and from ( 2 . 1 9 ) and ( 2 . 2 9 ) yields
$
h
-b
d ( s ) > +
0. Hence the continuous function -+
=
=
*
i(s)> = $(t,O)
=
T
+
$J(t,T) is constant so that
0 for any fixed
t.
Since
$E@
is
n
+
arbitrary this implies that U(t,s) E 0 i n LZ(6; F ) . This entails uniqueness + + for ( 2 . 1 6 ) since if u ( t ) were any solution of (2.16) with u ( 0 ) = 0 and
24
R. W.
+ T(*)u(t) = $(t,.)
CARROLL h
E Lm(O; F)
+
then t h e corresponding
s a t i s f i e s (2.19)
U(t,s)
in
Lz(6; F ) .
+ U(0,s) = 0
h
Second p r o o f :
h
be any s o l u t i o n of ( 2 . 1 9 ) as above w i t h
$(t,s)
Let
+ h
and s e t
(0
5
5
t
h
T $ ( T , t , s ) = C ('r,t,-is)$O(s)
E
bm
s e e Remark 2 . 5 f o r t h e n o t a t i o n
T;
+
b0
for
6
8"
a r b i t r a r y and
fixed
T
-T
(GT(T,t,-is)= C ( T , t , s ) ) .
Then from
( 2 . 2 9 ) we have
so that in
@,$(T,t,s)
satisfies
h
Consider now
- t
At = t
For
A
-+
e ( t , ~ )= < $ ( t , s ) , @ ( ~ , t , s )E> F
which makes s e n s e f o r
fixed.
t
sequential
Using a g a i n t h e h y p o c o n t i n u i t y r e l a t i v e t o bounded s e t s i n
L r ( b ; F)
of t h e map h
(8,;)
: Ls(b; F) x
+
A
h
h
+
F
h
,$> = -
6+
PT(t,-is)$>
6
for A
+
= 0
Thus
A
= < i ? ( ~ , s$,(s)> ),
e(T,T)
"
h
any
h
+
(from ( 2 . 3 3 ) and ( 2 . 1 9 ) ) . h
= ~(o,T)=
o
h
-+
@o 6 $m
and s i n c e
=
h
i s c o n t i n u o u s and c o n s t a n t so
e(t,T)
et
b a r r e l e d one s e e s t h a t
+
is a r b i t r a r y it follows t h a t
0
U(T,s)
in
Lm(8; F )
for
QED
T.
Remark 2 . 9
The e q u i v a l e n c e of t h e s e two p r o o f s in t h e p r e s e n t c o n t e x t i s s e e n + + -+ T + from t h e f a c t t h a t $ ( t , T ) = < P ( t , T , - i s ) U ( T , s ) , @ ( s ) > = < U ( r , s ) , P ( t , T , - i s ) @ ( s ) > = A
+
8(T,t) =
+
i(t,T,s)>
where
$(s)
A
i s taken for
h
-+
io(s).
A
The t e c h n i q u e of *
t h e f i r s t proof however, as used i n C a r r o l l [12; 1 3 ; 141 w i t h s i n g u l a r o r d e g e n e r a t e problems, i n v o l v e s no r e c o u r s e t o F-valued d i s t r i b u t i o n s and could b e so a p p l i e d
.
h e r e d i r e c t l y t o (2.16).
Remark 2 . 1 0
Thus t h e e q u i v a l e n c e i s a consequence of c o n t e x t .
We n o t e t h a t t h e s t a t e m e n t of Theorem 2 . 4 ( F ) i s somewhat s t r o n g e r than
t h e theorem which would r e s u l t a s a c o r o l l a r y of Theorem 2 . 8 .
A s t r i c t corollary
25
EXAMPLES
would require
7
k * Dten(t,7,s)
need this statement for k 2.4(F).
=
E
1 C (Ls(6))
0< k,n < m - 1 whereas we only
for
0 as indicated in the formal demonstration of Theorem
This can be made rigorous following the formulation of Theorems 2 . 6 and
2.8 but we will omit an explicit statement of this fact.
1.3.
Examples. We will give first some results €or singular and degenerate problems
using the pairing of Definition 1.1 (cf. Carroll [ 1 2 ; 1 3 ; 141 and Carroll-Showalter [ S l ) and then turn t o some examples of general parabolic and hyperbolic systems
following Donaldson [ 4 ; 5; 7 1 .
The singular and degenerate problems considered do
n o t fall into the framework of Section 2 since they are not of the form (1.2)
(either A.(t)
is singular at
3
Example 3 . 1 .
t
=
0 or
Am(t)
vanishes at
Euler-Poisson-Darboux (EPD) equations.
t = 0).
As the basic equation we
consider
+ 2m+l ut t
2
-
(3.1)
u
where A
generates a locally equicontinuous group
A
2
=
A
in
tt
Rn
=
A u; u(0)
= u
0’
u (0)
= 0
t
T(x)
in
F.
The case where
has been of considerable interest and an extensive survey of such
equations appears i n Carroll-Showalter [ 8 ] (cf. also Carroll [ 3 3 ; 341 and CarrollSilver [9; 10; 111 for group theoretic aspects). > m-
-4
We will treat here only the case
and denote the corresponding solutions u
easily proved for Re m > -%
m u ; parallel results are
by
while for other values of
bit (cf. Carroll-Showalter [ 8 ] ) .
m
the theory changes a
In the study of such equations with
AL
=
A
various growth and convexity properties of the solutions were shown to be of interest and to handle this it was absolutely necessary containing objects of unrestricted growth (e.g.
F
=
E
=
to
work in spaces F
Cm(IRn))
.
Thus our
program of phrasing the theory of operator differential equations in general locally convex spaces F
has a very realistic motivation.
Banach or Hilbert spaces
will simply not suffice to naturally enrompass certain interesting types of solutions (using weighted Banach spaces suggests itself but this distorts the operator
R. W. CARROLL
26
realization A2
-
(cf. (1.2)
of
A).
Thus following the procedure of Section 1 we consider
(1.3)) 2m+l m +R t t
(3.2)
t:R
= Rm
Here G
has been replaced by
Rm(O,x) = 6(x);
*
xx'
Showalter [81 where R"
R
iyt+-
2mf1
Gm
where
=
t
0.
was called a resolvant. One can work with
ii"t +
FRm . The
=
in conformity with the notation of Carroll-
and Fourier transform (3.2) in x (3.3)
m Rt(O,x)
s2fi" =
Rm(t,-)
€
S'
to obtain
-m 0; R (0,s) = 1; $ y ( O , s ) = 0
solution of (3.3) is
Now a version of the Sonine integral formula can be employed here in writing for m >
-4
(cf. Carroll-Showalter [8]) where
1 [6(x 2
+
t)
+
6(x
-
= Cos(st)
@(t,s)
with
R-'(t,x)
=
ux(t)
=
Consequently
t)].
E'
where the integral is taken in
=
Ei. We set cm
= r(m
+ l)/r(+)r(m+ 4)
so
that (3.6) can be written (3.7)
m R (t,x) = c m
1
1 (1
-
-
c2)m-46(x
Et)dc
-1
and express a solution of (3.1)
as
1 (3.8)
um(t) =
(bracket of Definition 1.1).
T(*)uo>
=
(1 - E2)m-4<&(x
c
For m =
m
- [t),
T(x)uo>dE
-1
-4
(3.1) is a wave type equation and we
EXAMPLES
27
have already expressed the unique solution as 1.8).
u-&(t)
=
cosh At uo
(cf. Theorem
It is known that
which correspond to recursion relations
-ARmfl(t,x) 2 (mtl)
(3.11)
Rm(t,x) t
=
(3.12)
R:(t,x)
= -
in
€:(A
=
2m
2 Dx).
Similarly u
m
m-1 [R (t,x) - Rm(t,x)]
t
Consequently
2m
m-1
= - [u
t
t
- um ]
and combining this with (3.13) one obtains (3.1).
The calculations can all be justified as in Example 1.7 and we can state Theorem 3 . 2
A
Let
Then for m >
-3
generate
a locally
equicontinuous group in
(m >
-4)
resolvants Rm(t,T,x)
and
2mtl m m ytt + t yt =
6(x),
=
Sm(t,T,X)
P.ym
=
5b
For
satisfying in
0 < T
2
t
and constructs
E'
m yxx
m R~(T,T,X) = 0 ,
Carroll-Showalter [8] f o r details).
< t
E D(A2).
is answered by the following
argument (see Carroll [ 1 2 ] , Carroll-Showalter [a]).
wit11 R"'(T,T,~)
u
a solution of (3.1) is given by (3.8).
The uniqueness question for this problem
(3.14)
F with
the distributions Rm(t,'r,.)
sm(T,-r,x)
When and
T
=
0,
Sm(t,T;)
=
0 , and
St~ ( I , T , ~ ) = ~ c x ) (cf.
Sm(t,O,x) E 0 , while for
0
5
T
have supports contained In a
R . W. CARROLL
28
fixed compact set K < 2. of order -
=
{x : 1x1
5
xo}
and, with their x
derivatives in
Furthermore one has
(3.15)
m RT(t,T,x)
(3.16)
S (t,T,X) =
= -AS
m T
m
(t,T,X) 2m+l + __
m -R (t,T,X)
m
s (tJ,X)
and the constructions which follow all make sense as in Example 1 . 7 . be any solution of (3.1) with u ( 0 )
u E C2(F)
(bracket of Definition 1.1).
= u (0) =
t
Setting $(t,T) = R(t,T)
T
@(t,T)
-f
Theorem 3 . 3
Then, referring to Example 1 . 7 to justify the calcu-
(a(0)
S(t,T)
we have $(t,O)
is continuous with
$T = 0).
= 0 =
u(t)
since
u(t)
of ( 3 . 8 )
is the unique
We consider problems of the form
= 0)
s(t)ut
=
(3.16))
2 C (F).
Degenerate Cauchy problems.
+
$(t,t)
-
This proves
Under the hypotheses of Theorem 3 . 2 the
solution of ( 3 . 1 ) in Example 3 . 4
+
Thus let
0 and set
lations (cf. also Carroll-Showalter [ a ] ) , we can write (cf. (3.15)
(i.e.
D', are
+
Ar(t)u
(3.21)
utt
(3.22)
u ( 0 ) = ut(0) = 0
-
2
A a(t)n
+ b(t)u
= f
EXAMPLES
where A
29
is the generator of a locally equicontinuous group
T(x)
in F.
The
discussion here follows Carroll 113; 141 to which we refer for bibliography.
Such
problems are an extension of the classical Cauchy problem for the Tricomi equation in the hyperbolic region and thus are of interest in transsonic gas dynamics for example. Various abstract versions have been treated by Carroll [26], Carroll-Wang [29], Krasnov [I], Lacomblez [l], Walker [I], and Wang [l; 21 while some classical results can be found in Berezin 111, Bers [l], and Protter [l] (cf. also Barantzev
[I], Bitsadze 11, Brgzis [2], Carroll-Showalter [81, Friedman-Schuss [4], Schuss Following Section 1
[l], Showalter [2; 3 ; 4; 51, Smirnov [l; 21, Weinstein [l; 21.
-
(cf. (1.2) (3.23)
( .3))
Gtt
+
S(t)Gt
for G(t) E S i . with
a(0)
(3.24)
=
we replace A
-
r(t)Gx
-
by
-d/dx
+
a(t)Gxx
It is assumed that
in (3.21) and consider first
r, a, and
s,
0. Taking Fourier transforms x
ett + s(t)et
+
iyr(t)e
+
= 0
b(t)G
a(t)y2e
+
-f
b
are continuous while
a(t) > 0
(3.23) gives
y
b(t)e
=
0 A
and it will be convenient to eliminate the b(t)
term as follows. Let
G(t)
=
rt g(t)
exp joy(c)dc
(3.25)
y'
+
sy
where y
+ y2 + b
satisfies the Riccati equation
= 0;
y(0)
= 0.
In order to produce a suitable y we note that if one sets y
= ci'/ci
then
ci
satisfies (3.26)
a"
+
s(t)a'
+
b(t)ci
=
0
and we choose a
to be the unique solution o f (3.26) satisfying a ( 0 )
a ' ( 0 ) = 0. Then
y ( 0 ) = 0 and the continuous function y
< t < T < t some interval 0 -
where
then restrict our attention to
[O,T] since f o r
t
2T
a(t).
We can
the problem is not
degenerate and can be handled by the methods of Section 2. consider
1 with
will remain finite on
is the first zero of t
=
Thus on [O,T] we
30
R. W. CARROLL
i'' +
(3.27)
+
(s
2y)g'
+
(ay
2
+ iry)g^ =
0
which w e w r i t e as a system
?
We look f o r s o l u t i o n s
0<
where and
T
5t5
z^(t,T,y)
2
and
of ( 3 . 2 8 ) s a t i s f y i n g
By w e l l known theorems t h e r e e x i s t s o l u t i o n s
T.
of (3.27) s a t i s f y i n g t h e p r e s c r i b e d i n t i a l c o n d i t i o n s a t
and t h e s e f u n c t i o n s are c o n t i n u o u s i n 0
5
T
5t5
T
2T =
(3.30)
and
y E
(ay2
+
11 5 c
in particular
Lemma 3.5 0
5
T
5
t
ij
-Z
+
(s
+
+
-
T)
11 11
where
(c
Q ( T , T , ~ )= I
when
Q = (q. .)). 1J
?(t,T,y)
and
while for
Hence one can s t a t e
?(t,T,y)
e n t i r e a n a l y t i c f u n c t i o n s of
y
(t,T)
a r e continuous i n
fixed,
A
A
yY, yZ, Y t ,
of e x p o n e n t i a l t y p e
A
$
=
Cp
+
iJ, one h a s
+ ay 20 + yr@ = 0. and add t o o b t a i n
Cp"
+
(2y
+
s)$'
and
(t,T,y)
it
for
are
5 "c.
For v a r i o u s r e a s o n s we need t o e s t a b l i s h growth e s t i m a t e s on writing f i r s t
then
d e n o t e s t h e m a t r i x norm ( s o t h a t
h
y E
for
31 i f one w r i t e s t h e s o l u -
+
g ( t , - r , y ) = Q ( t , T , y ) g ( T , ~ , y ) where
1 5 11 Q((
and
y
T
f
2y)(T)f
By a n argument now i n Gelfand-Silov
The f u n c t i o n s
5T
fT =
iyr)(r)?;
exp c l y l ( t 14
and a n a l y t i c i n
t
I t i s e a s i l y shown a l s o t h a t
(c.
t i o n of (3.28) i n t h e form Q(t,T,y)
(t,T,y)
Y
(cf. Carroll [ 4 ] ) .
11
?(t,T,y)
+ ay 2 4 -
h
Y , 2, e t c . and
yrJ, = 0
with
M u l t i p l y t h e f i r s t ( r e s p . second) e q u a t i o n by
J,"
Cp'
t
EXAMPLES
31
This type of inequality can be handled by the use of Gronwall type lemmas (cf Carroll [ 4 ] ) . Thus set P on
[O,T].
ay21?12
We add
+
dc
r )y
Q
and
=
1
-
+
2(2Y
so that
s)
to the right side of (3.32) and set
r
=
191 5
[?,I2
+
that
so
t
+
T< 1
(3.33)
(a'
=
1
t
+
P1?I2dc T
dc.
T
By Gronwall's lemma (3.34)
< E(t,T)
where
E(t,S)
+ j:PIE\
'E(t,C.)dS
exp c(t - 5).
=
litl 2
Forgetting the
term in
and following a Gronwall type procedure we obtain for P
A
P
where
Lemma 3.6
2
+
r )/a.
Given
aE C
(a'
=
satisfying ?(T,T,Y)
=
2
for the moment
0
This yields 1
,
$> 0, and
b,r,s E C o ,
?
the solution of (3.27)
qt(~,~,y)= 1 it follows that for
0 with
y
real and
O < T z t< T (3.36)
I2 5
a(T)y21?(t,T,y)
i
E(t,T)exp
t
(r2/a)dS.
T
Let now
F(t,T)
=
exp(-[
t
2 (r /a)d<)
with
F(T)
=
F(T,T)
that
so
F(T)
5
F(t,T).
JT
Then since E(t,T) < exp ? T
Note that T > 0
F(T)
both
=
k
we have from (3.36)
may tend to zero as
F(T)
and
a(T)
T +
0 while
are positive.
a(T)
+
0
by assumption but for
Similarly one obtains from the above
inequalities
(3.38) where
I?,(t,T,y)
<
- , . Q(T)
=
k max a(t)
IYtt(t,T,y) I
I2a(T)F(T) on
<
[O,T].
Setting
G(T)
=
(a(T)F(T))'
from the differential equation. Now writing
one can bound c(t,T,y)
=
32
R. W .
i t follows t h a t
G(T)?(t,T,y)
and i s bounded u n i f o r m l y f o r that
?(t,T,*)
say,
IG(t,T,y)
lC(t,T,y)l
5
r e a l and
5T5
0
5
is a l s o of e x p o n e n t i a l t y p e
I
is bounded by c o n t i n u i t y i n
W(t,T,*)
0 <
i n a f i x e d compact s e t f o r
T
2
t
T.
2
ET
A s i m p l e argument shows
T.
(t,T,y)
y
r e a l with
(yI
2
R
w h i l e from t h e a b o v e
From t h e Paley-Wiener-Schwartz
F-%(t,T,y)
=
5
t
ZT. A l s o f o r
l y l > Ro.
i s bounded f o r
k'/lyl
5
i s e n t i r e of e x p o n e n t i a l t y p e
yij(t,T,y)
+
y
theorem i t f o l l o w s t h e n t h a t
Wtt
y
CARROLL
E
Ei
with
supp W
contained
Similar conclusions apply t o
w h i c h w i l l i n d e e d r e p r e s e n t t h e d e r i v a t i v e s of
W
in
€4
and
Wt
( c f Carroll 1 4 1 ) .
There r e s u l t s Lemma 3.7
W, Wt,
Under t h e h y p o t h e s e s i n d i c a t e d
s u p p o r t s c o n t a i n e d i n a f i x e d compact set f o r W(t,T,'),
W t ( t , T , * ) , and
W
tt
(t,T,.)
0
Co(F)
with
a bracket
z(t) = s(t) t 2y(t).
by
s(t)
2
f ( t ) E D(A )
while
5
T
5t2
Let
Ah(*)
and
b(t)
T.
Moreover
A h(.)
<W(t,S.*), T(*)h(c)> a s i n Section
t h e map
(S,0)
bounded s e t s i n
S E
E'
of o r d e r
<
< S , 0 > : E'x CL(F)
-f
C2(F).
2
+
Consider f o r
with F
belong t o
u'(t) =
T(
-
-f
€4
Co(F).
for
h(.) E
We d e f i n e
1 ( c f . Lemmas 1 . 3 and 1 . 4 , D e f i n i t i o n
supp S C
2
Here
compact a n d f o r
W(~,T,.)
0 E C2(F)
w i l l a g a i n be hypocontinuous r e l a t i v e t o T
> 0
Then t h e f o l l o w i n g c a l c u l a t i o n s :an b e j u s t i f i e d a s i n S e c t i o n 1.
(3.40)
( t , ~ )
a n d assume
1.1, and Remark 1 . 2 ) and t h e a r g u m e n t s p r o c e e d as i n Example 1 . 7 . corresponds t o
and h a v e
t e r m i n view o f ( 3 . 2 5 ) and
h(t) = f(t)/o(t)
2
€4
belong t o
a r e continuous with values i n
Going b a c k t o ( 3 . 2 1 ) - ( 3 . 2 2 ) now we o m i t t h e replace
Wtt
and
) h ( E ) >dS
EXAMPLES
1
t
Similarly A2u(t)
=
2
There is no trouble in passing to the limit as
f.
=
transform back to the original equation. Theorem 3.8 s
and
Let
+
Eu'
+ Aru
-
belong to
f
with
Co
= (a'
(aF)'
with
F(T)
=
exp(-(
+
2
T
* 0 and using
a(0) = 0 and
a E C 1; let
2 r )/a > 0; and choose T
(r /a)dc)
y
to
Hence
t > 0 with
act) > 0 for
t =
and we obtain u"
T(*)h(c)>dS
T
2 A au
6
33
and assume h
=
b, r,
as indicated.
f / i j E Co(F)
with
Let
Ah(*)
JT
and
F.
A2h(.)
E Co(F)
where
A
generates a locally equicontinuous group
Then, after modification by a factor expIty(6)dS,
u(t)
T(x)
in
given by (3.39) with
0
is a solution of (3.21) - (3.22) on [ O , T ] .
T = 0
A
To treat uniqueness we need some estimates f o r with
" Y
Set P
=
and arguing as before
we obtain
(a'
+
r2)y2
and
adding the term (3.45)
Z(t,r,y)
< a(T)y2
=
dS
+
1
1 - 2s as before with
x"
=
" 2 IZt1
+
ay21?I2
to the right side of (3.44) we obtain
t
T
There results
Q
P1.il2dt
+
?j
t^
T
dC.
so that
R. W. CARROLL
34
Lemma 3.9
z^
Given the hypotheses of Lemma 3.6 and
1t (T,T,Y) =
satisfying ~"(T,T,Y) = 1 with
the solution of (3.27)
0 it follows ;hat for y
real and
O c T c t c T t
Iz"(t,T,y) 1'
(3.48)
5
E(t,T)eXpj
(r2/a)dS 'I
which can be written as
F(T)[.?(t,T,y)l':
E(t,T).
iT and iT
Next one can establish estimates on
?
already obtained for and
i.
and
Reasoning as before we also conclude that
are entire functions in y
?T
from (3.30) using the estimates
of exponential type
5
A
yT
ZT and one can easily
prove then Under the hypotheses indicated Fs (T)Z = F4(~)F-lz", q(-r)ZT
Lemma 3.10
Ei
~(T)Z), and G ( T ) Y ~ belong to 0
for (t,?)
5
-f
5
T
F4Z
t
5
or
T.
The
T
with supports contained in a fixed compact set
derivatives can be taken in
GZ, aZT, and
QYT
(and
Ei
T > 0 and
for
Ei.
are continuous with values in
Now for uniqueness we extend the technique of Section 1 and define
is any solution of the modified equation (3.21) (i.e.
where u by
E(t)
s(t)
=
+
2y(t)
and
b(t)
=
0) with
f = 0.
s(t)
is replaced
By our lemma above (3.49)
and (3.50) make sense as do the calculations which follow (cf. Example 1.7). (3.51)
R
5
=
5'
Letting now $(t,c) (3.53)
$5
=
Tu>
+
=
Tu> =
=
R(t,c)
T(u"
+
+ S(t,E)
+ gu' +
-
r(E)ATu>
2
-
< Y , a(5)A
we obtain from (3.51) 2
rAu - aA u> = 0.
2
Tu>
-
(3.52)
Thus
EXAMPLES
Theorem 3.11 Let
F-'(T)U(?)
-f
u
0 and
a, b, r, s hold.
35
be any solution of the modified equation ( 3 . 2 1 ) such that ;-'(T)U'(T)
Then u
-+
0 as
T +
0 while the other hypotheses on
is unique.
6 1. 0
Remark 3.12 The condition
a' > -r2
or
allows a(t)
not to be monotone
(which had been assumed in some earlier work on such equations). t
about this and about the behavior of
2
Some remarks
as
F ( T ) = exp(-/ (r /a)dE)
T
-t
0 are
JT
given in Carroll-Wang [29]. In general the requirements of Theorem 3.11 regarding the growth of as
u'(T)
(3.21). T
+
T
-f
0 are somewhat strong although possible for suitable f
It is therefore of interest to consider the case when
and
U(T)
F(T)
+
in
0 as
and the relation of this to results of Krasnov [l] and Protter [l] has
0
been discussed briefly in Carroll-Wang [29]. u ( 0 ) = 0 and
Theorem 3.11 are only that
examine the feasibility of this let with
f
=
0, u(0)
=
0, and
u
In this event the requirements of
a-'(T)u'(T)
+
0 as
T
0. To
+
satisfy the modified equation (3.21)
u ' ( 0 ) = 0. Multiply this equation by the term
t
s(5)dE)
exp(1
and integrate to obtain
0
IM
on
[O,T], for any continuous seminorm
p
on
F
have
Now (Au
p(Au)
and
and 2
A u
2 p(A u)
belong to
will be bounded for a solution u Co(F))
and
E
2 C (F)
of (3.21)
we
36
R. W. CARROLL
whereas
fiad5
((jtadc)4)2.
=
Hence for
0
and
a-'(t)u'(t)
+
0 if
t a(c)dS)'
! ) t ( ' a
+
0. This condition is examined in
0
Carroll-Wang [ 2 9 ] and Wang [l; 21 and since oscillations in a(t)
are permitted
h
> 0 it is not automatically satisfied. by the stipulation P -
near
t
0 it is obviously valid.
F
Theorem 3.13
+
a
is monotone
Consequently F(T) > 0
Assume the hypotheses of Theorem 3.11 with
and suppose a-'(t)(jta(E)d()'
If
0 as
t
-f
0.
Then if u
on
[O,T].
on
[O,T]
satisfies (3.21) -
0
( 3 . 2 2 ) with
f
=
0 it follows that
u E 0
We go next to some examples illustrating Section 2 and recall briefly some con-
"
structions of Gelfand-Silov [l; 2; 31 (cf also A. Friedman [l]). Example 3 . 1 4 (3.59)
5
IxkDP@(x)I > A
for
The space S B y B consists of ,A
functions @
satisfying
-k-p ka PO c(A,i,@)A B k p
> B.
and
C"
For
6,p =
I,$,%,
...
the topology is defined in
terms of norms
G;Bm With this topology If a,B > 0 For
p > 1
with
a
FSa
c1
+
= S i
S
= S i
c1
= US
",A,,,
while
S:
where
Am,Bm
-t
-.
is nontrivial.
S1-'/' and i t consists of functions UP as entire functions satisfying
P
(c
is a Monte1 space and
B> 1 then FS B = -SB c 1 c 1
the space Zp
extendable to
Since
a,A
we have
is defined as
FZp P
=
zpP
= Zq
q
where
1 P
-
+ -41 = 1 .
Multipliers
37
EXAMPLES f : Sa6
-+
Sa6
are determined by functions f
is continuous and if
q > p.
for any
5
If(z)I
@
such that the map
c exp(alzlP)
then
f
+
f@ : S t
is a multiplier
+
Sa6
Zq + Zq 4
9
Zp. We mention
Evidently polynomials are multipliers in any
also that the spaces Zp are sufficiently rich in the sense P with iIg@ldx < m , for all @ E Zp implies g E 0. P The space 0
Remark 3.15
FU
= 2
k
W
K
is denoted by
in Gelfand-Silov [l; 2 ; 31 and
is the space of entire functions of exponential type satisfying
1s $(u
+
i.r)I
5
s
for
c exp alTl
=
u
+
i?.
Now to work with equations of the form ( 2 . 1 6 ) let of
and let
t
for 1
max Re Xk(s)
Q
so
that
hk(s)
max clq j
so that
and for
where
jk
follows.
5
1J
p
5m
I 5 (1
Ip..(-is)l
5
k
2
1 1 1
5
CClqjkl
2
11 011
.
P(A)
= 0.
be independent
Set
A(s)
=
the operator norm of a matrix
and we let
for large
=
The entries in the matrix P(-is)
= Zcijk(-is)k
clslp
-XI)
det(lP(-is)
and denote by
pij(-is)
05 t < T <
po
5 2
will have the form k
be the roots of
P(t,A)
p
Then
Is/.
be the highest index
IllP(-is)II(
cllslp
+
c2
m
is called the reduced order of the system and is determined as
If one writes det(lP(-is) - h I )
=
(-lImXm
+
p 1(s)Xrn-’
+ ... + p,(s)
8
with degree p (s) m
=
p
m
then po
is defined
as
max pk/k
Definition 3.16 The system is called hyperbolic if for
A(s)
5
in which case
11
(3.63)
+ lull
p(m-1)
clsl
+
.
11 5
c(1
+
1 s1
p(m-l)ebt
Is[)
The system is called parabolic if
which case ( 3 . 6 2 ) holds with
11
s = u
+
iT
with
11
exp t P(-io)
5 c2
cl; A(u)
exp t P(-is)
15 k 5 m.
for
exp t P ( - i o )
11 5
c(1
+
A(u)
5
h -cIuJ
+
c1
lul)P(m-l)exp(-atlul
in
h
).
11
< -
38
R. W. CARROLL
In the case of a hyperbolic system the elements of i(t,s) h
will be multipliers in @ = 2 the elements of
E(t,x)
< t < T < set for 0 -
@ = K =
so
D
=
exp t IP(-is)
can be taken in Section 2.
Actually
will be distributions with supports in a fixed compact + + and U(t,x) = E(t,*)*T(-)u in Section 2 is defined for
m
(t,*) are of bounded order ij -t M and T(*)C0 will belong to C (F) when the elements of uo belong to D(AM). + will be the evaluation E[E(t,.) *T(S):~] Hence for suitable assumptions on u locally equicontinuous groups T(x).
The entries g
I
appropriate in Theorem 2 . 6 (see A. Friedman [l] or Gelfand-Silov [ 3 ] for further details in this direction). > 1
For parabolic systems one knowns p
0< h
and
5
po
even while for so called Petrovskij parabolic systems h
are integers with
=
p
8
= 2’’
5
1.
= p.
If
h
p1 > po
s o @ = Zql with P1 41 is indicated. Now there is another index associated with parabolic
then the elements of 1 +
i(t,s)
will be multipliers in
1 = 1 p1 1 systems called the genus, !J, where
1 - (po
-
h)
systems this can be defined directly from A(s)
1 ~ 51
the domain
k(l
+
(01)’
one has
h(s)
5
2
!J
as the largest p -cIuI
and various useful properties of the e ements of
h
If p > 0 for example and
+ cl.
E(t,x)
h
this.
For constant coefficient
p = po/po-!J with
such that in
It can be shown that
can be established from Poh > 3‘ where 0 =
e
b^Tpo/p then (3.65)
for
I\ E(t,x) 11 5
t E [O,T].
c exp
Thus the entries in E(t,x)
are functions with exponential can be taken in the classical
growth as indicated and convolution E(t,x) *T(*)C0
A
sense for continuous function elements in for suitable bo.
T(*)Zo
of growth bounded by
exp bolxlP
Similarly there will be estimates A
(3.66)
with
11
D:D:E(t,x)
c and sj
p
(1 5
cqi exp(-~lxlP)
depending on
t
(becoming
m
as
t
-f
0).
Thus €or t > 0
EXAMPLES
* T(O):~
one can differentiate E(t,*)
T(*)Z0
39
in the classical sense for continuous
of suitable growth so that
E[E(t,*) *T(-)Zo] + To insure continuity of E(t,-)*T(*)uo
Theorem 2 . 6 .
of differentiability on
Remark 3.17
with
as
t
+
0 suitable hypotheses
have to be imposed (not s o for Petrovskij
T(*):,
parabolic systems however). c exp(-a]xI$)
will be appropriate in
Analogous remarks hold for
= h/h-v
v2
0
where
,x) 11 2
11 E(t
and estimates of the form ( 3 . 6 6 ) are valid.
Using the notions just discussed of hyperbolic, parabolic, etc. and
the formalism of Section 1.1 (cf Theorem 1 . 6 ( F ) ) Hersh [l] gives versions of the following theorems where
F
is a Banach space.
-
equation to a system transpires as in (2.8)
Here the passage from a single
(2.10) and the results will also
follow from the technique of Section 1 . 2 as indicated in the discussion just concluded following Remark 3.15. P(Dt, -Dx)
or
Recall also that the Hersh technique involves
in (2.8); one can retain the notation i(t,T,s)
P(Dt, is)
however and simply use P(is),
P.(is), J
etc. instead of B(-is),
the discussion concerning the roots Xk(s).
etc.
P.(-is),
etc. in
J
An equation is said to be hyperbolic,
parabolic, etc. if the corresponding system has these properties.
-
constructed then as in ( 2 . 3 ) (with P(Dt, is)
cf. ( 1 . 3 ) )
The
Gn
are
and the solution is
represented in the form ( 1 . 4 ) . Theorem 3.18
Let
F be a Banach space and P(Dt. -D
a strongly continuous group and blem ( 1 . 2 ) has a solution for Theorem 3.19
One notes here that
p^
> 1
M exp wlxl
t 6
E
Dm(A)
(-m,
be hyperbolic with
&(t,x)
k
given by ( 1 . 4 ) .
m)
(cf. ( 3 . 6 5 ) ) .
11 unll
P(Dt, -Dx)
t < T
decays like
exp(-klxlY)
On the other hand it is well known that
(cf Hille-Phillips [l])
so
A(0)
<
c
for
11
t < T.
where
T(x)unll:
everything fits together in (1.4).
We mention also that a system of the form (2.20) where called Petrovskij correct if
be
Then ( 1 . 4 ) represents a solution of (1.2) for for fixed
T(.)
nD (A) for example. Then the pro-
=
Under the other hypotheses of Theorem 3.18 let
parabolic of positive genus.
y =
u
)
u
i(t,s) = exp t lP(-is)
real in which case one has
is
R. W. CARROLL
40
1)
5
t(t,o)ll
u
C(1
+
IUl)h
where
such that in the domain
p(m-1).
The genus is defined as the largest
I T [ 5 k(l + 1 ~ 1 ) '
for example one has then for
t
and this leads to estimates on the G (t,.)
5
h
C
one has
A(s)
5
c.
For
!.I > 0
T
11 G(t,x)(l
permitting compositions in (1.4) of
with suitably restricted T(*)un
(cf. A. Friedman [l], Gelfand-
V
Silov [31).
It is pointed out in Hersh [ l ] (cf. also Bragg [I]) that the techniques
Remark 3.20
of Section 1.1 can be extended to situations involving several commuting operators Ai
generating groups T (x) = exp A.x. i
(The case of several commuting operators
has also been treated in Hilbert space by spectral methods in Carroll [ 4 ; 20; 2 1 ; 2 2 1 and we refer here to section 1 . 4 . )
A
and
(3.68)
(0
5
i
B
Thus let u s take two commuting operators
to illustrate the matter and set
Bk
P(Dt, A, B) = Icijk D:A'
Lm).
As in (1.3) let the distributions Gn
Then for suitable u
(0 < n< m - 1)
satisfy
the function
will formally satisfy
The bracket in (3.70) is a suitable pairing as before between Gn(t,*,*) which will be
and vector functions TA(x)TB(y)un appropriate selection of the u
.
as
E
",y
smooth as necessary upon
The proof follows the lines of (1.6) exactly.
A slightly different approach to the same kind of problems is indicated in Bragg
EXAMPLES
41
[l] where the formula
is exploited. Let for example P(D) order form
=
P(D1,
...,D
be an elliptic operator of
)
2k with real constant coefficients where the highest order term has the (-l)k-'Po(D)
degree
2k.
with
a positive definite homogeneous polynomial of
One considers the Cauchy problem
and following Rosenblum (3.74)
Po(<)
v(5,t)
[i]a
solution
K(S-U,
=
to
(3.73) is given by
t)f(U)dU
L n where
K
is a Green's function for which growth estimates are obtained.
Symbol-
ically (3.74) says
Let then
r$
(3.76)
u(t)
nD(Aik)
6
with
T.(x)
=
exp A.x
=
K(-u,t)
1
as above and consider
.
n =
exp[tP(A)]r$
Ti(Ui)r$dU.
1
L.n
This is a formal generalization of (3.72) with replaced by
Ti(Ci)
f(a)
=
nTi(ai)@
(exp aiCi
is
in (3.72)) and will agree with the corresponding result
obtained via the Hersh technique. Remark 3.21 A variation on the Hersh technique is presented in Donaldson [31. One considers a Banach space group.
A solution Y(t)
(3.77)
F
in which
A
generates a strongly continuous
of the equation
fj(tA)AjD:-jY j=O
is required. Set m consider
=
max m. J
(0 5 j
5
n)
and
y
=
m
+
n.
Let
I$ E D(Ay)
and
42
R. W. CARROLL
(3.78) over
a
Y(t)
= j:F(x)T(xt)Qdx
suitable contour from a
to
B
in t
such that
B (3.79)
{AnP(P(x),
= 0
T(xt)$ll
a where
and
P
Q(D)
is a suitable bilinear concomitant defined below (cf. Ince [l]). is the formal adjoint of
(3.81)
so
F is a solution of
d
P(u,v)
=
vQ(D)u
-
Q*(D)
=
C Ca xn-jDP
P
and
P
Thus
is defined by
uQ*(D)v
that in the present situation one can write explicitly
n
"'j p-1
1 1 1
(-l)kap{DP-k-l[T(xt)m]}D
k n-j [x F(x)]
j = O p=O k=O
Theorem 3 . 2 2 Proof:
Under the hypotheses above Y(t)
We insert Y(t)
For finite intervals
is a formal solution of ( 3 . 7 7 ) .
given by ( 3 . 7 8 ) into ( 3 . 7 7 ) to obtain
(a,B)
of
IR
there is no problem in justifying the
43
OTHER TECHNIQUES
calculation; for infinite intervals one needs some growth condition on general contours in
(c
one requires that
T(x)
F(x);
for
have a suitable analytic exten-
sion.
QED
Various applications of this formalism are given in Donaldson [3] but they are covered here by other techniques. 1 . 4 Other techniques.
I n this section we will sketch some results involving
differential equations of the form (4.1)
Ln(Dt)v(t)
=
0
We will not deal with the case n
=
1 as such since the equation v
t
+ Av
=
0 is
essentially the classical linear semigroup problem which is covered exhaustively for example in Carroll [4], Hille-Phillips [l], Yosida [l], etc.
Such results as
are needed for dealing with higher order equations will simply be cited. seem worthwhile to include here some results for the case stant Ak
and variable A (t) k
n = 2
It does
both for con-
since a lot of this has not yet appeared in book
form and we will draw upon Baiocchi [ 3 ] , Bruk [ l ] , Carroll-State [25], DaPratoGiusti [ S ] ,
Gearhart [l], Goldstein [2; 3 1 , Kato [ 6 ; 7 1 , Kleiman [l], Krein [l],
Levine [l; 2; 3 1 , Levitan [ 4 ] , Lions [4; 91, Mazumdar [4], Raskin-Sobolevskij [ l ] , Sobolevskij [2; 3 1 , Torelli [l], Treves
[a;
9 1 , Vigik [l], Vigik-Ladyyenskaya [ 3 1 ,
Yakubov [l; 2; 3 ; 4; 6 ; 7 1 , Yosida [2], etc. (cf. also Lax-Phillips [l; 2 1 ) . n > 2
For
there are methods due to Abdulkerimov [l], Bairamogly-Khalilova [l],
Carroll [la; 19; 20; 2 1 ; 221, Chazarain [ l ] , Djedour [ l ] , Dubinskij [l; 21, Gasymov [l] , Gorba$uk-Ko&bei
131, Mazya-Plamenevskij [l], Raskin "21, Yakubov
[ 5 ; 91, etc. which we will discuss and indicate results.
Finally we mention some
interesting new results on operator differential equations with noncommuting operator coefficients due to Hersh-Steinberg [ 7 ] (cf also Grabovskaya-Krein [ l ] and Krein-Fikhvatov [2]).
In fact we will begin the section with this latter
material since it seems to be of more current interest.
R. W. CARROLL
44
The Hersh-Steinberg [ 7 ] theory goes as follows.
N (4.3)
Lu
k t
1 Pi(A)DtN-i u;
=
D u(0)
i=o where
P.(A)
=
05 k
Consider for
5
N- 1
Uk
is a polynomial sum of ordered products of generally noncommuting
One densely defined linear operators A in a Banach space F (1 5 j 5 n). j are a basis for a finite dimensional Lie algebra of operators A assumes the A j over IR which is "integrable". (Standard elementary facts about Lie theory are assumed here and we refer to Helgason [ Z ] and Varadarajan [l] for details.) Explicitly
(4.4)
[Ai,Ajl
" k 1 cij\
=
k= 1
and to explain the word integrable we recall that if sentation of a Lie group then
operators on g'(0)
=
d.rr
induces a map
TI
F. Here if
aE
on
G
:IR
g
G
g :
IR
-f
G
on G.
and an operator L that
in IRn
into densely defined linear
TI
a
= g'(0)
E
9
G
of
on
g(0) = e
and
into a nbh of
The right translation Rh
so
then
that the diffeomorphism
a
such that
one defines for
A
h
j
=
E G
is defined by
is said to be right invariant if
e e
F
9.
is seen to be right invariant. Now if
local coordinate system at 0
S
is a smooth curve with
form a basis of
as above and
for smooth functions f
so
G
and a representation j
f (gh-')
is a repre-
5 . The algebra A is said to be integrable if
for all j where the a
Now with
L(F)
then
there is a Lie group
Lf(g)
-f
+
(i.,e. a continuous homomorphism into L(F))
taking the Lie algebra
is defined on a dense set of
d?T(a.) I
F
: G
TI
s =
-1
Rt,
LRhf(g)
.
\f(g) =
( s ~ , .. ,sn)
@(s)
is a
takes a nbh of
has an expression as a first order differ-
ential operator in these coordinates. Take the basis
a
of
in the form
=
45
OTHER TECHNIQUES
ak
=
ISzo
a/as,$(s)
and one has then
Indeed in a nbh of
e
one can write
f
=
f
@
0
tion of euclidean coordinates and one writes To evaluate
notation.
P,i(s)
so that
of
@-I.
(P,i(s))
m
g.
clearly n(g)y
y
One knows that
A
where
E
F
defined by
L
Cm(G)
(note that
=
'f, Lh>
a*
=
-a + c
extend to be operators on Definition 4.1
If
Q(T)S
=
for where
Cm(n)
is a func-
in standard
$;I'
denotes the
?(@-'(@(r)$(s)h)) ith coordinate
is
.rr(g)y
dll
A.A. j i
G
F and
so that
n(g)y
E
F
valued
D(A.)
while
J
a . r i ( g ) = A.n(g) J
3
DR
a i2 j
as an
so
of right invariant in this correspondence).
D(G),
E'(G),
E(G),
denote the right invariant Haar measure
L
: D(G)
f ,h E D ( G ) c
=
Cm
-f
D(G)
where
is a constant.
a (formal) adjoint is
< f , h> = IfKdu.
Evidently
a
One shows
and
5"
E'(G).
T E €'(G)
operator with domain
(4.8)
Let
is a linear operator
then easily that
a^i/as,
0 we consider
is dense in
Cm'(n)
D ' ( G ) , etc are all well defined.
Then if
where
=
such that
One has a standard notion of distribution on
G.
@-I
for
is isomorphic to the Lie algebra
differential operators on
on
r
0
A simple calculation then yields
6 D(aj).
n(g)Cm(n)
7
=
is clearly invertible.
C (IT)be the set of
function of
I$-'
af/ask
at
Irzo
a/ark@il(@(r)$(s)h)
=
The matrix
Let next
that on
a/arkf(@(r)@(s)h)
0
C
the quantization
F
Q(T)
is defined as a linear
by the rule
IT(g)n(g)Fdu. J
c This is an abuse of notation of course and means valued
E' - E
bracket relative to dli. w
easily shown that for
To
seP
this consider
€ C
(n)
ir(g)t>
for a vectnr
This conrept is very useful since it is
R. W. CARROLL
46
The property (4.9) allows one to transform (4.1) into a partial differential equation on
from (4.3) replace A
Now in Lu P(T,U)
=
G.
Let P
LPi(U)~N-i.
terms of order N)
N
j
(T,U)
by
uj
and
Dt by
T
be the principal part of
to obtain a polynomial (i.e. the
P(T,U)
and define (4.3) to be of hyperbolic type if for all real
0 the equation P ( T , U ) N further
U
=
0 has
N distinct real roots T.(U).
Define
J
I
Theorem 4.2
If (4.3) is hyperbolic then
L
is hyperbolic as a partial differen-
tial operator in local coordinates s * @ ( s ) . The proof is straightforward and we omit details. Next recall that for T,S
E'(G)
E
one defines (with abuse of notation)
Then by patching together local solutions and using standard facts one can prove
If (4.3) is of hyperbolic type the Cauchy problem
Theorem 4.3 (4.13) has is
a
Lf.(t,g) 3
=
k 0; D f (0,g) t j
unique solution with
ern
f.(t;) J
in
=
E
(0 5 j, k
5
N-1)
Sjk6(g) E'(G).
ib
If
E Cm(G)
t.
Finally using (4.9) it is easily seen that for
5
E Cm(Trr)
then
I
fj(t,g)Q(g)dp(g)
OTHER TECHNIQUES
47
and t h e n e x t theorem f o l l o w s immediately from t h e p r e c e e d i n g r e s u l t s .
Let ( 4 . 3 ) be of h y p e r b o l i c t y p e w i t h t h e
Theorem 4 . 4
a l g e b r a of an i n t e g r a b l e a l g e b r a
A.
Then f o r
u
k
Pi(A)
i n t h e enveloping t h e problem ( 4 . 3 ) h a s
E Cm(7i)
a s o l u t i o n g i v e n by N-1
14-15>
1Q ( f j ( t , g ) ) u j .
u(t) =
j=O
I t i s a s s e r t e d i n Hersh-Steinberg
Remark 4.5
171 t h a t i f t h e a l g e b r a g e n e r a t e d by
A t i s a l s o i n t e g r a b l e t h e n t h e s o l u t i o n (4.15) of (4.3) i n unique. I t seems J p r o b a b l e t h a t a u n i q u e n e s s theorem based on t h e t e c h n i q u e s of S e c t i o n 2 could a l s o the
be d e v i s e d and we w i l l r e t u r n t o t h i s l a t e r .
The r e s t r i c t i o n t o h y p e r b o l i c equa-
t i o n s above i s o n l y t o i n s u r e t h e e x i s t e n c e of s u i t a b l e
f
j
such t h a t a l l b r a c k e t s
and i n t e g r a l s make s e n s e ; o t h e r s i t u a t i o n s can e a s i l y b e e n v i s i o n e d .
Example 4 . 6
A number of i n t e r e s t i n g examples and a p p l i c a t i o n s a r e g i v e n i n Hersh-
Steinberg [7].
The Weyl-Heisenberg a l g e b r a h a s a s a b a s i s t h e skew a d j o i n t opera-
tors
(4.16)
A
=
j
a/ax.; J
B~ = i x k ;
iI
each of which g e n e r a t e s a u n i t a r y group i n
(4.17)
where
tA. Jf(x) = f(x
e
ek
+
te.); J
e
t iI
o f t h e form
L2(Rn)
f(x) = e
it
itx
tB
f(x);
d e n o t e s an orthonormal b a s i s i n IRn.
e
kf(x) = e
k f (x)
Any l i n e a r p a r t i a l d i f f e r e n t i a l
e q u a t i o n L i t h polynomial c o e f f i c i e n t s can be w r i t t e n a s an o p e r a t o r e q u a t i o n of However t h i s i s n o t always t h e b e s t a l g e b r a t o
t h e form (4.3) u s i n g t h i s a l g e b r a . take.
(4.18)
It i s noted f o r example t h a t t h e e q u a t i o n u
tt
=
-U
2 xxxx + x uxx + 2 x u
4
1 4
- x u + - u
i s n o t h y p e r b o l i c i n t h e u s u a l s e n s e and would n o t l e a d t o a h y p e r b o l i c o p e r a t o r e q u a t i o n i n terms of t h e Weyl-Heisenberg a l g e b r a .
(4.19)
A
2
=
ix ;
B =
xnx +
1 -;
2
c
= i D
2 Y
However i f one t a k e s
R. W . CARROLL
48
t h e n ( 4 . 1 8 ) becomes sees that
[A,B]
A
tt
+ B2 +
= (A2
-2A,
.
which w i l l b e h y p e r b o l i c . [B,C] = -2C
while
Here one a r e formally
A,B,C
F u r t h e r t h e y e x t e n d t o u n i q u e skew a d j o i n t o p e r a t o r s on
g e n e r a t i n g u n i t a r y groups. and by Simon [l] A
= fA,B,C}
2 C )u
[A,C] = 4B, and
Cz(lR)
skew a d j o i n t on LL(IR)
=
u
is a t w o f o l d c o v e r i n g of
The H e r m i t e f u n c t i o n s a r e a n a l y t i c v e c t o r s f o r
is i n t e g r a b l e .
.
SL(2,lR)
I n f a c t t h e a s s o c i a t e d group
G
A n o t h e r a s p e c t o f c h o o s i n g a "good" o p e r a t o r
v e r s i o n of ( 4 . 1 8 ) i s t h a t one o b t a i n s n a t u r a l l y an e n e r g y
y i e l d i n g good a p r i o r i e s t i m a t e s .
with e l l i p t i c
L
A f u r t h e r a p p l i c a t i o n t o e q u a t i o n s of t h e form
is a l s o given.
We g o now t o some s e c o n d o r d e r e q u a t i o n s and w i l l s k e t c h v a r i o u s m e t h o d s .
f i r s t t h e e q u a t i o n ( 1 . 1 4 ) i n a Banach s p a c e w'(0) = ul. -f
(4.21)
Assume
w1 = w
Setting
wt = A:;
[ 1; 1;
+ w(0)
w2 = w '
and
=
A
=
F
with i n i t i a l d a t a
A
p(A) = t h e r e s o l v a n t s e t of
and
w(0) = u o
one h a s
[0
1
A
).
0
generates a s t r o n g l y continuous group
A
Consider
A
one e x p e c t s
T(*)
in
0 E
and a s s u m i n g
F
t o generate a strongly continuous
g r o u p g i v e n by
(4.22)
[
exp t A = cosh A t
I 0 I
]
-k
A t A.
A-'sinh
D(A)
I n f a c t i t is proved i n G o l d s t e i n 131 t h a t i f
t h e n t h i s i s indeed t r u e i n t h e s p a c e (but not i n F
X
F).
F x F
since
For v a r i a b l e
1
A- s i n h A t A
A(t)
i f € o r any
Isl,It/ < T
(1
'r
> 0
B ( t ) - B(s)II
x
F
where
First for
t h e r e is a c o n s t a n t
5
mlt-sl.
D(R)
D(A2)
X
D(A)
h a s t h e g r a p h norm
d o e s n o t d e f i n e a bounded o p e r a t o r i n
w i t h c o n s t a n t domain
t h e o r e m i s proved i n C o l d s t e i n [ 2 ] .
Lip(E,F)
D(A)
is t a k e n t o b e
Also i f
D(A(t))
R(t) m
E =
D
=
L(E,F)
m(T)
I)(A) C D(B)
t h e following
B(-) E
one says
such t h a t f o r with
A-1
E L(F)
49
OTHER TECHNIQUES
then
BA-l E L ( F )
Theorem 4 . 7
from
0
Let
2
(B(t)x,x)
by t h e c l o s e d g r a p h t h e o r e m .
F = H
be a H i l b e r t space with
c ( t ) I I x1I2
for
x E D(B(t))
o n bounded i n t e r v a l s .
(4.23)
+
w"
6
with
w
Here
D($(t))
E
t i o n term
w(0)
B(t)w = 0;
and
Assume
t
+
(= D ( A ( t ) ) = D
self adjoint satisfying
c(t) > 0
where
B(t)B-l(O)
i s bounde d away
E Lip(H).
2 w E C (H).
6
is automatically constant with
P(.)
with
The n t h e p r o b l e m
w ' ( 0 ) = w1
has a unique s o l u t i o n
w1 E D
P ( t ) E L(D,F)
wo;
=
6
=
B(t)
E Lip(D,F)
s k e t c h some o f t h e s t e p s of t h e p r o o f .
c a n b e added t o
and a p e r t u r b a B(t).
We w i l l
A p r e l i m i n a r y lemma b a s e d o n K a t o [ 2 ; 81
( c f . a l s o M i z o h a t a [l]) s t a t e s
Lemma 4 . 8
Let
F
b e a B anach s p a c e w i t h
t i n u o u s semigroup i n L(F)
with
Q(t)C(t)Q
Q(-) E Lip(F). tive
i
L(F)
-1
(t)
i ( - )E
d o ma i n i n d e p e n d e n t of
t.
an d o n e s u p p o s e s
(4.24)
dU
dt
: B+
Lip(F),
F
- C(t)U;
P r o o f of Theorem 4 . 7 :
R-l(t)
-
E(O))-'
w(t) E F = D x H
R(*)
:' R I
E L ( F ) , a nd
-+
L(F)
L(F)
Q-'(t)
€
and
with strong deriva-
E ( t ) = R(t)C(t)R-'(t)
has
E L(F)
From t h e s e h y p o t h e s e s i t f o l l o w s t h a t t h e r e i s
satisfying
U(0)
=
f
E
D(C(0)).
( 4 . 2 3 ) is equivalent to
-f
wh e r e
be such t h a t
Then
D(*) E Lip(F). +
Q ( t ) E L(F)
g e n e r a t i n g a c o n t r a c t i o n semigroup i n
D ( t ) = (I - E ( t ) ) ( I
u
Let
Suppose t h e r e i s a f u n c t i o n
such t h a t
a unique
t > 0.
for
t h e g e n e r a t o r of a s t r o n g l y c on-
C(t)
with
D(A(t)) =
8x0.
Let
Ft = F
w i t h t h e norm
50
R. W. CARROLL
and t h e n f o r f i x e d
11
w i t h norm
A(t)
t,
g e n e r a t e s a u n i t a r y group i n
I( A ( t ) f l (
fll =
(D
has the
argu me n t t h a t t h e r e i s a n o p e r a t o r Qo(t) : Dt
+
Set
2 = Qo(t)
To(t)
norm).
Q ( t ) E L(D)
is u n i t a r y , and
D = D
Do
considered i n
: D
Qo(t)
Do
with
To(t)
T > 0
Let
6
w i t h doma in
D
=
E L(D ),
T o ( t ) = B-'(O)B(t). (this takes a
T ( * ) E Lip(D ) [O,T]
CI
5
To(t) < c-lI
A standard formula ( c f . C a r r o l l [ 4 1 , Yosida
[I]) a l l o w s o n e t o w r i t e now f o r
1
Q,'(t)
so t h a t
b e f i x e d and t h e n o n
Do.
s e l f a d j o i n t on
Dt
is p o s i t i v e s e l f a d j o i n t .
D
+
Let
One shows by a r o u t i n e
such t h a t
Th e h y p o t h e s e s o f Theorem 4 . 7 w i l l i m p l y now t h a t
little calculation).
L(Ft).
f E D = D
m
(4.26)
Qo(t)f =
s4(sI
+
To(t))-'To(t)fds.
0
U s i n g t h i s f o r m u l a o n e c a n show t h a t p o s s i b l e t o a p p l y Lemma 4 . 8 .
Then
and
Q(t)
unitary.
Q-'(t)
He n c e
A(t)
First with
belong t o
Q(t)A(t)Q-'(t)
Q ( t ) e x p s A( t) Q - ' (t )
Q o ( * ) E Lip(D)
L(F),
F = D
Q(*)
H
X
( c f . Kato [ 4 ] ) . ( D = Do)
E L i p ( F ) , and
g e n e r a t e s a u n i t a r y group i n
(s E W ) .
Next s e t
R ( t ) = I.
g e n e r a t e s a u n i f o r m l y bounded g r o u p i n
L(F)
One knows with
I t i s now
define
: Ft
Q(t)
F
+
is
L ( F ) , namely
1 E p(A(t))
D(A(t))
=
5
x
since
D.
Hence
i n Lemma 4 . 8 o n e c a n w r i t e
a n d t o show
D(*)
E
Lip(F)
i t s u f f i c e s t o show
f o l l o w s w i t h a small c a l c u l a t i o n .
A(.)A-'(O)
E Lip(F)
which
Hence Lemma 4 . 8 applies a n d Theorem 4.7
is
QED
p rov e d .
We c o n s i d e r n e x t some v a r i a b l e domain p r o b l e m s f o l l o w i n g C a r r o l l - S t a t e
[25].
The
b a s i c problem w i l l be phrased a s
(4.28)
w"
+
B(t)w
=
f
i n a separable Hilbert space
H
and w e w i l l all-ow t h e doma ins
D(B(t))
to really
51
OTHER TECHNIQUES
vary i n time.
T h i s i s a c c o m p l i s h e d by u s i n g a t e c h n i q u e o f C a r r o l l [ 4 ; 3 6 ; 3 7 ; 381
t o d e s c r i b e t h e v a r i a t i o n o f s u b s p a c e s i n terms o f o p e r a t o r s ( c f . a l s o C a r r o l l Cooper [ 2 4 1 , Cooper [l; 2 1 , Mazumdar [l; 2 ; 3 ; 4 1 ) .
Other a b s t r a c t techniques
f o r v a r i a b l e domain p r o b l e m s a p p e a r i n B a i o c c h i [ l ; 2 1 , B e r n a r d i [ l ] ,B e r n a r d i B r e z z i [ 2 ] , G e a r h a r t [ l ] , K a t o [ 6 ; 7 1 , L i o n s [ 4 ; 5 ; 10; 11; 1 2 1 w h i l e some r e c e n t
[3],
work w i t h a p p l i c a t i o n s t o s c a t t e r i n g t h e o r y c a n h e f o u n d i n Bandos-Cooper Cooper [ 3 ; 4 ; 5 ; 6 1 , Cooper-Medeiros
[ 7 ] , Cooper-Strauss
[ 8 ; 9 1 , Inoue [ l ] ,
Strauss [ 3 ] .
We w i l l d e a l w i t h a weak v e r s i o n of ( 4 . 2 8 ) f i r s t .
H
f a m i l y of H i l b e r t s p a c e s d e n s e i n t h e a n t i d u a l s o one can w r i t e b e d e t e r m i n e d by
D(S(t))
=
V(t) C H C V ' ( t ) .
(x,y) = ((R(t)x,y))t
details).
one t o one o n t o
The o p e r a t o r s
H
for H
with
L 2 ( V ( t ) ) = { u E L (H) product
on
Let
x E H and
R(t)
and
: H
Let
V'(t)
V(t)
be
(into)
Then
y E U(t). R -4( t )
S(t) =
+
be a
R-l(t)
maps i t s domain (see Carroll [ 4 ] f o r
( ( x , ~ ) =) ~(Sx,Sy)
were c a l l e d " s t a n d a r d " o p e r a t o r s a t o n e t i m e b u t
S(t)
p e r h a p s a b e t t e r name would b e c a n o n i c a l
2
V(t) C H
with continuous i n j e c t i o n s .
is a p o s i t i v e s e l f a d j o i n t o p e r a t o r i n V(t)
Thus l e t
[O,T],
( ( U , V ) )= ~ (T(Su,Sv)dt.
V(t)
u ( t ) 6 V(t)
related operators.
Let
2
Su E 1, (HI)
a.e.,
W =
with s c a l a r
One makes h y p o t h e s e s w h i c h i n s u r e t h a t
J
11
S-l(t)II
5
c
with
t o be measurable f o r
;-'(*)h
measurable i n
k E H).
Then
H
W C L (H).
c o e r c i v e , s e l f a d j o i n t s e s q u i l l i n e a r form on
cII
xllt
11
yllt
and
b(t,x,x)
2
for
2
2
all x l l t .
Let
V(t)
x
h E H
(i.e.
b(t,.,.) V(t)
(S-'(-)h,k)
be a c o n t i n u o u s ,
with
Ib(t,x,y)I
2
Such a form i n d u c e s a t o p o l o g y on
V(t)
e q u i v a l e n t t o i t s o r i g i n a l t o p o l o g y a n d we i n t r o d u c e a new H i l b e r t s t r u c t u r e on V(t)
with s c a l a r product
B(t) : V ( t )
+
antiduality).
V'(t)
b(t,x,y).
Clearly
b(t,x,y)
i s l i n e a r and c o n t i n u o u s ( < , > t
= t
denotes
T h i s d e t e r m i n e s a n unbounded o p e r a t o r i n
where
V(t) - V ' ( t )
H , d e n o t e d a g a i n by
B ( t ) , w i t h domain
and f o r
x E D(B(t))
one h a s
b(t,x,y) = (B(t)x,y).
Since
b(t,x,y)
= h(t,y,x)
is
52
R. W.
we s e e t h a t
B(t)
Next we l e t
O(t) : V ' ( t )
is a s e l f adjoint operator i n
<x,y> = ((€J(t)x,y)), S
-2
(t)
-+
B(t)
V(t)
so t h a t
when r e s t r i c t e d t o
work w i t h t h i s
CARROLL
be t h e i s o m e t r i c isomorphism determined by
B ( t ) = O-'(t)
H.
H , c l o s e d , and d e n s e l y d e f i n e d .
Thus
in (4.28).
2
i s s e l f a d j o i n t and we w i l l
B(t) = S ( t )
S u i t a b l e p e r t u r b a t i o n s of
be p e r m i s s a b l e ( c f . C a r r o l l - S t a t e
O(t) =
and one shows e a s i l y t h a t
B(t)
w i l l of c o u r s e
[25]).
Now a weak v e r s i o n of ( 4 . 2 8 ) i s t o f i n d
w E W , with
w'
E
and
L2(H)
w(0) = 0 ,
so that
(4.29)
for a l l
v
E
W
with
v'
E
LL(H)
and
v(T) = 0.
I n g e n e r a l a term (wl,v(0))
w i l l be added t o t h e r i g h t s i d e o f ( 4 . 2 9 ) b u t we assume
(wl
city
for
C1 (S
-1
-
~ ' ( 0 ) ) . We assume t h a t
h , k E H)
S-l(.)
i s weakly
here f o r simpli-
w1 = 0 C1
(i.e.
i n which c a s e t h e r e a r e s e l f a d j o i n t o p e r a t o r s
11
( t ) h , k ) ' = (8-'(t)h,k),
S-l(t)
11 5
Lipschitz continuous ( c f . Carroll [ 4 ] ) .
11
cl,
i-'(t)
11 5 c 2 ,
and
(S
-1
(*)h,k)
8-'(t)
with
S-l(-)
is
Now i n making ( 4 . 2 8 ) o r ( 4 . 2 9 ) i n t o a
f i r s t o r d e r system an a d d i t i o n a l term a p p e a r s i n o r d e r t o t a k e i n t o account t h e v a r i a t i o n o f domain.
when
0
( ),
w;
Thus s e t
makes s e n s e .
w
Let
w = S -'wl
=
2 ), [z:]€H=L(H 2 )xL(H
and c o n s i d e r t h e problem of f i n d i n g
++
-(w,v')
(4.30)
for a l l
-t
v
E
W
+ );,;(A
with
+
+
v'
E
H
and o b s e r v e t h a t i n
-L
w E H
D'(H)
w = W x W ,
+ f =
such t h a t
+ + ++ (Aw,sv) = ( f , v )
and
+
v(T) = 0.
Here a
A
term is added on f o r
convenience i n t h e argument; i t does n o t a f f e c t e x i s t e n c e o r u n i q u e n e s s .
To s e e
t h a t ( 4 . 3 0 ) i s a c o r r e c t t r a n s p o s i t i o n of ( 4 . 2 9 ) w e w r i t e o u t t h e e q u a t i o n s a s
is
OTHER TECHNIQUES
(4.32)
53
(wl,Sv2)dt
Th u s t a k i n g setting
v 1 = S-'@
X = 0
X = 0
define
(0,T)
on
w e have
v '1 =
and
i S-'@'
( 4 . 3 1 ) says
w2 = (S-'w1)'
wh ic h means
@ E D(H)
for
=
w
-1
D'(H).
w2
so that
w1
S
=
in
+ w
Hence g i v e n a s o l u t i o n =
w'
and
(wl,Sv2)
of ( 4 . 3 0 ) w i t h
= (Sw , Sv2) = b ( t , w , v
).
2
Then ( 4 . 3 2 ) s a y s
(4.34)
+
-IT(w',v;)dt '0
wh ic h i s ( 4 . 2 9 ) w i t h
d e f i n e d as L'
D(L)
with
L'u
=
=
v2 = v .
As i n C a r r o l l - C o o p e r
Lu = u '
/Tb(t,w.v2)dt ' 0
W'
[24] = {u E
-u'
2
= L (V'(t))
=
O-lW
2
W ; u ' E L (H): u ( 0 )
with
D(L')
=
proved t h a t
Ls = L = ( L ' ) *
Co op e r [ 2 4 ] , n a me l y i f
=
S-2(.)
L
.
i s mo n o to n e i f f o r
(rake here
F
Similarly
2
L (H);
S-l(.)
is w e a k l y
C1
then
D(Q)
D(Q) = F
W'
Both
L
C'
by is
a nd
i t is
T h i s is a v e r y Q : F
We r e c a l l n e x t t h a t a map
-f
u,v E D(Q),
dense we s a y
Q
When
Q
is l i n e a r
i s ma xima l monotone i f i t i s n o t t h e A (nonlinear)
Q : F
+
i s bounded i f i t t a k e s bounded s e t s i n t o bounded s e t s a n d hemi-
c o n t i n u o u s i f i t i s c o n t i n u o u s from l i n e s i n is c o e r c i v e i f
W'
-t
b e i n g weakly
Ls = Lw.
p r o p e r r e s t r i c t i o n of a n o t h e r l i n e a r monot one o p e r a t o r . with
u ( T ) = 01.
t o b e a r e f l e x i v e Banach s p a c e f o r e x a m p l e ) .
and monotone w i t h
: W
L'
-f
A c t u a l l y a s t r o n g e r r e s u l t i s proved i n C a r r o l l -
u s e f u l c r i t e r i o n and t h e p r o o f i s n o n t r i v i a l . F'
E
a r e d e n s e l y d e f i n e d a n d a s a c o n s e q u e n c e of
-
0).
=
{u E W; u '
L : W
and one d e f i n e s
F
t o t h e weak t o p o l o g y of
F';
Q
F'
54
where
R. IJ. CARROLL
$(x)
-f
x
-+
($(x)
m
can b e n e g a t i v e f o r
[ 2 4 ] , by a r e s u l t of B r g z i s [ Z ]
Carroll-Cooper Ls = Lw.
as
m
x
small).
Now, f o l l o w i n g
i s maximal monotone s i n c e
L
The f o l l o w i n g s p e c i a l c a s e of a r e s u l t of Browder [8] i s then needed
f o r o u r development.
Lemma 4 . 9
Assume
i s a r e f l e x i v e Banach s p a c e .
F
d e n s e l y d e f i n e d l i n e a r maximal monotone map and t i n u o u s bounded c o e r c i v e map.
L
Then
+Q
Let
Q : F
maps
D(L)
L : F +
F'
C
F
+
F'
be a closed
a monotone hemicononto
F'.
Consider now t h e r e g u l a r i z e d s t r o n g p a r a b o l i c problem a s s o c i a t e d w i t h ( 4 . 3 0 ) ( c f Lions [ 5 ] f o r p a r a b o l i c r e g u l a r i z a t i o n )
Problem 4 . 1 0
(4.35)
E '
w
Find
+
LzE +
such t h a t
6 W
P;"
+
+
EKZ'
= f
Let u s check t h a t a s o l u t i o n o f ( 4 . 3 5 ) s a t i s f i e s ( 4 . 3 0 ) w i t h a s u i t a b l e E term + + added, L e t v E [U w i t h :' 6 H and v(T) = 0. Then t D(L') i n an obvious (0 - Id'
n o t a t i o n and t a k i n g
brackets i n (4.35) with
+ v
we o b t a i n
Indeed t h e i n d i v i d u a l e q u a t i o n s i n ( 4 . 3 5 ) a r e
(4.38)
L
(4.39)
L
E
WE
+
hw
WE
+
xw
s l
s 2
1 E
2
+
0
and
s
v E V(t),
x
F
s w1
+ swE1 +
and one n o t e s t h a t , f o r V'(t)
-1 -1.-1
Ee
E
H
-1
- sw; +
EO
-1
w
E
1
=
0
E
w2 = f
and
v E V(t),
<x,v> = (x,v)
< x , v > = O OX,^)) = (SHx,Sv).
while, for
x E
Hence i n a p p r o p r i a t e s p a c e s
OTHER TECHNIQUES E ' + +E = (Aw , S+v ) ,
e t c . so t h a t
GE
t o find solutions
'
E '
55
E'
= (Sw
o f (4.35) w i t h
+ , Sv) ,
I[
E4 Ew '
and
etc.
[Iw
The p r o g r a m now i s E4ISw +E
=
Iff
bounded.
Then by weak c o m p a c t n e s s w e c a n t a k e weak l i m i t s i n ( 4 . 3 7 ) which l e a d s t o a s o l u t i o n of ( 4 . 3 0) . Q = XI
for
+P+
EK
x E V(t).
F = IU
T h i s i s a c c o m p l i s h e d by t a k i n g
i n Lemma 4 . 9 w i t h
and making t h e h y p o t h e s e s
GE
X > B one o b t a i n s
Then t a k i n g
s o l v i n g (4.35)
lGElff
r o u t i n e c a l c u l a t i o n t o v e r i f y c o e r c i v i t y ) and t h e e s t i m a t e s E
4ISw E'
IH 5
follow e a s i l y .
Theorem 4 . 1 1
Let
V(t)
V(t)
X
Assume t h e c a n o n i c a l
weakly
with (4.40).
for
and p u t on V(t)
h E H
V(t)
Examples a r e p r o v i d e d t o show t h a t
V(t)
V(s) C V ( t )
2
,
( 4 . 4 0 ) h o l d s , and
a bounded o p e r a t o r i n
H.
exists a solution
w
(Sw)' E W ' , u = eXtw
w' E W , and
then
u'(0) = w'(0)
w E
u"
+
+
2
S u
Xw(0).
S-'(-)
S
-1.-1 S
of
w"
w" E =
+
H
eXt[w"
2Xw'
E
with
f ' E ff
+ X 2w + s2w
=
f
and
0
=
2
s.
using f u r t h e r
w(0) = w ' ( 0 ) = 0.
+
+ X 2w +
2
S w]
.-2
with
S
-1
S , extends t o be
S
f(0) = 0
satisfying
while 2Xw'
t
I n p a r t i c u l a r h e assumes
S, o r equivalently
f E H
Then i f
for
need n o t be c o n s t a n t .
Mazumdar [ 4 ] w o r k s w i t h ( 4 . 3 5 ) d i r e c t l y f o r
C
satisfy
!251 t h a t ( 4 . 4 0 ) i s e q u i v a l e n t t o
h y p o t h e s e s and some r e s u l t s o f Mazumdar [ l ; 2 ; 31. i s weakly
S(t)
Then t h e r e e x i s t s a s o l u t i o n of ( 4 . 3 0 ) .
and t h i s l e a d s t o t h e r e q u i r e m e n t t h a t
Remark 4 . 1 3
t h e corresponding Hilbert
related operators
I t i s shown i n C a r r o l l - S t a t e
Remark 4.12
and
be a family of continuous s e l f a d j o i n t c o e r c i v e
structure. C'
< c -
Consequently one h a s
b(t,.,.)
s e s q u i l i n e a r f o r m s on
( a f t e r some
there
Sw E W,
Note h e r e t h a t i f
u ( 0 ) = w(0)
and
Examples show t h a t t h i s i s a l s o a t r u e v a r i a b l e domain
situation.
Reamrk 4 . 1 4
I n K a t o [ 6 ; 71 e v o l u t i o n e q u a t i o n s of t h e t y p e
w'
+
A(t)w = f
are
56
R . W. CARROLL
s t u d i e d where t h e
-A(t)
generate
n e c e s s a r i l y a n a l y t i c semigroups.
s e m i g r o u p s i n a Banach s p a c e
Co
F
but not
Such e q u a t i o n s a r e t h e n s a i d t o b e o f h y p e r b o l i c
t y p e and h a v e some a p p l i c a t i o n s t o symmeyric h y p e r b o l i c s y s t e m s of d i f f e r e n t i a l e q u a t i o n s ( c f . C r a n d a l l [ 4 ] , F r i e d r i c h s [ 1 ] , P h i l l i p s [l; 2 1 ) .
The domain
i s a l l o w e d t o v a r y b u t i t is assumed t h e r e is a d e n s e l i n e a r s u b s p a c e
D(A(t))
D C D(A(t))
on which
a c t s "smoothly".
A(t)
The R u s s i a n s c h o o l h a s t r e a t e d s e c o n d o r d e r p r o b l e m s of " p a r a b o l i c " t y p e e x t e n s i v e l y and e x t e n d e d t h e t e c h n i q u e s t o h i g h e r o r d e r e q u a t i o n s . K r e i n [ l ] we l e t a Banach s p a c e
(4.42)
Since
E
D(A(t)) = D
and h e n c e
(4.43)
u(0)
t
with
6 L(E)
A-l(t) E L(E).
i t follows t h a t
-
If
A(0)u'
u ( 0 ) F- D
from
i s continuous.
with
A ( O ) u ( * ) E C'
= w(t)
A(O)u(t)
v' = A(t)v
u(o)]
+
(A(O)u(t))' = A(O)u'(t)).
A ( * ) y E C1(E)
B(t)A-'(O)w;
for
L e t now
so t h a t
w' = A(O)v
S o l u t i o n s of ( 4 . 4 2 ) c o r r e s p o n d t o s o l u t i o n s of assumes
1'
is closed
u(t) E D
and
A(O)A-'(t)
u'(0) = u
uo;
=
i n d e p e n d e n t of
c o n t i n u o u s and A(0)
be closed densely defined l i n e a r operators i n
B(t)
and c o n s i d e r
A(O)[u(t)
u' = v
and
u" = A ( t ) u ' i- B ( t ) u ;
Suppose
A(.)u'
A(t)
Thus g o i n g t o
y E D
with
A t)
).
x'
=
+
c(t)x.
The f i r s t t e c h n i q u e
s u i t a b l y a p p r o x i m a b l e by Y o s i d a
approximations
A ( t ) = -nA(t)RA(t)(n)
(recall
R (1) = ( A A
-
hI)-')
=
while
A(t)(I -
R(t)A
1 -1 ; A)
-1
(0) E L(E)
with
B(-)A-'(O)
strongly
57
OTHER TECHNIQUES
differentiable.
Then
i s thought of a s
Q(t)
A(t) t 8 ( t )
where
A(t)
(resp.
The p r o b l e m i s v i e w e d as a
8 ( t ) ) h a s z e r o s i n t h e s e c o n d ( r e s p . f i r s t ) column.
+ p e r t u r b a t i o n of
= A(t)y
and a u n i q u e s o l u t i o n
using r e s u l t s f o r f i r s t order equations. y E D
11
while
RA(t)(A)II
5
l/l+h
If
A(.)y
h > 0
for
uo,ul E D ,
s obtained f o r E
1 C (E)
1 y E C (E)
for
for
( i n s t e a d of t h e approxi mat i on
p r o p e r t y ) and t h e o t h e r h y p o t h e s e s are u n c h a n g e d t h e n t h e s a m e c o n c l u s i o n f o l l o w s The t h r u s t of t h e m a t t e r i n
by i n v o k i n g o t h e r r e s u l t s on f i r s t o r d e r e q u a t i o n s .
t e r m s o f o p e r a t o r s t r e n g t h is d i s p l a y e d f o r c o n s t a n t o p e r a t o r s i n t h e f o l l o w i n g C
t h e o r e m w h e r e we s a y t h a t a n o p e r a t o r
11
and ql/
AXII
Cx)Iz
cII
Ax11
x E D(A).
for
is subordinate t o
if
A
I f i n addition f o r small @,
f o r a c o ntin u o u s convex f u n c t i o n a l
then
B
D(C) 3 D(A)
11
ri
+
C x I I I @,(x)
is s a i d t o be completely
subordinate.
Theorem 4.15
Let
and
A.
completely subordinate t o
A.
A
=
-BA-l
g e n e r a t e a n a l y t i c semigroups with D(B) n D(A)
uo
Then f o r
*O
u1 E D(A)
and
there
e x i s t s a u n i q u e s o l u t i o n of ( 4 . 4 2 ) .
One a l s o s p e a k s of a weak s o l u t i o n o f ( 4 . 4 2 ) i n r e q u i r i n g and
B ( * ) u E Co
o n l y on
(O,T]
i n s t e a d of
4.15 such s o l u t i o n s e x i s t uniquely f o r b e a n a l y t i c i n some s e c t o r c o n t a i n i n g
Theorem 4 . 1 6 and s a t i s f y Re h > w
and
A(t)
and
Let
(0,m).
Ao(t) = B(t)A
IIRA(t)(A)ll(M/IA-wl
.
E D(A)
A'(t)A-'(t),
and
-1
u1 E D(Ap)
for
p >
CY
2
,
A(-)u' E Co,
I n t h e s i u t a t i o n of Theorem and
u1 € E
and
u(t)
will
S i m i l a r l y one h a s
(t)
h a v e d o m a i n s i n d e p e n d e n t of
)IRno(t)(X)IICMo/lA-w
Ao(t)Aol(0) a n d A o ( t ) A - a ( 0 )
h e bounded and s a t i s f y a H 6 l d e r c o n d i t i o n .
[O,l]
and
Let
u
[O,T].
u E C
Then f o r
uo
I
for
f o r some f
Re (Y
t ).
> w
E
D(A) n D(B)
t h e r e e x i s t s a u n i q u e weak s o l u t i o n o f ( 4 . 4 2 ) .
I t i s i n t h i s s p i r i t t h a t Gne f i n d s c e r t a i n r e s u l t s f o r h i g h e r o r d e r e q u a t i o n s .
F o r e x a m p l e f o l l o w i n g Yakubov [ 5 ] w e c o n s i d e r f o r c l o s e d unbounded d e n s e l y d e f i n ed
%
58
R. W . CARROLL k D u ( 0 ) = uk
with
(0
t
[O,T]
( 4.45) on
on
[O,T]
k
5
n-1)
u E Cnhl(E)
for
[ O,T] ,
while
=
1.
u E Cn(E)
A1(-)Dt
( O ,T] , and A,(.)D:-k~
on
n- 1 u E Co(E)
and i s
(O,T]
on
The f o l l o w i n g t h e o r e m s are p r o v e d by w r i t i n g (4.45)
[O,T].
1 satisfy
S o l u t i o n s of t y p e
i s a Banach s p a c e h e r e ) w h e r e a s s o l u t i o n s
(E
on
1. 2
n
A.
wh er e
u E Cn(E)
with
of t y p e 2 s a t i s f y Co(E)
5
L
E
1
on
as a f i r s t o r d e r s y s t e m
and i n v o k i n g r e s u l t s on f i r s t o r d e r e q u a t i o n s .
Theorem 4.17
(6 > 0).
Let
S uppos e Assume
[O,T].
D(Al(t))
D
=
A l ( * ) A i l ( 0 ) E C 2 (Ls(E))
uk E D
with
11 RAl(t)(A) 11 5
dense with
f = f(t)
and
%(*)A;l(O)
a function i n
C1(E).
l / ( X + 6)
1 > -6
for
E C1(Ls(E))
on
Then ( 4 .4 5 ) h a s a
unqiue type 1 solution.
Let
Theorem 4 . 1 8 Re
> 0.
(k = 2 ,
Let
...,n)
k < n- 2
with
D( Al( t) )
A1(*)A-'(O)
E C1(Ls(E))
u
n-1
E D(A')
with
f o r some
c1
+ A ( t) Dnt - lu
( 4.47)
=
F(t,u
with s u i t a b l e d a t a
[Al(-)A-'(O)]' [ O ,T] . and l e t
Let
c/(l+
and
for
A,(t)A-'(O)
uk E D
f = f(t)
1x1)
for
0 <
satisfy a
,... ,Dn- 1u ) L e t u s m e n t i o n i n p a s s i n g t h e p a p e r of
[ 11 wh er e t h e a b s t r a c t Love e q u a t i o n
+ A(t)[u + Vu"]
u"
RAl(t) ( X ) I I (
are o b t a i n e d i n t h e framework
f
b u t we w i l l n o t b e more s p e c i f i c h e r e . Kharitonenko-Yur&k
> 0
I'
Then ( 4 . 4 5 ) h a s a u n i q u e t y p e 2 s o l u t i o n .
Theorems f o r more g e n e r a l
DYu
dense with
s a t i s f y i n g a HElder c o n d i t i o n on
HElder c o n d i t i o n .
( 4.46)
= D = D(A)
u(0)
=
and
f
u'(0)
is t r e a t e d i n a H i l b e r t space.
W e go n e x t
t o a t e c h n i q u e of R a s k i n [ 2 ] ( c f . a l s o D u b i n s k i j [l; 21, Gasymov [ l ] , G r i s v a r d
[Z])
f o r e q u a t i o n s o f t h e form
where t h e conditions
Ak
a r e c l o s e d d e n s e l y d e f i n e d o p e r a t o r s i n a Banach s p a c e
k Dtv(0 ) = vk
(0
5k 5
n-1)
a r e required f o r suitable
E.
v,.;.
Initial Assume
OTHER TECHNIQUES
E
L(E)
and set
-Bk
=
2
n
(1 5 k < n)
-%A&
59
with resolvant %(A)
= (A1+Bk)-l
satisfying
where
n- 1 < 6
i%
=
as before).
(A-XI)-'
(note we are taking
Let
Ln(A)
=
ZXn-k4,
RA(A)
(XI -A)-1
=
here instead of
and the main tool in this work is
the following lemma. Lemma 4.19
1'
11 x I I
BkX!I
for x ED(Bk)
commutes with > u1 Re X -
Proof:
for k
Sk
and
11
R < Bo
c > 0 and
For some
Lil(A)ll
=
=
6 - (n - 1)
n D(Bk+l)
let
11
...,n -
k = 1,
and
2,...,n. Then Ln(X)
Bk+l~I(5
1.
Assume
-1 has an inverse Ln ( A )
B1
=
A1
for
5C~~XI-~.
One writes first
Since
BkBk-l
B1
where
il > i2 >
=
..- >
4,
it follows that
i . Hence writing m
we can express (4.51) in the form
One can establish the estimate and hence
11
K(X)ll(
1
11
K(A)
11 5
c2
I I
for sufficiently large
guarantees the existence of
-1 Ln ( A )
13-Bo for
IAl
Re X
(since
2
o1
B < Bo).
(see below) This
from (4.53) which we can write in the form
R. W. CARROLL
60
with a norm estimate as indicated. To obtain the estimate on
1x1
that for
11 K ( X ) I I
observe
large
i > 1 it follows that (note 1
Hence for any
Rn(X) for Similarly Bi Bi R1(X) 1 2 Rn(X) Rn(X) = R3(X) B2B1R1(X)
il > 2
0 . .
..*
-
XR2(X
D < 1)
can be estimated in terms of * a *
R,(X)
-
hB2R1(X)
Rn(X)
and using (4.56) it follows that
II K ( X ) II
-
QED
sni
n-i-1 Let now v
E
i
D(As),
=
.
0,. . ,n- 1, and write
Then there results Theorem 4.20 Under the hypotheses of Lemma 4.19 the function (4.59)
is a
v(t) Cm
=
I
o+im exp XtL~l(X)$o(X)dX Zni o-im
solution of (4.48) for
t
> 0.
n
Theorem 4.21
Let
"
(7
while
D(Bs))
I,!J~(X). for
Then t > 0
v(t)
with
v
6
for
D(As)
i 1 vk+2,...,~ n-1
i
= 0,.
=
v
j
with
n- 1 v ~ + ( ~n D ( A s ) )
u
1 satisfy the condition used in constructing
given by (4.59) is k + l D:v(O)
..,k
for
0< j< k
times continuously differentiable
+
1.
61
OTHER TECHNIQUES
Somewhat more a b s t r a c t l y and g e n e r a l l y D u b i n s k i j [ 2 ] c o n s i d e r s t h e m a t t e r a s f o l l o w s ( c f a l s o A t k i n s o n [ I ; 2 1 , Browne [ I ] , C a r r o l l [ 4 1 , C h a z a r a i n [l], C o r d e s
[ 2 ] , DaPrato-Grisvard
[ 2 ] , I c h i n o s e [l; 2 ; 3 ; 4 1 , KsllstrGm-Sleeman [I]).
[:I,
[ 3 ] , D u b i n s k i j [ l ] ,B . F r i e d m a n
Gasymov 111, G r i s v a r d
[ l ] ,Mohammed [ l ] , Roach-Sleeman
Let s
(4.60)
.
S
1 A.DJ*
P(Dt) =
1 AjX j .
P(h) =
j =O
j = J~ t '
The i d e a i s t o c o n s i d e r f i r s t a g e n e r a l o p e r a t o r e q u a t i o n
s (4.61)
.
1 A.BJu
P(B)u =
j=o
= h
J
A. J
with closed densely defined operators that the spectra
o(B)
and
(B-A)x=O
f o r some
B
i n a Banach s p a c e
r
a r e d i s j o i n t with
i f 3 x E D(P) = nD(A.) J
s e p a r a t i n g them. 'n = o ( P ) if
o(P)
and
E.
Suppose
a s u i t a b l e contour i n
(c
P(A)x = 0; A E o(B)
such t h a t
x E D(B) - o n l y p o i n t s p e c t r a a r e b e i n g c o n s i d e r e d .
Then f o r m a l l y a s o l u t i o n t o ( 4 . 6 1 ) w i l l b e
where
k
i s a n i n t e g e r c h a r a c t e r i z i n g t h e g r o w t h of
P-l(A)
at
m.
This is an
e x t e n s i o n o f m e t h o d s of G r i s v a r d [ 2 ] ( c f C a r r o l l [ 4 ] and s e e a l s o C h a z a r a i n [ l ] , DaPrato-Grisvard
Ichinose [ l ; 2 ; 3; 4 1 ) .
[3],
curve i n t h e halfplane to
lAl;
r.
Then I :
let
S+
( r e s p . S-)
Assume
O(P) C S - ;
111:
Re A > 0
D(A.B) J
going out t o
El
-f
c(l
and
E
D(P)
be
+
: E
F
1-1
ixl)k-j
Theorem 4.22
-f
-t
with a r c length proportional
m
=
D(BA.) J
with
E2
satisfying
E2
where
)I ( B
F = E
ans a n a l y t i c with E2
1
X
$
1
n E2; further for
E
and
: F
F
Let t h e above h y p o th e se s h o ld .
-f
+
El.
El
and
C E
(B - XI)-'
S+
- AI)-lll> c / ( l
A.P-l(A) J -f
E
Let
11:
A . B = BA: J J'
Assume t h e r e a r e Banach s p a c e s
i n b o t h norms
b e a smooth
b e t h e r e g i o n l y i n g t o t h e r i g h t ( r e s p . l e f t ) of
tinuous i n j e c t i o n s s u c h t h a t f o r a n y (B - XI)-'
r
More p r e c i s e l y l e t
: El
+
]XI)
1
S_
and
11
c
U(B)
E2 C E +
S+
and
w i t h conis 1-1 w i t h
D(B)
i n b o t h norms let
P-l(A)
: E2
-f
AjP-'(A))IZ
Then i t i s I m m e d i a t e t o v e r i f y
'Then f o r a n y
h E D(Bk+')
with
62
R . W . CARROLL
Bk+'h
E F
k+l.
by
C l I AjBju1IE2
F u r t h e r one h a s
then (4.62) gives t h e s o l u t i o n with
Wk+l
S e t now
of ( 4 . 6 1 ) g i v e n by ( 4 . 6 2 ) w i t h
u
there exists a solution
k
cII
B
k+l
Bkh E F
If in addition
hllF.
replaced
unchanged.
u E D ( B k+l P ( B ) ) ; Bk+'P(B)u
= (u;
k
E F}
ik+L = ( u ;
and
A B'uED(B k+l ) ;
1
Bk+lA.Bju E F I . J Theorem 4 . 2 3
u
The s o l u t i o n
is dense i n
un E Wk+l
Then it f o l l o w s t h a t
of
P(B)u = h
i n t h e sense t h a t f o r any
Wk+l
such t h a t
u
-
Wk+l
t h e r e is a sequence
u E Wk+l
B ~ + ' P ( B )*~ B~+'P(B)~ ~
and
u
+
as a b o v e is u n i q u e p r o v i d e d
E.
in
T h e s e t h e o r e m s c a n b e r e f i n e d somewhat b y making f u r t h e r a b s t r a c t h y p o t h e s e s b u t we o m i t t h i s .
I n t h e c a s e when
E = H
with s p e c t r a l measures
G C H
with a s c a l a r product
(h
C
p);
11
11
EX
As = I
, )F
(
h
and f o r
AjPp-l(X)II
5
+
C(1
is a s e l f a d j o i n t o p e r a t o r i n a H i l b e r t s p a c e one c a n r e f o r m u l a t e t h e m a t t e r a s f o l l o w s .
h l l a = j( d E Ah , dEhh)F <
I holds with
1-1 w i t h
B
E
F
one d e f i n e s
=
(h E H;
(EX - E ) h E G
1J
where t h e i n t e g r a l i s o v e r
a]
O(B)
IXl)k-j
assume t h e map i n the
F
+
H
P-l(X)
norm.
If
U(B).
: F
+
Assume
D(P)
is
Then from s t a n d a r d
s p e c t r a l c o n s i d e r a t i o n s there r e s u l t s
Theorem 4 . 2 4 u E D(P(B))
(4.63)
u =
Under t h e a b o v e h y p o t h e s e s i f of
P(B)u = h
Wk
For u n i q u e n e s s l e t
=
k
'
u
+
h
a s above a r e u n i q u e .
let
X
B h
F
then a solution
i s d e t e r m i n e d by t h e f o r m u l a
Zll A j B J ~ l I H (
B A.B'un J
+
c(ll hlIF+
k
11
{ u E D ( P ( B ) ) ; B P ( B ) u E F} u E Wk
Assume f o r a n y
BkA.B3u E F ) . J and
k
with
P-'(X)dEhh JO(B)
and t h e r e i s a n e s t i m a t e
u
h E F
k
'
B A.BJu J
in
k
B h(lF).
Gk
and
t h e r e is a sequence H
( j = 0 ,...,s ) .
=
unf
{u
f
wk
D(P(B)):
such t h a t
Then s o l u t i o n s o f
P(B)u =
To a p p l y t h e s e c o n s i d e r a t i o n s t o d i f f e r e n t i a l e q u a t i o n s
b e a H i l b e r t o r Banach s p a c e a n d d e f i n e
63
OTHER TECHNIQUES
H (s,P)(X) q
Let
-1
<
for y
H (s,P,y)(X) 4
u(t)exP(-Yt)
{u : I
=
+
D(P) = nD(A.): I
Y
A
+
=
y
11
is 1-1 with
norm denotes Y
+
h with
=
A
is called y-regular if o(P)
and there is a Banach space Y
D(P)
an interval;
u
q,Y
Consider now (4.60),and equations P(Dt)u
Re
R
such that k . Further let H(q,k,I,y)(X) = {u; CII DJ(u(t)exp(-yt))1Iq 0 = L (X) = Lq(X) with weight function exp(-yt)).
(note H ( q , O , I , y ) ( X )
P(Dt)
C
a real number be the space of
H (s,P)(X). q
Definition 4.25
I
AjP-'(A)II
5
X
C
I.
lies outside the line
such that for Re X
+
c(y)(l
=
=
y, P - l ( X )
..., s )
where the
(j = 0,
IXl)k-j
:
X.
It is straightforward then to verify the following theorem. Theorem 4.26 _____
Let
P(Dt)
=
t
h
and
in L 2 (IR, X).
he the spectral measures for -iDt the equation P(D ) u
Y
be y-regular with
X
Hilbert spaces.
Then for any
has a solution u E H2(s,P,y)(X)
Let
EX
h E H(Z,k,I,y)(Y)
given by
.m
If
= {u E
A u
D(P);
3
is dense in
Yl
E
______ Definition 4.27 The operator P(Dt) that
O(P)
C
{A;
Re
<
is 1-1 and analytic for c(a)(l
+
IXl)k-j
(0
t
Definition428
P(D ) t
is called parabolic if there is a
Re X > yo s)
while if
with norm Y
Re X
+
X.
2
11 AjP-'(A)
a > Yo,
P(Dt)
:
Y
yo +
such
D(P)
[I 5
is called backward
is parabolic.
is called hyperbolic if it is simultaneously parabolic and
backward parabolic. Thus there exists yo < Re strip yo -
then this solution is unique.
y 1 and on some Banach space Y CX, P-'(X)
I j5
parabolic if P(-D )
D(P)
X < yl while for Re
of Definition 4.27 holds.
5
y1
X < a < y
such that and
Re A
O(P)
1a
lies in the > y1
the estimate
64
R . W.
y
o r closed) s t r i p <
+
c(a ,al)(l
2
5
(0
y1
w i t h norm
s)
5
yo < a.
while f o r
5
j
O(P)
Y
+
X.
l i e s o u t s i d e some (ope n
5
Re h
11 AjP-'(A) 11
a l < Y1
is c a l l e d quasi-
P(Dt)
l i e s o u t s i d e of a f i n i t e number of (open o r c l o s e d ) s t r i p s
G(P)
2 Y2p+l
1
Re
5
< Re A o -
IAOk-j
hyperbolic i f
YZp
is called quasielliptic i f
P(D ) t
Definition 4.29
CARROLL
( p = O,..,,N;
1) w h i l e i n e a c h o f t h e s e " r e s o l v a n t "
N
s t r i p s t h e e s t i m a t e above holds.
T h e r e a r e a number of t h e o r e m s f o r t h e s e t y p e s of o p e r a t o r s t h e p r o o f s of w h i c h a r e a l l s h o r t and s t r a i g h t f o r w a r d .
Theorem 4 . 3 0
Let
h ( t ) E H(q,k
u ( t ) E H (s,P,y)(X)
t
a n d one may a ssume
Proof: u(0)
=
Let
0).
E
=
B
and
with
D u(0) = 0 t
T(D t )
a nd
for
0
5
k
5
be Parahas a
s - 1
i s t h e H e a v y s i d e f u n c t i o n (* d e n o t e s c o n v o l u t i o n i n
> 0
w i t h o u t l o s s of g e n e r a l i t y ) .
yo -
L (R+,X) 4.Y
with
Bu
=
Dtu
a nd
S i m i l a r l y f o r t h e backward p a r a b o l i c
y < a < -y
1
P (Dt )
a solution
D(B) = {u; u ( t ) E H(q,l,T,y)(X):
(E = E
Then Theorem 4 . 2 2 c a n b e a p p l i e d
y < -yo; t h e n f o r
Y > Yo
for
k
given by t h e formula
9
yo < a < y
whe r e
1, I , y)(Y)
P(D ) u = h t
b o l i c ; then t h e equation solution
+
First
let
a nd
E =L (R+,Y)). 2 4rY
h(t) E H(q,k+l,I,y)(Y)
u ( t ) E Hq(s,P,y)(X)
of
a nd
P(Dt)u = h
i s g i v e n by
Theorem 4.31 generality. P(Dt)u
=
h
Let Let with
t h e n a s o l u t i o n of
P(D t )
be h y p e r b o l i c and t a k e
h ( t ) E H(q,k+l,T,y)(Y) Dku(0) t
=
0
P(Dt)u = h
for
0
5
(same
k h)
with
5
y,
5
0
5 y1
w i t h n o loss o f
y > yl. Then a s o l u t i o n of
s - 1 i s g i v e n by ( 4 . 6 5 ) .
If
Y
<
Yo
w i t h no i n i t i a l c o n d i t i o n s i s g i v e n by
(4.66).
R e s u l t s f o r q u a s i e l l i p t i c and q u a s i h y p e r b o l i c
P(D t )
a r e a l s o given i n Dubinskij
65
OTHER TECHNIQUES
[ 2 ] where c o r r e c t problems a r e i n d i c a t e d . s u c h as
(A)
L
i n Lemma 4 . 1 9 o r
P(A)
For t h e s t u d y of o p e r a t o r " p e n c i l s "
i n t h e D u b i n s k i j work and some r e l a t e d
t o p i c s we m e n t i o n f o r e x a m p l e Abdukadyrov [l], A l l a k h v e r d i e v - G a s a n o v
[ll,
Corbaruk-GorbaFuk [ 2 ] , Gorbaruk-Vainerman
[ 4 ] , Markus-Mereutsa
M a t s a e v [l], Vornpaeva-Maslov 111, Yakubov-Balaev
Remark 4 . 3 2
v
[I; 2 1 , Corbacuk
[ 4 1 , Gohberg-Krein
[ 2 ; 31, Gasymov [ I ; 2 ; 3 1 , Gasymov-Maksudov
[l], Burk
[ l ] , Virozub-
[S].
The s p e c t r a l t h e o r y of C a r r o l l [ 1 8 ; 1 9 ; 20; 2 1 : 221 h a s a l r e a d y b e e n
s k e t c h e d i n book form i n C a r r o l l [ 4 ] s o we w i l l n o t d e v e l o p i t i n d e t a i l h e r e . However a few comments a r e a p p r o p r i a t e h e r e f o r c o m p l e t e n e s s and some m o t i v a t i o n was p r o v i d e d i n t h e i n t r o d u c t i o n .
4\
i D k = (-A) Ca
a
ja
(t)Dx
\
the
\
+
(-A)'
and
A
problem w i t h
a
hk
and
hi;'
(1
5
L(H).
E
5
ak,
N)
one can w r i t e
i s p o s i t i v e and s e l f a d j o i n t w h i l e
H
Thus l e t
t.
be a s e p a r a b l e H i l b e r t space
a family of p o s i t i v e s e l f a d j o i n t o p e r a t o r s with
(1 5 j < M)
b. E L(H) J and
ak
2
L (Rn)
F o r s i m p l i c i t y we t r e a t t h e c o r r e s p o n d i n g a b s t r a c t
i n d e p e n d e n t of
ja
Let
and w i t h t h e by t h e
a l l commute.
k
A = 1
where
=
a r e c o n t i n u o u s so t h a t i n p a r t i c u l a r
where t h e R ie sz o p e r a t o r s
= Zajka(t)RnAk
F
Recall t h a t i n
/ik.
A
Let
b . , and t h e i d e n t i t y
b e t h e c o m m u t a t i v e Bdnach a l g e b r a g e n e r a t e d We w i l l u s e a few f a c t s from Banach a l g e b r a
e.
be t h e j o i n t spectrum of t h e g e n e r a t o r s OA
(il($),
i
5
N
+ M)
of t h e
on
u n d e r t h e map
ak
y
OA.
EM).
in
EM
Clearly
ak
(resp.
b.)
( a A ( b ) = prM(OA) w h e r e zk =
a,(@) 2 0
in
OA
hi:
thus
=
15k
zk, and
(2' =
U A C (c
N+M
uA i s t h e image of
G ( $ ) ; then
v($)
< N -
OA
=
i s compact and
(resp.
z.
1'
N + l
5
aA(b) denotes t h e j o i n t spectrum
prPI d e n o t e s t h e p r o j e r t i o n of
a n d we s e t
Xk = z
the equation
is t h e c h a r a c t e r i s t i c polynomial
Let
d e f i n e d by
-f
@(x)
The s p e c t r a l v a r i a b l e s
correspond to
bi
and
= OA
..., g N ( @ ) , cl($), ...,c M ( @ ) ) w h e r e
homeomorphic t o
--
b e o p e r a t o r s commuting among t h e m s e l v e s
t h e o r y h e r e and r e f e r t o C a r r o l l [ 4 ] f o r t h e n e c e s s a r y d e t a i l s .
the c a r r i e r space
ak
( z ~ + ~ , . ,. z.
N+M
))
-1
.
EN+M
Associated t o
66
R. W. CARROLL
(4.68
(cf.
1 . 2 . 1 0 ) and the associated framework).
Here one thinks of ( 4 . 6 7 ) as a
c'
first order system as in Section 1 . 2 of the form
a matrix like (1.2.10) with bottom row of the form
generally let u s take a compact set
roots of
-
and write $(A,z')
Q(C,X,z')
+
it is I
on which
IcN+M
=
correct if for
2'
zk
Cs(h,z'),
max Re 5
b, while $ ( A , z ' ) < c -
P
is and
is real and
s = 1,.
. . ,m
z
> 0;
k-
be the
(X.2').
hyperbolic if for z ' E
$(A,z')
5
A > c real. The equation k- k h -alXI + b for real A > c * k - k'
E prMI, $ ( A , z ' )
2
c
parabolic if for z ' E prM 2 ,
is
0 where
...,Pm-,(A,b))
(Po(A,b),
The equation ( 4 . 6 7 ) will be called
$(x,z') 5 alAl
prM=,
zC
but this is not assumed. Let
Definition 4 . 3 3
=
Now one can develop the theory here in terms of aA but more
1 replaced by -1.
usually aA C
+P(A,b)u
for
for real
Xk > c k' +P(A,b)c
Now the spectral form of the Green's operator for the system
is 5 ( t , ~ , A , z ' ) = exp[-(t - T ) P ( X , Z ' ) ] and one wants to show that domain of a suitable functional calculus relative to
5
=
0
is in the
I with suitable
(t,T)
dependence. Then a matrix G(~,T) with entries in L(H) will arise yielding a + -+ This involves considerable solution of ( 4 . 6 7 ) in the form u ( t ) = G(t,T)u(T). analysis which we omit here (cf Carroll [ 4 ; 18; 19; 2 0 ; 2 1 ; 2 2 1 ) .
A typical
theorem is cited for completeness however. Thus we will say that 2 i s
5 $
convex if for any Ip(z)I p
< 1
for
one has
11
z E
I. We say I is
p(a,b)llc
a
Theorem 4.34 Let
there is a polynomial p
he a
p
spectral for A
p
c suplp(z,z')l
over
convex p
such that
=.
L
p(<)
=
1 and
if for any polynomial
Then
spectral set for A
with ( 4 . 6 7 ) a
parabolic equation. Then there exists a weak Green's operator 9 ( t , T ) (t,T) +G(t,T)
E
Co(Lw(H
m
)),
G(t,T) E C1(Ls(Hm)),
while 9
Remark 4 . 3 5
P(X) = A
( A -k B)u = h
Taking
in the form
G(t,sfi(s,~) =G(t,-r), t
+P(A,b)G
+
A in
=
p
G(T,T)
=
I, and
with t
0.
( 4 . 6 2 ) one obtains the solution of
+
z
67
OTHER TECHNIQUES
k
for
=
0
r).
( w i t h s u i t a b l e t r a v e r s e of a s u i t a b l e
T h i s i s t h e f o r m u l a of
G r i s v a r d [ 2 ] and h a s b e e n s y s t e m a t i c a l l y e x p l o i t e d i n D a P r a t o - G r i s v a r d
[3].
They
d i s t i n u g i s h two main t y p e s o f p r o b l e m s a s f o l l o w s .
D e f i n i t i o n 4.36
The o p e r a t o r
are invertible for
X > 0,
for
parabolic i f
ZA
= {A:
X > 0
with
...,
with
m = l,2, (A - A )
larg X I
5
TI
+
L = A
M
- OA1
CB
and
A
independent of
(B - A )
and
is hyperbolic i f
B
(A
and
-
{A:
5
larg X I
OBI
n -
(B - A)
and
The o p e r a t o r i s
m.
CA
are invertible in sectors =
A)
CB w h e r e
and
RA
with
+ BB
< n,
while i n these respective s e c t o r s
Thus f o r
L
above w i t h
hyperbolic
OA,OB < n/2
- A)
(A
and
11
and
(B - h )
( A - X)-'l(
5
are invertible in sectors as with
M/Re X
11
Thus t h e h y p e r b o l i c c a s e is a l i m i t of p a r a b o l i c c a s e s where limit.
T i m e d e p e n d e n t e v o l u t i o n e q u a t i o n s of t h e f o r m
-u'
+
(B -
OA
=
n
A(t)u(t)
=
f(t)
+
B
and r e s u l t s a n a l o g o u s t o t h o s e o f K a t o [ 2 ; 5 : 7 1 , Kato-Tanabe
[ l ] a r e obtained.
a r e s t u d i e d a l s o i n t h e C o n t e x t of noncommuting
For noncommutative
A
and
B
can a l s o be u t i l i z e d under s u i t a b l e hypotheses, e f f e c t i v e l y w i t h a s o l u t i o n of with
B
r e p l a c e d by
B - 5
I n t h e commutative c a s e
+
OB
u"
A ( t ) u ( t ) - hu = f ( t )
X)-'II 5 M/Re A.
S
r,
o n e a s s u m e s f o r example t h a t
Lu - i,u = h
(L
(c
- C)-'
(B - A ) - '
and A
and
[ l ] , and Tanabe
a formula s i m i l a r t o (4.69) One c a n work i n e i t h e r c a s e most =
A
+
R)
(for technical reasons).
=
in the
e x p r e s s e d by ( 4 . 6 9 )
Thus set f o r
5 > 0
whereas i n t h e noncommutative c a s e i f D(A)c D ( A )
for
A t p(B)
with
68
f o r suitable
R. W. CARROLL
4
will be small f o r
(0 = arg A ,
5
8'
large and
=
arg z )
(f - 5 ) -1
ity a s in Kato 16; 71 are also utilized.
(E - 5)s
then = S
5
(1
+ R$-l
=
5
.
1
+
R
where
5
R
5
Hypotheses of s t a b i l -
CHAPTER 2
TRANSMUTATION
2.1.
Preliminary remarks and examples.
The idea of transmutation of differential
operators will be useful in dealing with related abstract differential equations and we will discuss this from various points of view following at first Delsarte [I], Delsarte-Lions 1 2 ; 31, Levitan [l; 3; 51, and Lions [ Z : 3: 41 (cf. also Andros'c'uk [l], Braaksma [l], Braaksma-deSnoo 121, Carroll 1351, Carroll-Showalter [81, Chebli [l], Gasymov-Magerramov 151, Helgason [l], Hersh [3], Hutson-Pym [ I ; 21, LeBlanc [l], L&fstrgm-Peetre [l], Marrenko [l; 2; 3 1 , Povzner [l],
Thyssen [ I ; 21). polynomials
Let
P(D)
or distributions.
into
we have
u(x) in
onto
-1
u = B
E
F
Q(D)
Q(D)Bu
X
from
v
and P
Q
=
Y
0 so
v
=
then from
F and
P(D)u(x)
=
then setting v(y)
solutions of
F Q(D)v(y) ODE
=
=
Q(D)B
Bu
Y
and
of functions
B, usually an integral operator, transmutes =
BP(D).
Then for example if
satisfies Q(D)v
Q(D)v
=
0 we get
=
(Bu)(y)
g(A)v(y).
0. If
P(D)B-'v
0. More generally if
g(A)u(x) =
and consider (linear) differential
need not have constant coefficients nor be of the
if formally
satisfies P(D)u
so that in
a/ay
acting in suitable spaces X
Q(D) and
or
We s a y that an operator
same order. P(D)
D = a/ax
u
=
B
P(D)u
=
0
is an isomorphism
B-'Q(D)v
=
0 so
is a vector function with
for a suitable operator function
g(A)
acting
we have
It is in this spirit that one can connect
with operator coefficients and one s p e a k s then of related
differential equations. Let u s cite a few results on transmutation first to gain some perspective.
X
=
Y
be the space
H
of analytic functions of one complex variable
the topology of uniform convergence on compact subsets of
69
(c.
Let
z
P(D)
Let with
=
70
R. W. CARROLL
m
1 aj(z)D'
where
j =O
a.(.)
D = d/dz, and
H,
I
am(z)
=
1. Then Delsarte-Lions
[ Z ] prove
Theorem 1.1 BP(D).
There exists a continuous isomorphism B : H
Explicitly
B
H
-+
such that DmB
=
has the form p+km (p+km) !
'
m A s a corollary it follows immediately that if
Q(D)
1 b.(z)Dj
=
j=O
b (z) m
and
=
1 then there exists an isomorphism
: H
with
bj(-)
E
Q(D)B
=
H
J
H
-+
such that
The formal verification of (1.1) is straightforward and one need only con-
BP(D).
firm the convergence in order to prove Theorem 1.1.
Such results are however
special for analytic situations and the general question of just what be transmuted into what
Q(D)
is unsolved;
P(D)
can
it cannot always be done (cf.
Levitan [l]). Remark 1.2
Delsarte [l] was also concerned with generalizing the idea of trans-
lation and this is developed further in Levitan [ l ; 31. order differential operator for example of the form
A@
with
@(O,A)
= 1
and
@'(O,A)
= 0.
Lf
Thus let =
Write formally
@
f"
-
pf. k
=
m C
L be a second Let
L@
A @k with
=
L@, =
0
$k-l
and
so that
L@o = 0
L
Then the generalized translation operator associated with
for
f E Cm.
f(x)
while
Since bk(0)
@i(O)
0
=
=
0 for
for k
2
translation follows upon taking xn/n! that
s o that
LxT:f
(x)
DQn =
=
L TYf (x) Y X
k
1 with
@o(0)= 1
0 implies D TYf(x)ly=O Y X L = D, @(x,A) = exp Ax
= =
is defined by
we have
Tof(x)
=
0. The analogy with n n
I A x /n!, and
"k k and TYf(x) = ED f(x)y / k ! = f(x + y). 0 (x) while since LXT', = Z,@,(y)L,k+lf
an(x)
We note also
=
PRELIMINARY REMARKS AND EXAMPLES
71
For further comments in this direction see Remark 3.18. Now following Lions [2] we consider the Cauchy problem, for
f.
for suitable
priate space and define B P(Dx)@(x,y)
so
-
of order
m
(1.4) has a unique solution in some appro-
by the rule
(Bf)(y)
=
@(O,y).
-
-
Setting +(x,y)
=
we have
Consequently, given (1.3)
-
Assume that (1.3)
Q(D)
B
defined as above and the uniqueness of solutions of
(1.4) and (1.5) - (1.6), it follows that
that
B
transmutes P(D)
into Q(D).
We will utilize precise versions of
this later and for now simply state (cf also Remark 2.18) Theorem 1.3(F). and define
B
Example 1.4
Assume (1.3) - (1.4) and (1.5) as indicated.
where
=
(1.6) have unique solutions
BP(D).
Let u s apply this technique to (1.1.3)-(1.1.4)
in Example 1.1.10. find
Then Q(D)B
-
Thus let
P(D)
=
D2
and
Q(D) = D
with
toproduce (1.1.6) as g(A)
= A
2
.
To
we look at (cf. (1.3) - (1.4))
B
f
is a function in say
Remark 2.18).
s'
which we take (or extend) to be even ( c f .
A s with (1.1.12) one can take Fourier transforms in
x
to obtain
R. W. CARROLL
72
$t + s
2-
Q,
2
0, $ ( s , O ) = ? ( s ) , from which &(s,y) = F(s)exp(-s amd recalling that f~1 K(y,x) = exp(-s 2 y) we have =
(1.9)
Q,(x.y)
=
y)
where
&
FQ,
=
m
1
- C)d5
K(y,S)f(x -m
rm
a formula which makes sense even if
is of say exponential growth as 5
f(5)
Now apply this to the vector valued functions P(D)w
Thus
=.
2
A w
Q(D)Bw
= BP(D)w
BA2 w
=
f
(1.10) (cf. a l s o (1.1.13)).
5
-f
= A
Bw;
-
(1.10) corresponds t o the condition
u = Bw
we obtain ( 1 . 1 . 6 ) from the form of
w(S)
One should emphasize that
and be integrable against
m
2
in (1.8)
0. Consequently setting
=
t
of (1.1.3) - ( 1 . 1 . 4 ) .
w
and
K(y,c)
so we must take
must make sense a s
T(S)
to be a quasi-
equicontinuous group for example (local equicontinuity is not enough). Remark 1.5
With Example 1 . 4 in mind let us look at some other aspects of the
transmutation question.
Suppose we assume
DB
=
BDL
B
where
is an integral
operator of the form
I
where
is
or
(-m,m)
suitable behavior at
I
For
(0,m).
= (0,~)
we obtain formally, assuming
a),
1
m
(1.11)
(BD2f)(y)
=
b(y,x)f"(x)dx
=
0
+ bx(y,O)f(0)
j:bxx(y,x)f(x)dx
-
b(y,O)f'(O)
m
whereas
(I)Rf)(y)
=
[
b (y,x)f(x)dx.
Hence i E
'0
and K(y,x)
f'(O) =
=
(ny)
m.
and
the use of even functions w (0)
u
+
0 we will have
-L2 2 cxp(-x /4y)
DD
=
BD2
b
Y
acting on s u c l l
=
b
f.
satisfies these requirements.
xx
with
bx(y.O)
Clearly For
I
b(y,x)
= (-m,m)
=
0
=
thc,
AND EXAMPLES
PRELIMINARY REMARKS
s i t u a t i o n i s similar e x c e p t t h e c o n d i t i o n
I
intervals
2 A w
w (0) = 0 , and w e want t o f i n d
with
by a f o r m u l a
u
u(y) = j.b(y,x)w(x)dx.
satisfying
Then
u
=
on
(1.1.3)
A2u
b
if
Y
0 s i n c e from (1.11)
hx(y,O) = 0
f
s a t i s f i e s (1.1.4),
w
On t h e o t h e r h a n d s u p p o s e
m
w
d o e s not a r i s e ; t h e trjo
f'(0) = 0
a r e e q u i v a l e n t i f we c o n s i d e r e v e n f u n c t i o n s
f ' ( 0 ) = 0.
that
73
=
Y
(-m,a)
so
namely
wxx =
connected t o bxx
with
m
A2u(y) =
(1.12)
(y,x)w(x)dx
The i n i t i a l c o n d i t i o n a s is the case with
F
t > 0
i s a c h i e v e d when
b(0,x) = 6(x)
(one s i d e d )
b(y,x) = K(y,x).
Let us consider (1.l.S)
Example 1 . 6 in
u ( 0 ) = w(0)
u (y Y
=
when
A
g e n e r a t e s a n equicontinuous group
and f o l l o w t h e p r o c e d u r e o f S e c t i o n 1.1.
Thus f o r
--LD
< x <
and
a'
we a r e l e d i n t u i t i v e l y t o t h e h a l f p l a n e D i r i c h l e t p r o b l e m
It i s e a s i l y checked t h a t
r a t h e r t h a n a n i m p r o p e r l y p o s e d Cauchy p r o b l e m . G(t.x) =
1
-
2
H(t,x) = t n - l ( t 2
+
x2)-'
i s a s o l u t i o n o f ( 1 . 1 3 ) and we w r i t e m
(1.14)
1
z ( t ) = <- H ( t , . ) , 2
T(')u~>
= -
r"
=
I
I_.,
H(t,x)T(x)uodx
I-
H ( t , x ) c o s h Axu
'0 to o b t a i n (1.1.8)
formally.
Since
p(T(x)uo)
5
q(uo)
for seminorms
i t follows t h a t t h e b r a c k e t , or i n t e g r a l , i n (1.14) makes s e n s e f o r (since
IH(t,x)l
d o m i n a t e d hy
5
2t
;-(x
c(t)/xn
2 (n
+ 2
2 -1
t )
).
Similarly
2)
as
x
+ m
Ht,
Hx,
and
Htt
=
p
and
t > 0
-H x x
w i l l be
and o n e c a n d i f f e r e n t i a t e u n d e r t h e
i n t e g r a l s i g n i n ( 1 . 1 4 ) and i n t e g r a t e by p a r t s .
Thus
q
74
CARROLL
R. W.
(0,m)
( w h e n ' w o r k i n g on
note that
Hx(t,O) = 0
Thus w e c a n
wx(0) = 0 ) .
and
s t a t e ( l e a v i n g open t h e q u e s t i o n o f u n i q u e n e s s )
Theorem 1 . 7 while z
w
Let
g e n e r a t e an e q u i c o n t i n u o u s group i n
A
is t h e unique s o l u t i o n of ( 1 . 1 . 4 )
g i v e n by (1.14)
uo 6 D(A2 )
with
from Theorem 1.1.8.
Then t h e f u n c t i o n
(or e q u i v a l e n t l y (1.1.8)) i s a s o l u t i o n of (1.1.5).
L e t u s l o o k a t Example 1 . 6 from t h e p o i n t o f v i e w of t r a n s m u t a t i o n .
Remark 1.8
F o l l o w i n g Formal Theorem 1 . 3 we c o n s i d e r f o r
$xx
(1.15)
F
+
@lYY
and
Q(D) = -D 2
@(X,O) = f ( x ) .
= 0;
$ (x,O) = 0
Imposing a c o n d i t i o n
P(D) = D2
Y
would l e a d t o a n i m p r o p e r l y p o s e d Cauchy p r o b l e m
s o w e want t o f i n d some o t h e r way t o p r o d u c e u n i q u e s o l u t i o n s o f ( 1 . 1 5 ) . f
t o b e a n e v e n f u n c t i o n on
(1.15) t o o b t a i n as
y
+
$yy
+
2s @ = 0
i n s u c h a way t h a t
$(s,y) = P(s)exp(-sy).
(x2
-
(-m,m)
y2)-l
But
( c f . Remark 2.18) and F o u r i e r t r a n s f o r m
with $(s,y)
$(s,O) = ? ( s ) . +
F ?1 H ( y , x )
0 =
$
exp(-sy)
where
1 7 H(y,x)
$(x,y)
+
i s g i v e n by =
y~
-1
and h e n c e
z = Bw
a s i n (1.14).
The s p e c i f i c a t i o n o f Cauchy d a t a i n ( 1 . 4 ) i s o n e way o f a s s u r i n g a
unique solution B.
I f one r e q u i r e s
a s w e l l then a unique
A p p l y i n g t h i s t o v e c t o r f u n c t i o n s a s b e f o r e we o b t a i n
Remark 1 . 9
Extend
@
of ( 1 . 3 ) - ( 1 . 4 ) s o a s t o d e t e r m i n e as w e l l d e f i n e d o p e r a t o r
O b v i o u s l y a n y s p e c i f i c a t i o n of d a t a a t
uniquely determine
@
y = 0
p l u s growth c o n d i t i o n s which
w i l l s e r v e t o d e f i n e a t r a n s m u t a t i o n o p e r a t o r ( c f Remark
2.18).
Example 1.10 z
of ( 1 . 1 . 3 )
L e t u s e x a m i n e now t h e f o r m u l a (1.1.7) c o n n e c t i n g s o l u t i o n s
and ( 1 . 1 . 5 ) .
Setting
P(D) = D
and
Q(D) = -DL
f o l l o w i n g Theorem 1 . 3 ( F ) t o c o n s t r u c t a t r a n s m u t a t i o n o p e r a t o r
u
and
i s n a t u r a l and
B
one c o n s i d e r s
0
75
PRELIMINARY REMARKS AND EXAMPLES
w i t h some f u r t h e r s t i p u l a t i o n n e c e s s a r y i n o r d e r t o i n s u r e a unique s o l u t i o n . want t o produce a s o l u t i o n i n v o l v i n g as a k e r n e l which i t s e l f s a t i s f i e s
Ix = I
(1.1.8) i n
x
has t h e form (is)'
=
lfT
A(s) = 0
w e have G(s,y)
(1 + i ) s 4
=
$
I ( y , x ) where
Fc
A
=
,
G(s,y)
F
and
Thus
exp(-(is)'y)
I(y,Ix[)
I"
for
x
5
=
0
B
an i n t e g r a l o p e r a t o r w i t h k e r n e l
+ m
b(y,x).
z(y) = =
while
b(y,O)
=
0
u
= A
2
u
r O
0
I ( y , 1 x l ) e x p i s x d x = FI-.
Remark 1.11 I t i s i n s t r u c t i v e t o c o n s t r u c t
should have w e assume
FcI
and
z = Bu
x
as
y
+
then
m
d e n o t e t h e F o u r i e r c o s i n e and s i n e t r a n s f o r m s re-
Applying t h i s to v e c t o r f u n c t i o n s w e o b t a i n
as
0
exp(-s 4y / f i ) S i n s'y/fi
exp(-s 4y / f i ) C o s s4 y / f i = F,I(y,x),
I(y,x)exp(-isx)dx =
b(y,x)
+
Hence we t a k e
t o be t h e d i s t r i b u t i o n
z and assume
Now F o u r i e r t r a n s f o r m i n g
A
and i f w e s t i p u l a t e s a y t h a t
s p e c t i v e l y ( c f . E r d e l y i [l]).
I-
2
- is@= 0 w i t h @ ( s , O ) = f(s) t h e s o l u t i o n of which @YY A ( s ) e x p ( i s ) 4y + B ( s ) e x p ( - ( i s ) 4 y ) . C o n s i d e r i n g s real
is indicated.
But i t i s known t h a t
I(y,x) = 1 ~ r - ~ y x - ~ / ~ e x p2 (/4x) -y
( c l a r i f i e d below).
YY
A
W e
iFsI for
and i f w e t a k e x > 0
Consequently
which i s (1.1.7)
i n Example 1.10 by assuming i t is
W e can i n f a c t go d i r e c t l y t o
b(y,x)u(x)dx. and t h e n i f
it follows t h a t
FcI-
then
u
and
To s e e what p r o p e r t i e s b(y,x)
h a s s u i t a b l e growth
76
R. W . CARROLL
Thus
and we a r e l e d t o l o o k f o r a d i s t r i b u t i o n o r f u n c t i o n
(1.22)
b
YY
-
b
=
x
0;
b(0,x) = 6 ( x ) ;
Weknow a s o l u t i o n o f ( 1 . 2 2 ) i s Laplace transform (1.22)
6
(1.23)
where
YY
(x
b(0,s)
= sb;
b ( y , s ) = Lb(y,x) =
,
+
=
b(y,O)
=
b(y,x) = I(y,x) s)
b(y,x)
such t h a t
0 b u t t o d e r i v e t h i s we c o u l d
t o obtain
1
b(y,x)exp(-sx)dx
( s e e D o e t s c h [l], S c h w a r t z [l],
Sneddon [l], Widder [l], Zemanian [ l ] f o r L a p l a c e t r a n s f o r m s ) . (1.23) is
$ ( y , s ) = exp(-ys')
r e a l (the function
i f we r e q u i r e say
exp(ys4)
-
f o r Laplace transforms
+
as
0
But
LI(y,x)
and
u(t)
Let
s
for
(cf.
b(y,x) = I ( y , x ) .
g e n e r a t e an e q u ic o n tin uous group i n
F
a s follows.
with
u
b e t h e u n i q u e s o l u t i o n of (1.1.3) g i v e n by Theorem 1.1.11.
g i v e n by ( 1 . 1 . 7 )
Proof:
A
+
e x p ( - y s 4)
=
For c o m p l e t e n e s s we w i l l f o r m u l a t e a t h e o r e m r e l a t i v e t o ( 1 . 1 . 7 )
Theorem 1 . 1 2
y
c a n b e e x c l u d e d a l s o by g e n e r a l g r o w t h r e q u i r e m e n t s
c f . S c h w a r t z [l]).
E r d e l y i [l]) which y i e l d s
b(y,s)
The s o l u t i o n of
E D(A2)
Then
z
is a s o l u t i o n of ( 1 . 1 . 5 ) .
One n e e d s o n l y t o c h e c k t h a t t h e v a r i o u s i n t e g r a t i o n s , e t c . , make s e n s e
1 / 2 n4
We n o t e f i r s t t h a t w i t h
c
while
i s a seminorm i n
I
seminorm
= Ix.
YY
q
and
If
p
=
F
then
p(w(t))
5
q(uo)
f o r some
PRELIMINARY REMARKS AND E W L E S
where since
i,
h ( x ) = (TX)-'
u
=
2
A u,
(cf. Erdelyi
ill)
exp(-t
p(ux(x))
2
5
and
Similarly
2
h(x)q(A u o ) .
( c f . Theorem 1.1.11).
Consequently, from ( 1 . 1 . 7 )
s o t h a t ( 1 . 1 . 7 ) makes s e n s e .
w e l l d e f i n e d and by ( 1 . 2 5 ) t > O
by ( 1 . 1 . 6 )
/4x)dt
77
Ix(y,x)u(x)dx
Similarly
Similarly, t > 0
for
I(y,x)ux(x)dx
is
j'b
can be checked by n o t i n g t h a t f o r
n > 2
zt
and
ztt
can be o b t a i n e d by d i f f e r e n t i a t i n g under t h e i n t e g r a l
QED
sign in (1.1.7). Remark 1.13
The t h r e e formulas ( 1 . 1 . 6 )
-
( 1 . 1 . 8 ) , w i t h ( 1 . 1 . 9 ) - (1.1.111,
examined t o g e t h e r i n a c o n c r e t e a n a l y t i c a l s e t t i n g by Dettman [ 3 ] .
They a r e a l s o
[ 6 ; 7 ; 81 and Dettman
d i s c u s s e d s e p a r a t e l y i n v a r i o u s c o n t e x t s i n Brags-Dettman [l].
are
C l a s s i c a l L a p l a c e t r a n s f o r m and complex v a r i a b l e methods a r e used.
The
matter i s t a k e n up a g a i n i n Dettman [ 2 ] ( c f . a l s o Dettman [ 4 ] ) i n g e n e r a l s p a c e s u s i n g t h e H i l b e r t t r a n s f o r m and we w i l l s k e t c h some of t h i s .
For t h e c l a s s i c a l
H i l b e r t t r a n s f o r m see B u t z e r - T r e b e l s [ l ] , C o t l a r [ l ] , Horvgth [ 2 ] , O k i k i o l u [l], Orton [l; 2 1 , S t e i n [ l ] , and Stein-Weiss [ 2 ] .
The i d e a i s t o r e p l a c e t h e c l a s s i c a l
Hilbert transform
(1.29)
tf(x - t)dt
Hf = l i m -
by a formula ( c f a l s o Westphal [l])
(1.30)
where
Hf
T(t)
=
1 lim y+o
w
J-,
tT(t)fdt t
+y
i s an e o u i c o n t i n u o u s group of o p e r a t o r s i n
F
is t a k e n t o be a complete, b a r r e l e d , s e p a r a t e d LCS o v e r IR. be proved a s i n t h e c l a s s i c a l c a s e where
2
H f
=
-f
g e n e r a t e d by Then
F
A.
-2 H f = -f
( s u i t a b l e h y p o t h e s e s on
will
f
7%
R. W. CARROLL
are i n d i c a t e d below).
-
blems (1.1.3)
s o t h a t t h e pro-
A
(1.1.5) are uniformly w e l l posed i n t h e sense t h a t f o r D
dense subspace
One wants h e r e enough h y p o t h e s e s on
t h e r e e x i s t s a unique s t r o n g s o l u t i o n f o r
i n i t i a l v a l u e s i n a s t r o n g s e n s e , and such t h a t i f
u
CI
0
-+
t
in
i n g s o l u t i o n s converge uniformly t o z e r o on compact s u b s e t s of
3
u
in a
0 , assuming t h e t h e correspond-
D
+ lR .
Also t h e
problem
should b e u n i f o r m l y w e l l posed f o r t h i s t h e o r y . r e s o l v a n t s e t of setting
S(t)
=
A
and
A
g e n e r a t e s an e q u i c o n t i n u o u s group
1
cosh A t = $ T ( t )
a l s o equicontinuous (note
+ T(-t)]
v ( t ) = lJ(t)v
0
I n particular i f
i t follows t h a t
for (1.31)).
T(t)
is in the in
F
then
U(t) =
Under t h e s e assumptions
a l l of t h e problems a r e uniformly w e l l posed ( c f B a l a k r i s h n a n [ 2 ] , Bragg-Dettman [6; 7 1 , Dettman [l; 2 1 , F a t t o r i n i [ l ] , G o l d s t e i n [ 6 ] and o u r p r e v i o u s d i s c u s s i o n
Now d e f i n e
of t h e s e problems).
Then
@(y)f
is defined f o r
and
f E F
Y(y)f
e x i s t s as a Cauchy p r i n c i p a l
value i n t h e sense t h a t
(1.33)
I” %
Y(y)f = l i m N m
dt
-N t + y
1 l i m 7T
=
N*
IN 0
t“T(t) -T(-t)lf 2 2 t +y
-
T(-s)]ds = 2
dt
where
a(t) =
:1
[T(s)
is equicontinuous.
S(n)dn
and
Mote h e r e t h a t R(A)
J:
s i n h Asds
a t)
i s dense.
=
2[T(t/2) - T(-t/2)]U(t/2)
On t h e o t h e r hand
where
79
PRELIMINARY REMARKS AND EXAMPLES
0
The f i r s t t e r m on t h e r i g h t s i d e i n (1.34) t e n d s t o
is equicontinuous. Theorem 1 . 1 4
+
= since
and
For
f E D(A)
y > 0
and
for
0
case since
15 p <
m.
(Of)" = -A Of
and
f = u ).
f E D(A).
is defined f o r a l l
6
an i n v e r s i o n formula f o r
2 f E D(A )
Now f o r
(Yf)" = -A I f ; i n p a r t i c u l a r
is e a s i l y seen t o be w e l l defined f o r if
A.
2
( 1 . 1 . 5 ) and y i e l d s t h e formula ( 1 . 1 . 8 ) ( f o r
operator then
and
A = -d/dx
However t h e c l a s s i c a l t h e o r y i s n o t a s p e c i a l
i s n o t i n t h e r e s o l v a n t s e t of 2
evidently
one h a s
( Y ( y ) f ) ' = -AO(y)f.
The c l a s s i c a l Cauchy Reimann e q u a t i o n s are e x h i b i t e d i n t h e case F = Lp(-=,-m)
a(t)
By a s t r a i g h t f o r w a r d c a l c u l a t i o n one now o b t a i n s
[Cauchy-Riemann e q u a t i o n s ) .
( O ( y ) f ) ' = A.Y(y)f
N
as
let
f E F
g = Hf
z = Of
fi
and assume
satisfies
The H i l b e r t t r a n s f o r m
Further i f
and
one h a s
A
is a continuous
i s continuous. Hg
exists.
To p r o v i d e
Then f o r
f' E
F' < f ' , T(-x)Y(y)g>
as
y
-+
-+
I f one shows t h a t
0.
w i l l follow.
-2 < f ' , T(-x)H f > -2 < f ' , T(-x)H f > = - < f ' , T ( - x ) f >
-2
then
H f = -f
To do t h i s c o n s i d e r
I
m
$(x,y) = y
(1.36)
TI
--m
m
so t h a t
< f ' y T2( - t ) 2f > d t ; (x-t) + y (x-t) dt 2 2
$ ( x , y ) = < f ' , T(-x)O(y)f>
and
$(x,y) =
< f t r
T(-x)v(y)f>.
Y
As
+
5
$(x,y)
-+
< f ' , Tf-x)f>
while
$(x,y)
+
< f ' , T(-x)Hf>;
bounded harmonic f u n c t i o n s i n t h e h a l f p l a n e
y > 0.
further L e t now
@
and
$
are
0
80
R. W. CARROLL
which i s a bounded a n a l y t i c f u n c t i o n f o r
y > 0
and h a s t h e r e f o r e a Poisson
integral representation
Now l e t
R*(z) f
@*(x,y) = < f ' , T ( - x ) @ ( y ) g > and
=
+
@*(x,y)
i$*(x,y)
g = fif.
r e p l a c e d by
so t h a t
$*(x,y) = < f ' , T ( - x ) Y ( y ) g >
h a s a P o i s s o n i n t e g r a l r e p r e s e n t a t i o n l i k e (1.38) w i t h
Then
R e R*(z) = ImR(z)
so that
ImR*(z) = -Re n ( z )
to
w i t h i n a c o n s t a n t which must b e z e r o s i n c e
Hence
$*(x,y) = -@(x,y)
(1.40)
lim $*(x,y)
=
P O
and
consequently
= - 1 i m @ ( x , y ) = - < f ' , T ( - x ) f > . Y+0
T h i s p r o v e s t h e i n v e r s i o n formula
Theorem 1 . 1 5
Remark 1 . 1 6
If
if
-2 -f = H f .
has a Hilbert transform then
These i d e a s a r e extended i n Dettman [ 4 ] t o g e n e r a l i z e d H i l b e r t
t r a n s f o r m s r e l a t e d t o t h o s e of Heywood
[I], Kober [ l ] , and O k i k i o l u [I]. He
c o n s i d e r s t h e problems
(1.41)
u"
2v +u' t
(1.42)
w"
+
t
2
= A u;
2 w ' = -A w;
u(0) = f ;
w(0) = f
u'(0) = 0
(u
< 1/2)
a g a i n i n some complete b a r r e l e d s e p a r a t e d LCS e q u i c o n t i n u o u s group
T(t)
with
0
3.14.
where
i n t h e r e s o l v a n t s e t of
d i s c u s s e d (1.41) i n Example 1 . 3 . 1 (where m = -1/2)
o v e r IR
F
2u
=
2m
+
1
with
A.
g e n e r a t e s an
A
We have a l r e a d y
R e m > -1/2
or
and i t w i l l come up a g a i n i n Example 2 . 6 , Example 3 . 1 1 , and Example
I t p r o v i d e s a n i c e example of a v a r i a b l e c o e f f i c i e n t o p e r a t o r
s u s c e p t i b l e t o v a r i o u s t e c h n i q u e s of i n t e r e s t .
The problem ( 1 . 4 1 )
D
2
2 m + lD + __ t
i s a Cauchy
81
PRELIMINARY REMARKS AND EXAMPLES
p r o b l e m f o r t h e EPD e q u a t i o n w h e r e a s ( 1 . 4 2 ) i s a n a b s t r a c t D i r i c h l e t t y p e p r o b l e m f o r t h e e q u a t i o n of g e n e r a l i z e d a x i a l l y symmetric p o t e n t i a l t h e o r y ( c f . Wei nst ei n The s o l u t i o n t o ( 1 . 4 1 ) i s g i v e n by (1.3.8) when
[3]).
A
generates a l o c a l l y
e q u i c o n t i n u o u s g r o u p and t h i s f o r m u l a a l s o h o l d s of c o u r s e f o r
A
i n which c a s e t h e problem ( 1 . 4 1 ) w i l l b e u n i f o r m a l l y w e l l posed. (1.1.3)
equicontinuous The p r o b l e m
f o r t h e a b s t r a c t h e a t e q u a t i o n serves a s a l i n k b e t w e e n ( 1 . 4 1 ) and ( 1 . 4 2 )
( c f . Theorem 1.1.11 f o r t h e s o l u t i o n o f ( 1 . 1 . 3 ) g i v e n by ( 1 . 1 . 1 3 )
which a l s o
s p e c i a l i z e s t o t h e case o f a n e q u i c o n t i n u o u s g r o u p ) ; t h e p r o b l e m (1.1.3) w i l l b e a l s o u n i f o r m l y w e l l p o s e d as i n Remark 1.13.
Now l e t t i n g
( 1 . 1 . 3 ) one c a n w r i t e ( c f . B r a g g [ 3 ] , Bragg-Dettman
v
be t h e s o l u t i o n of
[ 6 ; 7 ; 81, D e t t m a n [ l ] )
and composing t h e s e f o r m u l a s o n e o b t a i n s
where
a(.,.)
is the beta function.
I n p a r t i c u l a r when
V = 0
u ( 0 ) = c o s h Auf
and ( 1 . 4 4 ) c a n be w r i t t e n t l - 211
(1.45)
When
w(t) =
R(%-
F = Lp(-m,m),
u,%)
Im J-m
1< p <
m,
T(U)fdU
2
2 1-1J
and
T(o)
( t +o )
i s t h e t r a n s l a t i o n g r o u p t h i s becomes
which i s t h e " P o i s s o n f o r m u l a " f o r t h e s o l u t i o n o f t h e g e n e r a l i z e d a x i a l l y symmetric p o t e n t i a l problem i n t h e h a l f p l a n e f(x)
(cf Weinstein 131).
The f u n c t i o n
t > 0
I$(x,t)
i n t e r m s o f boundary v a l u e s
i s t h e r e a l p a r t o f a pseudo-
a n a l y t i c f u n c t i o n i n t h e upper h a l f plane with conjugate f u n c t i o n
82
For
R. W. CARROLL
-4 < u
<
4
one h a s
which, e x c e p t f o r a c o n s t a n t , is t h e g e n e r a l i z e d H i l b e r t t r a n s f o r m of Heywood [ l ] , Kober [l], and O k i k l o l u [ Z ] .
u
-112
(1.49)
(1.50)
With t h i s background Dettman [ 4 ] sets f o r
112
Q (t)f
u
=
m
t l - 2lJ @(+-lJ&
-m
T((J)fdU ( t2+(J2 ) 1-lJ
YP(t)f =
and d e f i n e s a g e n e r a l i z e d H i l b e r t t r a n s f o r m as t h e s t r o n g l i m i t (when i t e x i s t s )
(1.51)
H f = lim \Y ( t ) f . t + O lJ
Using t e c h n i q u e s s i m i l a r t o t h o s e of Remark 1.13 i t is proved t h a t Theorem 1 . 1 7
If
f E D(A)
t > 0
then f o r
(Y,,(t)f)'
=
-t2'AQ
lJ
(t)f
and
AYlJ ( t ) f = t 2 1 J ( Q p ( t ) f ) ' .
Theorem 1.18
Theorem 1
If
. If
Remark 1.20
f E D(A)
H f
lJ
then
H f
e x i s t s and
lJ
h a s a g e n e r a l i z e d H i l b e r t t r a n s f o r m then
and
L1
L2
(H f ) = - f .
P
of second o r d e r a r e s a i d t o b e
connected o r r e l a t e d i f t h e r e a r e smooth f u n c t i o n s
,...,x,),
-lJ
Pagkovskij [l; 21 examines an i d e a of r e l a t e d e q u a t i o n s a s f o l l o w s .
Two p a r t i a l d i f f e r e n t i a l o p e r a t o r s
(xl
H
such t h a t f o r any
u , v E C'
ai(x)
and
bi(x).
x =
one h a s
n
(1.53)
!L2vL1u = LluL2v
+ ~Dit$i(x,u,v,uX,vX) 1
(1.54)
!Llu = Eb D.u i l
+
n
L1
!L2v = l iD iv
0
Boundary v a l u e problems f o r s a i d t o be r e l a t e d i f
la
b u;
L1
and
and L2
L2
+
a 0v .
i n a region
G
w i t h boundary
a r e r e l a t e d and whenever
u
and
v
aG
are
satisfy
83
FURTHER EXAMPLES
a p p r o p r i a t e homogeneous boundary c o n d i t i o n s n = 2
Taking
ja,Lm,(.,u,v,u
I,,
f o r convenience t h i s means
b e t h e formal a d j o i n t i t i s e a s y . t o s e e t h a t
L?L2 = IIfL2.
if
L v = 0 2
Thus i f
then
w
- 4 2 dy
aldx
and
L1
II2v
=
x
, v ) c o s nxido = 0. x
Lz
Then l e t t i n g
= 0.
are r e l a t e d i f and o n l y
L2
A number
L*w = 0.
satisfies
1
of s t r a i g h t f o r w a r d c a l c u l a t i o n a l r e s u l t s are proved.
2.2.
FURTHER EXAMPLES.
The p r e c e e d i n g s e c t i o n h a s g i v e n a n i d e a of t h e t y p e s
of r e s u l t s one can o b t a i n about t r a n s m u t a t i o n and r e l a t e d e q u a t i o n s and we w i l l F i r s t l e t u s t a b u l a t e what
compile some f u r t h e r i n f o r m a t i o n h e r e i n t h i s r e g a r d .
h a s been l e a r n e d about some s i m p l e o p e r a t o r s and e n l a r g e t h e list of examples. From Example 1.4 and Remark 1 . 5
From Example 1 . 6 and Remark 1 . 8
2 2 -D B = BD ;
(2.2)
(Bf)(y) =
The f i r s t e x p r e s s i o n f o r
a
satisfying
g(A)B if get
-
w
D
2
Bf
g(A)a
CY =
= A
in
2
zt t
and
wt t
c
ly.
The second e x p r e s s i o n s f o r
apply t o z = Bw
CY
(2.1) and (2.2) a p p l y t o any v e c t o r f u n c t i o n =
or
6
=
B.?
defined f o r
as b e f o r e f o r
Ba
then s a t i s f i e s
2
w
and
w, u , z
1.10 and Remark 1.11 one h a s
=
B6
B
6
=
5
or
Similarly taking
r e s p e c t i v e l y then
5 2 0
DR = g(A)B
a(<) must be d e f i n e d f o r
= A
= -A2z.
H(y,S)f(-S)dc;
-m
and
and
= Bz
m
B(0) = a ( 0 ) ; however
with
i s any s o l u t i o n o f
Gt
1
$
=
B6
g ( A ) = -A2
2-A v
and
<
-m
ctt
2 -D B =
or
5 <
a.
Thus
r e s p e c t i v e l y we if = A
ztt 2
6
= -A2?
respective-
i n (2.1) and (2.2) are r e f i n e d v e r s i o n s which
satisfying
at ( 0 )
=
0
a s i n the Introduction.
and we g e t
u = Bw
or
F i n a l l y from Example
R . W. CARROLL
84
The f o r m u l a s from s p e c i a l f u n c t i o n t h e o r y c o r r e s p o n d i n g t o ( 2 . 1 ) (setting
g(A) = A L
(2.4)
e -A
2
-
A
with
-
(2.3)
are
iX)
m
J 0exp(-T 2 / 4 t ) c 0 s
= (Tt)-’
ATdT;
The f i r s t two f o r m u l a s a r e w e l l known F o u r i e r c o s i n e t r a n s f o r m s a n d t h e l a s t f o r m u l a c a n b e o b t a i n e d f r o m a t a b l e of L a p l a c e t r a n s f o r m s ~ ( s / C ) ~ K + [ ~ 4( ]S ~ and )
K4(z) = ( ~ / 2 z ) + e x p ( - z )
-
(L(x-~” exp(-c/x))
=
c f . Magnus - 0 b e r h e t t i n g e r - S o n i
[11)* Example 2 . 1
QX = $yy
Thus one c o n s i d e r s t h i s becomes
fyy
?(s)exp(-(-is)
4y )
+
F I(y,x)
+
+
I (y,x) = 0
since
with
(-is)’
$(s,y)
+
0
x < 0.
Hence i f
2 Du = A u satisfies
then D
2
2
Bu =
= -A
2
= ?(s).
Q(D)
We t a k e as a s o l u t i o n
=
D2
.
;(s,y)
y
-+ m
where
( c f . Example 1 . 1 0 ) .
I+(y,x) = I(y,x)
for
e x p ( - ( i s ) 4y )
But x
2
0
and
which y i e l d s
I(y,S)f(-S)dS.
w
2.
satisfies
D2G = A
However t h e v a l u e s
2
6
while i f
~ ( - 5 ) or
2
Dv = -A v
v(-c)
then
which a r i s e
from ( 2 . 5 ) may i n t r o d u c e c o m p l i c a t i o n s .
A f e w o t h e r s i m p l e e x a m p l e s are w o r t h m e n t i o n i n g h e r e .
Q(D) = -D 6(x
-
y)
so o n e c o n s i d e r s
* f(x)
$x = -by
with
F i r s t let
$(x,O) = f ( x ) .
so t h a t m
6(s - y ) f ( - L ) d E
$(O,y) = ( B f ) ( y ) = -m
=
(1 - i ) s 4 and t h i s w i l l b e u n i q u e u n d e r
$(x,y) = I (y,.) * f ( * )
1,
and
= D
and a f t e r F o u r i e r t r a n s f o r m a t i o n
+
Thus
( B f ) ( y ) = @(O,Y) =
fi as
+
for
$(x,O) = f ( x ) ;(s,O )
=
i F s I ( y , x ) = FI ( y , x )
(2.5)
? = Bv
with
is; = 0
the stipulation that =
P(D)
The k e r n e l I w i l l a r i s e a g a i n i f we t a k e
=
f(-y).
P(D)
Clearly
= D
and
$(x,y) =
FURTSER EXMlPLES
Da
Applied t o vector functions if B(y)
=
a(-y).
Example 2 . 2
where
$
6 =Ba satisfies Df? = -A2f? and
then
A more interesting example is Let
P(D)
F'$.
=
A 2a
=
85
Thus
m
=
D3
and
$(s,y)
e-isxei s 3yds
(2.7)
so we consider
Q(D) = D
=
F(s)exp
is3y
=
K(3y.x)
=
and one knows that 4
(3y)-'j3Ain(x(3y)
3
)
-co
where Ain(z) = Ai(-z)
denotes the Airy
function
(cf. Roetman [l], Magnus-Oberhettinger-Soni *f(*)
[l]).
Consequently @(x,y)
K(3y,.)
and
By results of Roetman [ l ] this will make sense '->rcontinuous f with c exp kl El 3/2
-' as
5
+
m
and
@(x,y)
will satisfy ( 2 . 6 ) .
can state
.
g(A)
be a well defined operator function in F with
D(g(A)).
u(t)
satisfies D 3u
Assume p(u(t))
= g(A)u
for
-m
I f(c) I 2
In particular we
Theorem 2.3 Let
while
=
< t <
u
6
m
with
u(0) = u
F.
Then
v(t)
0
3/2 -F.
5
kp exp cp)tl
for any seminorm p
in
=
m
KOt,S)u(-E)dS
=
(Bu)(t)
satisfies Dv
=
g(A)v
with
v(0)
=
u
-m
The formulas of Example 2 . 2 are perhaps better exploited in solving the problem Du
=
-A3u
(t > 0)
Thus we assume A Gt = Gxxx
with
with
u(0)
=
uo E D(A3)
generates a group
T(.)
by the technique o f Section 1.1. and replace A
By (2.6) - (2.7)
G(0,x) = 6(x).
C(t,x)
solution of the abstract Cauchy problem is m
(2.9)
u(t)
=
= O
j
-m
K(3t,c)T(c)uodS
by
-D
= K(3t,x)
to obtain and a
R. W. CARROLL
86
which makes s e n s e s i n c e
p(T(C)uo)
- 6)
T(E)uo(x) = uo(x
t h e group
u(t) = u(t,x)
solves
5
I f we let
q(uo)e"lCI.
A = -Dx
i n a s u i t a b l e s p a c e of f u n c t i o n s
ut = uxxx
with
u ( 0 , x ) = u (x)
generate then
F
a s i n Roetman [ l ] .
Summarizing
Theorem 2.4 u
3
6 D(A ) .
Remark 2.5
A
Let
Then
g e n e r a t e a q u a s i e q u i c o n t i n u o u s group u(t)
d e f i n e d by ( 2 . 9 )
T(*)
ut = -A 3u
satisfies
in
with
F
with
.
u(0) = u
E v i d e n t l y t h e t r a n s m u t a t i o n method of r e l a t i n g s o l u t i o n s of a b s t r a c t
d i f f e r e n t i a l problems i n v o l v i n g a n o p e r a t o r e r a t i n g a group.
i s q u i t e independent of
A
A
gen-
Groups a r o s e i n Example 1.4. Example 1 . 6 , and Theorem 1 . 1 2
because of t h e f a c t t h a t Ij(t,x)w(x)dx
w ( t ) = cosh A t u
wtt
a r e known s o l u t i o n s of
and t h e c o r r e s p o n d i n g =
A 2w
and
= A2u
u
u(t) =
respectively
t
when
g e n e r a t e s a group.
A
Example 2.6
An important example of t r a n s m u t a t i o n was f u r n i s h e d by Lions [2: 3: 41
in treating
D2
and
L = D m
i n Section 1 . 3 we take Q(D) = Lm(D) = Lm.
m >
2
+
-4
together with related generalizations.. A s
D
h e r e f o r convenience and c o n s i d e r
Thus we l o o k f o r
@
R e f e r r i n g t o Example 1 . 3 . 1 w e s e e t h a t
and
satisfying
OY(X,0) = 0.
@(X,O) = f ( x ) ;
Qy:
P(D) = D L
@ ( x , y ) = Rm(y,x)
* f(x)
so that (cf.
(1.3.6) )
where
c
m
=
I'(m
+
w ( t ) = cosh Atuo
l)/r(l/2)r(m then
+
1/2).
um = B w
m
formula ( 1 . 3 . 8 ) from (2.11) ( i . e . a c t u a l l y c o n s t r u c t s an isomorphism for
L
m
If
2 2 D w = A w
satisfies u m ( t ) = 2c Bm
with
w (0) = 0 t
so t h a t
2 L um = A um and we r e c o v e r t h e
*J'
(1
-
m O with inverse
2 m-+i
E, )
Bm-1
cosh ActuodC).
Lions
i n s u i t a b l e spaces
and v a r i o u s g e n e r a l i z a t i o n s b u t we w i l l n o t s p e l l t h i s o u t h e r e ( c f .
FURTHER EXAMPLES
87
Examples 3.11 and 3.14 f o r o u r v e r s i o n of t h i s ) .
Remark 2 . 7
Various t r a n s m u t a t i o n o p e r a t o r s can be e n v i s i o n e d f o r h i g h e r o r d e r
s i t u a t i o n s such a s
P(D) = (-l)n+1D2n
Q(D) = D
with
s o t h a t one c o n s i d e r s
A f t e r F o u r i e r t r a n s m u t a t i o n t h i s becomes
@y + s
(2.13)
with s o l u t i o n
2 n ~
(0 = 0 :
G(s,O) = ? ( s )
$(s,y) = ?(s)exp(-s
2n
s p a c e of e n t i r e f u n c t i o n s s a t i s f y i n g clexp(-alxlP) [ 2 . 31.
for
I t is known t h a t
$ ( s , y ) = exp(-s with
real so
x
2n
2n
y) E ZZn
q = 2n/2n-1 = 1
@(x,y) = $(x,y) * f ( x )
+
Z;
‘-
FZ - Z P
=
P
A s i n S e c t i o n 1 . 3 we d e n o t e by
y).
-
5
(g(z)I
’-”’
slip 2
9
- S
c exp b l z l P
with
P
the
5
i n t h e n o t a t i o n of Gelfand-Silov
1-UP
where
1 P
-
+ -q1 =
corresponds t o a transmutation kernel 1/2n-1.
If(x)I
Zp
Thus f o r s u i t a b l e
1 and t h u s @ ( x , y ) E 2;
f , which can b e s p e c i f i e d ,
w i l l yield a transmutation operator
(Bf)(y) = @(O,y).
T h i s w i l l b e p e r f e c t l y r i g o r o u s b u t a n e x p l i c i t d e s c r i p t i o n of
$(x,y)
g e n e r a l l y n o t be a v a i l a b l e from a t a b l e of F o u r i e r t r a n s f o r m s .
Thus s i n c e h i g h e r
will
o r d e r d i f f e r e n t i a l problems do n o t u s u a l l y a r i s e i n p r a c t i c e anyway we w i l l n o t concern o u r s e l v e s w i t h t h e q u e s t i o n of t r a n s m u t i n g g e n e r a l
P(D)
(which is n o t always p o s s i b l e i n any e v e n t a s a n example w i t h Q(D) = D’
-
q(x),
q
not a n a l y t i c , i l l u s t r a t e s
-
into
Q(D)
P(D) = D3
c f . Levitan [ l ] ) .
and
Rather l e t
u s c o n c e n t r a t e on second o r d e r d i f f e r e n t i a l polynomials where e x p l i c i t f o r m u l a s can be o b t a i n e d .
Some g e n e r a l r e s u l t s f o r t h i s s i t u a t i o n a r e developed i n
L e v i t a n [ l ] and L i o n s [ 2 ;
3 ; 41 but s i n c e w e a r e concerned p r i m a r i l y w i t h opera-
t i o n a l c a l c u l u s and r e l a t e d e q u a t i o n s w e w i l l proceed r a t h e r d i f f e r e n t l y v i a some examples of i n t e r e s t i n a p p l i c a t i o n s and d e v e l o p g e n e r a l theorems i n t h a t context.
Example 2 . 8
A n example of r e a l t e d non-homogeneous e q u a t i o n s connected by a
g e n e r a l i z e d S t i e l t j e s t r a n s f o r m is provided i n Bragg 141 ( c f . a l s o Dettman [ l ]
88
R. W. CARROLL
and see Remark 2 . 1 0 for further information on non-homogeneous related equations). The discussion in Bragg [ 4 ] takes place in a Banach space F an equibounded Co
group
T(t);
Consider the problems
(2.14)
utt
-
a 2 t A u = f:
u(0) = u'(0) = 0
(2.15)
v
+
t 0A 2v = g;
v(0)
where
and
f
[ (a+2)T/21 2/o+2
UTT
(2'17)
' T T
where etc.
+
2
0, 0
2
0)
0
= [(a
+ 2 ) ~ / 2 2/a+2 ]
in ( 2 . 1 4 ) and
One
t =
in ( 2 . 1 5 ) to obtain
&UT -
2
A U = f*
2 VT + A V = g * .
+
U(7) =
=
(a
are
are suitable continuous functions with values in D(A2).
g
makes a change of variables t
(2.16)
generates
suitable extensions to more general F
clearly possible but we omit this.
tt
A
where
U([T] ( a +
2)T 2/0.+2 ) ,
f*(T)
=
(2/(af2)T)2"/a+2f([~]2/a+2),
The equation
(2.18
W
2 -AIJ=h
is taken as an intermediate link and in keeping with a general formalism indicated below (see Bragg [ 3 ] and Remark 2 . 1 0 ) one takes E,
in ( 2 . 1 8 ) to obtain
where
a
=
a+2
provided that
.
One can then write
<
= (1/4).r2
in (2.16)
and
T =
89
FURTHER EXAMPLES
Here
Lil{
ting
T =
where
means t h e i n v e r s e L a p l a c e t r a n s f o r m w i t h
<-'
b =
i n ( 2 . 1 7 ) and
T = 1/45
-+
5.
S i m i l a r l y set-
i n (2.18) one o b t a i n s
B / B + 2 , provided
Eliminating
W
between (2.20) and (2.22) t h e r e r e s u l t s
11
B+2)+ K(a,O) = C ( a , B ) i ( a + 2a) -( o
where
s
T h e c o r r e s p o n d i n g r e l a t i o n between
with
and
f
g
is
Now t o produce a s a t i s f a c t o r y theorem from t h e s e c a l c u l a t i o n s Bragg [ 4 ] f i r s t constructs explicitly a solution with
f ( t ) E D(A2).
u(t)
One w r i t e s f o r
t o ( 2 . 1 4 ) when
f E Co(F)
on
[O,m)
I$ E D(A2)
a+2
~
where
m(o,t) = 2t
where
K-1 ( a )
i1(0)
=
'$, and
=
<'/(a+2)
2 B(-, a+2
a+l a+2
--).
ai(0) = 0
and
Then while
U1(t)$
=
i,(t)
U2(t)@= i 2 ( t )
s a t i s f i e s (2.14) with s a t i s f i e s (2.14) w i t h
f
=
f = 0,
0,
R. W. CARROLL
90
&(o)
= 0,
Now let G
Jill
4.
A solution to (2.14) is then given by
be the space of
dt <
f(t)II
-
$;(O)
and
Theorem 2.9
on
f E Co(F)
with
[0,m)
2 f(t) E D(A )
and
There results
m.
Let
f
and a >
G
E
f by (2.26)
defined in terms of
8.
Then a solution v
of (2.15) with
is given by putting the u(t)
g
of ( 2 . 2 9 ) i n
(2.24). Remark 2 . 1 0
A class of related differential equations connected to generalized
hypergeometric functions are studied in Bragg [ 3 ] and this of course will yield corresponding transmutation formulas (which we mercifully will not spell o u t ) . Let 4 (2.30)
for
6
Ol(q,8,t,Dt) =
(a,, ...,Bq)
tDt
=
1
-
(tDt + B
j
1)
and P
(2.31)
02(p,cx,t,Dt)
t7T (tDt + a i ) .
=
1 Let
P
= P(x,D)
k1
orders are
= Ca (x)Dx
with
x
and
!L2
Q
=
Q(x,D)
o
t
e same form; suppose
:ir
t
respectively. Equations of the type
are considered and here it i s seen that
P and
Cj
could be suitable more general
abstract operators in a space F with no essential change in technique. Laplace transforms are used and some typical theorems are Theorem 2.11
2
with
p
max(p
+ R1,
Let
q, ai
q
+
u(x,t)
and
be a solution of
8 . positive, J
k 2 ) , and
u
u
E
Cr
in
(x.t)
for
and all derivatives of order
5
t > 0
where
r bounded.
r =
Let
FURTHER EXAMPLES
c(
P
91
> 0; then t h e f u n c t i o n
s a t i s f i e s (2.32). then
v(x,t)
+
If
u(x,t)
as
$(x)
t
+
+
Let
ai
-P
0
I$
with
bounded and c o n t i n u o u s
S i m i l a r l y one h a s
6.
and
t
0.
The proof is s t r a i g h t f o r w a r d .
Theorem 2.12
@ ( x ) as
be positive with
J
Bq
> 1.
Let
IJ(x,t)
be a
s o l u t i o n of
t > 0
for
If
V
exists,
V
then
s
and set
V E C r , and
Theorem 2.13
Let
p
a negative integer.
for
t > 0.
s a t i s f i e s (2.32) f o r Re s > k > 0
for
with a l l derivatives o r order
V
t > 0
such t h a t
then a s
5
q
Let
+ u
t
+
>
1, E
U(x.t)
0
U(x,l/s)
CY
P
V
s a t i s f i e s (2.32) f o r
i s bounded and a n a l y t i c i n
equal t o zero or
satisfy
$(XI
t > 0
Rl,...,Bq-l
> 0 , and no
as
0
$
with
t
+
1
- 0)
a Then
a r e bounded
l i m V ( x , t ) = $ ( x ) = U(x,O).
cr-l
-P
If
< r -
continuous.
Define
V
-a -1 U(x,ta)do.
and
Numerous a p p l i c a t i o n s of such formulas a r e g i v e n i n Bragg [ 3 ] and t h i s formalism
i s extended t o nonhomegeneous problems i n Bragg [ 2 ] .
We c i t e a few t y p i c a l
r e s u l t s r e l a t i v e t o Theorems 2 . 1 1 - 2 . 1 3 where by (2.32) = f [SO, - P02]u = f .
f o r example we mean
R. W. CARROLL
92
Theorem 2.14 by ( 2 . 3 4 ) .
b e a bounded s o l u t i o n of (2.33) = f
U
Let
V
Then
s a t i s f i e s (2.32) = g
and l e t
V
b e given
provided
a -1 (2.39)
g(x,t) =
Let
Theorem 2.15 e x i s t s as
t
U
0.
-+
r-'(a
P
re-ao
)
f(x,to)da
'0
be a bounded s o l u t i o n of (2.35) = f
Let
be given by ( 2 . 3 6 ) .
V
V
The
such t h a t
l i m U(x,t)
s a t i s f i e s (2.32) = g
provided
1-E
2-8 (2.40)
Theorem 2.16
U
Let
6 - a > 1.
and
q
where
qr(6q)Li1{s
g(x,t) = t
Gq =
P
(Bq
Example 2.17
s a t i s f y (2.37) = f
Then
P
- a
P
qf(x,;)ls+t.
-
and d e f i n e
s a t i s f i e s (2.32) = g
V
l)r(Rq)/r(ap)r(Oq
The S c h r o d i n g e r e q u a t i o n
-
ut
V
by (2.38) w i t h
a
P
> 0
provided
ap).
iu
=
V
viewed i n t h e Gelfand-Silov
xx
framework ( c f t h e d i s c u s s i o n around D e f i n i t i o n 1 . 3 . 1 6 and Theorem 1 . 3 . 1 9 ) l e a d s x(s) = -is2
to
and
Petrovskij correct. t
2
-is $
=
A(s)
=
T = 0
is zero f o r
Under t h e F o u r i e r t r a n s f o r m
which l e a d s t o
( E ( 0 , s ) = 1).
20-r
Since
A(s)
2
2 ( t , s ) = exp(-is t )
F
so t h a t the equation is
f o r a fundamental s o l u t i o n
1 ~ 51 k
i s bounded i n t h e domain
16(q)(t,~)5 l cq(l
+
we have i n f a c t
u =
(1
+
lOl)-'
the
genus
p = -1.
Q > 0
t h e one d i m e n s i o n a l Cauchy problem is w e l l posed i n t h e c l a s s of f u n c t i o n s
u(x)
Further
101)'
and one can s a y t h a t f o r any
h a v i n g , t o g e t h e r w i t h t h e i r d e r i v a t i v e s of o r d e r
of o r d e r
2
Q
-
2
(while
u(x,t)
5
9,
+
2 , polynomial growth
h a s polynomial growth of o r d e r
5 a).
Regard-
i n g t h e a b s t r a c t Schrodinger e q u a t i o n we c o n s i d e r
(2.42)
2 ut = i A u;
u ( 0 ) = uo
and t h e Hersh t e c h n i q u e l e a d s t o shown ( c f . T r e v e s [ 3 ] ) t h a t
Gr = i G
xx
with
G(0.x) = 6 ( x ) .
It i s easily
93
FURTHER EXAMPLES
Consequently
u(t)
=
T(x)uo>
is given by
T(x) uodx
(2.44)
u(t)
where A
generates a suitable group
=
T(*)
in a space F.
IJe note that if ;(t)
is the solution of the abstract heat problem (1.1.3) then ;(it)
u(t)
=
and
.
formally ( 2 . 4 4 ) can be written (2.45)
u(t)
K(it,x)cosh
=
Axuodx.
We note that there are some errors in Polyanskij-Fedoryuk [l] where the formula (n/p)’-
exp(-clp’)
=
1,
2
T-’exp(-
!& - pT)dT
is used (cf. Bragg-Dettman [ 6 ; 7; 8 1 ) .
Their formulation is however easily modified to give a connection formula between the u
of ( 2 . 4 2 ) and the
z
of (1.1.5) using spectral hypotheses on
A
2
.
Remark 2.18 The construction of transmutation operators by the method of Theorem 1.3(F) usually involves the extension of f(x)).
f
to
in some manner (eg
(-m,r))
f(-x)=
The resulting transmutation operator will depend on the extension and
the following discussion and remarks should clarify the matter.
First, there are
a number of results available concerning the solution o f half ulane or quarter \r
plane differential problems (see eg. Dikopolov [ 1 1 . Dikopolov-Silov [ 2 ; 31, Hersh V
[4; 5; 61, Palamodov [ 2 ] , Pavlov [l], Sarason [I: 21, Silov [l], ZarnitskayaSelezneva-Eidelman [l] and cf. a l s o Fattorini [3], Fattorini-Radnitz [4], Gasymov [3]) and we will give some sample theorems to indicate natural frameworks. Let u s say that for some space
Q
is P-admissible if
P(Dx)$
=
O(Dy)@,
with
Y , has a unique solution in some space @
@(x,O)
where
@
=
f(x) E Y
will usually
involve some conditions Di$(x,O) = 0 and/or growth conditions (see Section 2.3 Y also where this is subsumed under condition C). For such a problem to make sense one will usually have t o extend the space y - 0.
f(x),
given for x
2
0, to
(--,O)
0 deals with functions o r distributions in the half space
Then
Bf(y)
=
$(O,y)
determines the transmutation operator B
and -m
< x < m,
such that
94
R. W. CARROLL
QB = BP.
Note t h a t i f
Ek E L(Y,Yk)
{ O , m - l } , is g i v e n , where
c
P
+
Q
same e q u a t i o n f o r
6
$(O,y) = g(y) E A
h a s a unique s o l u t i o n i n s a y -m
i n d i c a t e d and one d e f i n e s for
$
< y <
m,
3
require global solutions i n
t = y
E =
(cf (1.6)
P(Dx)$ = Q(D ) $ Y
5
where
PR = KQ.
g(y)
with
@
is l i k e
but
y < 0
to
i s now
The h a l f s p a c e problems v
and t h e s t a n d a r d Gelfand-Silov t h e o r y t < T <
f o r example i s n o t immediately a p p l i c a b l e (cf S e c t i o n 1 . 3 where a r i s e s i n fixing spaces).
then c o n d i t i o n s
S i m i l a r l y working w i t h t h e
x > 0: an e x t e n s i o n of
R g ( x ) = @(x,O) w i t h
Yk
k E
BEf(y) = $(O,y),
i s Q-admissible i f
P
Ekf(x),
=
P(Dx)Ek = EkP(Dx)
and s e e L e v i t a n [l; 31 f o r simple v e r s i o n s of t h i s ) .
d e f i n e d over a r e g i o n
Y and
Y
for suitable
a s b e o f r e provided
one can s a y
Dk $(x,O)
s a y and d a t a
@, and t h e o p e r a t o r
would change of c o u r s e , as would
( E k ) , would transmute
m
i s of o r d e r
Q
m
often
Some m o d i f i c a t i o n s of t h i s t h e o r y a r e p o s s i b l e and w e v
c i t e f i r s t a few t y p i c a l r e s u l t s f o l l o w i n g Dikopolov-Silov
[2].
Consider f o r
ex ample
m-1 (2.46)
DTu =
1 Pk ( i D x ) D kt u
k=0 where t h e
Pk
a r e polynomials and we work i n t h e r e g i o n
--m
< x <
with
m
t > 0.
I n o r d e r t o r e l a t e t h i s t o o u r format l e t Pk be c o n s t a n t f o r k > 0 w i t h m-1 Po(iDx) = P(Dx). S e t t i n g Q(Dt) = DY - X P Dk w e have Q(Dt)u = P(Dx)u. Let m- 1 1 k t k C PkX = P o ( a ) , o r d e r e d by t h e r u l e A . (a) be t h e r o o t s of Am J 1
-
(2.47)
R e X,(o) <
... -< Re
r e a l , and l e t
for
U
G1 2
... 2 Gm.
Let
G
H
d e r i v a t i v e s (computed i n
1
Xm(u)
be t h e s e t of such
be t h e space of
P') so that
m u l t i p l i e d by a r b i t r a r y polynomials.
where
R e ),.(a) J
f u n c t i o n s on R
L2 H =
0
FH
c o n s i s t s of
5
0
so t h a t
together with a l l L2
f u n c t i o n s on R
These a r e c o u n t a b l y normed s p a c e s i n t h e
Y
s e n s e of Gelfand-Silov [I: 2 : 31 and one w r i t e s f o r example i s t h e H i l b e r t space w i t h s c a l a r product
H = UHp
where
Hp
95
FURTHER EXAMPLES
h
A sequence
fn
Similarly
fn
+
0
+
Fn E L 2 , and
in
0
Fn
H
H
in 0
+
i f f o r some
p
?n
€
and
Hp
(1
an(/p+ 0
i f f o r some d i f f e r e n t i a l polynomial
in
L
2
.
P(D),
A t y p i c a l theorem r e l a t i v e t o (2.46)
n
as
+ m.
f n = P(D)Fn, is
v . ( u ) be g i v e n on G where v is t h e p r o d u c t o f J j j 2 a polynomial times a f u n c t i o n i n L ( G . ) (one writes v . E H(G.)). The e q u a t i o n J J J ( 2 . 4 6 ) h a s a s o l u t i o n u ( x , t ) w i t h [D:-'u(x,O)]~ = vk(a) on Gk (1 5 k 5 m). Theorem 2.19
For
0
t
Let functions
u(.,t) E
H
and, as
no f a s t e r t h a n some power of and depends c o n t i n u o u s l y i n
Example 2.20 gives
=
H
i o with
+
G2 =
X2
- 0'
=
@.Thus
0
v (a)
1
X,(o)
= -1~11
(0
5
k l m-1)
grows i n
H
i s u n i q u e i n t h e c l a s s of s u c h f u n c t i o n s
so t h a t
X2(o) = -io.
and
u
on t h e d a t a
Cauchy problem f o r t h e wave e q u a t i o n . where
k Dtu(-,t)
-,
This
t.
2 2 Dtu = D u
Consider
X,(u)
t
Thus
v
k
E H(Gk).
Po(iDx) = -(iDx)2 GI = G 2 =
X,(o)
can be p r e s c i r b e d b u t n o t
=
= 0
2
2
2
D u = -D u = (iD ) u t
1u1, one h a s
v2(U)
+ u2
IR and we have t h e f a m i l i a r
On t h e o t h e r hand f o r with
X2
and
G
1
=IR
with
and w e have a w e l l posed
D i r i c h l e t problem f o r t h e Laplace e q u a t i o n .
I f more r a p i d growth is allowed
Remark 2.21 placed by
GC = {Re X.(o) < c ) ri 3 -
specified for
5
Is1
a r e nonempty f o r some works l o c a l l y i n
t
p o n e n t i a l growth and
(O(tPexp c t ) )
the
and i n t h e L a p l a c e example above
G
j
can be re-
v2(s)
can be
On t h e o t h e r hand f o r any e q u a t i o n (2.46) t h e s e t s
c. c
and t h e r e w i l l b e n o n t r i v i a l c o r r e c t problems.
GC
j
I f one
t h e Cauchy problem i t s e l f w i l l be c o r r e c t f o r s u i t a b l e ex-
0 < t < T
( c f . t h e end of S e c t i o n 1 . 3 ) .
Various r e f i n e m e n t s of such r e s u l t s a r e p o s s i b l e b u t we w i l l n o t e n t e r i n t o a d i s c u s s i o n of t h i s . Y
Remark 2 . 2 2
L e t u s mention now € o r completeness some c r i t e r i a of S i l o v [l] f o r
q u a r t e r p l a n e problems and s p e c i a l i z e h i s r e s u l t s t o some second o r d e r e q u a t i o n s for simplicity.
Consider
96
R. W . CARROLL
u(0,t) = w (t);
(2.49)
Dxu(O,t) = w ( t ) 1
where t h e number of c o n d i t i o n s imposed a t
=
0
may v a r y h e r e a s i n d i c a t e d
F i r s t o r d e r terms i n (2.48) a r e o m i t t e d h e r e f o r convenience.
below.
Xj(s)
a t the roots
A
(2.51)
2
=
( j = 0,l)
branch p o i n t s .
of
a
Thus i f
s =
?(-a/y)'
=
where t h e
X.(s)
have a l g e b r a i c
J
s = ? i ( a / y ) !5
y have t h e same s i g n we have
and
whereas i f t h e y have o p p o s i t e s i g n s
5B
One l o o k s
+ YS 2
a
and t h e r e are c r i t i c a l p o i n t s
Ims
t
s = ?(-a/y)'
is r e a l .
In e i t h e r case
la/yI4 f o r b o t h r o o t s and one works i n t h e h a l f p l a n e
t h e r o o t s a r e s i n g l e valued and d i s t i n c t .
A number
r
Ims >
(r = 0,1,2)
8
where
i s then
determined s p e c i f y i n g how many c o n d i t i o n s (2.50) a r e allowed, such t h a t , f o r
Ims >
6,
R e X.(s) 3
l e a s t f o r some permitted.
(2.52)
s
2
0
0 < j < r-1
for
with
>
Ilns
One w r i t e s (2.48 2 t
D u
= C~U
-
B.
r
For
and f o r =
2
j > r
Re A.(s) > 0 J
at
both c o n d i t i o n s i n (2.50) a r e
i n t h e form
2
y[D u - w 6 - w 6 1 1 1 0
and t h e q u a n t i t y
w i l l be of importance i n t h e t h e o r y .
for
u > r
(r = 0,l).
T h i s i n t e g r a l i s defined f o r
provided f o r example t h a t t h e One l o o k s f o r s o l u t i o n s
Indeed s e t
u
wi(t)
Ims >
B
and
Re Xv(s) > 0
have a t most polynomial growth as
i n a space
/ft
t
+
m.
d e l i m i t e d by c o n d i t i o n s of t h e form
rCO
exp(-ZBx) l I q u ( x , t ) I2dx < JO
The space
13
H+ =
FH:
m
where
Iqu
d e n o t e s a s u i t a b l e " p r i m i t i v e " of
on t h e o t h e r hand c o n s i s t s of f u n c t i o n s
r$(s), analytic
u.
97
FURTHER EXAMPLES
Ims > f3,
for
B
Ims >
(s = 0
Theorem 2.23 Gv(s)
+
2
A
wo,
q , uniformly i n
T =
-
s
(2.50) p r o v i d e d t h e Ims > f3
t o the half plane
a s func-
B
Let
+
f o r some
H+-
2
R e i(O
m
There e x i s t s a unique s o l u t i o n of (2.48)
Example 2.24 obtains
<
iT).
can be c o n t i n u e d a n a l y t i c a l l y i n
tions in
and
2
I @ ( s ) I /lsI2‘do
such t h a t
=
iT)
y = -1
and
s o t h a t (2.48) i s t h e wave e q u a t i o n .
-s , X
= is,
h
< 0
while
R e A ( s ) = Re[-i(5
= -T
wl,
0
CY =
u
and
1
f3 = 0 , and f a r
-is,
=
1
+
Ims > 0
Re Xo(s)
= 7 > 0.
iT)]
One =
r = 1
Hence
can be p r e s c r i b e d which i s a s t a n d a r d mixed problem f o r
t h e wave e q u a t i o n .
Example 2.25
h
2
Re
= s
2
, X
X 1( s )
Let
CY
= s,
h
=
1
and
f3
-s,
y
=
=
0,
1 which makes ( 2 . 4 8 ) a Laplace e q u a t i o n where and f o r
Ims > 0
( n e i t h e r of which i s always n e g a t i v e ) .
= 0
defined f o r
0
=
Res > 0
Res < 0
and
Re h ( s ) = Thus
r = 0
-0
while
and t h e
GV(s),
respectively, are
Y
An a n a l y s i s of t h e Cauchy Riemann e q u a t i o n s i n t h e p r e s e n t c o n t e x t ( c f S i l o v [l]) G (s)
a l l o w s one t o conclude t h a t t h e functions W+(x,t)
W+(t) = wl(t) for
x > 0
and
w
a r b i t r a r i l y when both
Remark 2.26
+_
iwi(t) t > 0.
and
w i l l be s u i t a b l y e x t e n d a b l e p r o v i d e d t h e
can b e extended t o b e a n a l y t i c € u n c t i o n s However n e i t h e r
w1
II
nor
u1
can be p r e s c r i b e d
are t o be s p e c i f i e d .
Consider a g a i n t h e wave e q u a t i o n
@xx = Q t t
with
O(x,O)
=
f(x)
Ii
and
Qt(x,O)
=
0
x > 0.
for
,
;
’
Extend
Qt
by
0
and
f(x)
by
T(x)
for
x
c’
0
R . W.
= f(0)
and
CARROLL
a t least
(?(XI = f ( x )
7 [f(t) +
Q(0,t) =
1 = 2 f(t)
?(-t)
I f w e c o n s i d e r t h e q u a r t e r p l a n e problem of Example 2.24 w i t h
h(t).
i n t e r c h a n g e d i t is s e e n t h a t
h(t)
could b e p r e s c r i b e d a l o n g w i t h ~
@,(x,O) = 0.
and
x
for
1 -
-
2h(x)
f(x)
Hence t h e r e is an e x t e n s i o n
such t h a t
B f ( t ) = h(t) g
o p e r a t o r determined by t h i s e x t e n s i o n .
f
B
where
of
f , namely
0).
Then
+
g(t)l =
and
x
t
@(x,O) = f ( x )
-
f(-x) = g(x) =
denotes the transmutation
g
W e note a l s o t h a t the extension
g = f
= Bf is i n t h i s c a s e a g e n e r a l i z e d t r a n s l a t i o n g o p e r a t o r of t h e t y p e s t u d i e d by L e v i t a n [l; 31 ( c f S e c t i o n 2 . 3 ) . Formally l e t us
corresponds t o
h = f
f E Y
consider then
o=
where
here.
f(0) = 0
if
B (f) =
g
Y
+
Bg
such
1
7f. is
and
with
B
I$E 0
Note f i r s t t h a t g
B
g
+
Ag : Y
+
h =
Thus
1
7
f
while
B-f(f) = 0,
$(O,t) E 0
-
h = 0
g = -f
and
B f ( f ) = f , and
is t h e zero operator then
Bg(g) = 0 = g
and
Further
T h e r e f o r e t h e o p e r a t o r d e f i n e d by and
g
-
g = 0
appear admissible.
In particular i f
1-1.
a s i n Remark 2.18 and l o o k a t
L(Y)
is
A (f) = B (f) g g
1-1 s i n c e
A
g
=
-
Bo(g)
implies
0
is linear, A (g) = g
g
A
g
- -21 g
E
L(Y'), 1 2
= - g = 0.
B ( f - g ) = Bof f o l l o w s A ( f - g) = Bo(f - 8 ) . From g g 1 1 1 A (f) = - [f g ] - - g = - f of c o u r s e we s e e t h a t A = is t r i v i a l b u t i n g 2 2 2 g 2 p a r t i c u l a r A E L(Y) ( c o n t i n u i t y ) . Thus g B : Y + A(") where A ( " ) denotes g g c o n t i n u o u s a f f i n e o p e r a t o r s Y + Y and B ( f ) = f i m p l i e s f = g. I t could b e g of i n t e r e s t t o examine s u c h mappings i n more g e n e r a l i t y f o r o t h e r e q u a t i o n s (cf Note a l s o t h a t from
+
+
Section 2.5).
2.3.
S e p a r a t i o n of v a r i a b l e s and t r a n s f o r m s .
Let u s f i r s t make some o b s e r v a t i o n s
about t r a n s m u t a t i o n and s e p a r a t i o n of v a r i a b l e s which l e a d t o a s y s t e m a t i c formal-
i s m f o r h a n d l i n g many problems of i n t e r e s t i n a u n i f i e d manner.
The p r e s e n t a t i o n
99
SEPARATION OF VARIABLES AND TRANSFORMS
w i l l b e e s s e n t i a l l y f o r m a l a t f i r s t b u t w i l l b e i l l u s t r a t e d by numerous e x a m p l e s t o j u s t i f y i t and a more r i g o r o u s t r e a t m e n t w i l l f o l l o w .
A preliminary version
3; 41).
a p p e a r s i n C a r r o l l [35] ( c f . a l s o L e v i t a n [l; 51 and L i o n s [ 2 ; s u p p o s e we c a n t r a n s m u t e
P(D)
with appropriate conditions solution
C
into
Q(D)
(0,m)).
w i l l denote
i Y
for
The c o n d i t i o n s
i E {l,m-
f E Y
i n g on f u n c t i o n s
B
by
(Bf)(y) = @(O,y).
f o r s u i t a b l e b r a c k e t s on
11
C
(a,-) (<,
or
(0,m)
W e assume >+
( c f Remark 2.18) w i l l i n v o l v e some combin-
Q or suitable derivatives plus stipulations that
a t i o n o f g r o w t h r e q u i r e m e n t s on
D @(x,O) = 0
v i a a s o l u t i o n of
( u n s p e c i f i e d a t t h e moment) t o g u a r a n t e e a u n i q u e
ip(x,y), which d e t e r m i n e s t h e n
(Bf)(y) =
Thus
where
m =
order
Q(D).
Suppose Y
f o r some r e a s o n a b l y l a r g e s p a c e
QB = BP
act-
and c o n s i d e r t h e
eigenvalue equations
If
h ( * , A ) E Y’
then
an eigenvalue f o r h(x,A)B(O,X)
QBh = BPh =
Q(D).
-A 2 Bh
Conversely c o n s i d e r
( c f . L e v i t a n [l]).
If N(B)
O(0,A) = 0
(Bh)(y,A) = < b ( y , x ) , h(x,X)>
is
@ ( x , y ) = w(x,y,X) = @ ( x , O ) =
c
Given c o n d i t i o n s
u s u a l l y b e p r o v i d e d by c o n d i t i o n s on h(O,A)e(y,X).
so t h a t
8, w e o b t a i n
for this
@,
which c o u l d
w(O,y,X) = $ ( O , y ) = ( B h ) ( y , h ) =
Consequently
or
(null-space)
m
we g e t nothing whereas
w h i c h is
fO}
if
B
h(0.X)
is 1-1.
=
0
corresponds t o
I n p r a c t i c e we w i l l u s u a l l y
work w i t h
h ( 0 , X ) = e(0,X) = 1 and summarize t h i s i n
Lemma 3 . 1
Under t h e c o n d i t i o n s i n d i c a t e d t h e e i g e n f u n c t i o n s o f
c h a r a c t e r i z e d t o b e o f t h e form
h E
R(y,h) = < b ( y , x ) . h ( x , h ) >
Q(D)
are
( f r o m now on
h(0,h) =
R. W. CARROLL
100
O(0,A)
=
1 is assumed unless otherwise stated).
Now for P(D) = Cp.(x)DJ
the formal adjoint P*(D)
J
Z(-l)’D’(pj$) (3.5)
is defined by
P*(D)$
=
and we let P*(D)R(x,A)
= -A
2
Q(x,A).
Consider then the formal expression
where P*(Dx)B
Theorem 3 . 2
QB
=
Q(D ) B Y
and define Bf(y) =
The operator Bf(y)
=
f(x)>
f(x)>.
Since
transmutes P
into Q ; i.e.
= BP.
Given
B
defined via (3.1) it will be of interest to know when
proceed as follows. Let
(3.7)
P(Dx)U(x,5)
satisfying U(x,O)
=
=
U(x,c)
and we
be the solution of
P(Dg)U(x,5)
f(x);
(or growth conditions).
B = B
conditions D>(x,O)
The function U(x,[)
= 0
may be imposed for uniqueness
will be a generalized displacement
or translation operator in the sense of Levitan (1; 31 which could be written
5 U(x,F,) = Txf(x)
(cf. Remark 1 . 2 ) and in general we will have U(O,C) = f(E,).
Explicit constructions via Riemann functions are often possible and examples will be given below.
Then we form the generalized convolution
and observe that formally
101
SEPARATION OF VARIABLES AND TRANSFORMS
Suppose now t h a t
i s chosen s o t h a t
R(x,A)
shown t o be n a t u r a l from t h e examples). f(x)
s o , by d e f i n i t i o n of
provided conditions f E Y'
Y'
with
Theorem 3 . 3
Then
@(x,O) =
a r e s a t i s f i e d (which we assume).
r e a s o n a b l y l a r g e w e can conclude t h a t
B
<6(<), U ( x , < ) >
=
( t h i s is U(x.0) =
B,
Under t h e c o n d i t i o n s i n d i c a t e d i f
the kernelof
Remark 3 . 4
C
B(0,x) = = 6 ( x )
is p r e c i s e l y
Since t h i s holds f o r
b(y,x) = B(y,x).
Hence
B(0,x) = < n ( x , A ) , 1>= S(x)
B(y,x) =
and
B
=
then
B.
G e n e r a l i z e d c o n v o l u t i o n s such as (3.8) s h o u l d be u s e f u l i n s o l v i n g
a p p l i e d problems and w e w i l l comment f u r t h e r on t h i s below ( c f . Theorem 3.7 f o r T h i s shows t h e importance of s o l v i n g e q u a t i o n s of t h e t y p e ( 3 . 7 ) a s i d e
example).
from t h e t r e a t m e n t of Hutson-Pym [ l ; 2 1 , Leblanc [ l ] , Povzner [ l ] , e t c . ( c f . Section 2.5).
Let u s d e f i n e now t h e t r a n s f o r m s ( s e e Appendix 2 f o r g e n e r a l t r a n s f o r m s ) .
(3.11)
O_f(X) = = f ( A ) .
We suppose t h a t
from ( 3 . 4 ) PO(y,*)
b ( y , x ) = O(y,x)
G(y,A) = < f i ( y , x ) , h(x,X)> (y
Q
=
PO(y,*)
i s e s s e n t i a l l y a parameter h e r e ) .
v a r i e t y of c h o i c e s of that
so t h a t taking
Q
be chosen s o t h a t
h(0,X) = e(0,A)
=
1 w e have
w h i l e by d e f i n i t i o n s
B(y,x) =
Now t h e r e w i l l u s u a l l y be a
f o r which t h e above p r o c e d u r e s a r e v a l i d and we a s k
9
i n (3.11) i s an isomorphism
H
-+
for suitably
102
R. W . CARROLL
large spaces
H
fi.
and
?
Then f o r
F),
E
PIP?=
f(y)> =
?.
Next w e
can w r i t e
and f o r =
g.
g = B*f = P Q f =IP? EIPfi
Hence
9 +PR.
: IPG
-+
2
we have IPPg =
and s i n c e PIP; =
; there
f ( y ) > = <8(y,x) f ( x ) >
r e s u l t s a l s o IF = P-'
:
We s t a t e t h i s a s
Theorem 3.5
Remark 3.6 that
P =IP
-1
If
B(y,x) = b ( y , x )
It f o l l o w s a l s o t h a t
(B*)-l = Q-lP : IPG
+
t h e n under t h e c o n d i t i o n s i n d i c a t e d IP = P-'. q-'F'B*f
=
f
from (3.12) and Theorem 3.5 s o
H.
We show next t h e r e l a t i o n of (3.8) t o a g e n e r a l i z a t i o n of a p r o c e d u r e of L e v i t a n [ l ] f o r solving (3.1).
Now f o r
$(x,X) =
F i r s t l e t u s w r i t e (3.8) as
U(x,E)>
w e have
P(D5)U(x,S)> =
solutions with
5
P(Dx)$ =
2
U(x,E)> = -A $.
P(Dx)U(x,S)> =
Ph =
-A 2h
has unique
h(0,X) = 1, perhaps s u b j e c t t o some growth c o n d i t i o n s , then
$(x,X) = $(O,X)h(x,X)
with
q(0,X) =
U(O,S)>
=
f(S)> = F(h).
Using t h i s w e can w r i t e from (3.13)
We can a r r i v e a t t h i s formula in a n o t h e r way by g e n e r a l i z i n g a p r o c e d u r e of L e v i t a n
[l].
Thus a s a s o l u t i o n t o (3.1) t r y
-1 Given IP = P
a s i n Theorem 3 . 5 , from t h e r e l a t i o n
103
SEPARATION OF VARIABLES AND TRANSFORMS
it is suggested t h a t
and t h i s w i l l b e c o n f i r m e d i n numerous e x a m p l e s b el o w .
Using t h i s we g e t formally
from (3.15)
Q(Dy)c = < Q ( D y ) @ ( x , y ) , Q ( X , U ) > = < P ( D x ) @ ( x , y ) , n ( x , U ) > = < @ ( x , Y ) ,
Then
If
P*( D x)R (x, p)> = - p 2 c .
2
Q0 = -u 0
p e r h a p s s u b j e c t t o some g r o wth c o n d i t i o n s . i t f o l l o w s t h a t whe re
F(p) = c(0.p)
< f ( x ) . Q(x.u)>
=
( 3 . 1 5 ) we o b t a i n ( 3 . 1 4 ) a g a i n . Theorem 3.7 O(x.y)
Let
and
from ( 3 . 1 7 ) .
in
and s u p p o s e
B(0.x)
c a n b e w r i t t e n i n t h e form ( 3 . 1 4 ) .
If
=
6(x) h ( x ,A )
so t h a t is unique
S i m i l a r l y i f (3.16) h o l d s
i s u n i q u e a s i n d i c a t e d t h e n ( 3 .1 4 ) r e p r e s e n t s t h e s o l u t i o n o f ( 3 . 1 ) .
e(y.h)
Remark 3 . 8
c(y,u)
Thus
F(h) =
@
c(y,p) = F(p)B(y,p)
Putting t h i s
g i v e n by (3.8) r e p r e s e n t s t h e s o l u t i o n o f ( 3 . 1 ) .
a s indicated the
e(0.p) = 1,
has unique s o l u t i o n s with
The q u e s t i o n o f i n v e r t i n g a t r a n s f o r m s u c h as
f ( x ) = H(x,X).
F(X)>
P
i n (3.9) b y a f o r m u l a
is s t u d i e d i n Mer cer [ l ; 2 ; 31 ( c f . a l s o T i t c h m a r s h [l]).
The g e n e r a l e i g e n f u n c t i o n e x p a n s i o n t h e o r y d u e t o Weyl, K o d a i r a , T i t c h m a r s h , e t . a l . w i l l be e x p l o i t e d l a t e r ( c f . T i t c h m a r s h
"21).
B e f o r e d e v e l o p i n g more g e n e r a l r e s u l t s o r r e f i n i n g t h e ab o v e l e t u s i n d i c a t e how t h e s e f o r m a l t h e o r e m s work i n a p p l i c a t i o n s : t h i s w i l l p r o v i d e some u s e f u l g u i d e lines. Example 3 . 9 277
exp(-iXx).
Take f i r s t Let
Q(D)
P(D) = D
2
= P*(D)
with
h(x,X) = exp(ihx)
and
n(x,h)
b e a r b i t r a r y s u c h t h a t t h e r e q u i r e m e n t s i n d i c a t e d ab o v e
a r e s a t i s f i e d and we w i l l s a y i n g e n e r a l t h a t s u c h
Q
a r e P-admissible.
Take
=
R. W. CARROLL
104
e(0,X)
=
B(y,x) =
1 and from (3.6) w e have
1 5
exp(-iXx)>
while i t is W
Recall here that
B(0,x) = 1 <exp(-iXx), 1> = 6 ( x ) .
known t h a t
2T
j-_exp (-iXx) dX
= l i m ( S i n Nx/~rx) = 6 ( x ) . Hence Theorem 3.3 a p p l i e s and from (3.4) w e have NO(y,X) = < B ( y , x ) , exp(iXx)>. The formulas (3.9) - (3.10) w i t h Theorem 3 . 5 t h e n y i e l d t h e F o u r i e r i n v e r s i o n formulas a s a by p r o d u c t .
T h i s cannot r e a l l y be con-
s i d e r e d a s a new d e r i v a t i o n however s i n c e a t r a d i t i o n a l d e m o n s t r a t i o n u s e s t h e formula
6(x
- 5)
l i m ( S i n N(x - E)/TT(XN” B(0,x) = 6 ( x ) .
equivalent t o
=
5))
( c f . Sansone [ l ] ) and t h i s i s
Let us define
Remark 3.10
( t h i s t r a n s f o r m w a s used i n Theorem 3 . 7 ) .
Then under t h e c o n d i t i o n s of Lemma 3 . 1
and Theorem 3 . 3 we can w r i t e
and t h i s i s a kind of P a r s e v a l formula. and a f = F - l f
1
where
-1 F[f(-x)] = F f
‘FQ,
FI)>
=
Let
F denotes the Fourier transform.
by p u t t i n g
P(D) = D2
and
@(y,X) = i m ( y , X )
f(-x)
i n p l a c e of
and
(3.20)
O(y,X) = (B h)(y,X) = Fxbm(y,x) = R
F(X) =
SO
*m
C can b e t a k e n as D @ ( O , h ) Y
’$(O,x) = 6 ( x )
Pf = F f , Z f = F - l f , f(A)
with
=
f
and
and we t a k e F
even
f(x)
i n (3.19). Let
h(x,X) =
by ( 3 . 4 )
m
as i n Example 2 . 6 and Thus
Evidently $f
a s i n Example 2 . 6 .
Q(D) = L,(D)
exp(iXx)
and c o n d i t i o n
Pf = Ff
s o we can i d e n t i f y w i t h t h e P a r s e v a l formula w r i t t e n a s
2n
Example 3.11
Indeed f o r Example 3.9 we have
=
(Y,X) 0
which h o l d s ; h e r e
by Example 3.9 when
LmBm = BmD
2
1 Q(x,X) = 2ll e x p ( - i h x ) .
B(y,x) = < n ( x , x ) , B(Y,X)>+
SO
t h a t if
105
SEPARATION OF VARIABLES AND TRANSFORMS
where
H
d e n o t e s t h e Hankel t r a n s f o r m .
m
or its dual Schwartz
HA
The Hankel t r a n s f o r m works on f u n c t i o n s
( s e e Zemanian [l] and c f . a l s o Dube-Pandey (1; 2 1 , Lee [ l ] , A .
[I], T r i o n e [l]).
Hence f o r s u i t a b l e
f
such t h a t
h-m-'IF(A)
can compute t h e Hankel t r a n s f o r m i n (3.21) w i t h i n v e r s i o n formula
Let u s w r i t e ( r e c a l l
h
g
Rm
on
i s d e f i n e d by ( 3 . 2 2 ) ; s e t t i n g
F(A) = Sm(RmF)( A )
and
F(A) = p f ( X ) = F-lf
?FSm.
0)
km = 1 / Z Z m r 2 ( m + 1 )
we have a formula
( c f . a l s o Levitan-Sargsyan f = FF =
1
7 FS% m f
so t h a t
[2]).
8m
Since
= B-'
m
=
There i s a t e c h n i c a l p o i n t involved h e r e i n t h e i n t e r p l a y between b r a c k e t s
( 0 , ~ ) and
work w i t h even
(-m,m)
which r e q u i r e s
f
F
and
Example 3 . 1 4 ) . D
2
Bm
=
Now t h e o p e r a t o r
EmLm
Smg(A)
even i n
and i n t r o d u c e d a f a c t o r of
t i v e l y we could work s o l e l y on
that
we
0
for suitable
1
2
Hi
F(X) = @ f ) ( A ) )
Then an i n v e r s e S m f o r l e t for
(A
E
( 0 , ~ ) with
Rm
X; 112
w e have chosen t o
in (3.21).
h(x,X) = 'd(x,A) = Cos Ax
Alterna(cf. also
should be a t r a n s m u t a t i o n o p e r a t o r such
and w e can check t h i s d i r e c t l y ( c f Lions [2: 3 ; 41 and Example
106
R. W. CARROLL
3.14 below).
Thus l e t
g(y)
b e given f o r
y
2
0
(g'(0) = 0
is n o t n e c e s s a r y )
and from t h e d e f i n i t i o n s 2(Bmg)(x) = (FS"g)(x)
Li$
=
2 D $
-
km<exp iXx, <(Xy)
2 2 DmBmg = -A Bmg.
we o b t a i n immediately
where
=
(2m
+
On t h e o t h e r hand
Here one h a s
l)D(y-'$).
2nd-1-m R (y,A), g(y)>+>
LG[(Xy)
2m+l~m R (y,X)]
(hy)2m+1Lmim(y,h) and a s i m p l e c a l c u l a t i o n shows t h a t t h e terms a t from i n t e g r a t i o n by p a r t s w i l l v a n i s h ( r e c a l l
2m+l > 0
or
Re(2m
=
y = 0
+
arising
1) > 0 ) .
Hence we can s t a t e ( c f . a l s o Example 3 . 1 4 )
Theorem 3.12
$FSm
23
Bm =
satisfies
D
2
Bm
= RmF-'
LmBm = BmD2
while
Bm
Bil =
=
The P a r s e v a l r e l a t i o n (3.19) w r i t t e n i n t h e form
= BmLm.
2a =
satisfies
F f ( * ) > e s t a b l i s h e s t h e c o n n e c t i o n between (3.21)
and (2.11) and t h e l a t t e r can b e r e w r i t t e n i n terms of
The formula (3.26) i s l i s t e d f o r comparison purposes l a t e r ( c f ( 3 . 6 1 ) ) .
For
p e r s p e c t i v e we w i l l o r g a n i z e h e r e a t a b l e of t h e v a r i o u s t r a n s f o r m s coming i n t o play.
Thus
Remark 3.13
We t a k e
h
so t h a t i f t h e r o l e s of new
h.
Take
R
with
and
Q
were r e v e r s e d t h e n
a s i n (3.5) s o t h a t
Theorem 3.5 holds. (3.27)
P
as i n ( 3 . 2 ) - (3.3) w i t h
0
and
Q*(D)fi =
Then t a k e
8
B(0,x)
=
6(x),
0
h(0.X) = B ( O , A )
B(y,x) = b ( y , x ) , and
so t h a t
I n t h e c a s e of Example 3.11 w e have
km(Xy)2mc1~m(y,A). R e l a t i v e t o
P
w e have t r a n s f o r m s
1
would c o r r e s p o n d t o a
-A28
= 6 ( x ) .
=
8(y,X) =
107
SEPARATION OF VARIABLES AND TRANSFORMS
while the corresponding
t r a n s f o r m s are
Q
I n Example 3.11 t h e t r a n s f o r m s i n (3.29) appear with
Bm
involve
< ,>+
and some f a c t o r s of
1/2
i n c o n n e c t i o n w i t h o u r u s i n g even f u n c t i o n s b u t t h i s s h o u l d
c a u s e no c o n f u s i o n l a t e r .
Now i f e v e r y t h i n g f i t s t o g e t h e r w e s h o u l d have
whereas i n Example 3 . 1 1 s i n c e t h e k e r n e l s are a l l f u n c t i o n s of
XA
or
yA
(i.e.
F o u r i e r k e r n e l s ) we have i n a d d i t i o n
21 Bm
Since
IP
=
Q.9by
F while
(3.23)
(resp
= Q-'IP-'
(Bi)-'
P&(resp. and
=
4
=
=
Sm s o
BG = I P Q
by ( 3 . 1 2 ) ) we have
QP as i n Remark 3 . 6 ) . Bil
=
51 FA
2Bi1
=a-'Q-'=
I n Example 3 . 1 1 $ =
as indicated.
F-I
W e refer also t o Sections
2.5 and 2 . 6 f o r examples of i n t e r p l a y between t r a n s f o r m s ( a l s o Appendix 2 ) .
Example 3.14
L e t now
w e l l d e f i n e d by (3.1)
im(x,A)
(3.32)
and, s e t t i n g
P(D)
=
(such
+
with
Q(D)
w i l l be c a l l e d
Q
e(0.A)
O(y,A) = 2mI'(m
Lm(D)
=
1,
Q(D) = D2
with
t r e a t e d a s even i n
A)
P
admissible).
8(y,A) = < b ( y , x ) ,
Am R (x,A)>+.
Let
B
h(x,X) =
Thus
1)X-m-4Mm[x-m-4b(y,x)l
and t h i s is c o n s i s t e n t w i t h t h e formula f o r when
unspecified but such t h a t
O(y,X) = Cos Ay. with kernel
Ern i n Example 3.11 which a r i s e s
Indeed from
B,(y,x)
w e have
Bm
=
+Sm
(with
Smg(A)
is
ioa
R. W. CARROLL
!;(XX)~'COS
= km~m!,
Xy Jm(Xx)f(x)dxdh
m
xm3""f (x) lHm[i&Cos
where
(3.34)
E
m
= 2mr(m
+
1) and
€,Bm(Y,X)
k
m
=
E
hyldx
-2
.
m
xm3* Mm[X%os
=
Hence we have f o r m a l l y
Xyl
which i s e q u i v a l e n t t o ( 3 . 3 2 ) f o r
B(y,A) = Cos Xy
d e s c r i p t i o n of such d i s t r i b u t i o n s
B,(y,x)
i n t h e c a s e of g e n e r a l
(3.35)
so t h a t
P
admissible
Q
and
E~B(O,X= ) x&lHm[X&]
we take
R(x,h) = km (Ax)
6(x)
and
6(x2)
so t h a t the kernel
with
cf.
Bm(O,x) = 6 ( x ) .
Bm(y,x)
of t h e t y p e reD
2
Bm
= BmL,
-4
Bm(y,x)
=
-
and b e g i n s by s e t t i n g
-1 < R e m <
B,(Y,X)
must b e used
He works d i r e c t l y w i t h t h e t r a n s m u t a t i o n problem
m
R (x,X)
m 6(x) must make s e n s e ( c f . T r i o n e [l] f o r a
For t h i s i t c l e a r l y s u f f i c e s t o show
quired i n ( 3 . 3 4 ) .
(3.38)
2m+lAm
= E
Thus l e t u s summarize L i o n s ' c o n s t r u c t i o n of k e r n e l s
and t h e n f o r
Now a p r e c i s e
i s worth d i s p l a y i n g , e s p e c i a l l y s i n c e
similar formula where a r e l a t i o n between
f o r more g e n e r a l
Bm.
e(y,A)>+ = kmcmxm4* Hm [ x ~ % ( ~ , x ) I
~ ( y , ~ = )
a l s o Bochner [ 2 ] ) .
b =
2 bmy(y
(note t h a t
Bmf(0) = f ( 0 )
-1 < R e m <
-4
h a s t h e form
or
2 -m-?, x )+
m (0,x)
x
=
~
6(x)).
f o r comparison w i t h ( 3 . 3 4 )
~
l
One h a s a formula t h e n w i t h
109
SEPARATION OF VARIABLES AND TRANSFORMS
(cf. Erdelyi et. al. [l]). B,(y,x)
in m
Starting here one wants to analytically continue
to establish formulas for all Re m >
-4.
To do this we write
first
and define recursively
Then for -1 < Re m < n - 4
the continuation of
B,
is given by y) dt
where
c^
m
=
+
(-l)nbm/(2m
1)(2m
-
1)
...
(2m
-
(2n - 3)
.
We note that for
n,l
... 3 - 1 =
since 2"r(m
+
(2n-1)(2n-3) 211-3 3/2
)/r(m - T ) ,
= ( - l ) & ' ~and ,
r(-m
...
2nr(~+k)/r(%, (2m+1) (2m-(2n-3)) = 211-1 2n-3 + ---)r(m 2 - 2) = (-~)~-~+'n, r ( m + 3 / 2 ) r ( - m - + )
the integral in (3.42) equals
T(m
Let u s give a different expression €or the kernel
+
l)r(-m
Bz(y,x)
+ ~211-1 ) / 2 r ( n + 4). of
8"m
as
follows.
It is easy to see that
where
cnl
=
(2n - 1) (2n - 3 )
written for m < n - 4
as
.. .
3
B
-1
=
2?(n
+ 4)/I-(%)
so
that Rnm
can be
R. W. CARROLL
110
(3.44)
B;f(y)
=
cm
n+l
-m+-
- 3) f(k-l)(ty)dt
k=1
- n+l 1 cnky-2n+lJ~x2dk(y2 - x
= c m
(2n
-m+-
(211-3)
1
f(k-l)(x)dx
k=1
Thus we have (3.45)
f(x)>
Summarizing we have Bm
=
2 and D 8
defined for all m > -$
m
Bt(y,x)
is a distribution given by ( 3 . 4 1 ) or ( 3 . 4 5 ) with
-1 < m <
-4 we have by
B;(O,x)
and
1 by Bmf(y)
example express Bmf(y) 1
gm(x,A)
=
=
1
f(x)>
1. 1.
6(x).
For
-m
x J (Ax) m
=
Amg (x,A)
m
Hence for
s o that ( 3 . 4 6 ) holds with
8 2 makes sense for -1 < m < $ and hence
But
( 3 . 4 6 ) ; note here that
(3.47)
=
The kernel
B m (f) is equal to
any of its determinations via formulas ( 3 . 4 1 ) or ( 3 . 4 5 ) for n
Bm replacing Em.
BmLm.
(3.39)
f(x) = X-~J~(AX) is an analytic function of x
Now
=
so
does
where
2-
n
so that ( 3 . 4 6 ) can be written
If
E
one knows m
= Cm[O,m)
+
Em is holomorphic with values in L(E)
Lions [2]) while for Re z > 0 both ( 3 . 4 8 ) holds for
when
P,(y,x)
-1
C
Re m <
4
T(z)
l/I'(z)
and
(cf.
are holomorphic. Hence
and continuing we obtain ( 3 . 3 4 ) for Re m >
d
is identified with a suitable determination B;(y,x).
-4
SEPARATION OF VARIABLES AND TRANSFORMS
We remark here also that the function U(x,c) L,(D,)U(x,S)
= Lm(Dg)U(x,S)
U (x,O)
5
=
where w2
=
indicated in (3.7) satisfying
U(x,O) = f(x)
(and U ( 0 , S ) = f(0,
Ux(0,5)
=
0) can be written explicitly in the form
x2
=
+ 52 +
(cf. Copson-Erdelyi 111)
2xct
Using a P admissible Q
Remark 3.15 B(0,x)
with
111
such as Q(D)
=
D2 and knowing that
the inversion theory for P
+ = 6(x)
in Theorem 3.5
is
equivalent to the Hankel inversion formulas and seems to represent a new derivation. However a certain amount of information about Hankel inversion is implicit in the fact that
B(0,x)
=
6(x).
We will want to deal with formulas of the form (3.16) for example
Remark 3.16
and record here the following observations. Thus take h(x,X) 2m+lAm R (x,X) R(x,X) = km(Xx)
=
im(x,X)
and
to obtain by the Hankel formulas
'0
Similarly by symmetry (3.51)
h(x,X)>+
=
6
We note here also that formally the truth of (3.50) implies
.
=
I>+
= 6(x)
1 but the converse does not seem to follow immediatly by analytical
I
Example 3 . 1 7
The following differential equation arises in the study of sperical
type functions on pseudo symnetric spaces (cf. Carroll [33; 3 4 1 , Carroll-Showalter [B], Carroll-Silver [ 9 ; 10; 111) (3.52)
Rtt
+
[(ZB
+
[(p
+ +
1)tanh
0)'
+
t
X2
+
(2p
+
+ q(q +
2a
+
1 ) c o t h t]Rt 2
2B)sech t]R
=
0.
112
R. W. CARROLL
(3.53)
R (0) = 0 t
R(0) = 1 and
The s o l u t i o n s a t i s f y i n g
R = RP'q(X,t) =
9-6,
cosh-R-Pt F ( F ,
R = iX
where
c a s e where
+
+ 6 + 1. For convenience we -4, R = -21 + i h , and q = 0, which
f3 =
a = 0,
yields
(ch = e o s h ,
SL(2,
= F
P-m
&+iX (
Zmr(m
=
where
a), and
setting
mtri-ih 2,
7
,
+
l)sh-"t
m
m
1, th't)
1,
2 -sh t )
+
2
RmSo
=
Sm
this
Q
+
m
Sm(h,t)
(2m
+
1)coth t D
Setting
+ 4)2
We remark a l s o t h a t t h e f o r m a l a d j o i n t of
Q (D) = D
the function
satisfies
2 m Qm(D)Sm = -1 S ,
(3.56)
(3.57)
Qi(D)$ = $tt
-
(2m
+
l)D(q coth t )
m
(D) i s given by
+
I t i s worth n o t i n g h e r e t h a t i n analogy w i t h
(m
Lm
+ 4) 2 $. one h a s f o r
?(A,t)
=
t S"(X,t)
(3.58)
Q;(D)Qm(h,t)
= -)i
P(D) = D2
with
F i r s t take =
(3.59)
with
PI:+iX(cht)
(m
Q(D)
p = m
9, +
+
2m+l
c o r r e s p o n d s t o working
d e n o t e s t h e s t a n d a r d a s s o c i a t e d Legendre f u n c t i o n .
lJ
(3.55)
w i l l take the special
sh = sinh, etc)
S m ( h , t ) = ch-R-mt F ( F ,
(3.54)
p + a + l , t a n h2 t)
p = a
and
p
i n a symmetric s p a c e based on
sh
is
Q,(D)
with
B(y,X)
2 Sim(h,t).
h(x.X) = exp iAx = Sm(X,y).
B(y,x) = < Q ( x , A ) , B(y,X)> =
Then
and
1 R(x.h) = 21T exp(-ihx)
b(y.x) = B(y,x)
while
and (3.6) y i e l d s
113
SEPARATION OF VARIABLES AND TRANSFORMS
(note Sm(h,y)
is even in
X
by the second expression in ( 3 . 5 4 ) ) .
We are in a
situation here analogous to Example 3.11 and the same kinds of remarks relative to brackets and even functions should apply.
< ,>+
(chy
-
where
chx):-%
f(x) =
then (cf. Magnus-Oberhettinger-Soni [l])
Fc
(3.61)
Now it is known that if
denotes the Fourier cosine transform. @(y,x)
Consequently
=
4
and it is of interest to compare this formula with ( 3 . 2 6 ) .
We recall at this
point also the generalized Mehler inversion formulas
1
cn
(3.62)
g(cht)
=
f(X)PI:+iX(cht)dX
0
(3.63)
where
f(X) c(X)
=
c(X)
=
g(cht)PIc+iX(cht)shtdt
n-'AshnAr(m
+ 4+
iX)r(m
+ 4-
iX)
(cf. Lowndes [l], Magnus-
Oberhettinger-Soni [l], Oberhettinger-Higgins [l]). Now let
P(D)
and let
Q(D)
so
?(A)
=
Qm(D) be
P
with
=
admissible with
can be chosen as
from which follows
h(x,X)
Sm(x,X)
and
R(x,A)
=
E(X)shZmc1
x
Srn(x,h)
B ( 0 , h ) = 1. From ( 3 . 6 2 ) - ( 3 . 6 3 ) we obtain
114
R. W. CARROLL
5
Letting
+
w e have
0
+ = B(0,x) = 6 ( x )
f o r m a l l y which can b e
written
S i m i l a r l y t h e formula
B(0,x) = 6 ( x )
then
B(y,x) = b ( y , x )
of view ( c f a l s o Bochner [l]). Lu =
on a domain
1 7
1 -
-
A(x)
dx
Let
-
(3.63).
x > 0 , and
for
as
l i m A'/A
x
points
be an o p e r a t o r of t h e form
L
(A(x)u')
D(L) C L2(A(x)dx)
be i n c r e a s i n g w i t h
A
i n (3.65)
We mention h e r e some work of C h e b l i [l] which c o n n e c t s seve;al
Remark 3.18
A(x) > 0
$(A)
(3.63)
and t h e i n v e r s i o n t h e o r y f o r
P i n Theorem 3.5 y i e l d s t h e g e n e r a l i z e d Mehler i n v e r s i o n (3.62)
(3.68)
-
f o l l o w s from (3.62)
p)
In p a r t i c u l a r i f w e know somehow t h a t t h e c h o i c e of
(cf (3.16)). w i l l yield
-
+ = 6(X
over
(0,m)
(A'/A)(x) = a / x
A(x) +
as
m
x +
where
+ B(x) and
m
A E Cm(O,m),
with A'/A
A(0) = 0 ,
B E Co[O,m).
Further let
d e c r e a s i n g ; set
p =
These h y p o t h e s e s a r e s a t i s f i e d by t h e r a d i a l p a r t of t h e
+ m.
Laplace-Beltrami o p e r a t o r on noncompact Riemannian symmetric s p a c e s of rank 1. L e t
1 2 2 H = { u , u ' E L (A(x)dx)], H ={uEH1;
0
at
m}
2
L (A(x)dx)l
(note
J
Adx = Au';
I
LUG Adx =
Let
be
Do(lR)
dense i n
D(L)
o(L) C [0,m).
Cm
=
u';'
,1
-
even f u n c t i o n s on IR
i n graph norm and f o r
u'vA(x)dx : H1
-f
L = -(D2
1, $ ' ( O , h )
x
=
0
+ Ax
i s continuous
w i t h compact s u p p o r t ;
Do(R)
is
@ E Do(IR)
D ) = x
w r i t t e n as m
and
(c
I:(Au')';dx).
- -1 D
x x @(y,X)
(xDx).
Then t h e s o l u t i o n of
m
=
1 @k(y)Xk
k=0 (3.69)
+
+
R e c a l l now Remark 1 . 2 d e f i n i n g g e n e r a l i z e d t r a n s l a t i o n o p e r a t o r s
and t a k e f o r example
X@,@(O,X)
D(L) = {u E H I ; v
or equivalently
i n t h e norm of
2 2 L u E L (A(x)dx)), and D ( L ) = { u E H ; A ( x ) u ' ( x )
@(y,X) = J o ( y f i )
Ty = @(y,L)
=
1
kXk
k=0
2k(k!)
s o t h a t one o b t a i n s a f t e r some c a l c u l a t i o n
i s g i v e n by
L@ =
115
SEPARATION OF VARIABLES AND TRANSFORMS
2lr
1 1
(3.70)
TYf = 2T x
(cf (3.49)).
f ( d
2
+
y
2
+
2xy Cos 0)dB
0
A second way t o d e f i n e
Ty
is v i a t h e Cauchy problem
L u = L u, X
u(x,O) = f ( x ) , w i t h
u ( x , y ) = T Y f ( x ) = TYf
h a s a s p e c t r a l decomposition
L =
(cf. (3.7)).
Y
Now i n g e n e r a l i f
I
AdEX t h e n i n t h e s p i r i t of
L
Ty = @ ( y , L ) one
can d e f i n e
(3.71)
Ty =
I
@(y,A)dEX
which one can make p r e c i s e a s f o l l o w s f o r [ 2 ] f o r d e t a i l s and c f . S e c t i o n 2 . 4 ) .
1
Fourier transform
L
of t h e t y p e above ( s e e T i t c h m a r s h f E Do(W)
Let
and d e f i n e a g e n e r a l i z e d
by
m
?(A)
(3.72)
f(x)@(x,A)A(x)dx.
=
0 Then t h e r e i s a tempered p o s i t i v e measure
u
on
u(L)
such t h a t
C [0,m)
m
m
IfI’Adx
0 2
l?l2do
=
L (u)
with inverse
(3.73)
f(x) =
Further
and
f
+
?
e x t e n d s t o an i s o m e t r i c isomorphism
L2(Adx)
1,
D(L) = I f ;
d e f i n i t i o n of
?(X)@(x,h)du(X).
i
h
X21?(A)12d~(A) <
a}
and
r
.
Lf = Xf.
One checks from t h e f i r s t
Ty = @ ( y , L ) t h a t
s o t h a t i n a s p e c t r a l form we d e f i n e a s e l f a d j o i n t
Ty
by
A (3.74)
+
’0
Tyf(X) = $(y,X);(A).
Using t h i s background C h e b l i [l] o b t a i n s some r e s u l t s of i n t e r e s t i n p o t e n t i a l t h e o r y and harmonic a n a l y s i s ; w e w i l l n o t d w e l l on t h i s but mention only a few p r o p e r t i e s h e r e of g e n e r a l i n t e r e s t . bounded s e l f a d j o i n t o p e r a t o r i n
Theorem 3.19
Ty
Define
Ty
by ( 3 . 7 4 ) ;
then
Ty
is a
2
L (Adx).
i s sub-Markov i n t h e s e n s e t h a t i f
0
5
f
5
1 a.e.
then
116
0
5
R. W . CARROLL
TYf < 1 a.e.
Ty
(1 T y f ( b 5 (1
‘16.
(3.75)
* g) ( x )
i s bounded i n
If
*g
f
I
LP(Adx)
i s defined f o r
for
0
5
f , g E Z),(IR)
p
5m
(p
i n t e g r a l ) and
by
m
(f
=
TYf(x>g(y)A(y)dr
0
*
then
1 L (Adx)
extends t o
@(x,X) f o r
characters
X
which becomes a commutative Banach a l g e b r a w i t h
E C = {A =
The commutativity of
*
X1
+
2
X2
iA2;
5
4p
2
All.
is related t o t h e property
5 Txg(x) = Txg(s)
5
( c f . S e c t i o n 2.5 f o r f u r t h e r d i s c u s s i o n ) .
Remark 3 . 2 0
L e t u s expand t h e f o r m u l a t i o n of Remark 3.18 t o r e l a t e i t t o some of
t h e formulas developed e a r l i e r i n S e c t i o n 2 . 3 ( c f . a l s o S e c t i o n 2 . 4 where such examples a r e i n d i c a t e d i n a g e n e r a l manner). that the solution h(x,X)
p = A2
for
Lu
becomes
@ ( x , u ) of
= -Qu
2 -A 8,
8(O,A)
Now
L
h(x,X) = @(x,A ) ) .
Thus
= 1, and
J,
u
=
G(u,y)
brackets.
@‘(O,p) = 0
The e q u a t i o n
is precisely
P(Dx)u = Q(Dy)u
=
? ( u ) O ( y . 6 ) = ?(P)$(Y,u)
where
Q8
=
X 2 . Using t h e i n v e r s i o n formula ( 3 . 7 3 ) we o b t a i n
?(~)$(y,u)@(x,v)do(~).
is s e l f adjoint r e l a t i v e t o
w(A)h(x,X).
i n (3.68) s o
L = -P
and w e w r i t e
u(x,O) = f ( x ) .
U(X,Y)=
L@ = p @ , . @ ( O , u ) = 1, 2
(i.e.
since
(3.77)
Thus t a k e
A(x)dx
so
P* = P
and we can t a k e
n(x,h) =
Thus (3.77) h a s t h e form (1.14) w i t h s u i t a b l e d e f i n i t i o n of t h e We n o t e t h a t t h e c o n v o l u t i o n (3.75) i s t h e same o p e r a t i o n i n d i c a t e d by
o u r g e n e r a l i z e d c o n v o l u t i o n of (3.8) s i n c e can be w r i t t e n
U(x.5)
=
E
Txf(x)
from (3.7) and (3.75)
MORE ON KERNELS AND TRANSFORMS
More on Kernels and t r a n s f o r m s .
2.4.
117
T h e r e a r e now s e v e r a l a s p e c t s o f t h e f o r m a l
In
d e v e l o p m e n t i n t h e l a s t s e c t i o n which must b e i n v e s t i g a t e d i n more d e t a i l .
p a r t i c u l a r i t was assumed t h a t a l l o f t h e b r a c k e t c o n s t r u c t i o n s and o p e r a t i o n s A l s o no
upon them made s e n s e ( j u s t i f i e d f o r s p e c i a l c a s e s i n t h e E x a m p l e s ) . d i r e c t i v e f o r choosing that
P(D) = D
2
was g i v e n a n d w e w i l l e x a m i n e t h a t f i r s t .
Q(x,A)
+ p(x)D + q ( x ) .
Then i t i s w e l l known t h a t
!
!
a s e l f a d j o i n t form by w r i t i n g
P(D)
P(D)y = e x p ( - p ) ( y '
+
exp p ) '
2
P ( D ) h = -A h functions
R(x,A) 2
P(D)h = - A h
hfx,h)
2
R
@ = a$
P*(D)@ we f i n d t h a t
@ =
aI,b
=
P*(D)@ = 0
satisfying
+
$T'
2Q'T = 0
P(D)$ = 0.
given t h a t
where
T =
a' - p a .
Hence
Computing
T = k$-2
and
!I,b-'e-lp.
kye/'
i s a second s o l u t i o n of
P*(D)@ = 0
where
I n d e e d l e t us
But by a s t a n d a r d r e d u c t i o n o f o r d e r t e c h n i q u e i t i s known t h a t
y
of
R = ah
w h i c h had t h e form
and i t t u r n s o u t t h a t t h i s i s g e n e r a l l y p o s s i b l e .
t r y t o find
(4.1)
P*(D)R = - A
satisfying
b u t we r e f r a i n
Now i n t h e Examples 3 . 1 4 and 3 . 1 7 we u s e d
h ( 0 , A ) = 1.
satisfying
can be put i n t o
qy
f r o m d o i n g t h i s s i n c e it is c o n v e n i e n t t o d e a l w i t h s o l u t i o n s
Suppose
of t h e form
and h e n c e t h e r e i s a s o l u t i o n
P(D)Y = 0
!
@ = y exp
kI,b!$-2exp(-lp)
p
where
P(D)y = 0.
@
=
of
T h i s c a n a l s o b e con-
f i r m e d d i r e c t l y and w e r e c o r d t h i s s u r e l y w e l l known f a c t as
Lemma 4 . 1 of
Given
P*(D)@ = 0
R
y
+
p l y @ and
(2y'y 2-
2
with
-u Y (iJ2
+y
-
=
x2)yq exp
p(x)D
+
t h e r e is a s o l u t i o n
q(x)
p
!
y@"
=
-(2y'yp
I
p
p = [(y";
(y
-
=
=
2
p'
0.
y(x,A),
y;")
f o r t h e Wronskian t h e r e r e s u l t s
+y E
so we have
y exp
J
( P ( D ) y ) @ - yP*(D)@ =
a s above
@
-
y"@
(p
+
i n Examples 3 . 1 4 and 3.17 a r e o f t h i s f o r m .
and
p)exp
2
@ =
!
Y exp p
P(D)y = 0.
when
The f u n c t i o n s t h a t with
P(D) = D
+
+y
='
p )exp
i
=
p
Similarly i f
-
p(y'y
=
y(x,u))
-
yy*)lexp
-
Next w e o b s e r v e
Y"@ - y$"
while
y'b
+ p ( Y ' @ + Y@') + y@' =
2
P(D)y = -1 y then
(P(D)y)$
p.
Setting
and
w
-
P(D)y =
yP*(D)i =
=
-
y'y
-
-
yy'
118
R. W. CARROLL
Since
(W’
+
we obtain
pW)eIP = (Wefp)’
b
a f o r b r a c k e t s over an i n t e r v a l
Let
;
then
weight f u n c t i o n
Example 4 . 3
y
Let
*m
= R
- yv’.
W = y’?
h o l d s where
-A 2 y
P(D)y =
=
and
I
and
-
while
By ( 1 . 3 . 9 )
and
=
at
We summarize t h i s i n
9 exp
I
p.
x = a
Then ( 4 . 3 ) and
x = b
are orthogonal r e l a t i v e t o t h e
y and
p.
-
y
-m
= R
(x,!~) with
2-m-tl
P:(x,X)
$
with
MeJP = 0
L e t u s l o o k a t t h e s e formulas when (x,A)
-u 2-y .
P(D)y =
2P(D)y = -1-1 y
In particular i f
exp
The e q u a t i o n i n t h e second l i n e of ( 4 . 3 )
2 P(D)y = -A y
a l s o f o l l o w s d i r e c t l y from
Lemma 4 . 2
(a,b).
= -xh R
$
= x
(x,X)/2(m 4 1 )
P(D) = Lm(D) = D
2m+l-m R ( x , ~ ) since
2
2m+l D. +expjp = x
2m+l
so t h a t
We r e c a l l some formulas o f t e n used i n c o n n e c t i o n w i t h !lankel i n v e r s i o n ( c f . Sneedon [l]).
First for
m >
[PJm(XX)JmCl(ux)
=-
u
b =
a.
-
AJdl(Xx)
J,(~X)
1
a
-A e Jp = xZmf1, we o b t a i n from ( 4 . 4 )
Consequently, s i n c e
Clearly
-4
WeJP = 0
at
x = a = 0
I t i s known t h a t
from ( 4 . 4 ) and w e c o n s i d e r i t s b e h a v i o r at
x =
119
MORE ON KERNELS AND TRANSFORMS
w h i l e , f o r small
Im( 7 ALJ )
E,
exp (
% E
Now l e t u s d e f i n e
hm(A,p,-)
= 0
implies
We]'
so t h a t
(Ap)4Hm(A,u,0)
+
22mr2(m
6(A
=
1)( A P ) - ~ ( P ~ - A2)hm
-
11).
and
k
m
-
22mr'2(m + 1) (Ap)-m-4(p2
-f
and t h i s is p r e c i s e l y ( 3 . 5 1 ) ( n o t e
U
E +
2
=
-2m -2
r
t o s e l f a d j o i n t form.
-
[C
-
(b
1 a ' ) / a4 -
.
' ' 2
B2
4
-
(m
+
+
-
l)(Ap)-m-46(A
(h~)-~-' with
1)
5
Thus i f
Then s e t t i n g
a ' ] Y = 0.
2
(2m
+
y = Y exp(-
l)/x
and
y
i n ( 4 . 1 0 ) can b e i n t e r p e r t e d as
h(x,A) = y(x,A)
'I
x J (1x1 = Y ; s i n c e m
I
=
a-'dx
J
and
Y"
+
[q
w e have
Y"
+
[A2
exp(-
7
'i
p) = x-m-'
-
1
-
4
-
p
I
exp(-
then
'I
y
W = YY' = Y'? = W e x p ( p)
p).
2
5
6)
ay"
+ BD5 +
+
+
by'
c)y = 0
cy = 0
where
2 D Y
we w i l l o b t a i n
5
+
-
y" + p y ' + qy = P(D)y = 0 1 - p ' ] Y = P (D)Y = 0. I f p =
-
2 1 / 4 ) / x ]Y = 0
t h i s gives
-
y'y
2
(m2
2 P(D)y = -A y
Y
Q(x,u) =
I f we start with
a l s o t h a t under t h i s r e d u c t i o n i f -
(D
then
y = Y exp(-$
yields
p)
q = A*
?5
5
11)
The l a t t e r i s a s t a n d a r d form o f t e n used i n s t u d y i n g
s e l f a d j o i n t e i g e n v a l u e problems. then
p =
A2) (Ap)'IHm(A,p,O)
We r e c a l l h e r e i n p a s s i n g someother s t a n d a r d r e d u c t i o n s of
=
1
W exp
.
km(vx) 2mc1;(x,u))
f?
Hm(X,u,O) = 0
0
T h i s seems t o s a y t h a t
1 i y ( x , A ) q ( x , p ) x 2mc1dx = 2 2m r 2 (m
-2m-1
E
However from ( 4 . 3 )
which e q u a l s z e r o .
(4.10)
as
On t h e o t h e r hand
f u
A
so t h a t f o r
= Hm(A,p,O)
-.
x =
at
% )/m.Hence f o r small 2E
2E
where
Another r e d u c t i o n s t a r t s from
and
y = x
-m
P(D)). =
with a solution
W e note
J,(Xx).
-u 2
:I
with
y = Y exp(- - p )
and
+
on
-(py')'
Ry = Xry
7
W = 7y' =
[a,b];
120
R . W. CARROLL
setting
z =
get
+
-u"
u = ( r p ) J4y ,
Ix(r/p)'dt, a u = uu
on
and
p = cX
2
- c Q./r f o r
r, = O " / O
[ O , ~ r r ] where
c = 71 1
with
0 =
Levitan-Sargsyan [ 2 ] ) .
Remark 4 . 4
I t seems a p p r o p r i a t e a t t h i s p o i n t t o r e c a l l a l s o a few g e n e r a l
f a c t s about e i g e n f u n c t i o n expansions f o r second o r d e r l i n e a r d i f f e r e n t i a l o p e r a t o r s ( i n view of Lemmas 4 . 1 and 4 . 2 ) .
There a r e s e v e r a l a c c o u n t s of t h e t h e o r y i n
[l],
book form t o which w e r e f e r f o r m o t i v a t i o n and d e t a i l s ( c f . Akhiezer-Glazman Coddington-Levinson Section 2.5).
Let
let
L!
121, Naimark [l], Titchmarsh 121 and a l s o
A s t h e b a s i c e q u a t i o n one u s u a l l y t a k e s
@(x,X) and
o(o,x)'=
[l], Levitan-Sargsyan
O(x,X)
@ ( O , X ) = S i n a, @ ' ( O , X )
s a t i s f y (4.11) w i t h
O'(0,X) = S i n a. We work f i r s t on t h e i n t e r v a l
Cos u , and
+
R@(b,h)]Cos D
v a r i e s from
to
+
[O'(b,X)
I?
+
I?@'(b,X)lSin
describes a c i r c l e
As
Ctn
Cb
e i t h e r tends t o a l i m i t point o r a l i m i t c i r c l e .
-a
m
o r any p o i n t on t h e l i m i t c i r c l e then
If
$(x,X) = O(x,X)
One d e f i n e s then
and from n a t u r a l c o n s i d e r a t i o n s
f(x) = l i m [-
7 Ti1 l
~ ~ ~ ~ Q ~ x . ~ ) d A l .
R"
Define t h e n a n o n d e c r e a s i n g f u n c t i o n
(4.14)
a,
1 0 , ~ ) and
be d e f i n e d by
[O(b,A)
(4.13)
= -Cos
k(X) = l i m 6+0
and p a s s i n g
R
and
:1 6
[-Im m(u
+
k(X)
by
i6)ldu
t o t h e l i m i t s we o b t a i n
B
Cb
m(X)
=
in
0. E
and a s
b
-+ m
is the l i m i t point
+ m(A)@(x,X) E
2
L (0,a).
121
MORE ON KERNELS AND TRANSFORMS
'1
m
(4.15)
f(x) =
77
For t h e i n t e r v a l
-1,
@'(O,A)
and
m2(A)
=
m
O(x,A)dk(A)l @ ( y , X ) f ( y ) d y . -m 0 let
(-m,m)
O(0,A)
-1,
=
A
S(X)
=
<(A)
= lim
lim
Im[
6+o
A
6+0
j
=
and
<(A)
=
s a t i s f y (4.11) w i t h
0.
e(x,A)
@(O,A)
There e x i s t f u n c t i o n s
+ m,(A)@(x,A)
ml(A)
L 2 (-m,O)
a r e d e f i n e d by
-1 ml(u+i6) -m2(u+i6) -m 1(u+i6)m2(u+i6)
Im[
n(A)
while a function
Q1(x,A)
c(A)
O(x,A)
e'(0,X)
1, and
a s b e f o r e such t h a t
and n o n d e c r e a s i n g f u n c t i o n s
(4.17)
@(x,A) and
ml(u+i6)
- m2 (u+i6)
of bounded v a r i a t i o n i s d e f i n e d by
f o r t h e expansion theorem l e t
and t h e n i t f o l l o w s t h a t
There are s i t u a t i o n s when t h e formulas t a k e a s i m p l e r form. t e n d s t o a r e a l l i m i t when dn(u) = ml(u)dE(u)
and
Im A
dC(u)
=
+
0
so that
2 ml(u)dc(u)
'b1(x,A) E L'(-m,O) s o (4.20) becomes
m
(4.21)
f(x) = Ti
$l(x,A)dS(A)! -m
UJ1(y,A)f(y)dy. -rn
For example if we have
and
=
R. W ; CARROLL
122
On t h e o t h e r hand i f
q(x)
i s even
ml(A) = -m2(X)
q(X)
and
=
0 s o (4.20)
becomes
We n o t e a l s o t h a t (4.15) is o f t e n w r i t t e n i n a n o t h e r form by t a k i n g
(4.23)
i
x(x,hf =
X
1
m
Ffhf =
@(x,u)dk(u); 0
x(y,X)f(yfdy.
0
Then one h a s
Example 4.5
I n t h e n o t a t i o n of Remark 4 . 4 t h e c l a s s i c a l F o u r i e r formulas can be
w r i t t e n as f o l l o w s .
(4.25)
For
[O,m)
B(x,A) = cos CY Cos
The f u n c t i o n
x G +X%in
c1
must be a m u l t i p l e of
$(x,X)
Sin x d i ;
e ixdi
if
Im A > 0
t o be i n
2
L ( 0 , ~ ) and t h u s
Hence
2 I m m ( h ) = - f i / C o s CY
+X
e(x,X) = X%in
xfi;
Sin
2
for
CY
A > 0
a = n/2
(4.27)
with
(4.28)
-1m m(X) = A-,’
m(X) = -iA-’,
k(X) =
Hence from (4.23)
$(x,X) = Cos x f i
i
and
h u-’du
(X >
0).
0
-
(4.24) f o r example
and is
0
‘ f o r A < 0.
For
MORE ON KERNELS AND TRANSFORMS
(4.31)
Writing
f(x) =
71-
= A
S'
,
123
COS x f i d
we o b t a i n t h e s t a n d a r d F o u r i e r c o s i n e formula.
Note a l s o t h a t
(4.15) w r i t t e n d i r e c t l y i s t h e same, namely
(4.32)
f(x) =
dA ~ C O Sx f i x
0
On
Cos y f i f ( y ) d y . 0
we have s i m i l a r l y
(-m,m)
(4.34)
1-
m,(A)
=
i h4; $(x,A)
=
exp(-ixfl)
and (4.22) g i v e s t h e s t a n d a r d F o u r i e r i n v e r s i o n i n t h e form
Example 4.6
Consider t h e Bessel e q u a t i o n
(4.38)
+
on and
y"
2 (S
( 0 , ~ ) . Thus
2'YV(xs),
- v - k ) y = o 2 2 q(x) = (V
where
s i n g u l a r s o one t a k e s
(4.39)
Yu(z)
=
1/4)/x2 [Jv(z)Cos
a E (0,m)
TV
-
J
= s
-U
2
.
This has solutions
( z ) l / S i n TTV.
-
YV(xs)Jv(as)];
x?lJv(xs)
Both e n d p o i n t s are
a s a p o i n t t o s p e c i f y d a t a and t h e n
@(x,h) = 7 1 naSx4[J (xs)Y,(as) V
X
and
R . W. CARROLL
124
At
x = 0
0 < V < 1
and
(V =
4
is not singular).
Now f o r
x4J v ( x s )
are r e s p e c t i v e l y
L 2 (a,m)
iY,,(z)
V 1.1 and t h e l i m i t c i r c l e c a s e f o r
we have t h e l i m i t p o i n t c a s e f o r
and
(we a r e i n t h e l i m i t p o i n t c a s e a t
-sJ,',(as)/J (4.40)
-
(as)
V
$l(x,A)
The s o l u t i o n s i n
x % ~ ( x s ) where
x =
also).
Hv(z) 1
Hence
L2(0,a) =
m,(X)
Ju(z)
+
=
and
= x'a-'.J,,(xw)Ji'(as).
rn,(k)
Similarly
1 2a
-
W > 1
1 1 -sH ( a s ) ' / H ( a s )
=
V
V
We a r e i n t h e c a s e of (4.21) where
1 -2a
and
m,(A)
is real for
X r e a l and (4.21) y i e l d s
t h e Hankel formula
where
sL
=
X.
0 < v < 1 m2(X)
For
-V CS
m ( A ) = -s 1
(4.43)
where
c
X >
(4.45) while f o r
-V
is a r b i t r a r y .
(4.44)
For
cs
llJ,(x,X)
x4
a4
=-
is t h e same as above and
- sVJJV(as)
J;(as)
Jv(as) - sWJav(as) Hence
[
CJ (XS) - s 2v J - v ( ~ ~ ) V
cJv(as) - s2vJ-v(as)
0
[cJv(as)
1
-1m
-
h < 0
m2(A)
[m,(X)
-
=
- na c2
m2(X)]-'
-
- 2cs2vcos 7iv + s4u i s r e a l and f o r
c
0
i t is continuous.
Hence from (4.21) a g a i n 2v
(4.46)
f(x)
=
0
c
- 2cs2v-Cosvn+s
-
s
2v
J-v(~s)lf(~)d~.
125
MORE ON KERNELS AND TRANSFORMS
Letting
c
.+
we g e t ( 4 . 4 2 ) .
-m
F u r t h e r Bessel f u n c t i o n f o r m u l a s c a n b e f o u n d i n
T i t c h m a r s h [ 2 ] from w h i c h t h e s e w e r e t a k e n .
L e t now
-
= y"
P(D)y = y o '
-
Qy = y"
+
1 2 [qp 2
P*(D)q = -A q.
4.1,
py'
+
+ qy
=
-A 2y so
1 2 2 p ' - q]Y = -A Y .
P
.
and
rl
%
P(D) = L (D) m
In the case
r)
Set
=
y exp
I
p
satisfies
Po(D)Y
s o t h a t , as i n L e m a
theory with the s e l f adjoint theory
P-P*
we would h a v e
y
%
X - ~ J ~ ( A X ) ,Y
%
x 4J m ( A x ) ,
X ~ ~ ' . J ~ ( A Xf o) r e x a m p l e . m > 1 consider
More p r e c i s e l y f o r
+,
A = s 2 we h a v e
g i v e n by ( 4 . 4 0 ) s o f o r
2m+l-m R (x,s) R ( x , s ) = krn(sx)
(4.49)
NOW
Y = y eXP(7jp)
Then t h e r e l a t i o n
provides a n a t u r a l connection of t h e for
1
that
1
;dS(A)
2 Jm(as)ds
as
=
by ( 4 . 4 1 ) and d e n o t i n g
1
;d t
g e n e r i c a l l y by
we
dp
can w r i t e
w h i c h by ( 4 . 2 1 ) c a n b e i d e n t i f i e d w i t h For
[0,m)
for
A
take (-m,m)
again
=
with
0
set
s2
d p = - dk.
a = 0
h ( 0 , s ) = 1.
t h e b e h a v i o r of
'I
CY = n / 2
i n accordance w i t h Section 3. i n t h e d e f i n i t i o n of
@(x,A) and
h ( x , s ) = @ ( x , A ) e x p ( - + j X p ( < ) d < ) s o t h a t h ( 0 , s ) = 1; f u r t h e r 0 T h e r e are a l s o t w o s i t u a t i o n s w h e r e ( 4 . 2 1 ) may a p p l y . F o r
1
take
r e g u l a r t a k e now
5)
6(x -
with For
a1
and
$l(x,A)exp(-
7
p)
$l(x,A)exp(-
2
'I
p)/L(a,s)
+
7
h ( x , s ) = $,(x,A)exp((0,m)
p(E)dc)
and
d p =;
1
d<
w i t h 0 s i n g u l a r (as i n t h e Bessel case a b o v e )
1 exp(- z j p )
L(a,s)
II:
0
or
as 00
x
+
0
must b e a s c e r t a i n e d ; i f
t h e n w e set
( a s i n ( 4 . 4 8 ) ) and t a k e
h(x,s) = dp =
dc.
Then i n ( 4 . 4 9 )
so
126
R. W. CARROLL
R(x,s) = $,(x,X)
'I
exp(7
p) R(a,s)dp/ds
to produce (4.50). Similarly whenever dp/dX
is a function with
dp/ds
=
2s dp/dX we take R(x,s)
to be of this general form in all the cases indicated (i.e.
R
'I
= $,exp(Y
p)dp/ds, or
R
= $I
'I
exp(y
p)dp/ds).
1 Q = @,exp(pIp)R
dp/ds,
There results in an obvious
manner Theorem 4.7 The constructions of Remark 4.4 give rise to a variety of situations where P(D)h
= -s
2 h
and
P*(D)R
=
-s
2R with
= 6 ( -~ t);
(4.51)
Reading off
6(x)
=
<1, R(x,s)>
R(~,s)>
=
&(x
- E),
from (4.51) as before we have an abundent source
of models for the developments in Section 2.3 (cf also Section 2.5). Remark 4.8
It is worth collecting here some information about general transforma-
tions following Titchmarsh [l] and Mercer [l; 2; 31 (cf. also Braaksma-Schurtman [l], Dijksma-deSnoo [l], Eringen [I; 2 1 , Nasim [l], Walton [l]). and
First let
f
g be connected by the so called Fourier kernels
(4.52)
f(x)
=
1,
k(xu)g(u)du;
g(u)
=
and let Mellin transform be defined by
with inversion formula (4.54)
C+iWA f(s)x-sds. c-im
f(x) =
Then the condition for (4.52) is A
(4.55)
h
k(s)h(l
-
S )
= 1.
This situation is discussed extensively i n Titchmarsh 11 where numerous examp es
127
MORE ON KERNELS AND TRANSFORMS
are g i v e n .
There i s no concern t h e r e however w i t h t h e p o s s i b l e o r i g i n of s u c h
kernels
and
k
h.
Another approach followed by Mercer 11; 2; 31 examines t r a n s f o r m s of t h e t y p e
?(a)
(4.56)
=
L
K(X,x)f(x)dx;
1,
f(x) =
z(X)H(X,x)dX
p a r t l y w i t h a view of che p o s s i b l e o r i g i n of such k e r n e l s i n d i f f e r e n t i a l problems and we propose t o d i s c u s s t h i s .
(4.57)
[pK']'
P
+
[A2u
-
Thus f i r s t assume
Indeed l e t
(4.58)
- (PW')'
while f o r
1 P
[-v 2u -
+
W(w,K) = wK'
-
satisfies
v]K = 0
w i t h (4.56) i n e f f e c t ; t h e n t h e form of cases.
K(X,x)
H(X,x)
can b e determined i n c e r r a i n
V ] W= 0
w'K
there holds
m
(4.59)
O(X,p) = -pW(w,K)
= L(X)M(p),
10
In t h i s e v e n t m u l t i p l y i n g (4.57) by
w
and ( 4 . 5 8 ) by
K
r e s p e c t i v e l y and sub-
t r a c t i n g we o b t a i n upon i n t e g r a t i o n m
(4.60)
(A2
+ p2)j
p(x)u(x)w(LI,x)K(X,x)dx = L ( i ) M ( p ) . 0
Thus given t h e i n v e r s i o n of (4.56) t h e r e r e s u l t s f o r m a l l y
In view of t h e i n t e g r a l i n (4.61) p o s s i b l y d i v e r g i n g Mercer u s e s t h e i d e n t i t y
t o w r i t e (4.61) as
128
CARROLL
R . W.
t = A2
s = p2
and
(4.64)
one can w r i t e ( 4 . 6 3 ) as
= wo(s i ,x)
LLIHo(t’,x)]
where
LL
d e n o t e s an i t e r a t e d Laplace t r a n s f o r m
transform).
t
+
p
+
s
(or a S t i e l t j e s
A s t a n d a r d s o r t of argument ( c f . Sneddon [ l ] , Widder [ l ] ) y i e l d s
( c f . a l s o Theorem 4 . 1 4 )
Theorem 4 . 9 where
L e t (4.56) hold with
satisfies (4.58).
w(p,x)
Example 4.10
Take
L(A) = A,
Then v = 0,
K(A,x) = Cos Ax, and
Example 4 . 1 1
H(X,x) =;
Take
One o b t a i n s
p = x,
Then ( 4 . 6 6 ) d e t e r m i n e s
v = 0,
w(p,x) = exp(-px) 2
2
u = 1, v = m /x
L(h) = Am
K (z) = m
w(p,x) = exp(-px).
from ( 4 . 6 6 ) .
w e have
If
p = u = 1,
L(A) = 1 w i t h
M(p) = p ,
Cos Ax.
and
t h e i n t e g r a l i n ( 4 . 6 0 ) i s t o converge. f o r example o r
H(A,x).
I<(A,x) = S i n Ax, and
2 S i n Ax H(h,x) = T
M(p) = 1, and
and ( 4 . 6 6 ) y i e l d s
Km(px).
p = u = 1,
s a t i s f y i n g ( 4 . 5 7 ) and assume ( 4 . 5 9 )
K(A,x)
$ cscmn[I-m
(2)
2
,
M(p) = p-m Here
-
K(X,x) = J (Ax), and
m
and
Km(z) =
I (z)] m
where
R e m > -1
w(p,x) =
is required i f
cosh m t exp(-z cosh t ) d t
imn
I ( z ) = exp(- -)J m 2
m
(iz).
Then from ( 4 . 6 6 )
Using t h e i d e n t i t i e s
and
Jm(z) =
1 51 [Hm(z) +
2
H (z)] m
one can w r i t e ( 4 . 6 7 ) as
H(A,x) = x X J (Ax). m
I n a n o t h e r p a p e r Mercer assumes ( 4 . 5 6 ) p l u s t h e c o n d i t i o n ( f o r 0)
X > 0
and
Re p >
129
MORE ON KERNELS AND TRANSFORMS
f o r some f u n c t i o n
w ( p , x ) ; n o t h i n g i s assumed now a b o u t d i f f e r e n t i a l e q u a t i o n s .
Then d e f i n i n g
( 4.70)
Hs(t,x)
=;
i
i sn -
[w(e
t,x)
i t i s shown t h a t f o r s u i t a b l e
-
w( e
-isn 2 t,x)]
i ( t 2 i ( t ) E L'(0,y)
and
-2,f ( t ) E L1(y,m)
t
for
y E (0,m))
each
( 4.7 1)
1 -
[f(A
-
+
0)
?(A
J
l i m - J o l ( t )A ( : s+l
+
O)]
=
, x ) H s ( t ,x) d x d t
Theorem 4. 12
Under t h e c o n d i t i o n s i n d i c a t e d f o r m a l l y
Example 4 . 1 3
F or
K( h ,x ) = S i n Ax
and
H(t,x) = l i m - Hs(t,x). s+l
w ( p , x ) = exp(-ux)
L(A)
p = u . = 1,
s i n e f o r m u l a s and o n e n o t e s t h a t ( 4 . 6 0 ) w i t h
we g e t t h e F o u r i e r =
A , and
M(p) = 1
i s t h e same as ( 4 . 6 9 ) .
Anothe r d i r e c t i o n p u r s u e d by Mer cer i n v o l v e s t h e e x t e n s i o n o f Theorem 4 .9 t o The p r o o f i s d i f f e r e n t
k e r n e l s a r i s i n g from h i g h e r o r d e r d i f f e r e n t i a l e q u a t i o n s .
and somewhat n i c e r s o w e i n c l u d e i t ; t h e p r o c e d u r e u s e d f o r Theorem 4 . 9 i s somewha t more r e f i n e d however i n t r e a t i n g i n t e g r a l s s u c h a s ( 4 . 6 1 ) . d e n o t e a n y d i f f e r e n t i a l o p e r a t o r ( l i n e a r ) of e v e n o r d e r P;.
P(K,w)
(4.73)
+
[P
Anu]K
=
0;
[PG
unu]w = 0.
be t h e b i l i n e a r concomitant of
WP K
-
d
KP*w = - P(K,w) n dx
( c f . I n c e [l]) and assume
(4.74)
-
-P(K,w)
I
m
=
0
L(A)M(LI).
K
and
w
'n
with formal a d j o i n t
suppose
( 4.72)
Let
n
Thus l e t
d e f i n e d by
130
R. W . CARROLL
Theorem 4 . 1 4
Suppose ( 4 . 5 6 ) h o l d s and t h a t the b i l i n e a r concomitant is s e p a r a b l e
in t h e s e n s e e x p r e s s e d by ( 4 . 7 4 ) .
A s s u m e in a d d i t i o n t h a t f o r
tion
h a s no p o l e s f o r
Xn-lw(Xe-in'n,x)/M(Ae-in/n)
uniformly i n and
0
Proof:
5
arg
x > 0
and
X 5 2n/n.
From ( 4 . 7 2 )
-
p > 0
where
r
0
5
arg
x > 0
t h e func-
X 5 2 ~ / n while
is t h e c i r c u l a r a r c d e f i n e d by
Ihl = R
Then
( 4 . 7 3 ) one h a s
and u s i n g ( 4 . 7 4 ) with ( 4 . 5 6 ) t h e r e r e s u l t s
Consider now t h e r e g i o n
1x1
< R,
0 < a r g A < 2n/n
bounded by a c o n t o u r
C.
Then
under t h e hypotheses s t a t e d . t exp(Zni/n)
Writing
i n t h e l a s t i n t e g r a l and
s i d e of ( 4 . 7 9 ) becomes
But by ( 4 . 7 9 ) and ( 4 . 7 8 ) w e can then conclude t h a t ( 4 . 7 6 ) h o l d s . Examples 4 . 1 5 Then
Take
P 2 = ,:D
L(A) = 1 and
M(p) =
and ( 4 . 7 5 ) h o l d s , w h i l e H(X,x) =
u = 1, K(X,x) = C o s Ax, and
w(p,x) = exp(-ux).
u s o the integrand i n (4.75) is
i exp(ihx)
2A -7 Im(exp(-ihx)/iA)
2
= -
has no poles f o r C o s Ax.
0
5
QED
i exp(iAx)/(X
arg A
5
n.
2
Hence
+
2
p )
GENERALIZED TRANSLATIONS
Remark 4.16
The r e q u i r e m e n t t h a t t h e b i l i n e a r
131
concomitant b e s e p a r a b l e as i n d i -
c a t e d i s somewhat ad hoc and seems worthy of f u r t h e r i n v e s t i g a t i o n . p o i n t of view t h e i n t e r e s t of a formula such a s (4.76) i s t h a t f o r Pt(D)H = -AnH
with
Pn(D)K = -1°K
From our u(x) = 1
H(X,x)
satisfies
w h i l e t h e i n v e r s i o n (4.56)
holds.
T h i s f e a t u r e e s t a b l i s h e s c o n t a c t w i t h t h e formalism developed i n S e c t i o n
3.
2.5.
Generalized t r a n s l a t i o n s .
We b e g i n by d e s c r i b i n g some work of Hutson-Pym
[l; 21 which g e n e r a l i z e s t h e methods of L e v i t a n [ l ; 31 and Povzner [ l ] ( c f . a l s o E h r e n p r e i s [ 3 ] ) where g e n e r a l i z e d t r a n s l a t i o n o p e r a t o r s a r e o b t a i n e d by s o l v i n g equations 2 2 [-Dx + q ( x ) l u = I-D Y
(5.1)
on i n t e r v a l s
f(x), 0
+
q(y)]u
[O,n] f o r example o r
with i n i t i a l conditions
[0,m)
D u(x,O) = h f ( x ) , and s u i t a b l e boundary c o n d i t i o n s
Y
and
D u Y
+ Hu
=
0
at
x =
say.
TI
D u
Y
-
u(x,O) =
hu = 0
at
x =
Such problems can b e s o l v e d by e i g e n f u n c -
t i o n methods o r by t e c h n i q u e s of i n t e g r a l e q u a t i o n s (cf a l s o L i o n s [ 2 ; 3; 41) and t h e Hutson-Pym approach d e a l s w i t h more g e n e r a l and a b s t r a c t v e r s i o n s of t h e m a t t e r which r e d u c e t o and i n c l u d e t h e c l a s s i c a l s i t u a t i o n s as s p e c i a l c a s e s (some c l a s s i c a l f o r m u l a t i o n s are i n d i c a t e d l a t e r ) .
Let
h
and
v
b e two f u n c t i o n s i n s a y
m
Lloc (5.2)
and c o n s i d e r t h e i n t e g r a l e q u a t i o n U ( X , Y ) = V ( X , Y )+
7
j, j y
x+y-t h(s,t)u(s,t)dsdt
x-y+t for
( x , y ) EIR
and
( x + y , 0)
(5.3)
2
.
A(x,y)
If
is t h e t r i a n g l e with v e r t i c e s
t h e i n t e g r a l i n (5.2)
1 Hw(x,y) = -
hw(s, t ) d s d t
'A(x,Y) Formally a s o l u t i o n of ( 5 . 2 ) i s makes s e n s e .
Let
is over
(x,y),
and one w r i t e s
. m
u =
(I
-
p (f) = sup)f(x)) for
k
A(x,y)
(x-y, O),
H)-lv =
C Hnv w i t h
0 x E K, K
Ho = I
when t h i s m
compact, and g i v e
Lloc
R. W. CARROLL
132
t h e topology determined by such seminorms; f a r
En
Let
I f ( x ) Idx.
f u n c t i o n s i n IR
Cm
be
K
a D f
of uniform convergence of
and
2
a
0
h,w E LO3 (IR')
gence i n
O(a) = A(0,a) U A(o,-a)
and hence m
Lloc
converges uniformly on
CHnv
and t h u s
(I
-
H)-l
+
L
(-a,O),
( x , y ) E D(a)
(O,a),
implies
n a 2n /(2n)!
K
for
C
There r e s u l t s t h e conver-
K.
m
: Lm. loc
(a,O),
5 %(v)[%(h)]
pk(Hnv)
Some r o u t i n e c a l c u -
1
n
and
qk(f) =
w i t h t h e Schwartz topology
10.1 5 n.
and
loc
From ( 5 . 4 ) f o l l o w s
A(x,y) c O ( a ) .
o r IR2
be t h e s q u a r e with v e r t i c e s
O(a)
so t h a t
(0,-a)
O(a) = 0
let
u s e seminorms
on compact sets f o r
l a t i o n y i e l d s immediately f o r
Now f o r
L1 loc
i s w e l l d e f i n e d and c o n t i n u o u s
loc
since
S i m i l a r l y one p r o v e s e a s i l y t h a t f o r
(resp
H : E"
-f
E~+')
and
(I
-
h E
H)-l
:
LT~~
h 6 En)
(resp.
Eo
+
Eo
E"+'
(resp.
-f
m
H : Lloc
Eo
-+
E"+~) a r e
continuous.
The i n t e g r a l e q u a t i o n (5.2) w r i t t e n a s
u = v
+ Hu
is related t o a differential
e q u a t i o n similar t o ( 5 . 1 ) s i n c e f o r m a l l y
(5.5)
D 2u
-
2
D u
X
+
hu = D 2v
Y
-
D2V
X
Y
i s a s u i t a b l e combination
and if v
g(x
+
y)
+ k(x -
y)
with
h = -q(x)
+ q(y)
we have ( 5 . 1 ) . m
E4
Now t a k e l i n e a r
Ei
: LloefX)
+
Lloc(R)
(resp.
€"(a)
+
En(E)) , i = 1,2,
set
(5.6) mapping
Ef(x,y)
Elf(x
m
Lloc(7R)
+
-
+ 2
Lloc(IR )
y)
+ E2f(x (resp.
- y)
En(R)
+
En(IR2))
.
One d e f i n e s then a
and
GENERALIZED TRANSLATIONS
m
T : Lloc(lR)
generalized t r a n s l a t i o n operator
-f
LToc(?R2)
a s t h e unique s o l u t i o n of
+ HTf
Tf = Ef
(5.7)
f E D(T).
for
T = (I
Since
-
H)-lE
t h i s makes s e n s e .
n o t i o n of boundary o p e r a t o r s a s f o l l o w s .
D Hu(x,O) = 0 Y
(5.8)
2Elf
u
evidently
Hu(x,O) =
Bf(x) = D Tf(x,O) = D Ef(x,O) Y Y
IX
+
= Af
For c o n t i n u o u s
Tf E C L one w r i t e s
and f o r
Af(x) = Tf(x,O) = Ef(x,O);
so t h a t
Next one w a n t s a s u i t a b l e
Bf
and
-
2E2f = Af
fX
joBf.
One p r o v e s t h e n e a s i l y
JO
Theorem 5 . 1
Let
n
2
2
and
T :
+
with
A :
En
+
En
and
B : En
E'(~R')
f E E"
satisfying for
We r e c a l l now t h a t c o n v o l u t i o n of d i s t r i b u t i o n s i s d e f i n e d by t h e r u l e <S@T,
@(c+Q)>f o r
say
standard t r a n s l a t i o n .
E(iR2)
topologies.
(5.10)
for
E
E'
T*
and
T E
D'. 1
2
: E'(lR ) + E'(7R )
=
E'(R)
< f * g.$'
(5.12)
=
+ Q)
=
T'@(S)
5
is a
E(IR)+
i s c o n t i n u o u s i n t h e weak o r s t r o n g
T*(u@v)
and i s a map
E'
x
E'
=
+
E'
T;g(E)
continuous i n t h e s t r o n g topologies If
u
=
f
and
u = g
w e have
I
f(x)g(y)(T~)(x,y)dxdy.
Note t h a t i n a c a s e s u c h a s (3.75) w i t h
E T g(x)
@(€,
of Theorem 5 . 1 i s a l s o c o n t i n u o u s from
T
and i n t h e weak t o p o l o g i e s on bounded s e t s .
(5.11)
'Here
<S*T,@> =
T h e r e f o r e a g e n e r a l i z e d c o n v o l u t i o n i s n a t u r a l l y d e f i n e d by t h e r u l e
p*v
u,\)
S 6
The map
so i t s a d j o i n t
-t
Then t h e r e i s a u n i q u e c o n t i n u o u s g e n e r a l i z e d t r a n s l a t i o n
c o n t i n u o u s l i n e a r maps. operator
h E En-1(1R2)
TX r e a l and s e l f a d j o i n t s a t i s f y i n g Y
one can w r i t e f o r example
l ( T @ ) ( x , y ) g ( y ) d y = J(T:@(x)g(y)dy
=
IT;@(y)g(y)dy = j@(y)T;g(y)dy
134
R. W. CARROL
s o t h a t ( 5 . 1 1 ) becomes
= I$(Y)
T;g(y)f(x)dxdy
f
a l l o w i n g one t o i d e n t i f y
*g
with t h e integral
as i n (3.75)
Txg(y)f(x)dx i Y
(note here t h a t
I T l f g ( y ) f ( x ) d x = jT:g(x)f(x)dx
Now Hutson-Pym ( l o c . c i t . ) s t u d y t h e c o n v o l u t i o n of (5.10) t o s e e when i t i s commutive o r a s s o c i a t i v e f o r example and e s t a b l i s h a number of r e s u l t s some of which are i n d i c a t e d below.
First let
M(K)
K' =
=
(EO)'
be t h e Banach s p a c e of Radon measures w i t h s u p p o r t i n
W(K)
a s a set and t h e s t r o n g topology on
convex topology i n d u c i n g t h e norm topology on each
K'
is the finest locally T : Eo(R)
For
M(K).
c o n t i n u o u s a r i s i n g as i n Theorem 5 . 1 one can e a s i l y show t h a t f o r supports i n now
there is a constant
K
1
L (K) C M(K)
measure
dX
cK
1 1 LK = UL (K)
a s a set.
1
For
f E Lloc
f o r t h e a s s o c i a t e d ( p o s s i b l y unbounded) measure. one p r o v e s t h a t i f
then for
p E
p r o p e r t i e s of
Eo
+
Eo.
T : E0(N)
Eo n L1
K' El,
+
K'
.
2
Eo(R ) with Let
be t h e a b s o l u t e l y c o n t i n u o u s measures w i t h r e s p e c t t o Lebesgue
with
c o n t i n u o u s when
p , E~
11 p * ~ l l LcKll v I I 11
such t h a t
so t h a t
K
and
E2
v
+
topology and
L1
1 both
E LK
V*V
and
fh
or
fdh
Using p r o p e r t i e s i n d i c a t e d above
i s continuous with
E0(X )
has the
one writes
V*p
Eo
E
E
1' 2
:
Eo n L1
t h e topology of
belong t o
can b e a s s u r e d by s i m i l a r p r o p e r t i e s of
1 LK. A,
+
1 Lloc
The d e s i r e d B
for
A,B :
For commutativity and a s s o c i a t i v i t y one h a s f i r s t
P r o p o s i t i o n 5.2 Tf(x,y) = Tf(y,x)
E',
Convolution i n for a l l
(T@I)Tf = (I@T)Tf for a l l
f
E
in f
K ' , or
in
Eo.
or
E
Li
or
Fo.
Eo
i s commutative i f and o n l y i f
I t i s a s s o c i a t i v e i f and only i f
GENERALIZED TRANSLATIONS
Proof: Tf(x,y)
K'
so that
6x(f) = f(x)
For c o m m u t a t i v i t y c o n s i d e r
135
( f ) = G X @ 6 (Tf) = Y i s n e c e s s a r y i n E' or Y
6 * 6 x ( f ) = T f ( y , x ) s o symmetry of Tf Y 1 LK c a s e i s r o u t i n e . The c o n v e r s e i s obvious.
while
and t h e
one t h i n k s of
( T @ I ) ( g ) ( x , y , z ) = (Tg(.,z))(x,y)
*6
6x
For a s s o c i a t i v i t y
and l o o k s a t f o r example
( 6 x * 6 y ) * 6 z ( f ) = ( ( 6 x * 6 y ) @ 6 2 ) ( T f ) = 6 x @ 6 y @6 2 ( T @ I ) T f . More r e l e v a n t s i n c e it d e a l s w i t h t h e s t r u c t u r e of
Theorem 5 . 3 ing
T
Proof:
I f c o n v o l u t i o n is commutative t h e r e i s a f u n c t i o n
such t h a t
Suppose
vertices
(x,y),
h(x,y)
T
+ h(y,x)
=
i s d e f i n e d by (x+6, y+6),
h E LToc.
( x , y + 2 6 ) , and
t i o n of two v a r i a b l e s w r i t e
[g](x,y,b) = g(x,y)
I
hTf
y+6).
over
S(x,y,6).
[ E f l ( x , y , 6 ) = 0. S(x,y,6)
g(x,y) =
Since
Let
y = x
(Tf)- = Tf.
[Tf](x,y,6) =
I
hTf
over
-
:I,
hTf;
+
+
y)
(h
+ h)Tf
then
-
E2f(x
6))
+
=
g
g(x ,Y
a func-
+ 26)
[g](x,y,6)
y)
clearly
b e t h e r e f l e c t i o n of Thus i f c o n v o l u t i o n i s
[Ef](x,y,&) = 0
w e have
1 ~ - 6 , 6 ) = - - I-hTf =
-
S
Tf = (Tf)"
For
one h a s
A(x,y)
x
[Tf]-(x,y,6)
= -[Tf](y+G,
Hence i f
Then
g(x+6, y+6)
g(x,y) = g(y,x).
Now s i n c e
-
(x-6, y+6).
(= S ( y + 6 , x - 6 ,
and
determin-
be t h e square with
- X(A(x-6, y + 6 ) ) .
Ef(x,y) = Elf(x
s(x,y,G)
in the line
commutative
(5.14)
Thus i f
S(x,y,6)
Let
+ x(A(x,y+26))
g(x-6,
h E L" loc
0.
i ( ( A ( x , ~ ) )- x ( A ( x + 6 , y + 6 ) )
-
is t h e following
T
one o b t a i n s t h e e q u a t i o n
+
(h
1
-
2
h)Tf = 0
h(Tf)-.
s and hence
JS =
lh,Tf
0
a.e.
= jhZTf
Now
Tf(x,y)
s o modify
h
may b e z e r o b u t i f
hl
h2
i f n e c e s s a r y such t h a t
o n l y when
Tf = 0
h(x,y) = -h(y,x)
(h(x,x) = 0).
QED
Assume now c o n v o l u t i o n i s b o t h a s s o c i a t i v e and commutative w i t h Then f o r
f
such t h a t
(5.15)
(Il(x,y)
+
( T @ I ) T f E C2
h(y,z)
one h a s a . e .
+ h(z,x))(T@I)Tf(x.y,z)
in
=
R 0.
3
h E L;,,.
136
R. W. CARROLL
h E E'
To see t h i s l e t f i r s t
f E E2
and
and set
s o t h a t u s i n g P r o p o s i t i o n 5.2 one h a s immediately 2 (Dx
From Theorem 5 . 1 t h e r e r e s u l t s u(x,y,z) = u(y,z,x) 2 (Dz
and
(5.15).
-
D )u(x,y,z) X
For g e n e r a l
-
+ h(z,x)u(x,y,z) and
h
u(x,y,z) = u(y,z,x) = u(z,x,y).
2 D )u(x,y,z) Y
one h a s s i m i l a r l y
2
u(x,y,z) = (T@I)Tf(x,y,z)
(D:
+ h ( x , y ) u ( x , y , z ) = 0. S i n c e - D:)u(x,y,z) + h ( y , z ) u ( x , y , z ) = 0 by t h e same token.
= 0
as i n d i c a t e d (5.15) f o l l o w s by approximation.
f
T h i s r e s u l t can now b e f u r t h e r r e f i n e d as f o l l o w s . for all
(T@I)Tf(x,y,z) = 0 3
and
U = P IF
Take
I
i s open.
Uz
+ h(y,x)
h(x,y)
= 0
f o r almost a l l
( x , y ) E V.
uUz =IR
h(y,z)
2
a.e.
m
(z E
W)
F = {(x,y,z);
and
V x I C
T I ( X )=
Then
is closed
F
V C Uz
U.
1
Let
p E LK(lR)
I
h(x,s)dp(s).
(x,y,z) E V x I
b e compact.
From Theorem
h(x,y) = -h(z,x)
-
Hence
This leads t o
h E Lloc
Let
such t h a t
a . e . and hence f o r
-
Theorem 5.4
( x , y , z ) E Ul
= {(x,y);
z E I
Write
( T @ I ) T f E C2}.
~ ( 1 )= 1 and w r i t e
with
I
h(y,z) = h(x,z)
If
such t h a t
a compact i n t e r v a l w i t h
have s u p p o r t i n 5.3
f
Let
Adding g i v e s
w i t h c o n v o l u t i o n b o t h commutative and a s s o c i a t i v e .
then there is a function
m
TI E
Lloc
such t h a t
h(x,y) =
T I ( X )- ~ ( y ) a.e. In general
UUz
i s a t least dense i n IR2
i n a dense open s u b s e t of from Theorem 5 . 1 t h a t and
P2 w i t h
Af(x) = Tf(x,O)
61(f) = (Df)(O) = - 6 ' f .
t i o n i s commutative
TI
Then
and
h ( x , y ) = I T ( X )- ~ ( y ) l o c a l l y a . e .
depending on t h e nbh chosen.
R e c a l l now
and
6f = f ( 0 )
Bf(x) = D Tf(x,O). Y
A = ( I @ 6 ) T and
Let
B = (I@61)T.
I f convolu-
Tf(x,y) = Tf(y,x)
from P r o p o s i t i o n 5.2 s o
(D@I)Tf(O,O) =
( I @ D ) T f ( O , O ) which can be w r i t t e n as
6186(Tf) = (6@6,)(Tf)
(i.e. in
E'
AIAf = iSBf.
If
=
6*6
Theorem 5.5
),
1
I f c o n v o l u t i o n is commutative t h e n f o r
Tf E C1
c o n v o l u t i o n i s both commutative and a s s o c i a t i v e then f o r
Tf E C1
ABf = BAf.
137
GENERALIZED TRANSLATIONS
Proof:
For t h e f i r s t s t a t e m e n t one h a s
(5.16)
61Af = & , ( I 8 6)Tf = 618 6 ( T f )
=
6 8 6 1 ( T f ) = 6 ( I 8 J l ) T f = 6Bf.
For t h e second a s s e r t i o n one h a s t h e n f o r
(5.17)
x E IR
ABf(x) = A ( ( 1 8 D ) T f ( * , O ) ) (x) = T ( ( I 8 D)Tf(. , O ) ) (x,O) (T 8 D)Tf (x ,O ,0) = ( I 8 I @ D) (T 8 I) Tf (x,O,O)
=
BAf(x) = B ( T f ( * , O ) ) ( x ) = ( 1 8 D ) T ( T f ( . , O ) ) ( x , O )
(5.18)
=
( ( 1 8 D ) T @ I ) T f ( x , O , O ) = ( I @ D B I ) ( T @I)Tf(x,O,O)
=
(18D81)(18T)Tf(x,O,O) = (I8 (D@I)T)Tf(x,O,O)
= (I
8 ( I 8 D)T)Tf (x,O,O)
= (I 8 I
8 D) ( I 8 T)Tf (x,O,O)
= (1818D)(T8I)Tf(x,O,O)
Now f o r
-
h(x,y) = n(x)
n(y)
we s e t
2 Lf = D f
For convenience i n f o r m u l a t i o n assume h e r e
.
For any c o n t i n u o u s
S : E(R)
c e n t r a l i z e r f o r a convolution
while
also
*
if
+
h t E
E(R)
S'(p*v)
so t h a t
the adjoint
TI
E E
S'
for
= (S'p)*v
and
T : E(R) *
is c a l l e d a
p , E~ E ' .
Thus
f E E
for
S'
and (5.9) becomes
(L8I)u = (I8L)u.
(5.19)
E(R2)
+ nf
( S ' p ) * v ( f ) = S ' p @ v ( T f ) = ( ( S ' @ I ) ( u @ v ) ( T f ) = p @ v ( ( S @ I ) T f ) . Consequently
is a c e n t r a l i z e r i f and o n l y i f TS
+
Theorem 5 . 6
( S 8 I ) T f = TS;
*
if
i s commutative t h e n
( I @S)T. Let
S :
E
-f
E
commute w i t h
L,
A , and
B.
Then
S'
is a c e n t r a l -
R. W. CARROLL
138
izer. If S ' L'
i s a centralizer then
is a centralizer if and only if L commutes with A commutes with L we have for f
Proof: If S (5.20)
(L@I)(S@I)Tf (S @
where
=
and B.
E =
(LS@I)Tf = (S@I)(L@I)Tf
(I@L)Tf
(S8I)Tf
commutes with B
since Tf
(S@I)Tf
is a s o l u t i o n of (5.19).
satisfies (S$I)Tf(x,O)
=
SAf(x)
=
ASf(x)
If S =
commutes
TSf(x,O).
If S
one has
=
Thus
E
In particular
I) (I@ L)Tf = ( S @ L)Tf = (I @ L) (S @ 1)Tf
(L@I)Tf
with A
=
and B.
commutes with A
S
and TSf
SBf(x) = BSf(x) = D TSf(x,O) Y
satisfy (5.9) and by uniqueness are equal. The second
assertion follows by rearranging the above calculations.
QED
Next one examines sufficient conditions for commutativity and associativity and we will simply state some results of Hutson-Pym [ 2 ] without proof. Theorem 5.7
Suppose h(x,y)
Ef(x,y) = Ef(y,x)
+ h(y,x)
a.e. and Ef(-x,y)
= 0 =
a.e.,
Ef(x,y)
h(-x,y) = h(x,y) a.e.
a.e.,
Then convolution is
commutative. Let now h E E , Theorem 5.8
h(x,y) = T(X)
Let n
-
n(y)
be analytic with
analytic functions. Let
E, and L
2
+ n.
as
above with n
A
and B mapping analytic functions to
L commute with
A and
E
B with
= D
6B = 61A. Then con-
volution i s commutative. Theorem 5.9
Let n
E
E
and suppose L, A, B
all commute with convolution
commutative. Then convolution is associative. The smoothness requirements on
TT
can be relaxed but we omit this. A simple
139
GENERALIZED TRANSLATIONS
c o n c r e t e example of some of t h i s i s
Example 5.10 A
and
B
Let
f(-x)]
and
c o m u t e s with
L
and
A
-
Bf(x) = B [ f ( x )
when
B
IT
f(-x)].
Then
(Af)'(O) = 0 = Bf(O),
ndap a n a l y t i c f u n c t i o n s t o a n a l y t i c f u n c t i o n s ,
AB = BA = 0.
and
+
Af(x) = ct[f(x)
i s even.
Hence t h e
c o r r e s p o n d i n g c o n v o l u t i o n w i l l b e commutative and a s s o c i a t i v e .
Some a s p e c t s of t h e above t h e o r y are g i v e n a more a b s t r a c t form i n Hutson-Pym [ I ] and w e w i l l r e p o r t on t h i s h e r e .
L
with
and
One c o n s i d e r s t r a n s l a t i o n s
is o f t e n a compact o p e r a t o r h e r e ; however v e r y o f t e n n-th o r d e r
L
differential operators
have compact r e s o l v a n t s ( c f . Goldberg [ l ] ) and w i t h o u t
Dn
l o s s of g e n e r a l i t y one can assume
i s compact.
L = D-l
w i l l t h e n be i n d i c a t e d below i n Example 5 . 1 7 .
and
E'
B : E'
E".
bidual
-+
E ' , and
C : E"
+
E" E'
w i l l be w r i t t e n as
B
where
A
E"'
and
A*
C
as
+
A*
and
L
w i l l be a c o n t i n u o u s l i n e a r map
L
while
L* : E '
-f
Let
The c o r r e s p o n d i n g r e s u l t s be a normed s p a c e w i t h d u a l
E
A : E
One w i l l have t h r e e c o n t i n u o u s l i n e a r maps
t i o n of t h e c a n o n i c a l map
times
associated
T
E'
E
restricted t o
= C
+
E"
: E"'
+
E,
Thus some-
when no c o n f u s i o n w i l l a r i s e .
A
whose e x t e n s i o n t o
is again called
E"
according t o t h e conventions indicated.
B(E',E')
Let
d e n o t e c o n t i n u o u s b i l i n e a r forms w i t h i t s s t a n d a r d norm and one w r i t e s f o r t h e c l o s u r e of
8
means
@Ti).
For
E'@E'
in
B(E',E')'
A , B E L(E';E')
so t h a t
define
+
E'
with
+
(E' 6 E ' ) ' = B ( E ' , E ' ) .
T** : E"
admits
L)
(IBL**)T
if
or
and s i n c e
E' 6 E '
C
: B(E',E')
A*@B*
T : E
( L 8 I ) T = TL = ( I 8 L ) T . (L**@I)T** = T**L**
=
same and t h e meaning w i l l b e c l e a r from c o n t e x t . a E E',
A E L ( E ; E " ) , such t h a t
T
is
T h i s can mean
(IBL**)T**
T f ( a , * ) = Af
for
+
+
L
6E' (thus
B(E',E')
B(E',E')
we consider
B(E',E')'
One s a y s
E'
(E' @ E ' ) ' = B(E',E')
A t r a n s l a t i o n o p e r a t o r w i l l be a norm c o n t i n u o u s map
T* : B ( E ' , E ' ) '
E",
is t h e composi-
B
C = B*.
E ' , and
+
by
so t h a t
T* : E ' 6 E '
admissable (or
+
E'
T
( L * * @ I ) T = T**L =
but these are essentially the A B-condition
f E E.
Thus
is a pair T
(a,A),
satisfies
140
R. W . CARROLL
i f t h i s is t r u e or equivalently
(a,A)
Tf(a,.)
= Af
b e assumed compact u n l e s s o t h e r w i s e s t a t e d s o t h a t t e n s i o n of
i s compact.
t h e r e is a p r o j e c t i o n
Uo(L*)
r
where
r;
E"
to
L
X
surrounds
R(<,L*) = (L*
-
P?
-
for sufficiently large
vX
and
A : E'
+
t h e e x t e n s i o n of a projection
write (L
-
PA
XI)
Now i f
vx
Q
for
: E"
x
X
vx x
= 01
dim V h
E
u(L*)
E"
or in
c o n d i t i o n s which d e t e r m i n e
f E E"
(5.22)
and
T
P*.
and
Tx
and i f
Now t h e spectrum of
A
X
and f o r
E Oo(L*)
Q, = (Pf)*. U
x
=
there is
One w i l l
PXE" = { f E
El';
a
E
E'
Tf = T f ( a , * )
T
T
(a,,T)
and t h e
Thus g e n e r i c a l l y t h e r e w i l l b e
T.
and one wants f u r t h e r
L
(B)
u n i q u e l y ; f o r second o r d e r d i f f e r e n t i a l o p e r a t o r s
satisfies
(a,A)
F i r s t w e c o l l e c t some e l e m e n t a r y then
ALf(x) = T ( L f ) ( a , x ) = ( I 8 L ) T f f a . x )
xT
P*P* = 0 A P
then
s a t i s f i e s t h e B-condition
x E E'
=
Thus
f l~
and set
t h e s e t o o k t h e form of boundary c o n d i t i o n s .
for
i n s i d e o r on
Taking a d j o i n t s t h e same r e q u i r e m e n t s hold f o r
a s s o c i a t e d w i t h a given
f a c t s and o b s e r v e t h a t i f
A E
and f o r
v A as above).
c o l l e c t i o n of such B-conditions d e t e r m i n e s
L
I?
If
m.
then i t commutes w i t h
The d e f i n i t i o n s a y s t h a t
T
{O)
then
i s t h e same as
(f E E).
many t r a n s l a t i o n s
-
Uo(L*) U { O }
i s an L-admissable t r a n s l a t i o n o p e r a t o r set f o r
T
f E El'.
U(L*)
E" d e f i n e d a s above such t h a t
(same
Taf = T f ( . , a )
and
+
A
restricted to
Qx
f = 01
E"
to
L
XI)
L*
commutes w i t h
E'
be
w i l l now
d e f i n e d by
V = P*E'
If
(L*
{X
u0(L*)
L
i s compact, and t h e ex-
L*
w i t h no o t h e r p o i n t s of
E E';
VX =
Let
f E E".
for
Tf(a,L*x) = Af(L*x) = LAf(x).
commute wich
L.
Next one h a s
A
commutes w i t h
L.
Indeed
141
GENERALIZED TRANSLATIONS
P r o p o s i t i o n 5.11
A E ao(L*), TUX C Proof:
For
X
For
UA 8 U
f E El',
I n a similar manner
A'
x , y E E l , one h a s
TI? = ( P 8 I ) T
A
of
B(E',E')
For
f ,g
f
f 8 g(x,y) = f(x)g(y)
p r o p o s i t i o n w i l l f o l l o w by s h o w i n g
and i f
UX
Hence t h e r e are s c a l a r s
g l ( y ) , . . . , g ,(y)
'k)
such t h a t
gk E E"
fi(xk)
and l e t
for
U1,U2
E"
C
g t U2}.
h
E
f 8 g E B(E' , E l )
The l a s t a s s e r t i o n of t h e
such t h a t
so f o r each
i
k
Let
fl,
k
xk E VA
there are Then
...,f
be
y E E'
Then f o r
h ( * , y ) = Cgk(y)fk.
f k ( x k ) = 1.
and
is
i s t h e subspace
U1@U2
PA 8 P X B ( E ' , E ' ) .
UX
TPX =
The
fk
(P2xk =
gk = h ( x k , * )
so
and
=
Thus
= 0
PX o n e o b t a i n s
PA 8 P X B ( E ' , E ' ) C UA 8 UA.
a v e c t o r space b a s i s of
are l i n e a r l y i n d e p e n d e n t and i n
=
now an e l e m e n t
E"
{ f 8 g ; f E U1,
s p a n n e d by
2
PX
and s i n c e
X
(PA 8 I) (I 8 PA)T = ( P A8 P A ) T . d e t e r m i n e d by
TPA = ( 1 8 P X ) T = ( P A 8 1 ) T = ( P A 8 P X ) T . F o r
E Uo(L*),
gk E UA
One d e n o t e s b y
h(xk;)
=
and s i n c e XUX
gk.
h = Z fi. @ gi
the r e s u l t follows.
t h e s m a l l e s t v e c t o r subspace of
E"
Then by s i m p l e c a l c u l a t i o n a s a b o v e o n e c a n show t h a t i f a d m i s s i b l e t r a n s l a t i o n s and e i t h e r
(a) f o r each
E VX
implies
Tlf(x,y)
=
T2f(x,y)
or
E VX
implies
Tlf(x,y)
=
T2f(x,y)
then
is w e a k l y d e n s e ( i . e . weak* d e n s e ) i n
X
(b) f o r each
E"
T 1f
2
T2f
E
QED
containing every and
T1
o0(L*)
f E UX
A E oo(L*) for
f
E
T2
CUA.
f E E
a r e two Land
and
Thus i f
then under t h e s e hypotheses
x,y x,y XUX
T1 = T2 .
R. W. CARROLL
142
Let now Comm L* be the set of operators on E' which commute with L* and 2 2 Comm L* the set of operators commuting with Comm L*. Comm L* i s a commutative 2
ring with
Comm L*
Corn L*
C
and
P* E Comm L* h
X
for
E Uo(L*).
The following
basically algebraic facts are easily proved and we omit details. First the norm closure V
of
note that if says that
2
CVX is a module over Comm L* with each VX a submodule. Simply
x E VX
and
{xi; i E I}
C
V
A E CommLL*
then Ax = APtx = P?Ax E VX.
is a set of generators for the module V 2
{Axi; A E Comm L*; i E 11 is V.
closed linear span of
Then clearly if
{xi)
Conversely if a set ates V. 2
Comm L*
{xi}
generates V is such that
{P*x A i1
The index y
generates VX generates Vx
{PTx,}
if the of V
yx be the index of VX
the minimal cardinal of a set of generators. Let m).
Now one
is
(yX <
YX 5 Y.
so
then
{xi}
gener-
In fact one has y = sup y A with a little argument. Finally if A and
then A*f E CUX and
f E CUA
this note that A*
commutes with
L, T* commutes with
=
A*Txf (y)
L*
=
(A*@I)Tf x E E'.
PA and let
and A; hence Tx
(I@A*)Tf
=
Since Tx
commutes with
A*.
= TA*f.
To see
commutes with For y E E '
TxA*f (y) = TA*f (y,X).
The following theorems now require some proving and we will indicate the idea without spelling out all the details. Theorem 5.12 Suppose XUA tion T
is weakly dense in E".
can be determined by
Proof: Let
y <
ing B-conditions
m
y B-conditions.
and suppose T1 (ai,Ai),
and
2
that
(by previous remarks).
Y x = CBiai. Then 1
T2
1< 1< y, where
To show this one need only show that if T f(x,y)
Then an L-admissible transla-
are L-admissible translations satisfy{ ai 1
f E Ux
But for x
E VX
generates V.
and
x,y
E
Vx
Then T1 = T2.
then Tlf(x,y) 2
there are 3. E Corn L*
=
such
143
GENERALIZED TRANSLATIONS
Similarly y
C.
1C.a
=
J j'
Since T1
E
2
Comm L*, and one has
J
satisfies the B-conditions
(ai,Ai)
Tlf(x,y)
=
CCT CtB*f(ai,aj). 1 3 i
we have
v v
and the right side of (5.26) is independent of Theorem 5.13 Let 1 5i
5
s where
translation T
1
be an L-admissible translation satisfying does not generate V.
{ai]
W
over
2
Comm L
then there is an
(i A
E
s)
. . . ,PTas}
tion T
X
be such that for some does not.
E
satisfying the B-conditions
+
w
m
the obvious fact that if (ai,Bi) then T1
Now (cf (5.10))
(5.27)
6
E'
T
cannot be
2
Corn L
(al,A), (a2,0),
{x,
such that
,...,xs3 Axl 9 0
Secondly let generates V X
IP!ai} A f 0
First one shows
+ T2
T1, T2
...,(as,O).
is L-admissible and satisfies
so
if T
x - y = T*(x
8 y).
+
E'
It follows that
are L-admissible satisfying
one defines a multiplication on
T
E'
{ai}
{xi}
not a
but C
E'
but
and an L-admissible translaThis requires
some work but there are basically no new constructions in the proof.
m : E'
for
a linear map,
.
oo(L*)
Then there is an
w
l,.. , s ) , and
=
0 for 2 < i< s. The proof is routine algebra.
(1 5 i 5
QED
f.
Then there is an L-admissible
is a finite dimensional vector space, L~ :
set of generators of
{P!a2,
(ai,Ai)
$: T satisfying these B-conditions. Thus if y <
a set of generators for W
=
T = T2 acting on 1
This is proved using some lemmas which we summarize here.
that if W
Axi
so
y - 1 B-conditions.
determined by Proof:
T
TI
(ai,Ai
+
Finally note
(ai,Ai) Bi).
and QED
as a continuous linear map
is a translation operator we define
is L-admissible if and only if for x,y
E
E'
R. W. CARROLL
144
Indeed f o r
f E E"
one h a s
f(L*(x
y ) ) = Lf(x
y) = TLf(x,y)
f ((L*x) * y) = Tf (L*x,y) = (L 8 I ) T f ( x , y )
Thus
L*(x * y) = (L*x)
-f
X *
x * y (Ay).
commute w i t h
generates
Let
i f and o n l y i f
Aka
Proof:
T'
Define
L-admissible and T
= A*a j i
i j
by
CUx
f o r each
i s commutative i f and o n l y i f
T'f(x,y)
f E Ux
for
Tf(a ,a.) = T'f(ai,a.) i J J
f E E"
for
x
=
A ( x * y) =
A 1 , '1 i' i
El'. Then T
i
and
is.
5 s , where i s commutative
for
f E E, while
x,y E E ' .
Then
T'
f ( x * y) = T f ( x , y ) .
and t h i s h o l d s i f and o n l y i f
T = T'
x,y E E'
and
+
(i,j).
Tf(y,x) = T'f(x,y)
=
(a
is weakly dense i n
T'f(x,y) = Tf(y,x)
f(y.x)
(Ax) * y
A E Comm L*
.y
x
i s commutative e t c i f t h e product ( 5 . 2 7 )
T
Suppose
V.
T h i s shows t h a t
b e L-admissible and s a t i s f y
T
which i s t h e a d m i s s i b i l -
2
and hence f o r
L*
.
TL = ( L C4 I ) T
i f and only i f
One s a y s now t h a t
P r o p o s i t i o n 5.14 {a 1 i
y
The o t h e r e q u a t i o n i s similar.
i t y condition. y
-
while
by p r e v i o u s remarks.
Thus Tf(x,y) =
I t i s enough t o have
and h e r e
f(A*a.) = A$*f(a.) = T f ( a . , a . ) = T ' f ( a j , a i ) ; i J J 1 J
(5.29)
The r e s u l t €ollows.
QED
One shows e a s i l y t h a t T @ I : B(E',E')
f ( x * y,z)
-f
T
i s a s s o c i a t i v e i f and o n l y i f
i s d e f i n e d by
B(E',E',E')
(similarly for
Theorem 5.15
S1,Sz
: E
Szf(x,y) = (I@T)Tf(x,y,z)
for
admissible translation.
where
( T % I ) f ( x , y , z ) = f(T*(x,y),z) =
Under t h e h y p o t h e s e s of P r o p o s i t i o n 5.14
First define
(T63I)T = ( I 8 T ) T
( I @ T ) ) . Then one h a s
a s s o c i a t i v e i f and o n l y i € both
Proof:
is
A3a
=
1 j
+
B(E'
z E
A;ai
, E l f
E'
and
by
fixed.
This is c l e a r f o r
S1
T
i s commutative and
A*An i j = A*A* j i
for all
i,j.
Slf(x,y) = (T@I)Tf(x,y,z) Then
Si
while f o r
( i = 1,Z) S2
and
i s an L-
one can w r i t e
145
GENERALIZED TRANSLATIONS
(5.30)
( L 8 1 ) S 2 f ( x , y ) = S2f(L*x,y) = ( I @ T ) T f ( L * x , y , z ) =
Lf(x
Similarly f(A;y) a
i
z ) ) = (I@T)TLf(x,y,z) = S2Lf(x,y).
( I @ L ) S 2 = S2L.
so t h a t
. (aj
(y
a.
y) = A+A+y 1 J
.
Next o b s e r v e t h a t
y = AZy. while
If
( a i * y)
is commutative we have t h e n
T
-
f ( a i * y) = T f ( a i , y )
a
j
= a.
J
-
(ai
*
y) = A*A*y.
= Aif(y)
a i * (y * a j ) =
Now g i v e n
j i
=
T
commutative t h e f o l l o w i n g c h a i n of equ'ivalences shows t h a t a n e c e s s a r y and s u f f i cient condition for a s s o c i a t i v i t y is t h a t f , X , Y ,z
(5.31)
(T8I)T = (I@T)T
(I@T)Tf(x,y,z)
-
(I@T)Tf(ai,y,a.) J
A?A+y = A?A?y J 1 1 J Theorem 5.16 of
-
-
A?A+ = A4Ax. 1 J 3 1
Thus f o r a p p r o p r i a t e
(T@I)Tf(x,y,z)=
(T@I)Tf(ai,y,z) =
(=>
Tf(ai* y,a.) J
=
QED
A+AA = A?A?. J 1
111
Every c o n t i n u o u s nonzero homomorphism
@
of
E'
is an e i g e n v e c t o r
L.
Proof:
We can f i n d
Therefore
L@ =
y E E'
!&kd @. @(Y)
such t h a t
@(y)
0.
Then f o r
x E E'
146
R. W. CARROLL
We note also that if @ $(x)@(y)
= ($@@)(x,y)
is an eigenvector as above then T@(x,y)
T$ = $ 8 4 .
so
admissible translation, E' can embed
If T
@(x
=
=
is a commutative associative L-
is semisimple, and
XUX
is weakly dense in E"
into a function algebra on the ideal space I of E'
E'
y)
*
one
as i n the
classical Gelfand theory and some results in this direction are proved in HutsonPYm
[ll. To relate the preceeding to differential problems let A
Example 5.17
and suppose A
tor in E with domain D
has a compact resolvant for some
which without loss of generality can be taken to be A : E
+
E
A if
commutes with
T
is A-admissible if for
D
with
with A
A(Tf(x,-))
=
AD C D
fE D
(TAf)(x,.)
if and only if
A
be an opera-
and
0. Set L = A-l
=
for f E D .
AAf = AAf
and
x E E',
and
A(Tf(*,x))
commutes with
X
Tf(x,*)
and
L and for E
A translation
Tf(-,x)
= (TAf)(*,x).
and say
belong to
Then A
reflexive T
commutes
is AV
admissible if and only if it is L-admissible. Recall that UX = If; ( L - X I ) for large enough vx that for X
9
0 UX
=
and set Wx = {f; WA-l.
(A
-
V
AI)
xf =
01.
xf = O }
Then it is easy to show
The preceeding results applied to this situation
yield Theorem 5.18 Let
E
be a reflexive Banach space and
compact resolvant. Suppose XWu associated with (a,A)
(a E E'
is dense in E.
Then a translation operator T
A cannot be determined by fewer than and
A commuting with A).
which do determine T and
T
A an operator on E with
y = max dim
There are sets of y
W
u
B-conditions
B-conditions
is then both commutative and associative if and only
while A A = A A if A*a = A*a i j j i i j ji'
The continuous nonzero homomorphisms of
E'
are eigenvectors of A . There is a great deal of material available on the normed rings associated with generalized translation operators and second order differential equations (see e.g. Leblanc [l], Levitan [l; 31. Marc'enko [ 1 1 , Naimark [ 2 1 , Povzner [l] and references there).
Related spectral considerations arise in the inverse Sturm-
Liouville theory developed by Gelfand-Levitan [ 4 1 and in inverse scattering theory
147
GENERALIZED TRANSLATIONS
[ l ] , Faddeev
( c f . Agranovi?-Mar&ko Marzenko [ 2 ; 31).
(5.33)
(-m
2 Li(Dx)u = D u
L1(Dx)u = L2(D ) u ; Y
< x <
y
m;
0).
a l l y w i t h (3.1) f o r
-
q (x)u, i
i = 1,2,
One compares h e r e w i t h ( 5 . 1 ) f o r
P(Dx)
L1(Dx)
=
and
Q(D ) = L2(D ). Y
Y
$I,
t h i s p o i n t of view f o r s u i t a b l e c h o i c e s of i n a t r i a n g u l a r region
R(x,y,xo,yo)
and c o n s i d e r
u = u(x,y);
L e v i t a n [ l ; 31 ( c f a l s o Lions [ 2 ; 3 ; 41, Marzenko
tion
[ 2 ] , Levin [l] ,
We w i l l r e c a l l some of t h e b a s i c c o n s t r u c t i o n s h e r e f o l l o w i n g
Let
Marzenko [ 3 ] .
Lang [l], L a x - P h i l l i p s
111,
111,
$.
A(xo,yo)
L
= L2
and more gener-
The o r i g i n a l work of
Povzner [ l ] ) s t a r t e d from
One works w i t h a Riemann funca s before t o obtain a
formula
(5.34)
L e t now
Then
eo(X,y)
ixx
u(x,y) = e
-4
(I
I
eo(X,y)
+K
Yo)
+ $(xo -
Yo)]
2 L(Dy)eo = -1 e
[DxR(x,O,O,yo)
The o p e r a t o r
+
satisfies
2 (L(D ) = D Y Y
Y
K(yo,x) =
[UXo
satisfy
D u(x,O) = eixx)'
(5.36)
1
U(XO.YO) =
w i t h ' e (h,O) = 1 and
2
D u = L(D ) u Y
- q (y)).
with
u(x,O) = eixx
Put t h i s i n ( 5 . 3 4 ) w i t h
+ DyR(x,O,O,yo)]
t o obtain
e'(X,O)
( @=
x
= 0
=
f(x)
+
L(Dx)eo = -A
eihx
2
e
and set
4')
K(x,t)f(t)dt
w i l l b e c a l l e d a t r a n s m u t a t i o n o p e r a t o r i n t h e s p i r i t of S e c t i o n 2.3. of
D2eixx
=
-A2eixx
into the solution
eo(X,x)
s a t i s f y i n g t h e same i n i t i a l c o n d i t i o n s .
Applying t h i s f a c t t o
e fixx
and s e l e c t e d i n i t i a l c o n d i t i o n s we o b t a i n
I t maps
of
ih.
and
d e f i n e d now by
+ K)f(x)
the solution
=
R. W. CARROLL
148
Theorem 5.19 w(A,O,h)
1, w'(A,O,h)
=
(5.37)
Let w(X,x,h)
K(x,t,m) h
=
=
(resp. w(A,x,-))
satisfy L(Dx)w
(resp. w(A,O,e>) = 0, w'(A,O,m)
h
K(x,t) = K(x,-t);
:1
+ K(x,t) + K(x,-t) + h
K(x,t,h) [K(x,S)
-
= =
-A 2w with
1) and set
=
K(x.-S)]dS.
Then one has the formula (5.38)
w(X,x,h)
COS Ax
+
:1
K(x,t,h)Cos
ht dt
The corresponding operators (cE. ( 5 . 3 6 ) ) are denoted by as Volterra operators they will have inverses I
+ Lh
+ Kh and I + K, and I + La where for I
and
example
If one sets now K(t,S)
151 >
0 for
=
it can be proved from the formula
It1
(variation of parameters)
and ( 5 . 3 5 ) that K(x,t)
(5.42)
K(x,t)
= -
satisfies t
4(x + t)
+
q(u)du
/iq(u)
2 o Theorem 5 . 2 0
+ (x- u)
1
K(u,S)dSdu.
t - (x - u )
The integral equation ( 5 . 4 2 ) has the unique solution K(x,t)
is continuous. If g E Cn
then K E Cn+l
which
in both variables and satisfies
K(x,-x) = 0 with (5.43)
L(Dx)K
=
D:K;
K(x,x)
=
2
J:
q(E;)dE.
Similar theorems apply to the other transmutation kernels. For example L(x,t,h) satisfies
GENERALIZED TRANSLATIONS
2 D L = L(Dt)L;
(5.44)
- -l
L ( x , x , h ) = -h
149
X q(t)d+ 0
with
hL(x,O,h) = DtL(x,O,h).
Consider n e x t
f o r example and d e n o t e them by
w(A,x),
w(A,x,h),
K ( x , t ) , and
K(x,t,h),
L(x,t)
and
L(x,t,h)
for simplicity.
m
Write
C(A,f) =
f ( x ) C o s Axdx
f o r t h e c o s i n e t r a n s f o r m and
0
Ji
(5.45)
w(1.f) =
(5.46)
?(x) = f ( x )
+
= g(x)
+
Let
K
2
f(x)w(A,x)dx
2 K (a))
(resp.
1,
f(c)K(c,x)dc;
c
g(S)L(S,x)dS.
b e t h e s p a c e of
support (resp. with support i n of
K2
L1
Ig(A)I
A
for
The s p a c e
C
5
c exp alImAI.
Let
Z(0)
in
CK2 = UCK
2
(a); then
by d u a l i t y f o r
2'
Z C CK
f E Z
2
,
C(1.f)
of c o s i n e t r a n s f o r m s
be t h e even e n t i r e f u n c t i o n s
g E
Let
f o r real
with the standard
b e t h e d u a l of
Z'
Lh
g E L
2 = ~Z(CI)
T E 2'
Now from (5.38) and t h e analogue of ( 5 . 4 0 ) f o r
w i t h compact
and
with
Let
and
CK2(0)
[0,m)
A
g(A)
r e a l s a t i s f y i n g t h e same estimate.
topology and define
[O,o]).
c o n s i s t s of even e n t i r e f u n c t i o n s
such t h a t
f u n c t i o n s on
L2
Z
and
as
p l u s t h e d e f i n i t i o n of
and ( 5 . 4 5 ) t h e r e f o l l o w s from ( 5 . 4 6 )
One can prove now
Theorem 5 . 2 1
for
f , g E L2
t h e formula
There e x i s t s
R E Z'
of compact s u p p o r t ; R =
2
(1 + C(L))
where
such t h a t
R
i s connected w i t h t h e k e r n e l
L(x,t)
by
C(L) = C ( A , L ( x , O ) ) ,
T h i s i s a form of P a r s e v a l theorem where
R
is called t h e generalized spectral
R. W. CARROLL
150
function. The proof is based on constructing a sequence R ( A ) m,
jjn(h)w(h,x)w(A,y)dh
-t
-
6(x
y)
such that as n
(cf. Sections 2 . 3 and 2 . 4 ) and we omit
As a consequence of Theorem 5.21 one can prove easily that
details here.
In the case when q
and
h
are real (called the symmetric case)
one
has (cf
Section 2 . 4 ) Theorem 5 . 2 2
In the symmetric case there exists a nondecreasing function p(p>
such that for f E L '
where the Parseval formula
also holds. Remark 5 . 2 3
Let u s underline the fact that we are not dealing with generalized
translations here, These would arise if in ( 5 . 3 3 ) one took L1 = L2 and say
I$ = f with u(x,y) (5.51)
J, = 0
(growth conditions on
q
are also imposed).
can be obtained then via a Riemann function again and written as u(x,y)
=
Tzf =
[f(x
+ y) +
f(x
-
y)]
y)dt
(cf Levitan [l; 3 1 , Povzner [l ) .
The generalized convolution
can then be expressed in the form (5.53)
The solution
f . g = -2f * g +
More generally
fog = h
J,
~~(t,x,y)f(x)a(y)dxdy.
is that function which satisfies
+
151
GENERALIZED TRANSLATIONS
for an arbitrary Cm
commutativity and associativity
11
identity with norm under
@. Symmetry
even function
+ f (1
Xe
=
-
properties of R
+ 11 f 11
IX I
then L
+
an
is a commutative normed ring
In the spirit of Remark 5.23 and the spectral considerations preThis work illustrates and
ceeding this let u s cite some results of Leblanc Ill. refines some of the early Margenko-Povzner ideas. and consider Lu
+ X 2u
0 with
=
is written as w(X2,x) = aa(X corresponding to
2
u(0) = a
,x)
+ b6(X
a = 1, b = 0 (resp.
notation a(XL,x) = w(X,x,O)
while
2
and
Let Lu u'(0)
a
=
R(XL,x)
+ x2) I q(x)
@,(x)
the solutions of
Idx <
Lu
@;(x)
A = 0 and refers to Margenko [l] for the case a
9
a
9
0, A
q(x)u
A general solution
6)
is the solution
and showed that when
when
(a,A)
and
-
Povzner [l] studied
,x)
the associated Banach algebras into four classes
+0
2
D u
0 can be written as
=
is bounded and tends to zero with
the cases a = 0, A
=
Thus i n the preceding
w(X,x,m). 2
L(Dx)u
(resp
0, b = 1). =
=
= b.
a
,x) where
the associated Banach algebras with characters w(X
where
L1
=
0.
Remark 5.24
li(1
If L
more or less immediate.
lead to
x
Leblanc puts
+ m,
according as
0, A
a
=
0 or
0. Leblanc studies
=
+ 0, leaving the case
a = 0, A = 0
open since it seems less interesting. First consider a = 0, A
0 and make the
hypotheses (weaker than Povzner [l]) m
(5.55)
xlq(x) Idx < '0
m;
lim
ao
(0,x)
It is mentioned that the condition on above but it gives a better estimate at of the results without giving proofs.
Proposition 5.25
+
0.
X "
q
is less natural than Povzner's condition m
which is useful.
First (cf (5.39)
Given the hypotheses above
6(X2,x)
so
We will state some
B(t.x)
%
K(x,t,m))
may be written
152
R. W.
B(A 2 ,x)
(5.56)
_ Sin _ Ax
=
A
CARROLL
+ jxB(x,y)
dy
0
where B(x,y) i s d e f i n e d f o r 0 5 y 5 x (= 0 f o r y > x) and s a t i s f i e s rx rm B(x,y) Idy 5 K w i t h x l B ( x , y ) Idx 2 2Ky. The map U d e f i n e d by JY (5.57) sends
Set
Uf(x) = f ( x ) L
1
1
= L (xdx)
2 L2 = L (dx)
+
:1
B(t,x)f(t)dt
i n t o i t s e l f with
11 U(f)II 5
(K
+
1)
11
fll
.
now and r e c a l l a s t a n d a r d theorem due t o Weyl ( c f Titchmarsh
[21). There are a f i n i t e number of r e a l n e g a t i v e v a l u e s of
Theorem 5.26
i s bounded.
x-lB(A2,x)
Denote t h e s e by
2
An
and set f o r
A2
such t h a t
f E L2
'0
M(X)
Then i n
L2
=
1
+
loe-'"rv(t)o(A',t)df.
one h a s
t o g e t h e r w i t h t h e c o r r e s p o n d i n g Parseval-Planc-here1 formula. Define now a p r o d u c t
(5.60)
f og(x) =
f(u)g(v)dudv
I
u+v>x> u-v and s e t f o r
$ E L
Then t h e r e i s (5.62)
IM(X
provided
M(0)
I
1
I$,$E L1
90
such t h a t
( t h i s condition is equivalent t o
DtB(O,t)
9 0).
This requires
GENERALIZED TRANSLATIONS
some c a l c u l a t i o n which w e omit.
The p r o d u c t
S i n Xx/h.
with a
L1
of (5.60) p r o v i d e s
o
Banach a l g e b r a s t r u c t u r e whose c h a r a c t e r s a r e
153
Now i n v e r t (5.56) t o
obtain
so t h a t
H(x,y)
and d e f i n e f o r
(5.65)
2,
L(x,y,m)
from ( 5 , 4 0 ) .
Set
f E L1
Wf(x) = f ( x )
P r o p o s i t i o n 5.27
+
1,
H(t,x)f(t)dt.
is a map
W
L1
+
L1
such t h a t i f
M(O)
$.
11 W f I I
0
<
cII f l l
The proof i n v o l v e s u s i n g some f a c t s which r e l a t e v a r i o u s t r a n s f o r m s and we d i s p l a y some of t h i s . Note t h a t
Thus we have t r a n s f o r m s n
2 rB(h2,x)B(hn,x)dx
=
0
for
X
2
and
so
0
g i v e n by (5.58) and ( 5 . 6 1 ) .
%
can be r e p l a c e d by
H
H
in
0 (5.63) and t h u s
- 2 - 2 f(X ) = Wf(X 1,
(5.66) Since
- 2 wf(X ) = 0
(5.59) becomes
We n o t e h e r e t h e c o r r e s p o n d e n c e s (5.46). (5.61), (5.65)).
C(X,f)
The r e l a t i o n
%
^i,
w(X.f)
%
C(X.g) = w(X,i)
f,
Wg
2,
( c f (5.45)
-
of (5.47) h a s a n analogue
here in 0
(5.68)
=
Theorem 5 . 2 8
Wg.
Let
Then t h e image o f gives
I
E L1
B(A:,x)
and
The product
f
be t h e s p a c e g e n e r a t e d by t h e under
W
is equal t o
I.
' = E @ I. L
-
g = W[U(f)
a s t r u c t i i r e of rommutative Banach a l g e b r a which i s isomorphic t o
with t h e product
0.
The Gelfand t r a n s f o r m a l g e b r a s of
I
and
L1
0
U(g)]
L1
a r e t h e same
154
R . W. CARROLL
relative t o characters
B(A2,x)
and
S i n Ax/A
The proof i n v o l v e s examining t h e r e l a t i o n (5.68) p l u s 7
(5.69)
f
g
-
n
(5.70)
N
U(f)
=
o
Y
f = U(f).
Similarly using general characters Theorem 5.29
Let
One can p r o v i d e characters a r e @ EN
where
Po
2 w(X ,x)
2
(1 + x ) [ q ( x ) [ d x <
l o L ((1
+ x)dx)
+ x)dx)
-
one can prove and c o n s i d e r t h e case
+ bB(A 2 ,x)
f 0,
which i s isomorphic t o
has the product
*) w i t h c h a r a c t e r s
Miscellaneous t o p i c s .
a
A
f
0.
w i t h a commutative Banach a l g e b r a s t r u c t u r e whose
2 2 w(X ,x) = aa(A ,x)
Ll((1
t o be even f o r
2.6.
N
U(g) = lJ(f)U(g)
C o s Ax
f
+
0
g
+
1
f
*g
(prolong
L1((l f
+
and
x)dx)
g
S i n Ax/X.
We f e e l compelled t o make a few remarks h e r e about
t h e a p p l i c a t i o n of some of t h e t r a n s m u t a t i o n - t r a n s f o r m t h e o r y of S e c t i o n 2.5 t o t h e i n v e r s e Sturm-Lionville and i n v e r s e s c a t t e r i n g problems f o l l o w i n g Marrenko [ 3 ] I n a d d i t i o n t o r e f e r e n c e s a l r e a d y c i t e d we mention a survey a r t i c l e by DurovinMatveev-Novikov
[l] which should i n d i c a t e some i n t e r e s t i n g new c o n n e c t i o n s ( c f .
a l s o Maslov-Manin
We u s e t h e n o t a t i o n of MarFenko i n S e c t i o n 2.5.
[l]).
i n v e r s e Sturm-Lionville problem i n v o l v e s r e c o v e r i n g t h e problem u ' ( 0 ) = hu(O), from t h e s p e c t r a l f u n c t i o n
wants
q(x)
w(A,x,h)
given
R.
of Theorem 5.21.
We c o n t i n u e t o s u p p r e s s t h e
2 L(D ) u = -1 u ,
Thus b a s i c a l l y one
h
i n writing
w(h,x)
for
1 and
Q " ( 0 ) = -h
s o ( 6 . 2 ) makes
e t c . and write f o r m a l l y
I t i s shown t h a t i n f a c t
sense.
R
The
Further i f
0 E C3
f(X) E C K L ( U )
with and
O'(0)
=
< f ( A ) y ( X ) , R> = 0
for a l l
y(A) E CK'(0)
155
MISCELLANEOUS TOPICS
then
f(X) = 0. These two properties essentially characterize spectral functions.
One now derives an integral equation for
function w(X,x) (recall R
is a multiplier in
2 2 -= T I T I
+
(R
+:
-
: 1;
K(x,t)Cos
and
Z'
as follows. The
s o consider
(R
2
- ;)w(h,x)
htdt.
+ y, 0) + L(lt -
rK(x,t)[L(t
= K(x,t,h)
From ( 5 . 3 8 ) one has then
C(X, L(x,O))).
-1
2
K(x,t)
yI, 0)ldt.
0
Express now Cos Ax F(X)
=
through w(A,x)
by a formula analogous to ( 5 . 4 0 ) so that if
C(X,f)
Hence by Theorem 5 . 2 1 we have
I
m
Moreover
y2
w(X,x)>
=
rx
Thus
2
(R.- -)w(h,x)
2
%
71 [L(y,x) - K(x,y)]
f
>
=
F(h)[Cos 0
hx +
J:
K(x,y)CosXydyldh
and u s i n g ( 6 . 4 ) this gives
R. W. CARROLL
156
(6.8)
Thus for 0 5 y
(6.9) where
f(x,y) f(x,y)
one has the important integral equation satisfied by
5x
=
+ K(x,y) + 1
7
J:
= 0
K(x,t)f(t,y)dt
+ y, 0) + L( Ix -
[L(x
yI, O ) ]
which is easily seen
f(x,y)
Theorem 6.1 Given R E Z '
satisfying the two properties indicated and
-
in (6.2).
(6.2) the equation (6.9) has a unique solution K.
tinuous and has in both variables as many continuous derivatives as Thus from (6.9) K
K(x,y)
be the
KO
same as the expression for
defined by (6.1)
K
a,
K
is con-
@".
is expressed by means of the spectral function R
determines the Sturm-Lionville problem; the uniqueness of K
a unique Sturm-Lionville problem associated with R.
f
and
implies there is
The uniqueness proof is
based on showing that the homogeneous equation for any a (6.10)
g(y)
+ jag(t)f(t,y)dt
= 0
0
has only the zero solution and this follows from the first property of ed earlier.
If 0 E C4
2 DyK(x,y),
L(Dx)K(x,y)
(6.12)
q(x) = 2DxK(x,x);
That is q Theorem 6.2
In order that
sufficient that
0
2
y
2
x <
D K(x,O) = 0. Y
is defined by (6.12) and then appears in L(Dx).
problem L(Dx)u
=
-XLu,
R E Z'
u'(0)
(1) For any
= 0
R> = 0 for all y(X) C3 with
indicat-
one obtains from (6.9)
(6.11)
=
R
O'(0)
=
be the spectral function of a Sturm-Lionville
with continuous q
hu(0)
> 0 there is no nonzero E
Summarizing one has
2
CK ( u )
f ( X ) E CKL(0)
(2) The function 0
1. The function q
have as many continuous derivatives as 0"'
it is necessary and
and
such that
defined by (6.1)
(defined by (6.12)) will then h
=
K(0,O).
157
MISCELLANEOUS TOPICS
The proof r e q u i r e s some argument b u t we omit d e t a i l s s i n c e t h e i m p o r t a n t c o n s t r u c We w i l l g i v e n e x t a b r i e f i n t r o d u c t i o n t o some
t i o n s have a l r e a d y been d i s p l a y e d .
t e c h n i q u e s needed f o r i n v e r s e s c a t t e r i n g t h e o r y based on t h e p r e c e e d i n g c o n s t r u c T h i s w i l l s e r v e t o t i e t o g e t h e r some of t h i s m a t e r i a l and p e r h a p s m o t i v a t e
tions.
t h e r e a d e r t o p u r s u e t h e t h e o r y f u r t h e r ; i t a l s o c o n t i n u e s t o p r o v i d e examples of t h e t r a n s f o r m c o n n e c t i o n s i n d i c a t e d i n S e c t i o n 2.3.
The matter i s b e a u t i f u l l y
developed i n Marcenko [ 3 ] a t some l e n g t h and w e r e f e r t o t h i s f o r i n s p i r a t i o n . Consider again
L(Dx)u =
e(h,x)
h
e(X,x) = e i h x
K(x,x) =
21
m
u(0) = 0
<
while
m.
x
AS
-+
OJ
- u
and one i d e a is t o e x t r a c t i n f o r m a t i o n about
v
with
v.
i n t h e c l o s e d upper h a l f p l a n e t h e r e i s a s o l u t i o n
L(D ) e = -hLe,
of
(6.13)
with
For
2
-A
D v =
by comparing s o l u t i o n s a t
Proposition 6.3
with
2
t h e equation approaches q
-A 2u
+
e(h,O) = 0 , given i n t h e form
,/xK(X,t)e i h t d t
m
q(t)dt. X
K(x,t)
E x t i m a t e s on t < x
and
1
a r e a l s o o b t a i n e d b u t we omit t h i s .
s a t i s f i e s an e q u a t i o n a n a l o g o u s t o ( 5 . 4 2 ) .
K
Here
K(x,t) = 0
One writes
(I
for
+ K)f
m
f(x)
+
K(x.t)f(t)dt
with inverse
I
+
L
where
X
+
(6.14)
(I
(6.15)
L(x.y)
Again
K
+
j:
+ K(x.y) +
:1
L)f = f ( x )
L(x.t)f(t)dt
L(x.t)K(t,y)dt = 0 .
i s c h a r a c t e r i z e d by
2 L(Dx)K(x,t) = D t K ( x , t ) ,
l i m DxK(x,t) = l i m DtK(x,t) = 0
as
x
+
t
-t
m.
DxK(x,x) =
E s t i m a t e s on
-4 q ( x ) ,
e(h,x)
and
are a l s o
o b t a i n e d which we omit i n remarking t h a t e s t i m a t e s on t h e v a r i o u s f u n c t i o n s are however very i m p o r t a n t f o r t h e t h e o r y . provides f o r every x + o
h
a solution
Another f a i r l y r o u t i n e c o n s t r u c t i o n now
w(h,x,m)
of
L(D )w = -hLw
satisfying as
=
158
R. W. CARROLL
The f u n c t i o n
w
Im
is analytic f o r
X>
0
and e s t i m a t e s comparing i t t h e r e t o
are given.
S i n Ax/X
Proposition 6 . 4
Let
S ( h ) = e(-A,O)/e(h,O)
S(-A) =
=
For r e a l
S-'(-X).
0
one h a s
The f u n c t i o n
e(X,O)
can have f o r
Im h > 0
o n l y a f i n i t e number of z e r o s a l l of
which are s i m p l e and l i e on t h e imaginary a x i s . bounded i n some
nbh
one can show t h a t
of
1
-
FS1 E L1(-,m),
where
1,2,
...,n ,
-1 mk
=
11
0.
S(A)
e(iAk,x)II
mk-2
The f u n c t i o n
=
Similarly
1
2 + FS(x)
i s t h e F o u r i e r t r a n s f o r m of
e(X,O)
2 L..
in
suplFS 2
and
1
<
m
.
0 < hl <
o r d e r e d by
FS(x) = FS(x) Now l e t
X2 <
... <
iAk,
k
An
and
=
In fact
-er(iXk,0)e(iXk,0)/2iAk
-
u(X,x) = e(-A,x)
L(Dx)u = -ALu,
is
Xe(X,O)-'
T h i s t a k e s some p r o v i n g which we o m i t .
FS 2 E L 2 (-m,m),
b e t h e z e r o s of
The f u n c t i o n
u(0) = 0, with
S(X)e(A,x),
X
E (O,m), i s a bounded s o l u t i o n of
u(iXk,x) = ye(!.Xk,x).
I n f a c t such f u n c t i o n s
rm form a complete s e t of e i g e n f u n c t i o n s ; a Parseval r e l a t i o n
One a r r i v e s a t t h i s by o b s e r v i n g f i r s t t h a t f o r
-
fm
f(x)e(-X,x)dx where that f*(X) = J o form of f * . Indeed from (6.13) m
(6.19)
I f(x)e(-h.x)dx '0 =
Hence T
j
-f*(X)
f*(x) = (I
+
K*)f
it follows
denotes t h e ordinary Fourier trans-
m
=
f(x)[e-iXx
+
0
+
jjf'"'
-
u(X,x) = f*(h)
' 0
-
S(A)f*(-A)
with
f o r t h e r i g h t s i d e o f (6.18) and s e t
u ( i X k , x ) = mk
f*(x)exp(-Xkx)dx.
Write
159
MISCELLANEOUS TOPICS
1
m
(6.21)
F(f) =
F(x
+
y)f(y)dy.
0
Then
F
obtains
is a bounded s e l f a d j o i n t o p e r a t o r i n T = ((I
+ K)(I +
+ K*)f,g).
F)(I
2
and w r i t i n g o u t
L (0,m)
T
one
Hence t h e P a r s e v a l r e l a t i o n (6.18)
is e q u i v a l e n t t o
K(x,y) = 0
W r i t i n g t h i s o u t and u s i n g t h e f a c t t h a t
for
x > y
it follows t h a t
(6.22) i s e q u i v a l e n t t o
This i n t e g r a l equation f o r
Theorem 6 . 5
The k e r n e l
i s b a s i c i n t h e t h e o r y and one can prove
K
K(x,y)
s a t i s f i e s (6.23) f o r
0 < x < y <
-
where
F
is
g i v e n by (6.20) and from (6.23) f o l l o w s t h e P a r s e v a l r e l a t i o n ( 6 . 2 2 ) o r ( 6 . 1 8 ) .
The proof u s e s (6.17) and o t h e r f a c t s mentioned above.
Now a t y p i c a l problem i n
quantum s c a t t e r i n g t h e o r y i n v o l v e s r e c o v e r i n g a p o t e n t i a l of
Xk,
S(X),
and s e t t i n g
and
mk.
q(x) =
-4
t i e s r e q u i r e d of an e q u a t i o n
S-'(-X)
while
from knowledge
Theorem 6 . 5 a l l o w s one t o do t h i s by f i n d i n g
D K(x,x).
Xk,
S(X),
-A
2
and
mk
so t h a t t h e y r e p r e s e n t s c a t t e r i n g d a t a f o r
with
u(X,O) = 0.
The f u n c t i o n
S(X)
i s d e f i n e d on t h e r e a l a x i s w i t h
1
-
S(h)
-t
0
(1
-
as
K(x,y)
The q u e s t i o n r e d u c e s t o s p e c i f y i n g t h e p r o p e r -
u
L(D ) u =
Property 6.6
q(x)
1x1
+
m
The c r u c i a l f a c t i s
-
S ( X ) = S(-X)
and i s t h e F o u r i e r t r a n s f o r m of a func-
t ion
1
m
(6.24)
FS(x) = 2*
which i s t h e sum of For
x > 0
one h a s
S(X))e
iXx
dX
-m
1 FS(x)
I!
and
2 FS(x)
x ( F S ( x ) ( d x<
-.
=
with
1
FS 6 L1(a-.m)
and
2 FS
L
2 (-m,u?)
R. W. CARROLL
160
Another p r o p e r t y which s h o u l d b e mentioned i s t h a t t h e number
n
of e i g e n v a l u e s
i n d i c a t e d above s a t i s f i e s
(6.25)
n =
Theorem 6.7 K(x,y) E L
- log
l o g S(O+)
I f P r o p e r t y 6.6 h o l d s then (6.23) h a s f o r
1
The f u n c t i o n
(E,m).
q ( x ) = -$ DxK(x,x)
with
1- S(0) 4 -
S@+)
211i
and, w r i t i n g
+ Fa(f)
e(X,x)
XI
and
> 0
E
a unique s o l u t i o n
d e f i n e d by (6.13) s a t i s f i e s
q(x) Idx <
m
.
F(x)
Here
L(Dx)e =
-A
2
e
i s g i v e n by (6.20)
m
F(t
=
+
y
+
2a)f(t)dt, i f
I
+ :F
h a s an i n v e r s e f o r
a = 0
0
one can t a k e
E
= 0.
Another approximation t o a "best" theorem ( c f . Theorem 6.9) i s o b t a i n e d by d e f i n ing
I I
m
(6.26)
Fif =
FS(t
0
+ y)f(t)dt;
0
F-f
=
+ y)f(t)dt.
FS(t
-m
Theorem 6 . 8
Assume P r o p e r t y 6.6 and
the zero solution i n
L (-m,O).
Then t h e s p e c t r a l d a t a
a r i s e s from a problem
( k = 1 , .. . , n )
f
+ F+S f
2 L ( 0 , ~ ) (B) t h e e q u a t i o n
l i n e a r l y independent s o l u t i o n s i n
2
(A) t h e e q u a t i o n
L(Dx)u
=
-A 2 u ,
S(X),
g
= 0
-
has
F-g = 0
hk,
u ( 0 ) = 0 , and
S
n h a s only
mk
q
defined a s
above s a t i s f i e s
We omit t h e p r o o f s of t h e s e theorems b u t i n c l u d e them t o i n d i c a t e t h e c i r c l e of ideas.
F i n a l l y p u t t i n g a l l t h i s t o g e t h e r one h a s
I n order t h a t the s p e c t r a l data
Theorem 6 . 9 L(Dx)u
=
2
-A u,
u(0) = 0 , with
1,
xlq(x)Idx <
S(h), m
hk,
mk
a r i s e from a problem
i t i s n e c e s s a r y and s u f f i c i e n t
t h a t P r o p e r t y 6 . 6 hold w i t h ( 6 . 2 5 ) . For much more material i n t h i s d i r e c t i o n and a p p l i c a t i o n s t o modern work i n KdV
e q u a t i o n s s e e Margenko [ 3 ] .
We mention now a c o n s t r u c t i o n of t h e t r a n s m u t a t i o n o p e r a t o r f o r Sturm-Lionville
161
MISCELLANEOUS TOPICS
t y p e e q u a t i o n s of second o r d e r w i t h o p e r a t o r c o e f f i c i e n t s f o l l o w i n g Androgcuk [ l ] L e v i t a n [ 4 ; 6 ] , Rofe-Beketov
[2;3],
( c f . a l s o GorbaFuk-Gorbazuk Consider
u ( 0 ) = h E D(A), u ' ( 0 ) = 0 , and
(6.27)
2 LU = -D U
+
q(X)U
-
LU = XU
Au;
u ( x ) E H , a s e p a r a b l e H i l b e r t s p a c e , and
where
semibounded below which can be t a k e n as generality.
Assume a l s o t h a t
s e l f a d j o i n t and
x
+
+
E
is a self adjoint operator
A
A > 0
A4q(x)A-'
with
A-'
E L(H)
Co(Ls(H))
w i t h o u t l o s s of
( q ( x ) E L(H)
with
q(x)
Then one wants t o extend formulas s u c h
q ( x ) E Co(Ls(H)>>.
( 5 . 4 0 ) i n some manner.
as ( 5 . 3 8 ) and an a n a l o g u e of
Theorem 6 . 1 0
x
[ l ] , Solovev [ l ] ) .
Under t h e h y p o t h e s e s i n d i c a t e d t h e e q u a t i o n (6.27) can b e s o l v e d i n
t h e form
(6.28)
u(x,X) = h C o s f i x
+
IXK(x,t)Ah Cosfit d t ' 0
where
(x,t)
+
K(x,t)
Co(Ls(H))
with
K(x,t) = 0
for
t > x
The proof i n v o l v e s working w i t h t h e i n t e g r a l e q u a t i o n (by s u c c e s s i v e a p p r o x i m a t i o n s )
(6.29)
w(x,X) = C o s m x
+
q ( t ) w ( t ,X)dt. 0
Using t h e formula i n o p e r a t o r s
-x + t one o b t a i n s a formula
(6.31)
w(x,X) = C o s m
,(!:-
i
where then for
R ( x , t , - r ) E Ls(H) w(x,X)h q = 0
and
t R ( x , t , T ) C o s f i T C o s m t dTdt
and depends c o n t i n u o u s l y t h e r e on
s a t i s f i e s (6.27).
Now
C o s m xh
C o s m xh = h C o s f i x
+
g(x)
( x , t , ~ ) . If
h E D(A)
i s t h e s o l u t i o n of (6.27)
with
162
R. W. CARROLL
(6.32)
g(x) =
jxo
cosfitdt
S i n m ( x - t ) Ah
m
S i n c e i t i s known t h a t
J o ( f i w s ' )
(6.33)
one can e x p r e s s
g(x)
and hence
(6.32), (6.33). (6.30))). Theorem 6.11
Let
qfx),
CosJj; TdT
1
=
0
C o s m xh
e x p a t , and
A'q(x)A-', q(*) E C
i s e x p r e s s e d by (6.34) below where Ho
*
on
(H(x,t) = 0
Let
E(A)
+
2
.
H(x,t)
For
exp
h E D(A)
-at
is a weakly c o n t i n u o u s o p e r a t o r func-
,I
H(x,t)Au(t,A)dt.
2 2 f i xoA f ,
some f i x e d number); l e t
let
be t h e c o r r e s p o n d i n g n e g a t i v e norm s p a c e ( i . e . l e t for
u E H+,
Rf
i s t h e completion of
E
Hi
Ho
H+
be t h e r e s u l t i n g space.
and f o r such
i n t h e norm
g ( x , y ) = C o s f i xw(y,X)h
Set
f E H
11 i f [=[ s u P l ( f , u ) , l / l l 11 f II_= 11 if11 1. Now t h e
( a n t i d u a l ) and
tion (6.35)
A
A
2 2 A exp 2 f i xoA g)
(xo > 0
H-
be s t r o n g l y c o n t i n -
an i n v e r s e t r a n s m u t a t i o n
be t h e s p e c t r a l r e s o l u t i o n a s s o c i a t e d w i t h
( f , g ) + = (A'exp
(using
d e f i n e t h e scalar product
E(A)H
H-
h Cosa x
t > x).
for
h C o s A x = u(x,X)
(6.34) Proof:
H-
i n terms of
P u t t i n g a l l t h i s i n (6.31) g i v e s ( 6 . 2 8 ) .
uous o p e r a t o r f u n c t i o n s w i t h
tion
2 -T2
&-t)2-S
Ho = H SO
uII+(
and
(f,u)o =
11 f l l o -
v e c t o r func-
MISCELLANEOUS TOPICS
Let
Ho * H-
Here
d e n o t e t h e t r a n s p o s e of
and c o n s i d e r
Q < (O,y,x)
=
(T,Q,<)
Ho
so
(x-
5) 2 -
+
(exphi T)+ :
d e f i n e d by
go
9
For
H+,
<
i s a weak s o l u t i o n of t h e
The f u n c t i o n
E
i s t h e i n t e r i o r of t h e c o n e
( O , y , x ) <: ( t o , y o , x o ) , 0 <
with vertex
for
g o ( x , y ) E H-
( e x p a T ) : H+
( e x p a .r)+Af(S,n)dQ go(x,Y) = 2T 2 2 l QIC ( 0 - y, x ) / ( X - 5 ) ' (Y-rl) - T
(6.37)
0
TI+
(exphi
163
0
q(y)
the function
< x < x o , and g
n)'
Po = ( t o , y o , x o )
e q u a t i o n above f o r
g(x,y)
(y-
-
T~ =
fixed.
q = 0, i . e .
a b o v e i s a s o l u t i o n of t h e
i n t e g r a l equation
where
maps
Ho
into
H-.
U s i n g t h e form o f
f(x,y)
a b o v e and ( 6 . 3 5 ) o n e
obtains
(6.'41)
C O S x~w ( y , h ) h =
1
[w(x-y, h)h
+ W ( X + Y ,h ) h ]
+ ril( x , y ,t ) Aw( t ,h 1h d t . 0
Setting
y = 0
Remark 6 . 1 2
(6.34) r e s u l t s where
-
H(x,t) = Hl(x,O,t).
I t seems a p p r o p r i a t e h e r e t o make some comments a b o u t s e p a r a t i o n of
v a r i a b l e s and t e n s o r p r o d u c t s i n t h e s p i r i t of B e r e z a n s k i j [ L ] , Cordes [ 2 ] , Reed-Simon
QED
Carroll I271,
B. Friedrnan [ l ] , I c h i n o s e [l: 2 ; 3: 4 1 , Maurin-Maurin
[ 2 ; 31, S c h e c h t e r [ Z ] .
[ 3 ] . Mohammed [ l ] ,
F i r s t l e t u s r e v i e w some o f I c h i n o s e ' s c o n s t r u c -
t i o n s : m a t e r i a l on t o p o l o g i c a l t e n s o r p r o d u c t s , c r o s s n o r m s , e t c c a n b e found i n
R. W. CARROLL
164
LaMadrid [l], Grothendieck [l], Shatten [l], Treves [l].
E
F be Banach
and
a a "reasonable" crossnorm on E Q F (i.e. the dual norm a'
spaces and
.
fll
is a
E' 8 F'); we recall that a crossnorm is to satisfy a(e 8 f) =
crossnorm on
11 ell 11
Let
A reasonable crossnorm
tinuous map
E
ia F
(A,B) E L(E)
X
L(F)
Let A : D(A) C E
E
-+
ic F
is said to be faithful if the natural con-
ci
is 1-1: a crossnorm
11 All 11 BII
sup11 (A 8 B)ulla(
-+
E and B : D(B)
F
C
+
ci
is uniform if for any
11 u11,Z
where the sup is over
1.
F he closed densely defined linear
operators (the theory can be developed more generally but we are only interested in such operators).
Let
P(<,q)
=
C c
c'qk jk
P(A 8 I, I 8 B ) = C c 'A Q Bk Ik
(6.42)
with domain D(Am 8 Bn) = D(Am) Q D(An) in
5 and
follows
-
where m
respectively. Let A
rl
which makes sense if a
A
and consider
B
ka I)(I ia B)
if one assumes p(A)
be the closure of
A Q B
ia F
in E
is reasonable and faithful (which we shall assume in what
also for simplicity assume a
B = (A
P
are the degrees of
and n
&a B) (A
= (I
and
extended spectra Ue(A)
p(B)
and
is uniform).
It can be shown then that
6a I) ; similarly
'A
6a I =
(A
Oa
I)'
and
nonempty while it does not occur that one of the
oe(B)
contains 0 with the other containing
m
then (6.43)
Here oe(T) p(B)
I, I
P(A o(T)
=
c U C p(A)
with
such that
V)
- -I + large
15
G(T) = G(T) then
=
of P(o(A),
(resp.
c V C p(B))
?.
C
W
and
?
o(B))
there are nonempty open U
with "nice" boundaries
+ Iql)-'
Now an extension
p(A)
and
be the class of polynomials such that: For
(b) CR(5,A)
(c) P ( F , , n > ( l C ]
lql.
P(A,B)
in Ic
(a) P(U,V)
(resp
Bk.
for our purposes and we assume in general that
are nonempty. Let now
any open nhh W
U
B) = C c. 'A Jk
(resp qR(q,B))
aU
(resp.
aV)
is uniformly hounded in
is bounded away from 0 on
'?
(resp V)
of an operator T
x
for
is called maximal if
has no proper extension with this property; if T
The following spectral mapping theorem then holds
is closable
165
MISCELLANEOUS TOPICS
Theorem 6 . 1 2
P E P(A,B)
Let
=
U(?(A 8 I , I Q B ) ) .
T h i s means i n p a r t i c u l a r t h a t ( 6 . 4 4 ) h o l d s i f
0
U(P(A 8 I, I 8 B ) ) =
a
For
Then
o ( B ) ) = U(P(A 8 I , I 8 B ) )
P(U(A),
(6.44)
a b e any reasonable uniform crossnorm.
and
f a i t h f u l with
U(A)
9
4
i f and o n l y i f a t l e a s t o n e o f p(A)
and
p(B)
and
90
U(B)
U(A)
or
U(B)
P(A 8 I , I 8 B )
nonempty
while a r e empty.
is c l o s a b l e so
5
P
and f o r
=
P(a(A), U ( B ) )
ka
(Z c . A’
F(A 8 I , I 8 B) =
(6.45)
Ic
Jk
Bk)-.
Thus when i t d o e s n o t o c c u r t h a t o n e o f there results
other contains u ( F ( A $a I , I
6a
P(U(A),
11
p(T)
zR(z,T)
9
I 8a
k)
U(B))
=
o(B) o(P(A
T
0
contains
ga
1, I
Ga
11 5 M ( O ) ,
8 = arg
then. taking
A
j
z,
outside
is of t y p e
j = 1.2
(OT,
is s a i d t o be of t y p e
c o n t a i n s t h e complement o f t h e s e c t o r
c r o s s n o r m as a b o v e and (j
and
while the
B)) =
B)).
A closed densely defined operator
n, if
o(A)
S(8;)
S(8,)
0 < 8. < - J
Mj(8)),
f o r s i m p l i c i t y and s e t t i n g
5
a
If
8.
TI,
J
A1= A
l
OT <
8T 1
and
is a
+
8a
5
0
l a r q 21
= {z;
BT < 8 ; < n .
for
(ej,
M(8)),
Bk <
TI
B2 =
I,
A2.
T h e r e a r e a number of r e s u l t s c o n c e r n i n g when
C IPI = j
(In.)- ( c f .
also Carroll
J
[ 2 7 ] ) and we c i t e ( c f M i k h a i l o v [l])
Theorem 6.13 F
with
cc
adjoint or Tan
eA
Let
and
A
6
be m-accretive o p e r a t o r s i n H i l b e r t spaces
t h e s t a n d a r d H i l b e r t c r o s s n o r m and l e t e i t h e r (h) A
Tan 9 < 1. B -
and Then
B
(a) A
a r e m-sectorial with semiangles ( A 6e I
+
I 8 B)-
= A
6a
I
+ I 6a
eA *
B
and
or BB
B
E
and
is s e l f
such t h a t U(A)
+ U(B)
=
166
U(A
R. W. CARROLL
GO
+ I 8a
1
B).
Here an o p e r a t o r
is accretive i f
A
)I A + XI)-'ll
t h e l e f t half plane with
A
operator
operator w i t h
(Im(Ax,x)
< 1/Re X
eA,
and semiangle
1 5 Tan
8 Re(Ax,x). A
0) a uniform r e a s o n a b l e norm i n
(resp.
for
0.
< n/2
Now g o i n g back t o
E Q G
s a t i s f y t h e Gowurin p r o p e r t y ( f i n i t e ) set
E
Ei E
An m - s e c t o r i a l i s a n m-accretive
E,
F
K
of d i s j o i n t p r o j e c t i o n s and any s e t
and
G
f o r example and
G
which w e assume
The f a m i l y
i s there is a constant
C;
should c o n t a i n
F @ G)
(resp.
f a i t h f u l whenever n e c e s s a r y f o r a s p e c i f i c c o n c l u s i o n .
for
X
Re
0 < 8 - A
p(A)
E be a bounded Boolean a l g e b r a of p r o j e c t i o n s i n
Banach l e t
a
0
with vertex
i s d i s s i p a t i v e and
-A
E
is said t o
such t h a t f o r any
Ai E L(E,F)
one h a s
u E E @ F
T h i s p r o p e r t y i s examined i n I c h i n o s e [ 3 ] and w e omit t h e d e t a i l s h e r e . a l if
B : D(B)
C G
+
I n gener-
is a s c a l a r t y p e s p e c t r a l o p e r a t o r ( c f Dunford-Schwartz
G
[ l ] ) o r a s e l f a d j o i n t o p e r a t o r w i t h a c o u n t a b l y a d d i t i v e r e s o l u t i o n of t h e i d e n t i t y E(dX)
then t h e p r o p e r t y
g u a r a n t e e s t h e e x i s t e n c e of i n t e g r a l s
IF(X) @ E(dh)lv
(6.48) o(B) in
F
8B
G
E 8 G
v
when
F(X) : u(B)
+
L(E,F)
i s c o n t i n u o u s and u n i f o r m l y
Consider t h e n polynomials such as ( 6 . 4 2 ) and w r i t e
bounded. cm(rl)cm
for
+
... + c
(n)
where
cm(n)
0.
is a closed densely defined operator i n t h e l a r g e s t i n t e g e r such t h a t
q E u(B)
h a s an i n v e r s e
and t a k e
a =
B
Bore1 sets w i t h
for simplicity. E(UUv) = I
E
E
Let
L(E)
=
Then
w i t h domain
D(Am"l))
~ ~ ( ~ f f 0.n ) Assume c o n d i t i o n
P-'(A,n)
P([,n)
where
m(n)
(P) : P ( A , Q I
which i s uniformly bounded on
for
o(B)
uV be an i n c r e a s i n g sequence of bounded
and n o t e t h a t f o r example
is
167
MISCELLANEOUS TOPICS
for
u E D(Am) 8 D(Bn).
(= P-l
The i d e a now i s t o r e p r e s e n t
P-l(A 8 I , I 8 B) =
under c i r c u m s t a n c e s i n d i c a t e d above) i n terms of t h e i n t e g r a l , f o r
v E
E 8 G,
(6.51)
--1v = jLi(B) [P-l(A,n) P
@ E(dn) l v .
The d e t a i l s are s t r a i g h t f o r w a r d u s i n g e x p r e s s i o n s such as (6.50) and one h a s i n particular
Theorem 6.14
Assume t h e p r o j e c t i o n s
identity for
B
( 6 . 4 9 ) while (j,k).
Then
Let
P(A,q)
k j -1 rl A P (A,n) E L(E) :(A
G
have t h e p r o p e r t y
reasonable, f a i t h f u l ) .
E(dn)
as above f o r
E
and
c1 =
s a t i s f y t h e condition
(P)
indicated a f t e r
and i s u n i f o r m l y bounded on
$a
G
4
E = F
8 I , I 8 B) = (1 c . (A’ Ilk
v e r s e g i v e n by (6.51) mapping
a r i s i n g from t h e r e s o l u t i o n of t h e
k
iCc B ))-
jnto
D(?)
for suitable
has a continuous in-
and C
o(B)
(uniform,
flD(AJ
k
B )
w i t h norm
For polynomials as i n Theorem 6 . 1 2 , o r even l a r g e r c l a s s e s , t h e r e are s p e c t r a l mapping r e s u l t s and c l o s u r e theorems ( c f I c h i n o s e [ 3 ] ) .
There a r e obvious r e l a t i o n s
h e r e between f o r m u l a s such as (6.51) and some of t h e development i n S e c t i o n 1 . 4 (see e.g.
( 1 . 4 . 6 3 ) ) and w e w i l l expand upon t h i s f u r t h e r i n a H i l b e r t s p a c e con-
t e x t f o l l o w i n g B e r e z a n s k i j [I]. F i r s t l e t u s r e c a l l some c o n s t r u c t i o n s i n v o l v i n g
so c a l l e d n e g a t i v e norm s p a c e s and g e n e r a l i z e d e i g e n v e c t o r s . b e two H i l b e r t s p a c e s w i t h
11
ullo
5 11
uII+
H+
Thus let
H+
Ho
dense and t h e i n j e c t i o n c o n t i n u o u s i n t h e form
( a l l H i l b e r t s p a c e s w i l l be assumed s e p a r a b l e and of c o u r s e
16 8
R . W. CARROLL
(1 U I ( ~ (
can b e handled by renorming).
c ( ( u(I+
11
t h e norm and w r i t e
1)
fll-=
( a , ~ )is~ d e f i n e d
with t h e
a
s c a l a r product for
Ho
+
a
for
Ho
E H-
in
be t h e i n j e c t i o n
Then
H+.
can b e viewed
H-
u E H+
and
)I fll-5
and
coinciding
e x t e n d s t o a map II : H-
I
The map
E Ho.
Ho
(f,q)- = ( I f , Ig)+
i n t h e s c a l a r product
Ho
A s c a l a r product
fllo.
I = O* : Ho
+
Hi
(lf E
The completion o f
0 : H+
Let
H-.
so t h a t
(f,Ou)o = ( I f , u ) +
1) 5 )I f [Io .
Ilf
ifll i s denoted by
as t h e completion of
11
)I
(lf(u) = ( f ,do and
a s before with
w e have
f E Ho
For
+
h
H+ Ho
o n t o and one h a s +
Ho
D(K)
with in
and s e t
Ho
H-
to
(?I)';
K =
H+
=
(f,u)o = (If,u)+
R(K) = Ho
and
while
( u , v ) + = ( K U , K V ) ~ . Consider
H-
and form t h e c l o s u r e i n t h e s p a c e
(f,g)o = (Kf,Kg)-
D(K) = Ho.
and
I = 01 :
Let
is a p o s i t i v e s e l f a d j o i n t o p e r a t o r i n
K
then
(ff,g)- = ( l I a , g ) o .
with
II
One h a s
-1
M with
to g e t an o p e r a t o r
=KK
and
HO
a s a map
K
( f , K u ) o = ( K f , uI0
(note
and, f o r
f E Ho,
and 9 = K such as
-1
K-'K-I
s o t h a t II = JJ ,
H+
mapping s a y
K
-1 -1 K f = ?f
f = K
+
while
with
H
( K u , f ) o = (u,K*f)+
+
-1 (I K*Ioa,u)o = (K a , ~ )( f ~o r
garded a s a map w r i t e s X = K+
Ho
+
H-
and
t h u s e.g.
We r e c a l l n e x t t h a t an o p e r a t o r orthonormal b a s i s
A
ei
in
has a f i n i t e trace i f
H1
E(A)
A : H1
C H+ C
L2 C
D'
Hq
the series
ei.
Let
Ho
C
with
H
C
D'
where
K,
and
K+
can be re-
Consequently one taken i n
.
H
is Hilbert-Schmidt i f f o r (any)
111 Ae a,
A
K* = K
(f,Ku)o =
( ~ , K U =) ~(Ioa,Ku)o=
6 Ho)
112
< 03. A n o n n e g a t i v e o p e r a t o r H2 which a g a i n i s independent of
b e a s e l f a d j o i n t o p e r a t o r in
t h e c o r r e s p o n d i n g r e s o l u t i o n of t h e i d e n t i t y .
a s above w i t h c o r r e s p o n d i n g o p e r a t o r s
D
+
a
by II).
I
extends
K
so t h a t
identity,
=
u E H+,
t r ( A ) = C(Aei,ei)
t h e c h o i c e of orthonormal b a s i s with
. I
by e x t e n s i o n ( r e p l a c e
+ J =J ;
-1
3 = K
A f u r t h e r b i t of n o t a t i o n i n v o l v e s o p e r a t o r s
( K ~ , U )a s~ above; it f o l l o w s t h a t , s e t t i n g (K*Ioa,u)+ =
One sets
?f = I f ) .
Let
H
H+ C H C H -
be
K , e t c . ; one can a l s o l o o k a t a c h a i n
i s a dense n u c l e a r s p a c e ( f o r example
t h e s t a n d a r d Schwartz s p a c e ) .
D
The c o n t i n u o u s o p e r a t o r
C
H1
C
K-l :
MISCELLANEOUS TOPICS
Ho
+
Ho
2 Kullo
and
an orthonormal b a s i s i n
H+).
(by c o n s t r u c t i o n
e
for
so
j
i f w e assume t h e embedding
i s Hilbert-Schmidt
K* = K)
and
11
2 uII+=
169
11
p(A) = trO(A)
H+
+
H
H+
+
Ho
is quasinuclear
q u a s i n u c l e a r means
O ( A ) = K*-lE(A)K-l
Consider
f o r Bore1 sets
A
C
1 1 1e j l l i < (*
in
m
Ho
(JO,~). There e x i s t s an
r
$(A)
operator function
(6.52)
@(A) = JA$(h)dp(h)
such t h a t
and f o r
f , g E H+ = D(K)
( E ( A ) f , g ) = (@(A)Kf,Kg)
IAfP(h)Kf,Kg)dp(h). Roughly t h e o p e r a t o r corresponding t o
A.
K*$(X)K = K $(X)K
Thus f o r any c o u n t a b l e s e t of
$(X)K(A - XI)u = 0
one h a s
i s a p r o j e c t i o n o n t o an e i g e n m a n i f o l d
a.e. f o r
X
of i n t e r v a l s c o n t r a c t i n g t o
with
dp(X).
u E D(A) n H+
Note h e r e t h a t i f
a s above t h e n f o r
u
with
Av
Au E H
+
i s a sequence
f E €i
K ( A - XI)U, f ) =
The measure
(,)(A)
=
(E(A)u, K-lf)
and a l i t t l e argument f o r since
p(A)
a.e.
f
is absolutely continuous with respect t o
i n a c o u n t a b l e d e n s e s e t y i e l d s t h e remark above
t h e l i m i t i n ( 6 . 5 3 ) is z e r o .
To e x t e n d t h e t r e a t m e n t h e r e t o g e n e r a l i z e d e i g e n f u n c t i o n s c o n s i d e r f o r
where
P(X) =M$(A)K
Thus f o r
u , v E H+
p(A)
i s d e f i n e d from
(6.52) becomes
H+
to
H-
u , v E H+
and w i l l be H i l b e r t Schmidt.
170
R. W. CARROLL
D
Suppose
C
H+
4 E H-
An element
uous.
D
is d e n s e ,
D(A*)
C
(A*
Ho),
in
A* :
and
i s a g e n e r a l i z e d e i g e n v e c t o r of
A
D
-+
H+
for
X
is c o n t i n i f for
u E D
(6.56)
For symmetric
9 * H+ C+
-
(+, (A*
A,
11)~) = ~0.
* D'
D(A)
that A$.
A
C
C :
D
-+
A
i n (6.56) when
H+ t h e r e i s a c o n t i n u o u s
(C
w e have
(Af,u)o
i. Thus
from (6.56) a g e n e r a l i z e d e i g e n v e c t o r i s a s o l u t i o n of
P
F u r t h e r one can t a k e f o r
((u,v)) = (u,v)+
+
A :
+a, u ) =~ (a, C U ) ~ as b e f o r e and one d e f i n e s i= A* + + = (A* f , u ) o = (f,A*u)o = ( A f , u ) o f o r uEDCD(A*) s o
d e f i n e d by -,
On
can b e r e p l a c e d by
A*
Now f o r any c o n t i n u o u s
is continuous.
: H-
and
A* 3 A
4E
D+
so t h a t
(A*u, A*v)+
any e v e n t e v e r y e l e m e n t .
the space
R(P(X))
D(A*) = D+
k$ =
with s c a l a r product
i s a separable p r e h i l b e r t space.
satisfies
ic$
In
s i n c e ( c f . (6.53) and
=
remarks t h e r e )
=
(4 =
P(x)v,u
Theorem 6.15 and
A
ous.
(@(X)Kv, K(A
-
as i n ( 6 . 5 3 ) ) .
D
Let
C
H+
-
X I ) U )=~ (Kv, $(X)K(A
C
Summarizing some of t h i s we s t a t e Ho
C
H
a self adjoint operator i n
- C v' Ho
as i n d i c a t e d
with
There i s a nonnegative f i n i t e measure
D
C D(A)
dp(X)
d e f i n e d I I i l b e r t Schmidt v a l u e d o p e r a t o r f u n c t i o n h o l d s and
R(P(A))
XI)U)= ~ 0
.(H+
111 $,(A) (6.58)
11-
= 1
( E ( A ) U , V ) =~
,...,N x )
j
and f o r
+
continu-
H+
and a n almost everywhere P(A) : H+
+
c o n s i s t s of g e n e r a l i z e d e i g e n v e c t o r s of
(a = 1
quasinuclear)
Ho
A : D
and
One can f i n d a system of such e i g e n v e c t o r s , o r t h o g o n a l i n
2
+
A
H-
H-
such t h a t (6.55) (i.e.
i$ = A$).
such t h a t
u , v E H+
NA
1
vcc(A)dp(U
A 1
where say
ua(X) = ( + a ( A ) , u ) o .
Thus
-
( P ( X ) U , V )= ~ Z ua(X)va(X)
o r f o r m a l l y one can
I
171
MISCELLANEOUS TOPICS
Going now t o a n o p e r a t o r
E
b e r t space
(resp
terms of r e s o l u t i o n s assume
A
and
B
F)
C = A @
I
+
18 B
with
A
B)
(resp
acting in a H i l -
we can produce a r e s o l u t i o n of t h e i d e n t i t y f o r
E(dX1
and
F(dX)
for
A
and
are self adjoint for simplicity.
B Then
respectively. C
C
in
Thus
i s symmetric and
one h a s
Theorem 6 . 1 6
A r e s o l u t i o n of t h e i d e n t i t y i n
E 8 F
for
C = A 8 I
+
I @ B
p r o v i d e d by
We n o t e i n t h i s r e s p e c t t h a t a s a weak i n t e g r a l ( 6 . 6 2 ) c a n b e w r i t t e n
and p a r t of t h e p r o o f of ( 6 . 6 2 ) i s c o n t a i n e d i n t h e f o r m u l a f o r s u i t a b l e
which f o r m a l l y f o l l o w s from
e. f
is
R. W. CARROLL
172
If
is now a bounded Bore1 f u n c t i o n f o r example one h a s
f(1)
(6.66)
1
co
I,
f(X)dGA =
f(C) =
I
(f(A
+ pI)
8 I ) d p ( I 8 Fp)
-m
m
since
f(A
+ pI)
=
f(X)dEX-p.
Applying t h i s t o t h e r e s o l v a n t
RZ(A),
Imz
f 0, 6
-m
a l i t t l e argument shows t h a t t h e c l o s u r e of
C
i s s e l f a d j o i n t and h a s t h e r e s o l u -
t i o n of t h e i d e n t i t y g i v e n by ( 6 . 6 3 ) .
Let now
E C E-
E
+C
embeddings
E+
relative t o
A
E
and
and
B
+
F+ C F C F-
and F+
F.
-f
be chains a s before with quasinuclear
One o b t a i n s s p e c t r a l measures
and t h e s p e c t r a l measure
<(A)
for
dp(X)
and
do(X)
i s g i v e n by
m
(6.67)
<(A)
~ ( -1 p)do(p)
=
= (P
*
IJ)(A).
m A
Recall here
h
p(A) = t r ( J E ( A ) J )
f o r example where w e w r i t e
c o n s i d e r e d a s an o p e r a t o r i n
F-
J 8 J’
one h a s maps
1
Eo.
Then f o r t h e c h a i n
e t c . and from (6.62)
m
(6.68)
S(X) =
1
-m
A
,
(EA-pJei,
.
Jei)d
K
-1
= OJ =
,. when
J
E+ 8 F+ C Eo 8 Fo C E- @
MISCELLANEOUS TOPICS
173 Pt3 (A)
Similarly there is a generalized projection operator a s follows.
F i r s t by w r i t i n g o u t
(Gh(e 8 f ) , e ' @ f ' )
for
obtained
from (6.62) and u s i n g
(6.55) one o b t a i n s
=
1
P ( v ) t3 P'(P)dP(w)do(u)
W+V
where
P(v)
(resp.
Theorem 6.17
A
For
is a s s o c i a t e d w i t h
P'fp))
and
B
(resp.
A
Hence
B).
self adjoint the self adjoint
has spectral
measure g i v e n by (6.67) and p r o j e c t i o n o p e r a t o r s g i v e n by (6.70).
Further
and (6.70) can be e x p r e s s e d i n terms of a B o r e l m e a s u r e
c o n c e n t r a t e d on
the l i n e
w
+
p = h
dp ( v , p )
x
i n t h e form ( c f a l s o ( 6 . 8 5 ) )
The material s k e t c h e d h e r e on t e n s o r p r o d u c t s and s p e c t r a l t h e o r y i s c l e a r l y v e r y r e l e v a n t t o t h e f u r t h e r s t u d y of t r a n s m u t a t i o n o p e r a t o r s .
I t h a s been o r g a n i z e d
and i n c l u d e d h e r e w i t h t h a t purpose i n mind.
I n t h i s s p i r i t l e t u s i n d i c a t e some e s s e n t i a l l y formal r e l a t i o n s which s u g g e s t f u r t h e r study.
We c o n s i d e r o p e r a t o r s
and
P
P(Dx)@(x,y) = Q ( D y ) @ ( x , y ) and
@(x,O) = f ( x )
t h a t under s u i t a b l e h y p o t h e s e s
$(x,y)
< Q ( x , h ) , B(y,A)>
5 Txf(x)
and
U(x,E)
=
Q
a s i n Section 2.3 with
with
@(O,Y)
=
=
where
so t h a t ( c f . (3.78))
i s t h e generalized convolution associated with Similarly l e t
V(y,n)
satisfy
Q(D )V = Q ( D )V Y rl
s u i t a b l e o t h e r conditions f o r uniqueness
with
(Bf)(y).
Recall
B(y,x) =
(f*g)(x) = P
and
T.
V(y,O) = ( B f ) ( y )
(V(0.n) = ( B f ) ( n ) ) .
Denote by
and
Sy rl
the
R.
174
w.
CARROLL
generalized t r a n s l a t i o n operator associated with and s e t , f o r = -A
2 h(x,h)
h @ g(y) = .
Bf = h , and
2 Q(D )B(y,A) = -1 B(y,X).
f o r e l e t u s set
-A
of
(resp
P(D )H = p H
h(x,h)
and
and w r i t e
= p
Q(D
2
Y
=
O(y,-h ) = B(y,X).
suitably s e l f adjoint with Sy = O(y,Q)
)O
H(x,p)
pi))
V(y,n) = Sy(Bf)
n
Further recall that
P(Dx)h(x,X)
(resp.
O(y,p))
for the solutions
w i t h t h e same i n i t i a l v a l u e s ; t h u s
For convenience h e r e we t h i n k of
s o as i n Remark 3.18
o ( P ) = U(Q)
and f o r s p e c t r a l r e s o l u t i o n s
(n)
For u n i f o r m i t y of n o t a t i o n t h e r e -
Y
2
so t h a t
Q
dEX and
dFX of
P
H(x,-A
and
P
and
Q
) =
as
Q
TX = H(x,P)
2
and
respec-
t i v e l y one h a s
Thus
Pf(X) =
Hf(-X2) = Pf(X))
and (i.e.
t o i n v e r s i o n s such a s
(6.74)
corresponds t o
z(p) = Hf(p)=
Qf(X) = Tf(-X2) = Q f ( A ) ) .
-1
P=P
corresponds t o
Then ( c f (3.72)
-
(i.e.
f(p) = Tf(p) =
(3.73)) corresponding
of Theorem 3.5 one w r i t e s
f ( x ) = l?(U)H(x,p)dO(u);
h(y) =
I
h(v)O(y,v)dp(v). A
Formally one has
TX = H(x,P)
and
Sy = O(y,Q)
d
Syh = O(y,v)h(v) that
(thus eg.
H(x,p) =
@ ( x , y ) can be w r i t t e n a s
f ( c ) > =
c T 6
with
formally).
X
T f = H(x,p)Z(p)
We r e c a l l from Theorem 3 . 7
@ ( x , y ) = since
Q
=
h
and
where
h e r e (up t o a c o n s t a n t ) .
F(h) = This
corresponds t o a fornula
(6.75) Clearly
@(x,y) = @
O(y,p)H(x,p)i(p)do(p).
g i v e n by (6.75) s a t i s f i e s f o r m a l l y
$fx,O) = f ( x ) .
Again
I
S i m i l a r l y c o n s i d e r ( c f (6.86)
P(Dx)@ = Q(DY)$ and
+(O,y)
=
(Rf)(y).
P(Dx)@ = Q(D,)@
-
and
(6.87))
Hence by d e f i n i t i o n s and assuming
175
MISCELLANEOUS TOPICS
p r e s c r i b e d we have
uniqueness with
Bf(y)
Theorem 6.18(F)
Under t h e h y p o t h e s e s i n d i c a t e d t h e u n i q u e s o l u t i o n
P(Dx)$ = Q(Dy)$, (6.76).
$(x,O) = f ( x ) ,
Q(0,y) = ( B f ) ( y )
I n t h i s event t h e kernels
and
and
K
$(x,y)
of
i s g i v e n by e i t h e r (6.75) o r
below a r e i n v e r s e s w i t h
Bf(y) =
f ( x ) = .
K(y,x) = lO(y,ll)H(x,v)do(v) ;
(6.77)
Formally one can t h i n k of t h e formulas (6.77) i n t h e form ( c f ( 6 . 8 1 ) )
and one is g e n e r a l l y m o t i v a t e d t o e x p r e s s
B
s t r u c t i v e t o r e c a l l here t h a t i f
in
2
{u E L ;
f u n c t i o n and f
(thus
u ' ( 0 ) = 01
u" E L 2 ;
T:f(x)
=
-1 [f(x 2
Bf(y) = f ( y ) ) .
P = -D2 then
+
+
y)
Note t h a t
H(x,p)
f
*g
= JoT:f(x)g(y)dy
2 L (0,m) =
- y)]
i(x
CosGy
rm
and r e c a l l t h a t
i n terms of
T
with
and
S.
It i s i n -
D(P) =
C O S ~ X i s a g e n e r a l i z e d eigen-
T
where
i(u)
=
T
(cf (3.75)).
I,
i s t h e even e x t e n s i o n of
[r(x+y)
+
z(x-y)1Cosqxdx
A l l f u n c t i o n s are extended
t o be even h e r e and c l e a r l y
=
1
7 [g(x) + g(-x)l
It is i n t e r e s t i n g t o look a t
indicated since then
@
= g(x).
g i v e n by (6.75) f o r
P = Q = -D
in
L ~ ( o , ~ as )
176
R . W. CARROLL
(Ty6 * f ) (x) = T:f(x)
Note a l s o t h a t
by a simple c a l c u l a t i o n and d e f i n e
s o t h a t (6.78) is t o be i n t e r p e r t e d a s
L e t u s o b s e r v e n e x t a way of l o o k i n g a t k e r n e l s such as
Thus t h i n k of N(E)
where
F+ C Fo
C
F-
and IB = -Q
P = A C = A 8 I
+
I 8
IB
and l o o k a t
Let
E
+
C
E
o
C
E
-
and
as above and n o t e ( c f Maurin [ 2 ] , Maurin-Maurin [ 3 ] ) t h a t from
(6.72) t h e g e n e r a l i z e d e i g e n s p a c e
(6.82)
as an element of t h e n u l l s p a c e
)I
Ho = Eo 8 Fo.
acting In
based on Theorem 6.17.
K
H(A) =
8
E(v)
H(A)
can be w r i t t e n as
for
F(p)dph(v,p)
v+p=A where
E(v)
t o values H(A)
where
(6.85)
(resp
v
F(p))
(resp
a r e t h e eigenspaces f o r Suppose
p).
A
(resp
IB) c o r r e s p o n d i n g
dim E(v) = dim F ( p ) = 1; t h e n t h e e l e m e n t s of
have t h e form
d P (A - p,p)
i s d e f i n e d by
U A
ph ( h
-
T,T)
-
d
ph ( h , O ) = sgn T d y ( X ) ~
1
dp(u)do(v)
v+p
(for
T < 0
[O,T)
means
[T,O)).
F u r t h e r i t s h o u l d be noted t h a t (6.69)
(6.72) a r e v a l i d w i t h any s p e c t r a l measures s i n c e terms l i k e
P(v)dp(v)
-
etc
177
MISCELLANEOUS TOPICS
w i l l be i n v a r i a n t under changes of measure
d p ( v ) = M(v)d;(v)
Thus when
etc.
t h e measures are a b s o l u t e l y c o n t i n u o u s r e l a t i v e t o Lebesgue measure one can simply u s e
dv
e t c . t o make some c a l c u l a t i o n s , i n p a r t i c u l a r t o compute dppX(h - p , p ) .
or equivalently
on
Thus f o r example i f
( 0 , ~ ) one f i n d s e a s i l y from (6.85)
(0,h).
and i s z e r o f o r p
(-=,O) for
then
p E
Now f o r
d<(X) = dX
(4,
v
+
min(h,O))
p= 0
If on
dp(X) = dX
and i s z e r o f o r
we c o n s i d e r
d p
dp(X) = do(X) = dc(X) = dX
(X -
with
(0,m)
on
u , p ) = dp
u
[min(X,O),
E
Ae(v) = v e ( v ) gives
with
=
d p
U X
Take
2 IB = -Q = D ) Y
so
o(A) = ( 0 , ~ ) and p % A
and
U
and (6.83) becomes, f o r
(-m,O)
(6.86)
%IB
o(IB) =
O(y,v).
above.
X
For
=
0
(X -
];,p) = dp
The n u l l s p a c e
H(x,v) 8 O ( y , v ) =
(e.g.
(-m,O)
on
e ( v ) = H(x,v)
so
w i l l thus be associated with generalized eigenvectors
H(x,v)B(y,V).
p 6 (0,A)
m).
A = P
f ( u ) = O(y,-p)
for
da(X) = dX
and from (6.85) a g a i n
(-m,m)
w h i l e IBf(p) = -Qf(p) = p f ( p ) N(C)
that
dpA(v,p)
A = P = -D2
d p
lJX
(A -
and
p , p ) = dp
on
do(p) = M(p)dp
h =
-
Note h e r e t h a t t h e r e is a s h i f t i n n o t a t i o n for
%
c (pi) = M(p)(Bf)(p) h
corresponds t o t h e w e have, f o r
Cp
of
(B
p
(--f
0
from (6.75) = ( 6 . 7 6 ) !
is the transmutation operator) the
(6.76).
On t h e o t h e r hand s e t t i n g
w
h = -p
Now
of (6.86)
i n (6.86)
dp(v) = N(w)dw,
, A
which c o r r e s p o n d s t o (6.75) w i t h by
-p
i n the
HO
of ( 6 . 7 5 ) .
c,(-v)
= N(U)f(v)
since
p
could be replaced
T h i s s u g g e s t s t h e f o l l o w i n g a s s e r t i o n which we
s t a t e as a f o r m a l theorem w i t h o u t a t t e m p t i n g h e r e a complete p r o o f . Theorem 6.19(F)
In the s e l f adjoint case with
o ( P ) = O(Q)
unique s o l u t i o n s
Cp
178
of
R. W. CARROLL
P p = Qa,
@(x,O) = f ( x ) ,
w r i t t e n i n t h e form (6.86)
Remark 6.20
@(O,y) = ( B f ) ( y ) , can b e o b t a i n e d from ( 6 . 8 . 3 ) and
-
(6.87).
L e t u s n o t e t h a t (6.78) and (6.81) are i n keeping w i t h t h e r e q u i r e -
ment i n S e c t i o n 2.3 t h a t
= 6 ( x )
since
O
= h
h e r e and i n w r i t i n g
f o r example
Another way of w r i t i n g (6.78) which d i s p l a y s
(6.90)
S
and
T
is
K(y,x) =
K(x,y) = Remark 6 . 2 1
r
Sy6 TX6 dp.
The f o r m u l a t i o n s of s e c t i o n 2 . 3 a r e f u r t h e r combined w i t h t h e pre-
c e e d i n g s p e c t r a l c o n s i d e r a t i o n s i n C a r r o l l [ 3 9 ] and a t r a n s f o r m t h e o r y i s developed. 0
Thus given
PH = pH,
as above w i t h
P*R =
<si(y,p),
=
6(y)
QO = p 0 , and
we w r i t e
H(0,p) = O(0,p) = 1, and
Q*fi = p 8
satisfying
H'(0,p)
= O'(0.p) =
< O ( x , p ) , DU= 6 ( x )
and
179
MISCELLANEOUS TOPICS
where t h e s u b s c r i p t s
u
or
i t y w i t h o u r n o t a t i o n above.
p
i n d i c a t e s d i f f e r e n t s p e c t r a l p a i r i n g s i n conformThen i f e v e r y t h i n g f i t s t o g e t h e r one h a s
Observe t h a t i n S e c t i o n 2 . 3 ( e q u a t i o n (3.31)) w e used t h e symbol
Q
1p
for&
and
f o r d ; t h e p a r t i c u l a r n a t u r e of t h e k e r n e l s t h e r e made t h i s p e r m i s s a b l e . For
f u r t h e r r e s u l t s on t h e t r a n s f o r m t h e o r y see Appendix 2. V
t o r s see a l s o Marcenko [ 4 ] and Gasymov 161.
For n o n s e l f a d j o i n t opera-
This Page Intentionally Left Blank
APPENDIX 1
We w i l l o u t l i n e h e r e someof t h e b a s i c i d e a s from f u n c t i o n a l a n a l y s i s which are
The p r o o f s are a l l s t a n d a r d and can be found f o r example i n
used i n t h e t e x t .
Bourbaki [l; 2 1 , C a r r o l l [ 4 ] , Dunford-Schwartz
[l] , Garsoux [ l ] , Ilorvgth [ l ] ,
Kgthe [ l ] , Marinescu [l], N a c h b i n [ l ] , Reed-Simon
[ l ] , Schwartz [ l ] , T r e v e s [ l ] ,
[I], E h r e n p r e i s ( 2 1 , Rosinger 111).
and Yosida [ l ] ( c f . a l s o Bjorck
can b e thought of as a k i n d of minimal working background.
The material
F i r s t assuming known
t h e concept of a v e c t o r s p a c e w e d e f i n e a seminorm i n a v e c t o r s p a c e p : F +W+
t o be a function
If in addition
p(x) = 0
F
topology on
over
F
C
satisfying
x = 0
implies
p
then
i s c a l l e d a norm.
Now a
0
c o n s i s t s of t h e p r e s c r i p t i o n of a f a m i l y of open s e t s
satisfy-
ing
(0
UDcl E UI
(any i n d e x s e t ) ,
nocl E ID
(finite intersection),
b e i n g t h e empty s e t ) ;
space operations
F
-t
F
point
(a,x)
-f
F
F
0,
E
0
is a topological vector space
ax :
(c
x F
-f
(x,y)
F,
-+
x
+
are c o n t i n u o u s w i t h t h i s topology ( s e e below).
x E
0
and
(TVS) i f t h e v e c t o r
y : F x F
F, and
x.
x.
I n a l i n e a r v e c t o r space
i s completely determined by F
every set i n implies V E N(x)
nNi
E
N(0).
one h a s
N(x)
( f i n i t e intersection)
there e x i s t s
1-4 E N(x)
t h a t t h e r e i s a unique topology on
N(x) = x
Here, given a f a m i l y
containing a set in
N(x)
F
such t h a t F
+ N(0)
N(x)
i t s e l f belongs t o (c)
x E N
V E N(y)
such t h a t
N(x)
-x:
-+
Thus a s e t i s
open i f and o n l y i f i t i s a nbh of each of i t s p o i n t s and we d e n o t e by nbhs o f
x
A neighborhood (nbh) of a
i s any set c o n t a i n i n g an open s e t c o n t a i n i n g
F
-t
if
N(x)
s o t h e topology
satisfying N(x)
(b)
N E N(x), and
for all
y
the
E
(a)
Ni E N(x)
(d) i f
W, i t f o l l o w s
is t h e f a m i l y of nbhs of
R. W. CARROLL
182
x.
Indeed one simply p i c k s a s open sets t h o s e
A family
x E 0.
x
of nbhs a t
such t h a t
c o n t a i n s a set
N E N(x)
B E B(x).
covered as t h e f a m i l y of sets c o n t a i n i n g a s e t i n
x
E B(x)
(b) B1,B2
(d) above h o l d s f o r
f o r each
A family
R(x).
B ( x ) , pre-
(a) each
B1 n B2 3 B g E B(x)
implies
i s t h e n re-
N(x)
x E F, q u a l i f i e s a s a f s n , i f f o r example
s c r i b e d f o r each contains
0 E N(x)
i s c a l l e d a b a s e of nbhs o r a fundamental system
B(x) C N(x)
i f every
0
and
B E B(x)
( c ) axiom
B(x).
D e f i n i t i o n 1.1 A l o c a l l y convex TVS ( a b b r e v i a t e d LCS) is a t o p o l o g i c a l v e c t o r space
whose topology is d e f i n e d by a f a m i l y of seminorms.
F
5 €1
{ x E F; p,(x)
s e c t i o n s of t h e
then a fsn
V
with
(E)
Thus i f
VJE)
=
is taken t o b e t h e f a m i l y of f i n i t e i n t e r -
B(0)
+ B(0).
B(x) = x
We r e c a l l a l s o t h a t c l o s e d s e t s a r e t h e complements of open sets and t h e c l o s u r e
6
of a s e t
B , d e f i n e d as t h e smallest c l o s e d set c o n t a i n i n g
i z e d a s t h e set of
x
such t h a t e v e r y nbh of
d i r e c t i n g t h e nbhs
U
of
one can produce a n e t
5
where N >
S
N
E F
a; and given
a set
B
if
2
and
+
y
such t h a t
B
a
y
x.
N
2a
and
B
x E
1. 6 *
S N E B;
2 6). SN
F
A net
y
*
N
-
(S,?)
1. y;
is eventually i n
Sa
converges t o
-
xu E B n U
is a p a i r
F
1. 6, 6 L
Thus i n g e n e r a l a s e t
converges t o a p o i n t i n
Further,
then picking
a
(i.e. y
B.
intersects
Recall here t h a t a n e t i n
x.
with
e v e n t u a l l y i n e v e r y nbh of o n l y i f no n e t i n
by i n c l u s i o n , i f
d i r e c t s t h e s e t of
a.6 3
3 6
x
x
B , can be c h a r a c t e r -
x
is
SN
i s c l o s e d i f and
B C F
Hence c l o s e d s e t s , and
B.
c o n s e q u e n t l y open s e t s , a r e completely c h a r a c t e r i z e d by convergence p r o p e r t i e s . T h e r e f o r e t h e topology i s completely d e s c r i b e d i n terms of convergence. remark a l s o t h a t a t o p o l o g i c a l s p a c e x,y E F
3
open sets
U
3 x
and
V
F 3
We
i s s e p a r a t e d o r Hausdorff i f f o r any
y
UnV
such t h a t
= @;
a l l TVS w i l l b e
assumed s e p a r a t e d u n l e s s o t h e r w i s e s t a t e d .
Now t o c o n s t r u c t t h e Schwartz d i s t r i b u t i o n s i n IRn c r e a s i n g sequence of compact s e t s w i t h of
Cm
“K,
=Rn.
complex valued f u n c t i o n s w i t h s u p p o r t i n
let
Let Km.
K
m
Qm
C
K,el
be an in-
be t h e v e c t o r s p a c e
The s u p p o r t of
@,
written
183
APPENDIX 1
c l o s u r e of t h e s e t of
x
where
where
P1 Dp = D1
(1.4)
... Dn'n
1 V(Km, ,; s) =
urn;
1 -) m' n
p
D e f i n e now, w i t h
Np(@) 5 E
nV (K
f i n i t e intersections for
n = 1,2,
for
]PI
5
IpI = Cpi,
s}.
IpI < s , and t h i s p a r t i c u l a r s e t of
for
... and
s =
O,l,
...
B(0)
suffices t o define
Dm a s i n d i c a t e d i n D e f i n i t i o n 1.1. To v e r i f y t h a t t h e v e c t o r s p a c e o p e r a t i o n s
(a,$)
-+
a @ , (@,$)
Definition 1.2 i f f o r any
@
+
A map
+ $,
+
3
W E N(f(x))
@
and
f : F
s h o u l d be open f o r any open
G
-+
are c o n t i n u o u s we r e c a l l
-@
between two t o p o l o g i c a l s p a c e s i s c o n t i n u o u s a t
V E N(x)
such t h a t
f(V) C W.
+ N(0))
Dm.
and t h i s i s r o u t i n e now f o r
a countable f s n
B(0)
Equivalently
f-'(U)
U.
For l i n e a r maps between TVS one need o n l y check c o n t i n u i t y a t x
Dm by
D e f i n e seminorms i n
Dk = 3/axk.
with
V(Km,&,s) = { @f
Evidently
x
0.
f(x)
N (@) = suplDP@l "Ic,
(1.2)
in
$I E 0 ; e q u i v a l e n t l y i t i s t h e
i s t h e smallest c l o s e d set o u t s i d e of which
supp @,
0
W e note also that
(since
N(0)
in
N(x) =
Dm h a s
a s d e f i n e d and t h u s c l o s u r e and convergence can be d i s c u s s e d
i n terms of sequences ( n e t s a r e n o t n e c e s s a r y ) . which means t h e r e can b e d e f i n e d on
F =
Dm
Such a LCS w i l l b e m e t r i z a b l e
a metric f u n c t i o n
d : F .x F
+
R+
satisfying
and
d(x,y)
f s n at
x.
=
0
-
x = y
Vn(x) = f y E F; d ( x , y )
5
1
form a
Such a m e t r i c can e a s i l y b e c o n s t r u c t e d b u t we omit d e t a i l s ( s e e
C a r r o l l [ 4 ] , Kothe [l]). given any
such t h a t t h e sets
V E N(0)
3 8
We r e c a l l n e x t t h a t a n e t such t h a t
Sa
-
S E V Y
Sa for
i n a TVS is Cauchy i f
a,y
5 B.
Such a s p a c e
F
184
R. W. CARROLL
is complete if every Cauchy net converges. When if every Cauchy sequence converges. complete.)
F
is metrizable it is complete
(Unless otherwise stated all spaces will be
A complete metrizable LCS is called a Frechet space and a complete
normed space is a Banach space.
Dm
routine argument for completeness, the result that Let now
D
Clearly
Dm C
D
on
be the vector space of all
0
Dm now yields, with a
The construction of
and we denote by
is a Frechet space.
Co3 functions with compact support in the injection im :
im
the finest locally convex topology such that all
Dm .+ D.
Rn.
We shall put
im are continuous. Here
finest means strongest in the sense of having the largest collection of open sets or nbhs. We shall construct the nbh system explicitly and recall first that a set is disced if x E B
B
W
is convex if x,y E B
a TVS over
(c
let
(a1
and
and
5
0
Fo.
Then B
C
+
(1
functional as pB(x) {x E F; pB(x)
=
2
inf{h;
1).
C
-
in a TVS over
1)y E B.
F
for
is
i
is convex if it is convex in
1x1
when
F one defines the gauge of B
x E XBl
If F
with multiplication by
is called absorbing if for x t F x E XB
B
a convex disced absorbing set B
=
A < 1 implies Xx
be the same space over R
F
considered as an automorphism of Fo. A set
C
A set B
1 implies ax E B.
ho.
Given
or the Minkowski
X > 0. Then pB is a seminorm and
Thus if one has a TVS with a fsn consisting of convex
disced absorbing sets then the topology can be defined by a family of seminorms as in Definition 1.1 s o the space is locally convex. Now for 27 in
Dm
and write
r(UV ) m
for the convex disced envelope in
D
the smallest convex disced set in tion of such r ( W ) m
r(Wm)with im(Vm) im
containing B = W m
=
Vm
we see that
im
since
iil(V)
The sets B (y,x)
+
yx
=
0 =
for T' V n
r(UVm)
then V 3 r(U(V
Dm. Hence V
E
N(0)
D of UV,;
Let
and take it as a fsn for a topology T
A be the collec-
D. Since m'
on
Dm))
fl
on
D
and
with V n
for T and
T
Dm E
is a convex
V
N(0)
is continuous; hence the topology T
D
in
Dm
is stronger than T'.
are absorbing since they are taken as nbhs of
the finest locally co,ivcx topology on
this is
is continuous. On the other hand if
is continuous for a locally convex topology T'
disced nbh of
.
let Vm E N(0)
0
and
is locally convex and indeed
such that the
i m
are all continuous.
API'ENDIX
D
The s p a c e
1
185
w i t h t h i s topology i s c a l l e d t h e s t r i c t i n d u c t i v e l i m i t of t h e
and one e a s i l y checks t h a t i t i s independent of t h e d e f i n i n g sequence
uKm = l R n .
with
R
We remark a l s o t h a t i f
R
i s a sequence of compact sets i n
with
clRn is any open s e t and UKm =
R
Km
m'
Km+l
K C K m m+l
n(R)
t h e n one d e f i n e s
in the
same manner.
There are o t h e r e q u i v a l e n t ways of d e f i n i n g t h e topology of
D
( c f . Schwartz [ l ] ,
Hbrmander [ l ] ) b u t t h e above method is p e r h p a s t h e most r e v e a l i n g .
A crucial
p r o p e r t y of t h i s t o p o l o g y , which a l l o w s one u s u a l l y t o r e p l a c e n e t arguments by sequence arguments, i s determined by
Theorem 1 . 3
A l i n e a r map
restriction
f
= fl
:
f : 17
D,
+
+
F,
F
a L C S , i s c o n t i n u o u s i f and o n l y i f each
i s continuous.
F
*m
To prove t h i s simply n o t e t h a t V E B(O) f-'(V)
in
E N(O)
F. in
fm i s continuous.
D'
-
Take
D
V
is continuous
f
f-'(V)
convex and d i s c e d s o t h a t
f-l(V)
n Dm
E N(0)
in
D,
f-'(V)
E N(O
in
( s i m i l a r l y one o b t a i n d
D'(R)
+
for
for
i s a l s o and t h e n
which i s e q u i v a l e n t t o s a y i n g
We now d e f i n e t h e Schwartz d i s t r i b u t i o n s i n lRn
of c o n t i n u o u s l i n e a r maps
D
a s t h e space
Ic; t h i s i s c a l l e d t h e ( t o p o l o g i c a l ) d u a l of
R
CIRn
open).
That i t i s n o t a t r i v i a l s p a c e
f o l l o w s from t h e Hahn-Banach theorem below b u t one can a l s o produce e l e m e n t s of
D'
directly.
If
f E LfOc
(1.7)
For example t h e
then
f
6
" f u n c t i o n " is d e f i n e d by
d e t e r m i n e s an element
f E
D'
( @E
D)
by t h e r u l e
=
The i d e a i n c o n s t r u c t i n g d i s t r i b u t i o n s i s p a r t l y t o have a s p a c e where one can d i f f e r e n t i a t e che o b j e c t s a t w i l l s o t h a t a t h e o r y of d i f f e r e n t i a l o p e r a t o r s can be developed. if
f
C1(iR)
That t h i s i s p o s s i b l e and n a t u r a l f o l l o w s now from t h e f a c t t h a t f o r example t h e n
D
186
R. W. CARROLL
Then i f
T E D'(Rn)
we d e f i n e
= -
(1.8)
D
where
=
k
a/axk.
D Q> k
Q
T h i s makes s e n s e s i n c e t h e map
Q
c o n t i n u o u s so t h a t
: 0
+
+
+
DkQ : 0
0 is o b v i o u s l y
+
is c o n t i n u o u s and d e f i n e s t h e r e f o r e a
(E
distribution.
W e w i l l d e f i n e some f u r t h e r s p a c e s of d i s t r i b u t i o n s now and t h e n g i v e some a d d i t i o n a l g e n e r a l i n f o r m a t i o n about f u n c t i o n a l a n a l y s i s which should be of u s e i n t h e text.
T E
F i r s t for
for a l l
Q E D(n)
port i n
R.
s u p p o r t of of
x.
Rn
Let
s = O,.,
E
E.
K
m
N
P
supp T).
f u n c t i o n s w i t h compact sup-
Cm
E
supp T
T
if
and
S
0
be t h e space of
V(Km,
E.
for
1 n'
s),
m = 1,2
any polynomial i n t h e
Q(D)
bounded f u n c t i o n on lRn (k = O , l ,
...;
for
5
for
IpI IpI
2
m
q = 1,2,
and
E
...;
m
Q
E S.
=
O,l,
(Q) = s u p P ?k
n
a c o u n t a b l e f s n and
S
=
E w i l l be
is a F r e c h e t s p a c e .
l/lxl.
Thus i f
then
P(x)Q(D)Q
0
A s a f s n of
in
S
1x1
P(x)
+ m,
along
i s any
is a continuous
we t a k e t h e sets
...I
x E En. Thus one is d e a l i n g w i t h seminorms
where
m
when
Dk
m
n = 1,2
d e c r e a s i n g as
f u n c t i o n s on lRn
Coo
VK
,... ,
,...,
T h i s i s a c o u n t a b l e f a m i l y and
w i t h a l l d e r i v a t i v e s , f a s t e r t h a n any power of polynomial and
i n any open nbh
as b e f o r e b u t r e f e r r e d t o func-
V(Km,E,s)
m e t r i z a b l e ; s i n c e i t is a l s o e a s i l y s e e n t o be complete Next l e t
0
(no r e s t r i c t i o n on s u p p o r t s ) w i t h topology d e f i n e d
V (K ,E), P m
as a fsn at
x
Thus
Then t a k e t h e sets
...,
if
be a sequence of compact sets a s b e f o r e w i t h
C Kmel
(a),
R CRn
i n an open s e t
i s open and i t s complement i s d e f i n e d t o b e t h e
T = 0
where
T = 0
i s t h e space of
E = Cm(lRn)
and d e f i n e
Q
D(R)
(written
T
Let now
tions
one s a y s
where
W
a s follows.
D'
(1
+
1x1 2 ) k I D P
$1
for
N
m,k
x ERn.
i s e a s i l y s e e n t o be complete; hence
s
= sup n
P ,k
Again we have
is also a
187
APPENDIX 1
Frechet space.
E'
The two s p a c e s (recall
F'
p a l y an i m p o r t a n t r o l e i n t h e t h e o r y of d i s t r i b u t i o n s
i s t h e s p a c e of c o n t i n u o u s l i n e a r maps
E' C D'
course t h a t prove t h a t
S'
and
E'
S'
and
D'
C
F
-+
One must show of
C).
( i t does r e q u i r e p r o o f ) and t h e n one can
i s t h e s p a c e of d i s t r i b u t i o n s w i t h compact s u p p o r t .
The s p a c e
S'
on t h e o t h e r hand p r o v i d e s a n a t u r a l s e t t i n g f o r t h e F o u r i e r t r a n s f o r m d e f i n e d as
6 E
follows.
For
(1.10)
?(@I=
s
set m
$(x)exp i<s,x>dx
$(s) =
-m
where
Cs,x> = 1 x . s 1
The i n v e r s i o n formula i s ( c f Titchmarsh [l] f o r p r o o f )
i'
.m
F
I t is e a s i l y s e e n t h a t (< , >
P a r s e v a l formula
valid for functions in : S
+
C
< ,>
where
: S
S.
i s an isomorphism of LCS.
T E S'
Let
denotes
This extends
S
Next we r e c a l l t h e
i s simply i n t e g r a t i o n )
t h i s d e t e r m i n e s an element of
.
-t
and c o n s i d e r t h e l i n e a r map
S - S' S'
duallty.
Since
which we d e n o t e
F t o a map S '
+
S'
by
F: S FT
+
S
so t h a t
d,
-+
< T , Fd,>
is continuous
=
which w i l l b e an a l g e b r a i c isomor-
phism.
I n o r d e r t o d e f i n e c o n v o l u t i o n , and f o r o t h e r a p p l i c a t i o n s i n C h a p t e r 1, we need t h e i d e a of t e n s o r p r o d u c t .
Thus l e t
t o be t h e s p a c e of b i l i n e a r forms u n t i l otherwise indicated.)
ZBk(fk, 8), indices.
=
Let
C(aR A.
A
x C
=
) with Ca.(f. J ~ " j
l i n e a r combinations s p a c e s t r u c t u r e on
Let
F
A
+
by
F
and +
G
3
E
ge)
where t h e
C)
b e t h e set of a l l formal f i n i t e
a.
Then l e t u s d e f i n e a v e c t o r
aXa.(f g . ) = Caaj(fj, gj) J j' J
6,)(fR,
B(F,G;
(The c o n s t r u c t i o n s a r e a l g e b r a i c
t.
A ( F , G)
a.
be LCS and d e f i n e
R
and
Za.(f g.) J j' J
indices include the
j
+ and
be t h e s u b v e c t o r s p a c e o f f i n i t e l i n e a r combinations w i t h
k
188
R. W. CARROLL
C
coefficients in (f,gl
+
-
g2)
of e l e m e n t s of t h e form
(f,gl)
-
F 8 G
tensor product
and s i n c e
b
a ( f , g ) , and
b : A
t o d e f i n e a l i n e a r map
F 8 G
+
b : F 8 G
i t d e f i n e s a b i l i n e a r map
+
A.
v a n i s h e s on
A/Ao
C
L(H; K)
-
H
+
+
-
(f,wd
-
(f2,g),
a(f,g).
A/Ao.
If
Then t h e
b E B(F,G; C)
b(Caj(fj,gj)) =
w e can f a c t o r i t through t h e q u o t i e n t
Similarly i f
C.
b(f,g)
d e n o t e s l i n e a r maps
(fl,g)
by t h e r u l e
C
=
b ( f 8 g)
K.
i s a l i n e a r map
b
from
B(F,G; E)
one h a s a 1-1 correspondence ( a l g e b r a i c ) between where
f2,g)
is d e f i n e d t o be t h e v e c t o r s p a c e
we can e x t e n d it t o b e a l i n e a r map Cajb(fj,gj)
-
(af,g)
(f,g2),
+
(fl
F x G
to
Thus
E.
L(F 8 G ; Ic)
and
The t e n s o r p r o d u c t can a l s o b e
c h a r a c t e r i z e d i n terms of a u n i v e r s a l f a c t o r i z a t i o n p r o p e r t y ( s e e e . g . C a r r o l l Next l e t u s r e c a l l t h a t i f
[4]).
i s a TVS t h e r e e x i s t s a unique (up t o i s o -
E n
morphism) complete TVS denoted by
E
such t h a t
E
i s isomorphic a s a TVS t o a
A
dense subspace of F
and
E.
There a r e v a r i o u s n a t u r a l t o p o l o g i e s on
due t o Grothendieck [ l ] .
G
perty that
gTIG ;
L(F
The
1~
where
C) = B(F,G; C)
w e w i l l n o t have o c c a s i o n t o u s e t h e
TI
F 8 G
f o r LCS
topology is c h a r a c t e r i z e d by t h e pro-
B
and
L
topology.
r e f e r t o c o n t i n u o u s maps;
The
F 8 G
topology on
E
is t h e topology of uniform convergence on p r o d u c t s of (convex, d i s c e d ) e q u i c o n t i n uous s e t s i n t h e set
E
V E N(0) F 8E G
F'
and
the E
E
if
Here a set
G'.
X
nh'-l{lzI in
in
E
F'
<
€1
for
F
topology and when
E
GTI F + E 6
F
+
0
is e q u i c o n t i n u o u s i f g i v e n
i s a nbh of
€1
0
f' E A
uniformly f o r
Thus a n e t
h ' E H.
for
and
g'
B
GE
G.
In general the
fJi
d e f i n e d by
one can i d e n t i f y
<W, @ ( x , y ) > = <Sx,
Y'
Sx 8 T
Dy;
C)
0
A C
when
F.
Most o f t h e d i s t r i b u -
Y
with the d i s t r i b u t i o n
Now f o r
W E
@ ( x , y ) > >= > f o r Y'
Here one n o t e s t h a t t h e r e i s a n a t u r a l map L(Dx 8
-+
i s a so c a l l e d n u c l e a r s p a c e t h e c a n o n i c a l i n j e c t i o n
is an isomorphism i n t o , f o r e v e r y LCS
T E p' Y
and
e
topology i s s t r o n g e r t h a n
TI
t i o n s p a c e s a r e n u c l e a r b u t we w i l l not e x p l i c i t l y u s e t h i s p r o p e r t y . S E
3
E; equivalently
in
are convex, d i s c e d , e q u i c o n t i n u o u s sets; t h e completion i n t h e
topology is denoted by E
h' 6 H
h'(V) C { I z I <
such t h a t <ea, ( f ' , g ' ) >
B C G'
H C L(E; C) = E'
d e f i n e d by e x t e n d i n g t h e map
Di
8
< S 8 T.
into
(nx 8
D
Y
D i ,Y
6E
)* =
@ 8 $> = <S,Q> and
X9Y
.
189
APPENDIX 1
D
continuous
Dx
and s i n c e
(c
-f
XYY
(Dx 63
in
Sx 63 Ty
acts l i k e
W
thus
uy)*.
uy
8
But
can e a s i l y b e shown t o be
W
8 T X
0
in
1
ox 8 uy
which g i v e s a n a l g e b r a i c map
X,Y
Y
w e have XSY
1
W = S
D
is c l e a r l y dense i n
I
ux,y.
into
The
Schwartz k e r n e l theorem, which w e do n o t n e e d , s t a t e s i n f a c t t h a t 1 , .
1
I
1
px e9E D
=
ox
Y topologies).
=
1
a l g e b r a i c a l l y and t o p o l o g i c a l l y ( f o r s u i t a b l y d e f i n e d
ux(uy)
9Y
Using t h e t e n s o r p r o d u c t one c a n now d e f i n e t h e c o n v o l u t i o n
*
S
T,
when i t makes s e n s e , by
1
(5,n)
1
ux, S E ox, (5 + n ) d o e s
$ E
for
-+
T
and
t
ux.
One must be c a r e f u l w i t h s u p p o r t s h e r e s i n c e
n o t have compact s u p p o r t i n g e n e r a l . 1
demonstrated t h a t i f
S E
1
E
and
T E
For s u i t a b l e f u n c t i o n s
defined.
*
S
*
*
T = D S k
*
T = S
and
S
DkT
=
S
*
more s p a c e s which p l a y a r o l e i n c o n v o l u t i o n . t i o n s on lRn fDp$ p
i s bounded f o r
and any
(OM c
FO,
such t h a t
f E S
<
Dp$
f E
and a l s o
OM.
*
Let
@
and
E
0,
m
0M
be t h e s p a c e of
fDP@a + 0
@a
and w e s a y
+
u n i f o r m l y i n lRn,
e v i d e n t l y ) and s t i p u l a t e t h a t
OM
T) =
We mention two
0
0,
in
S a E 0,
i s t h e s p a c e of n u l t i p l i e r s
-
C
func-
p).
Thus
i f f o r any
Define f u r t h e r
I
Then
is w e l l
converges
S
1 +
S
Fscl
0'
C
=
con-
1
and
0,
maps
1
1
S
Dk(S
makes s e n s e .
T
I
verges i n
ux
i s bounded by a polynomial (depending on
s
one h a s
T E
of c o u r s e
T
assuming
DkT
I
*
S
however t h e n T
*
D 6 k
F u r t h e r i t is e a s i l y v e r i f i e d t h a t
D 6 k
ux
I t can be e a s i l y
+
S
under c o n v o l u t i o n .
I t w i l l b e u s e f u l t o d e f i n e a l s o t h e i d e a of t h e o r d e r of a d i s t r i b u t i o n .
Dm be t h e s;Jace of IpI 5 m). Topologize
Clearly
Um (K
)
Cm
Dm
=
f u n c t i o n s w i t h compact s u p p o r t ( i . e . L,
Dm(Kn) where
Kn C Kn+l
i n t h e obvious manner and p u t on
vex topology such t h a t a l l
tion also applies for
i
d"(Q)
:
dn(Kn)
when
+
b"
Sl C Rn
Dm
open.
U
Kn = Rn.
t h e f i n e s t l o c a l l y con-
a r e continuous. ig
for
DP@ E Co
a r e compact w i t h
Let
The same c o n s t r u c -
One h a s a sequence of
190
R. W.
6""
c o n t i n u o u s maps
+
b"
D'm
and i n j e c t i o n s
d i s t r i b u t i o n i s s a i d t o be of o r d e r
I t c a n b e shown t h a t e v e r y
K
CARROLL
m
E
T E
-+
D'.
tives are in x 1 supp T
T =
If
and
T E
5
Ip1
m
Elm
=
Illm n
then
E'
' =D 1.
DpP
(lpl
P
@ E
and
If
lo
Em
5
T E m)
A
L(b";C).
Dfm-'.
but not t o
I f one w r i t e s
0 CO(Kn) = { @ E Co; s u p p @ C KnI
of measures such t h a t
P
Dim
i f i t belongs t o
d e f i n e d t o b e t h e s p a c e o f Radon m e a s u r e s on IRn(K
p
D V m=
where
i s of f i n i t e o r d e r f o r example.
f o r t h e s t r i c t i n d u c t i v e l i m i t of
f i n i t e family
D'm+l
K'
then
Dim
is
3
then
a
where t h e d e r i v a -
i s such t h a t
LIP@ = 0
for
0.
=
We m e n t i o n a l s o a n e l e m e n t a r y b u t u s e f u l f a c t a b o u t c o n v e r g e n c e o f v e c t o r v a l u e d functions.
R
limit
t E R
If
as
t
+
f a c t is t h a t i f
and
where
i f f o r any sequence
to
F
and
E
f(t) E F
are
LCS
l i n e a r map f r o m a d e n s e s u b s p a c e
with
A C E
f(tn)
to
TVS
tn
+
F
c o m p l e t e and
to
t i n u i t y t o a u n i q u e c o n t i n u o u s l i n e a r map
is e.g. a
F
then
F f : E
+
f
F.
then
R.
+
f
f(.)
has a
Another standard
i s a continuous
c a n b e e x t e n d e d by conNow l e t u s t u r n t o some
observations about d u a l i t y theory.
Theorem 1 . 4 .
E
(Hahn-Banach)
Let
p
b e a c o n t i n u o u s seminorm i n a complex
R
and s u p p o s e g i v e n a l i n e a r f o r m
that
lR(x)l
Eorm
5
on
p(x) with
E
In particular i f and e x t e n d
L
for
la(p)l
xo E E
to
x E F.
E.
5
for
R
such
F C E
L may b e e x t e n d e d t o a c o n t i n u o u s l i n e a r
Then
p(x)
define
d e f i n e d on a c l o s e d s u b s p a c e
LCS
x E E.
{xo} = { a x o ; ci E lc3
on
by
k ( a x O ) =u.p(xO)
E
T h i s g i v e s t h e e x i s t e n c e of n o n t r i v i a l el ement s of
so
l
one h a s
nontrivial duals
Now l e t u s d e f i n e a set of
0
B
E
for
E
in a
o r e q u i v a l e n t l y by a n y
a
B(0);
n o t e s t h a t Cauchy s e q u e n c e s a r e hounded. d u a l i t y i f t h e r e is a b i l i n e a r f o r m any G
x # 0
in
t h e r e is an
F
there is a
x E F
y E G
such that
its dual then t h e n a t u r a l p a i r i n g
Two
<*,.>
TVS
>
F
F x G
on
such t h a t
,
B C XV
t h i s means
< x , y > # 0. <
s i n c e c o n t i n u o u s seminorms e x i s t .
t o b e bounded i f i t i s a b s o r b e d by a n y
TVS
V
LCS
and
for G
Ihl
2
XO.
nbh One
a r e s a i d to he i n
w i t h t h e p r o p e r t i e s (1) f o r
<x.y> # 0
Clearly i f
( 2 ) f o r any F
is an
LCS
e s t a b l i s h e s a d u a l i t y between
y
# 0
and
F
in F'
and
APPENDIX 1
1
191
I
F
Let
on
F
1
F
be d u a l t o
F
with
F
an
LCS.
Then t h e s t r o n g t o p o l o g y
B(F , F )
1
i s t h e t o p o l o g y o f u n i f o r m c o n v e r g e n c e on bounded s e t s o f 1
pB(f ) = s u p
t h e form h e r e as a
F;
seminorms of
1
fsn
Il
for
f E B
bounded c a n b e u s e d .
t h e p o l a r s of c l o s e d bounded d i s c e d s e t s
Bo
of a d i s c e d
If
F
B
B C F
One c a n u s e
where t h e p o l a r
is
I
i s a normed s p a c e t h e n the s t r o n g t o p o l o g y i n 1
by
Ilf
11
h a s t h e norm d e t e r m i n e d
F
I
I
sup Il
=
/ I f 1 1 '1.
for I
t h e c o a r s e s t t o p o l o g y on continuous.
Thus
with a
of
F1
I
making a l l t h e l i n e a r maps
F
EF = IIFE
C
and
o(F'.F)
f
on
U(F',F)
The weak t o p o l o g y
: F
0
+
E
is t h e induced product topology 1
fsn
is
I
I +
F
c o n s i s t i n g of f i n i t e i n t e r s e c t i o n s o f s e t s
{f
'
E F ;
I
I l
5
A barrel in a
El.
is a c l o s e d , convex, d i s c e d , a b s o r b i n g s e t and
TVS
a space i s c a l l e d barreled i f every b a r r e l is
nbh
2
of
A Baire-space
0.
is a
t o p o l o g i c a l s p a c e such t h a t e v e r y c o u n t a b l e union of c l o s e d sets w i t h o u t i n t e r i o r p o i n t s i t s e l f h a s no i n t e r i o r p o i n t .
A complete m e t r i z a b l e space i s B a i r e .
Theorem 1 . 5 .
E v e r y Baire
is barreled.
LCS
A s a c o r o l l a r y we s e e t h a t e v e r y F r e c h e t o r Banach s p a c e i s b a r r e l e d a n d a f u r t h e r
P
e l e m e n t a r y a r g u m e n t shows t h a t
is a l s o barreled.
Now l e t
and
E
F
and c o n s i d e r t h e v e c t o r s p a c e
L(E;F)
run over a
composed o f c o n v e x c l o s e d d i s c e d sets a n d
fsn
0
of
in
F
of c o n t i n u o u s l i n e a r maps
o v e r a l l bounded c o n v e x d i s c e d s e t s i n
fsn
0
of
composed of sets
N(B,V)
=
E.
s t r n n g l y bounded i f
H C xN(B,V)
that
6 H II
y
'-1
of
for
1x1 5 xo
1 r y
'-1
I
y
€sn
0
o r equivalently
B C
F.
-t
LCS
Let
V
B
L(E;F)
run has a
The weak t o p o l o g y
composed o f f i n i t e i n t e r s e c -
N(x,V) = { u E L ( E ; F ) ; ~ ( x )C V: x E E ) .
t i o n s of sets
for
The s t r o n g t o p o l o g y i n
{u E L ( E ; F ) ; u(B) C V 3 .
o r topology of s i m p l e convergence h a s a
E
be
H
A set
C L(E;F)
i s then
I
o r equivalently
U
y ( B ) C XV
9
(V)
for
y
E H.
Thus i t i s r e q u i r e d
1
(V)
w e a k l y bounded
for Q
fI y
y
E H
'-1
a b s o r b e v e r y bounded set.
Similarly
H
C
L(E;F)
is
1
(V)
for
y E
H
absorbs every p o i n t .
A set
H C L(E;F)
192
R. W .
is equicontinuous if if H
y
1
'-1 (V)
for y E H
E
the other hand it is easy to see that if
H
C
disced nbh
0 in a
of
LCS
x
- y E V.
A set
V.
One knows that B
F
then the weak topology on
H
Theorem 1.6. (Banach-Steinhaus)
Let
Uo : E
converging simply to a map
formly on precompact subsets of
+
Uo
C
+
E.
E
in a TVS
a convex V
if
is said to be precom-
F
Q
+V
B
by sets xi
B
is precompact
is equicontinuous for
L(E;F)
and this leads to
F.
Then
If
Ua
E
Uo
to a map
U
a sequence
E L(E;F)
and
L(E;F)
Un
+
uni-
Uo
is a simply counded net con-
L(E;F)
uniformly on precompact subsets of
Recall next that a set A
C
be barreled and
E
verging simply on a dense subset E C E Ua
B
V
is equivalent t o the topology of
uniform convergence on precompact subsets of
and
Obviously
is small of order
is compact
and complete. It is easy to show now that if H and
A C F
as indicated there is a finite covering of
V
which are small of order
E
0 in E.
of
is barreled then every weakly bounded
one says a set
F
it follows that
x,y E A
pact if for any
LCS
nbh
is both strongly bounded and equicontinuous. Now for
L(E;F)
for any
is a
is equicontinuous then it is both weakly and strongly bounded. On
L(E;F)
C
I)
CARROLL
Uo : E
+
F
then Uo
€
L(E;F)
E.
is said to be relatively compact if A
E
is compact and compact means of course that a n y covering by open sets has a finite subcovering. A topological space is locally compact if every point has a compact nbh.
C(E;F)
means continuous maps E
+
F,
not necessarily linear; we cite
then Theorem 1.7. spaces).
(Ascoli)
Then H
C
Let
C(E;F)
E
h E
One can show that B
C
HI
-
F a TVS
E
H
is equicontinuous and for all
is relatively compact in
D is bounded
-
B
C
LCS
x E E
the
F.
Dm for some m and is bounded there.
Using the above theorems it then follows that bounded sets in D tively compact. A
(both Hausdorff
is relatively compact for the topology of uniform con-
vergence on compact subsets of set H(x) = {h(x);
be locally compact and
or
E are rela-
which is barreled and in which every bounded set is rela-
tively compact is called a Montel space so that
D and E are Montel spaces (as
APPENDIX 1
s).
a l s o is
193
These f a c t s a l l o w one t o a s s e r t t h a t i f a sequence
uous i n t h e s t r o n g t o p o l o g y .
D', El,
p l e t e and
E'
Montel; then E
e E E
into
s'
and
a r e a l s o complete.
I f t h i s map i s o n t o
t h e s t r o n g topology of
E"
reflexive.
v,
Next l e t u
The s p a c e s
u : E
F
-t
c o n t i n u o u s from
F'
EE' = IIEIIc.
E
S,
and
E'
-+
E
-f
<e,e'> : E'
and
to
a r e com-
(Mackey-Arens)
duality (i.e.
yields
where
T(E,E')
embedding
E
is c a l l e d
are
LCS.
t~ : F'
+
Then
E'
is
Here o n e w r i t e s
t~
and
U(F,F')
is continuous f o r
can be
o(F',F)
T
on
E
i f and o n l y i f
is compatible w i t h
T
E'
5(E,E') c T
as dual
E - E'
i s a t o p o l o g y of u n i f o r m
which a r e convex d i s c e d and
yields
he
U(E',E)
compact.
c
.
T(E,E'
Further
if a linear
u : E
tinuous f o r
'r(E,E')
be
and i n p a r t i c u l a r , u s i n g t h e T i c h o n o v t h e o r e m , t h e s t r o n g l y c l o s e d
T(E,E')
F
-+
is continuous f o r
and
unit ball i n the dual
E'
T(F,F').
If
U(E,E') E
of a Banach s p a c e
E
and
x g A
o(F,F')
t h e n i t i s con-
E
is weakly compact. A
A C E
A geometric
is a c l o s e d convex s e t i n a
t h e n t h e y a r e s e p a r a t e d by a c l o s e d h y p e r p l a n e .
quence e v e r y c l o s e d convex containing i t .
and
i s b a r r e l e d i t s i n i t i a l t o p o l o g y must
v e r s i o n o f t h e Hahn-Banach t h e o r e m a s s e r t s t h a t i f TVS
and
is t h e topology of uni form conver-
E
on
U(E,E')
compact s e t s i n ' E ' .
a s dual) E'
(J(E',E).
continuous f o r
A topology
convergence on sets covering
Ic
E'
f o r e x a m p l e i s t h e i n d u c e d t o p o l o g y from
u(E,E')
u
+
then
F
o(F',F)
Theorem 1.8.
E
E
and f r o m
u(E',E)
on
E
and i t s t r a n s p o s e
The Mackey t o p o l o g y
T
0'
and
are a l l r e f l e x i v e .
= <e, u(f')>
E'
i s contin-
with strong dual
o(F,F')
and
g e n c e on c l o s e d d i s c e d
is
is c a l l e d s e m i r e f l e x i v e and i f i n a d d i t i o n
t
Thus a t o p o l o g y
e'
TVS
to
I n f a c t any l i n e a r
r e p r e s e n t e d as
is a
E
c o i n c i d e s w i t h t h e t o p o l o g y of
U(E,E')
= <e,tu(f')>
o(E',E).
If
b e a c o n t i n u o u s l i n e a r map w h e r e
is c o n t i n u o u s from
D'
D'
a r e Montel s i n c e t h e s t r o n g d u a l o f a M o n t e l s p a c e i s
d e t e r m i n e s a c o n t i n u o u s l i n e a r map
E".
-+
We m e n t i o n a l s o i n p a s s i n g t h a t
s'
and
D'
Dk :
weakly c o n v e r g e n t t h e n i t i s s t r o n g l y c o n v e r g e nt ; a l s o
T. E J
A s a conse-
is t h e i n t e r s e c t i o n of t h e c l o s e d h a l f s p a c e s
S i n c e c l o s e d h y p e r p l a n e s a r e d e t e r m i n e d by
E'
t h e c l o s e d convex
194
R. W.
c o n v e x sets i n
are t h e same f o r a l l t o p o l o g i e s c o m p a t i b l e w i t h
E
I n f a c t t h e bounded sets i n
ity.
CARROLL
E
E
-
dual-
E'
a r e t h e same f o r a l l s u c h t o p o l o g i e s .
Next
w e mention
Theorem 1.9.
(Theorem o f b i p o l a r s )
t h e c l o s e d convex e n v e lo p e f o r
A C E
Let
U(E,E')
of
with
LCS.
is
Aoo
Then
0.
and
A
a
E
A few f a c t s and d e f i n i t i o n s a b o u t v e c t o r v a l u e d f u n c t i o n s and d i s t r i b u t i o n s w i l l
R
use
E
dual.
Eln by
C
Co(E)
E
and d e f i n e
K(E)
i n t h e same way; i f o n e h a s a compact
and
2'
E F'
(F
is a
o v e r IR)
LCS
c o n t i n u o u s w i t h compact s u p p o r t (we s p e a k t h e n o f < f ( x ) , z ' > : E +IR
d e n o t e s a l g e b r a i c d u a l ) w h i c h we c a l l
< /fdp,z'>
O(F,F').
=
If
K = supp f ,
1 fdp E
and l e t
f E K(E,F)).
f : E
Then
is c o n t i n u o u s w i t h compact s u p p o r t and o n e s e t s T h i s d e f i n e s a l i n e a r form on
< f ( x ) , z ' > dp. (*
1 o r IR
n
D e n o t e IR
i s t h e s p a c e of Radon m e a s u r e s .
i n s t e a d w i t h u n i f o r m convergence and measures a r e e l e m e n t s of t h e
p E K'
Let
w a s d e f i n e d e a r l i e r in d i s c u s s i n g t h e o r d e r o f a
K'
d i s t r i b u t i o n and i t s d u a l o r any open
K
The s p a c e
also be u s e f u l .
< /f(x),z'> h : E
p(f)
We n o t e t h a t
[0,1]
+
c
and
dp.
F'
/
or
i n d u c e s on
a compact s e t i s compact t h e n
F'*
F
t h e topology
f(E)
in
F'*
then
If t h e c l o s e d c o n v e x e n v e l o p e i n
p.
p(f) E F
=
h = 1 on
i s c o n t i n u o u s w i t h compact s u p p o r t ,
f o r any p o s i t i v e
@(z')
so t h a t
fdp
B(Ft*,F')
be
+
and h e n c e a n e l e m e n t o f
i s t h e w e a k l y c l o s e d c o n v e x e n v e l o p e of
p ( h ) C C F'*
x
F
-f
and t h i s h o l d s whenever
F
of
is c o m p l e t e
F
f o r example o r i n t h e weak t o p o l o g y o f t h e d u a l o f a Banach s p a c e .
$' E K(E)
Now f o r
12 p <
and
lower semicontinuous d e f i n e
I* gdp
= inf
now and such t h a t
I* hdp
Fp(E,F)
for
h
m
I* hdp 2
g
and
L
LP(E,F)/NP(E,F).
sup
1 @dp
(E,F)
-.
<
Let
t h e c l o s u r e of
We know i f
for
lQ(x)lPdp)l'P. @
lower semicontinuous.
(]*I I f ( x ) 1 Ipdp)l'p
complete b u t i t is n o t H a u s d o r f f . Fp(E,F)
=
1,
Np(@) = (
b e t h e s p a c e of a l l f u n c t i o n s
Np(f) =
P
write
E
With
NP(E,F) K(E.F)
f E K(E,F)
and
+
N
P
5
h.
Let
F,
For any F
F
g
h
2
>0
d e f i n e d e v e r y w h e r e on
as seminorm
Fp(E,F);
0
b e a Banach s p a c e
Fp(E,F)
b e t h e adherence of
in
For
then
i s Banach t h e n
0
in
Lp(E,F) = j f d p E F;
is
E,
195
APPENDIX 1
1 llfdlll I
further uous when
5
K(E,F)
-
p = p+
= Nl(f).
1
L
E,F)
L
with
p-
0,
f
Since
(E,F).
p-
0
* F is contin-
I f d p : K(E,F)
+
Jfdu E F.
to define
u+
Hence
1
h a s t h e topology of
extend t h i s map t o written
11 f ( x ) l Idp
is dense w e can
K(E,F)
S i n c e any r e a l measure can be
one can s p e a k of i n t e g r a t i n g w i t h
A func-
r e s p e c t t o any measure; complex measures can a l s o b e handled r o u t i n e l y . f : E
tion
* F i s s a i d t o be s c a l a r l y i n t e g r a b l e i f f o r any is integrable.
x +
K
For
f :E
Banach
K'
t h e r e i s a compact valently
f
a b l e and
b)
f(x) E
Ti
is
+
F
C
p-measurable
f o r any compact
D'(F) = L ( D ; F ) ,
i n s t r u c t i v e t o t h i n k of
p(K
n K') 5
F
D'(F)
a
E F'
2'
j f d p E F. and
K C E
c o n t i n u o u s on
x
+
H
C
F
E
> 0 Equi-
K'.
i s measur-
such t h a t
no topology w i l l b e needed h e r e b u t i t is
LCS;
= D ' 8E F.
in the
E
topology s i n c e i f
U
C
D"
=
Let
Dk = a / a x k
Dk 8 I
D and
and d e f i n e
D' 8E F.
in
V C. F'
in
Dk
This i s continuous
are equicontinuous,
@
U,
then
DkU
i s equicontinuous with
U.
C l e a r l y one h a s
and t h i s l e a d s t o t h e d e t e r m i n a t i o n o f = -
in
F
= D
or
T E D'(F),
DkT,
k
A few more g e n e r a l f a c t s s h o u l d be mentioned.
d e f i n e d on a dense l i n e a r set graph
f
f(K)
x E K.
as t h e e x t e n s i o n by c o n t i n u i t y of
and
satisfies
t h e r e i s a denumerable
D'(F)
f' E V
and
E
a ) f o r every
K C E
almost a l l
p
for
Define now
-
with
K
f
p-measurable i f f o r any compact
is
E F'
i s r e f l e x i v e Banach and
compact t h e n a s c a l a r l y i n t e g r a b l e
i s bounded f o r F
F
I f f o r example
2'
G(L)
closed i f
of G(L)
L
D(L)
C
E
is t h e s e t of p a i r s
i s c l o s e d and
L
(E
Let and
(e,Le)
is closable i f
in
L :E
D'. +
F
be a l i n e a r o p e r a t o r
can be general
F E
by e i t h e r o f t h e r u l e s
X
F
-
G(L)
for
LCS).
e E D(L).
L
i s a graph ( i . e .
The is G(L)
-
c o n t a i n s no e l e m e n t s
(0.f));
L
i s t h e o p e r a t o r w i t h graph
G O .
D(L*)
is the
196
R . W. CARROLL
f ' E F'
s e t of
e
such t h a t
t h e topology of
*
*C
< L e , f ' > : D(L)
is c o n t i n u o u s when
E.
Extending by c o n t i n u i t y w e g e t a n
= .
* L
7 = L** .
then
L
is c l o s a b l e i f and o n l y i f
A family
e' = L f ' E E'
has
with
is d e n s e l y d e f i n e d and
Banach, i s c a l l e d a s t r o n g l y c o n t i n -
T ( t ) E L(F), F
T(t+'r) = T ( t ) T ( T ) , T(0) = I ,
semigroup i f
D(L)
*
and
t
*
0 T ( t ) E C (Ls(F)),
0
5
t <
m;
o t h e r k i n d s of groups and semigroups a r e d e f i n e d i n Chapter 1 and w e need n o t d e a l w i t h them h e r e . defined f o r
x E D(A)
as
RX(A) = (XI t
+
m
(1.14)
by
-
Ax = l i m ( T ( t ) x
of
as
x)/t
-
A)
-1
Re X > w o
is defined f o r
is the l i n e a r operator
T(t)
t > 0, Dt T ( t ) x = AT(t)x = T(t)Ax
i s c l o s e d and f o r vant
A
The i n f i n i t e s i m a l g e n e r a t o r
t
0.
-f
for
where
D(A)
is dense,
x E D(A).
A
The r e s o l -
I IT(t)l I/t
oo = l i m l o g
and
RX(A)x =
Theorem 1.10.
r 0
exp(-At)T(t)xdt
(Hille-Yosida-Phillips-Miyadera)
t i o n for a closed l i n e a r operator c o n t i n u o u s semigroup
w i t h dense domain t o g e n e r a t e a s t r o n g l y
is t h a t t h e r e e x i s t
T(t)
1 1 (XI - A)-"l I 5 m/(X 1 I T ( t ) 1 I 5 m exp m o t .
A
f o r real
oo)
A n e c e s s a r y and s u f f i c i e n t condi-
m > 0
and
such t h a t
wo
X > oo and a l l i n t e g e r s
In t h i s event
n.
A c o r o l l a r y is t h e c l a s s i c a l Hille-Yosida theorem which s t a t e s t h a t a n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r a c l o s e d d e n s e l y d e f i n e d l i n e a r s t r o n g l y c o n t i n u o u s semigroup of c o n t r a c t i o n s i s t h a t
X > 0.
In a Hilbert space
A
t o generate a
A
I I (XI -
A)-'/
I
5 1/A
for
g e n e r a t e s a s t r o n g l y c o n t i n u o u s semigroup of con-
t r a c t i o n s i f and o n l y i f i t i s a c l o s e d maximal d i s s i p a t i v e o p e r a t o r w i t h dense
A d i s s i p a t i v e means
domain.
5
0
for
x E D(A)
and maximal d i s s i p a t i v e
is n o t t h e p r o p e r r e s t r i c t i o n of a n o t h e r l i n e a r d i s s i p a t i v e o p e r a t o r .
means
A
A map
A : E
+
F,
E
F Banach, i s c a l l e d open i f
and
sets o r e q u i v a l e n t l y i f
1 I el I 5 c
d e n s e l y d e f i n e d l i n e a r map
is closed
Re(Ax,x)
(b)
(f; < f,f'> =
R(A*)
o
for
E
+
is closed f ' E "A*))
F
lAel
I
€or
i t maps open sets t o open
e E D(A).
If
A
is a closed
then t h e following a r e equivalent: c) (f)
A
i s open
(d)
A*
i s open
R(A*) = i e ' ; c e ' , e > =
o
for
(a) (e)
R(A) R(A) =
e E N(A)I
APPENDIX 1
(here
and
N
R
Theorem 1.11.
197
d e n o t e n u l l s p a c e and r a n g e r e s p e c t i v e l y ) .
(Banach)
c o n t i n u o u s l i n e a r map
Let u : E
-f
F
and
E
be complete metrizable
onto
F,
TVS.
is open.
F,
u : E
A c o r o l l a r y i s t h e c l o s e d g r a p h t h e o r e m s t a t i n g t h a t a l i n e a r map ( i n t o ) i s c o n t i n u o u s i f and o n l y i f p l e t e metrizable).
G(u) = graph
u
is c l o s e d
We r e c a l l n e x t t h a t a Banach a l g e b r a
which i s a n a l g e b r a o v e r
Ic w i t h
I j x y l 1 5 1IxI I 1 I y I 1 .
t i v e and h a s a n i d e n t i t y
e
1 I el I
with
e x i s t s and o t h e r w i s e s i n g u l a r . with
x
-f
x-'
continuous.
1.
=
The g r o u p
An i d e a l
I
B/I
and i n f a c t
one w r i t e s
&
can w r i t e O(x)
of
set
p(x)
=
x
If
x E B
1 I &I 1
with
is t h e set of
-1
X
+
E
X
defined i n
is a n a l y t i c i n
@ : B
for
i s t h e set of
R X ( x ) = (Xe-x) RX(x)
A&
(c;
p(x),
=
QB
x
such t h a t where
x
p(x).
-
x = Xe Xe
o(x)
I
f o r i t s image i n
B
x
-1
is open
i s maximal B/I
and o n e
The s p e c t r u m
i s s i n g u l a r and t h e r e s o l v a n t
is regular.
m,
i s a c o n t i n u o u s homomorphism t h e n
The r e s o l v a n t i s a g a i n
and s a t i s f y i n g
i s a maximal i d e a l .
ker @ = I
o f maximal i d e a l s i s c a l l e d t h e c a r r i e r
is a w e a k l y c l o s e d s u b s e t o f t h e u n i t b a l l i n
@ ( x ) = 2 ( @ ) . QB
is commuta-
i s a n o n v o i d c l o s e d bounded set a n d
v a n i s h i n g of
QB
com-
F
p r o p e r l y and i s
I C B
i s a f i e l d i f and o n l y i f
&
F
is r e g u l a r i f
/ A 1 . T h i s i d e n t i f i e s k w i t h A.
The c o n v e r s e a l s o h o l d s and t h e s e t space.
B
W e assume
i s maximal i f
n o t contained i n any o t h e r i d e a l . B/I =
(E a n d
of r e g u l a r elements i n
B
+
i s a Banach s p a c e
B
An e l e m e n t
G
C
Then e v e r y
i s t h u s ( w e a k l y ) compact and t h e map
B'
x
+
and o n e w r i t e s
2 :B
-f
C(QB)
from
i n t o t h e s p a c e o f c o n t i n u o u s f u n c t i o n s on a compact H a u s d o r f f s p a c e i s c a l l e d t h e G e l f a n d map.
One s e e s t h a t
GB.
n(x)
Further
L e t now
f(z)
-1 x
e x i s t s i f and o n l y i f
is t h e set of v a l u e s t a k e n on by
be an a n a l y t i c function i n a
closed curve enclosing
O(x).
Then c o n s i d e r
nbh
of
^x
does n o t vanish on
;($)
for
@ E
O(x)
and
r
aB. b e a smooth
B
198
R. W . CARROLL
A
F(x) E B
Clearly
Theorem 1 . 1 2 . functions elements
f
Let
F ( x ) ( @ ) = f ( 6 ( @ ) ) . One h a s
6
b e a c l o s e d bounded set i n
a n a l y t i c i n some open
x E B
J x ( f ) = F(x)
and
~ ( x C ) Z
with
where
Rf
ubh
and f o r
(d) I f
x t D
nbh
K,
of
then
t i n u o u s spectrum
( X I = L)F
L)F
dense i n
(3)
uc(L)
i s n o t dense i n
F
(2)
A
may b e bounded, unbounded, v o i d , o r a l l of
U(L)
and a t
(1.16)
where
r
x
L,
A C U(L)
o(L);
A
namely
1
-
Ic.
A F
(XI
such t h a t
L)-l.
implies
Z,
of
fn
is not
f o r which
A1
-
L
i s c a l l e d t h e con-
-
L)
ur(L).
L.
We assume
Banach, i s
is 1
-
1 but
Similar consider-
L
For such p(L)
u(L)
i s n o t void how-
o(L)
nbh
of
one can w r i t e
S i m i l a r c o n c l u s i o n s t o t h o s e of Theorem 1 . 1 2
(fg)(L) = f(L)g(L).
=
is
f(X)R(A ,L)dh
t h e r e is a function
E(A)
by
XI - L
such t h a t
t h e s e t of
o ( f ( L ) ) = f(o(L)
which i s both open and c l o s e d i n
one writes
X
K
i s a s u i t a b l e smooth curve e n c l o s i n g
+2n i
B
is t h e set o f f u n c t i o n s a n a l y t i c i n some
H(L)
R(X,L) = R (L) = ( X I
For such
on
and
m
f(L) = f ( m ) I
hold f o r such subset
Then i f
+
Then ( a ) Jx
nbh
a t i o n s a p p l y t o c l o s e d d e n s e l y d e f i n e d l i n e a r unbounded
ever i n general.
rf.
L E L(F), F
i s t h e r e s i d u a l spectrum
F
b e t h e set o f
Jx : H ( E )
but not equal t o
t h e s e t of
B
C
converges i n norm.
X E u(L)
P
-
D
define
of a l i n e a r o p e r a t o r
1 - 1 i s c a l l e d t h e p o i n t spectrum u (L)
(A1
Let
the algebra of
F f x ) = e ( c ) f(X) =
Fn(x)
(1) t h e set of
divided i n t o t h r e e p a r t s :
1 - 1 with
H(6)
converges uniformly on a compact
fn(A)
I n g e n e r a l t h e spectrum u(L)
is
and
i s d e f i n e d by (1.15) w i t h s u i t a b l e
F
a n a l y t i c i n a n open
6.
of
a n a l g e b r a homomorphism (b) f ( h ) = 1 i m p l i e s F(x) = x
(I:
f E ff(L)
E(A,L)
= f(L).
The map
1 on A
{m}),
etc.
A
is c a l l e d a s p e c t r a l s e t .
u(L)
equal t o
U
+
A E(A)
and
0
elsewhere
i s an isomorphism
from t h e Boolean a l g e b r a of s p e c t r a l sets t o t h e a l g e b r a o f p r o j e c t i o n s
E(A)
and
199
APPENDIX 1
where
I'
surrounds
space
H
and l e t
A
b u t no o t h e r p o i n t o f be a commutative
B
*
L e t u s now go t o a H i l b e r t
o(L).
algebra in
L(H)
*
m u t a t i v e Banach a l g e b r a o f bounded o p e r a t o r s w i t h i n v o l u t i o n A
-+
* A
here).
Adjoin a u n i t
*
is isometrically C(OB)
-+
u
is a measure
isomorphic t o
A
on
dp(X,x,y) = dp(h,y.x)
A E B
and
r(f) =
C(QB)
+
-2)
f(L) =
]
E(u)
]
with
f(A)dE(h)
f(h)dE(h)
t i o n of
L.
f(L) =
f(h)E(dh)
111
r
and w e w r i t e
Then
B
:A =
By a theorem of R i e s z t h e r e
for
o E 8
and f o r each B o r e l set p(6.x.y) for
y i e l d a s p e c t r a l f a m i l y of p r o j e c t i o n s while
* (A
=
(E(6)x.y).
f E C(A)
Applied t o a bounded normal o p e r a t o r
sionally).
which is s i m p l y
such t h a t
a d j o i n t s p e c t r a l measure for
i s a com-
the i d e n t i t y operator, i f necessary.
( t h i s r e s u l t is due t o Gelfand-Naimark).
B
Further
e,
B
(i.e.
E(o)
One h a s
L
* (LL
* L L)
=
f
-+
f(L)
occa-
these r e s u l t s
v a n i s h e s on
E
p(L)
This is called t h e s p e c t r a l resolu-
f a c t f o r e a c h complex bounded B o r e l f u n c t i o n and
E(o)A = AE(o)
dE(X) = E(dh)
(we w r i t e
such t h a t
f E C(o(L)).
there is a s e l f -
*
i s then a continuous
f
on
o(L)
set
homomorphism of t h e L(H)
with
1 -t I
Let us s p e l l o u t a few more d e t a i l s f o r t h e s e l f a d j o i n t c a s e where
O(L)
is a
a l g e b r a of bounded B o r e l f u n c t i o n s (sup norm) o r and
h
+
(1.18)
L.
For
U(L)
into
s e l f a d j o i n t one h a s
L
E(a,b) = l i m 2i: EO '
jli:
[R(P-iE ,L)
-
R(u+ic ,L) l d ~
6+0
subset of
and
L
( 4 ,~)
may n o t b e bounded.
the identity or spectral resolution for
L.
o w ,
and
(f(L)x,y) =
I f(h)(E(dh)x,y).
Further
Let
E(dh)
Then f o r
f
be t h e r e s o l u t i o n of a B o r e l f u n c t i o n on
200
R , W. CARROLL
one can a l s o proceed somewhat d i f f e r e n t l y and, t a k i n g f o l l o w t h a t t h e r e i s a measure s p a c e
2
U :H
-
L (M,dp),
-f
x E D(L) If
and a real f u n c t i o n
2 f ( . ) ( U x ) ( . ) E L (M,dp)
f
h(L))
-f
M
2
h
$(hn)
+
0
define
from bounded Bore1 f u n c t i o n s i n t o
$(h)
(c)
strongly.
then
h(L) =
L(H)
if
hn(h)
Clearly
+
h(X)
X
h(h) =
(a)
if
ULU-l$!(A) = f(A)$(X).
-1 U T
a c t i n g as m u l t i p l i c a t i o n by
LL(M,dp)
* $(h) > 0
f i n i t e a . e . , such t h a t
$! E U(D(L)
and i f
homomorphism, norm c o n t i n u o u s , and s a t i s f i e s (b)
f i n i t e , a unitary operator
p
on
is a bounded B o r e l f u n c t i o n on IR
h
i s t h e o p e r a t o r on
$(h
(M,p),
t o be s e p a r a b l e , i t w i l l
H
is c y c l i c f o r
i s dense i n
H
1 h(A)dpx(A)
=
if
L
Lx = 5x
and i f
x
(x,h(L)x)
H = L ( B,dpx)
is c y c l i c and
L
B) = C"
IR and t h e f u n c t i o n
For
x E H
(x,Ps2x)
Borel f u n c t i o n
h
the p r o p e r t i e s (a) h(L)
Pn = x,(L)
-
x,
where
h(L)
U-lTh(f)U.
1
has
A vector
where dux
-1
M
1.
is determined by I n general
H
above is a union of and
i s t h e c h a r a c t e r i s t i c f u n c t i o n of
IR denoted by
by
If
d(x,Pix)
R.
and a bounded
1 h(h)d(x.PXx); t h i s s a t i s f i e s uniqueness of 0 must c o i n c i d e w i t h
(x,h(L)x) =
( c ) e t c . above and by
defined there a s
hn(A).
but
then
The s p e c t r a l p r o j e c t i o n s a r i s e i n a n o t h e r
i s a Bore1 measure on
determines
*
$ ( h ) x = h(C)x
above a r i s e s i n s e l e c t i n g a f i n i t e p
i n s e r t i n g Radon-Nikodym d e r i v a t i v e s . common n o t a t i o n as
then
6(h) = L
becomes m u l t i p l i c a t i o n by
f(A)
h(f)
Then t h e map
functions vanishing a t
decomposes i n t o a d i r e c t sum of c y c l i c subspaces s o t h a t c o p i e s of
T
I I hn\ Im 5 M
pointwise with
2
where
i s a (unique) a l g e b r a i c
corresponds t o
{h(L)x; h E C,(
U
h(f(X)).
t o be o b t a i n e d h e r e a s a l i m i t of s u i t a b l e bounded f u n c t i o n s
x E H
h(f)
is an unbounded Borel f u n c t i o n d e f i n e
h
then
and
Dh
i s dense i n
H.
Symbolically
Formula (1.18) h o l d s a l s o f o r unbounded
h(L)
=
1 h(X)dP x
Maurin [ l ] ) .
1"
H(X)dV(h)
which d i a g o n a l i z e s
This i s s i m i l a r t o w r i t i n g
dPA = dE(X).
L.
S t i l l a n o t h e r p o i n t of view i n v o l v e s r e p r e s e n t i n g
Hilbert spaces
and c l e a r l y
H
as
H L
a s a d i r e c t i n t e g r a l of (cf C a r r o l l ( 4 1 . Dixmier
LL(M,du)
above where t h e
111,
APPENDIX 1
Radon-Nikodym d e r i v a t i v e s r e p r e s e n t e d by e r a l l y let
f(A)
and one l o o k s a t
H = JIH(5)
+
I / x ( S )I 1
(1 I I I
i s measurable
5
is such t h a t
+
H(S)
separable.
xk
E
f
I
2
s t r u c t u r e on
meaning
H
x = 0 =
i s c a l l e d measurable i f
H
in
A
I* 5
H(<)dv. +
=
,/ I
x(5) = 0
A family
A(<)x(5)
space
H
on
w i t h spectrum
A
A
'b
*
HfC).
x
E
y E H
y E E
5 E 2
is
E
if
(c)
t h e closed
In particular
H(5)
K
is
be
This gives a p r e h i l b e r t
m.
a.e.
The a s s o c i a t e d H i l b e r t s p a c e
of l i n e a r operators i n
A(<)
x E H.
i s measurable f o r
A(5)
H(5)
An o p e r a t o r in
H(5).
A
If
@ A t h e n t h e r e i s a measurable f i e l d of H i l b e r t spaces
r e l a t i v e t o a ( b a s i c ) measure V
H = H =
such t h a t
I * H(<)dV
and
onto the algebra of diagonalizable
L"(A)
Applied t o a s e l f a d j o i n t o p e r a t o r
A
in
H,
p o s s i b l y unbounded, one
o b t a i n s a d i r e c t i n t e g r a l over a p o s s i b l y i n f i n i t e i n t e r v a l s u c h t h a t sonds t o m u l t i p l i c a t i o n by
H(5)
(b)
then
on
a l g e b r a of o p e r a t o r s i n a s e p a r a b l e H i l b e r t
t h e Gelfand map becomes an isomorphism operators.
for
H(5))
i s d i a g o n a l i z a b l e i f i t is r e p r e s e n t e d by f u n c t i o n s
i s a ( s u i t a b l e ) commutative
H(C)
(a)
x E E
2 Ix(L)l lSdV
Gen-
Hilbert,
The f i e l d
such t h a t f o r each equals
H(5)
are c a l l e d measurable v e c t o r s and w e l e t
11x1
with
E
xk(C)
t h e s e t of s u c h v e c t o r s w i t h
w i l l be denoted by
such t h a t
C H
d e n o t e s t h e norm i n
g e n e r a t e d by t h e
The e l e m e n t s o f
K
E
5 * H(<),
x ( 5 ) E H(5).
-f
( X ( < ) , Y ( ~ ) )i~s m e a s u r a b l e f o r a l l
t h e r e i s a (fundamental) sequence subspace of
5
dv.
a p o s i t i v e measure
V
i s a map
2
with elements
measurable i f t h e r e i s a v e c t o r subspace
5
a r e incorporated i n t o
be a l o c a l l y compact Hausdorff s p a c e and
2
A f a m i l y o r f i e l d o f H i l b e r t s p a c e s on
2.
201
A
corre-
5.
It is appropriate here t o i n d i c a t e a l s o t h e construction of generalized eigenfuncw
t i o n s i n t h e c o n t e x t of r i g g e d H i l b e r t s p a c e s ( c f . B e r e z a n s k i j [l], Gelfand-Silov Y
[ 3 ] , Smulyan [l]). one.
Thus l e t
H
There a r e v a r i o u s ways t o approach t h i s and w e simply choose b e a H i l b e r t s p a c e and
0
C
H
a d e n s e n u c l e a r s p a c e w i t h con-
t i n u o u s i n j e c t i o n ( e v e r y t h i n g s e p a r a b l e ) and t h i n k of tion.
l?
C
L2
as a model s i t u a -
We w i l l n o t d w e l l h e r e on n u c l e a r i t y ; a main f a c t needed is t h a t i n
bounded sets are r e l a t i v e l y compact.
For
c o n t i n u o u s and d e t e r m i n e s a n element
fh E 0'
h E H
t h e map
written
$
+
(h,$)
: 0
+
@
E
is
s o t h a t we have
R. W . CARROLL
202
0C H
C
0 ’ ; 0’ w i l l b e t h e a n t i d u a l o r c o n j u g a t e l i n e a r d u a l .
s p e c t r a l f a m i l y of p r o j e c t i o n s i n
xA E
f i x e d one c a n show t h e e x i s t e n c e of
t h e s e n s e o f a Radon-Nikodym d e r i v a t i v e .
0
t i o n a l on
xh
=
dE.,e/do.
xx
The
@ E H(e)
o(A)
f o r example and
H
0 ’ such t h a t Indeed
eh =
is of s t r o n g l y bounded v a r i a t i o n s i n c e
Now l e t
associated t o
be t h e subspace of
H(e)
H
0
=
dEX is a
If
(Ehe,e)
<xi,$>
=
e E H
for
d(E e , $ ) / d o h
in
c o n s i d e r e d as a func-
E e
x
i s n u c l e a r and one w r i t e s
g e n e r a t e d by t h e
el
=
Ehe.
are orthonormal and complete i n t h e sense t h a t f o r
e
n
(1.22)
E(@)e=
1
Xxd9(X).
A
can then be decomposed as an o r t h o g o n a l d i r e c t sum of s p a c e s
H
The s p a c e
and one h a s a s s o c i a t e d elements
Let now
@,$ E
a.
A
0’.
x
E(@-)
E
a’.
Then f o r any
be a c o n t i n u o u s l i n e a r o p e r a t o r
Then
A* : 0 ’ *
@’
a s e l f a d j o i n t extension t o in
A
H
A : 0
+
0
i s c o n t i n u o u s and e x t e n d s
@ E 0
with
A
on
there results
( A @ , $ ) = ($,A@)
0.
If
A
xA
as above.
Then
for
admits
then i t h a s a complete system of e i g e n e l e m e n t s
Indeed o n e s t a r t s from t h e a s s o c i a t e d r e s o l u t i o n of t h e i d e n t i t y and c o n s t r u c t s
H(ea)
xx
APPENDIX 1
SO
that
Ay = -y” satisfy
Axh
+
Axh.
=
qy
for
-xi +
relative to
For example i f one t a k e s f o r
set
real with say
q E Cm
qXh = A X h .
EX
If
U,(h)
2
Ay
A
t h e Sturm-Liouville operator
y‘(0) = 0y(O),
0
real, then t h e
i s a system of g e n e r a t o r s f o r
ea
= (EXea,ea)
i s t h e c l a s s i c a l s o l u t i o n of C ba(A)doa(X)
203
=
hy
and
x,(h)
(take here
=
ba(X)y(x,h)
0 =
D).
Set
H = L (0,~) where
y(x,h)
dci(A) =
m
and
F(A) =
y(S,A)$(S)d<.
Then
and t h i s can b e extended t o a n isomorphism between
2 L (0,m)
and
xb
2
L2 0
since
This Page Intentionally Left Blank
APPENDIX 2
N o t a t i o n of (2.6.91)
-
(2.6.100) w i t h a few minor changes i n expanding t h e frame-
work somewhat t o f i t t h e model s i t u a t i o n of Section 2.3).
Thus l e t i n (2.6.91) and ( 2 . 6 . 9 6 )
$3 : E
(2.1)
i'; J1:
+
and 4
with inverses
F
+
and
P ( D ) = Lm(D)
Q(D)
= D
2
(cf.
( c f . C a r r o l l [ 3 9 ] ; Remark 2.6.21)
Cp
g i v e n by (2.6.93)
and ( 2 . 6 . 9 8 ) .
Let i n (2.6.92)
and
(2.6.97)
with inverses P to
2:
P :I E s u ' ;
(2.2)
and
0'
and
F + F
given by (2.6.95) and ( 2 . 6 . 1 0 0 ) ( w i t h n o t a t i o n changed
Q
'lI
The " t w i s t i n g " t r a n s f o r m s
p').
and
$ '
g i v e n by ( 2 . 6 . 9 4 ) and
(2.6.99) w i l l i n g e n e r a l be unbounded o p e r a t o r s ( c f . below)
D' :
(2.3)
Gp
+
E;
:
i'
F.
+
Let u s u s e t h e model i n d i c a t e d above t o p u t t o g e t h e r a r e a l i s t i c p i c t u r e .
(2.4)
H(x,p) = "Rm(x,A) = 2y(ntk1)(hx)-mJm(Ax)
(2.5)
Q ( x , p ) = 2-2?'(mtl)-2(A~)
for
u
=
(2.7)
-A
2
.
Thus
2mt.1911 R (x,X)
Then
mt4
Mm[x
Pf(P) =
f ( x ) ] = ?(IJ)
2"r (mtl)
and s i m i l a r l y f o r
Now f o r s u i t a b l e we choose
where
P and IP, x m, 7Hm
E = I f ; xd7'f(x)
do = (-l)-m-'dL,/2v5.
-m-+
f(x)
f->
under Hankel t r a n s f o r m a t i o n .
XmfL2?(li)
2
w i l l be a n isometric isomorphism f
L21 w i t h Take now
Efl =
2 2 f ( u ) E LA) = L (-m,O;
rl;
E' = i f ; x
2
L (0,~) + L (0,~) and
-m-k
f(x)
E
L
2
1
=
IE
du)
( t h u s we do
206
R . W.
E
not identify 2 L (&,O;
do')
and
El)
where
CARROLL
=
and t a k e
(go)'
do' = (-p)*'dp/Zfi.
=
r?;
2
L (0,m;
=
1 :(p)z(p)dX.
dh),
For
Q
i' - (so)'
Note t h a t t h e d u a l i t y
f o r example i s expressed i n terms of t h e "basic" measure r u l e >:,<:
2
E LA} =
h&'f"(p)
we take
dh = d ~ / 2 6by t h e
2
F = L (O,m)
=IF
i n i d e n t i f y i n g t h e s e spaces w i t h t h e i r d u a l s .
" F" =IF
and
=
Then t h e f o l l o w i n g
diagram d i s p l a y s t h e s p a c e s and maps i n a c l e a r manner.
Theorem 2.1.
The diagram
2.9) i s a t y p i c a l model of a t r a n s f o r m t h e o r y based on
t r a n s m u t a t i o n and i n a d d i t on t o t h e formulas of Theorem 2.6.22 d i s p l a y s t h e identifications and
B*
=
$*
(PA)*
= P , a*= Q, !P* =
=
P,
and
g*
=
Q which y i e l d
B*
= (@)*
=Po,
$P.
A few f u r t h e r remarks a r e a p p r o p r i a t e h e r e and w e r e f e r t o C a r r o l l [39] f o r an extensive treatment.
F i r s t from
Bf(y) = < B ( y , x ) , f ( x ) > and
i t s d e t e r m i n a t i o n s ( 2 . 3 , 4 5 ) w e see t h a t
-4 <
m < n
-4
and t o s p e c i f y
so
D(0
B
f
s h o u l d have
w i l l n o t b e d e f i n e d on a l l of
is t o write
E.
n
B
g i v e n by one of
d e r i v a t i v e s when Another way t o s e e t h i s
APPENDIX 2
(2.10)
where
@f(y) =
i(P)
want a l s o
=
? ( p ) E Lf
A-m-4
( c f . (2.3.26)).
Amt”F(h)Cos
h m+g F ( h )
and
w e want
[ PG
G ( h ) E L:
hydh
from ( 2 . 7 ) w i t h
so
D(Q) =
207
1‘
n
F.
2
F t LA
.
In order for
Similarly for
i s given by t h e formula ( 2 . 8 ) ;
or
G E
.‘i
Hence
D(W) =
B
@f E L 2 = F
= P A w e have
thus for
? n 2.
PG
G =
to lie in
E
We n o t e a l s o t h a t
A f u r t h e r s t u d y of (more g e n e r a l ) f o r m u l a s s u c h as (2.6.90)
a l s o conducted i n C a r r o l l [ 3 9 ] .
we
is
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INDEX
Absorbing, 184 A b s t r a c t boundary v a l u e problems, 9 A b s t r a c t Cauchy problem, 4 A b s t r a c t d i f f e r e n t i a l problems, 3 A b s t r a c t D i r i c h l e t problem, 8 1 A b s t r a c t e v o l u t i o n e q u a t i o n s , 9 , 10 Abstract operator equations, 9 A c c r e t i v e o p e r a t o r , 166 Admissable, 93, 1 0 3 A f f i n e o p e r a t o r , 98 A i r y f u n c t i o n , 85 A l g e b r a i c d u a l , 194 A n a l y t i c c o n t i n u a t i o n , 9 7 , 109 A n a l y t i c semigroup, 57 A s c o l i theorem, 192 A s s o c i a t e d Legendre f u n c t i o n , 112 A s s o c i a t i v i t y , 134 B c o n d i t i o n . 139 Backward p a r a b o l i c , 63, 6 4 Baire space, 191 Banach a l g e b r a , 6 5 , 116, 151 Banach s p a c e , 184 Banach S t e i n h a u s theorem, 192 Barrel, 1 9 1 B a r r e l e d space, 191 Base of neighborhoods, 181 B a s i c measure, 201 Bessel f u n c t i o n s , 125 B i l i n e a r c o n c o m i t a n t , 42, 129 B i p o l a r s , 194 Boolean a l g e b r a , 198 Bore1 s e t , 199 Boundary o p e r a t o r s , 133, 140 Bounded o p e r a t o r , 53 Bounded s e t , 190
Canonical o p e r a t o r , 51 C a r r i e r s p a c e , 65, 197 Cauchy n e t , 183 Cauchy p r i n c i p a l v a l u e , 78 Cauchy problem, 4 Cauchy Riemann e q u a t i o n s . 79, 97 C e n t r a l i z e r , 137 C h a r a c t e r , 151 C h a r a c t e r i s t i c polynomial, 65 Closed s e t , 182 C l o s a b l e , 164, 195 Closed graph theorem, 49, 197 C l o s u r e , 182 Coercive, 51, 53 Commutativity, 134 Commuting o p e r a t o r s , 40, 65 Compact, 192 Compact r e s o l v a n t , 139. 1 4 6 Complete, 183, 188
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Complete orthonormal set, 202 Completely subordinate, 57 Composition, 2 Condition C, 99, 104 Condition P, 166 Continuous map, 183 Continuous spectrum, 198 Contraction, 196 Convergence, 182 Convex, 184 Convex (p), 66 Convexity, 25 Convolution, 133, 134, 187 Correct problem, 66 Countably normed space, 94 Crossnorm, 164 Degenerate problem, 28 Delta function, 185 Diagonalizable operator, 201 Direct integral, 200 Dirichlet problem, 73, 95 Disced, 184 Dissipative operator, 196 Distribution, 182, 184 Dual, 185 Duality, 190 Effete snobs, vii Eigenmanifold, 169 Eigenvalue, 99 Eigenfunction, 99 Elliptic operator, 41 Energy, 48 Envelope, 184 Enveloping algebra, 47 Equicontinuous, 188, 192 Equicontinuous group, 3 Euler-Poisson-Darboux equation, 25 Evaluation, 21, 38 Evolution equation, v, vi Exponential type, 37 Extended spectrum, 164 Extension by continuity, 190 Faithful crossnorm. 164 Formal adjoint, 42, 83 Fourier cosine formula, 123 Fourier sine formula, 129 Fourier transform, 187 Frechet space, 184 Fundamental system of neighborhoods, 182 Gas dynamics, 29 Gauge, 184 Gelfand map, 197 Gelfand Naimark theorem. 199 Gelfand transform, 153 Generalized axially symmetric potential theory, 81 Generalized convolution, 100, 116, 133, 173
INDEX
Generalized eigenfunction, 167, 169, 201 Generalized Fourier transform, 115 Generalized Hilbert transform, 8 0 Generalized hypergeometric function, 90 Generalized Mehler inversion, 1 1 3 Generalized projection operator, 1 7 3 Generalized spectral function, 149 Generalized Stieltjes transform, 8 7 Generalized translation, 70, 98, 100, 1 1 4 , 1 3 1 , 146, 174 Genus, 38, 39, 92 Global s o l u t i o n s , 94 Gowurin property, 166 Graph, 195 Green's function, 4 1 Green's operator, 66 Gronwall lemma, 3 1 Group aspects, 25 Growth properties, 25 Haar measure, 45 Hahn Banach theorem, 85, 190, 93 Halfplane problem, 93 Halfspace. 1 9 3 Hankel formula. 124 Hankel inversion, 111, 118 Hankel transform, 105, 205 Harmonic function, 79 Hausdorff space, 182 Heavyside function, 64 Hemicontinuous, 5 3 Hermite functions, 48 Hersh technique, 4, 10, 1 3 , 39 Hilbert crossnorm, 165 Hilbert Schmidt operator, 168 Hilbert transform, 77 Hille Yosida theorem, 196 Hslder condition, 157 Hyperbolic problems, 2 , 3 7 , 46, 47, 4 6 , 63, 66, 67 Hyperfunctions, 9 Hyperplane, 1 9 3 Hypocontinuous, 23, 32 Ideal space, 146 Improperly posed Cauchy problem, 73 Inductive limit, 185 Infinitesimal generator, 196 Integrable, 44 Integrable function, 195 Integral equation, 131, 159 Integral operator, 69 Inverse scattering, 146, 154 Iterated Laplace transform, 128 Joint spectrum, 6 5 KdV equation, 160 Kernels, 1 1 7
L admissable, 140 Laplace Beltrami operator, 114 Laplace transform, 76
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242
R. w. CARROLL
Lie algebra, 44 Lie group, 44 Limit circle, 120 Limit point, 120 Lipschitz map, 48 Local coordinates, 45 Locally convex space, 182 Locally equicontlnuous group, 3 Love equation, 58 Lower semicontinuous, 194 m accretive operator, 166 m sectorial operator, 166 Mackey Arens theorem, 193 Mackey topology, 193 Maximal dissipative, 196 Maximal extension, 164 Maximal ideal, 197 Maximal monotone, 5 3 Measurable, 195 Measurable field, 201 Measurable operator family, 201 Measurable vector, 201 Mellin transform, 126 Metric function, 183 Metrizable, 183 Microfunctions, 9 Minkowski functional, 184 Monotone operator, 53 Monte1 space, 192 Multiplication, 143 Multipliers, 15, 37 Negative norm space, 167 Neighborhood, 181 Net. 1 8 2 Noncommuting operators, 44, 67 Nonlinear evolution problem, 10 Nonlinear semigroup, 10 Norm, 181 Normal operator, 199 Normed ring, 146 Nuclear, 188 Nullspace, 196 One parameter group, 4 Open mapping theorem. 196 Open set, 181 Operator coefficeints, v Operator pencil, 65 Operator strength, 57 Operational calculus, viii, 10 Order o f a distribution, 189 Ordered products, 44 Paley Wiener Schwartz theorem, 32 Parabolic, 37, 56, 63, 66, 67 Parabolic regularization, 54 Parseval formula, 1 3 , 104, 149, 152, 159
INDEX
Perturbation, 4 9 , 57 Petrovskij correct, 39 Petrovskij parabolic, 38 Plancherel formula, 1 5 2 Point spectrum, 5 1 Poisson formula, 8 1 Poisson integral, 8 0 Polar, 1 9 1 Potential, 1 5 9 Potential theory, 115 Precompact, 1 9 2 Primitive, 96 Property G, 166 Pseudoanalytic function, 8 1 Pseudosymmetric space, 111 Quantization, 45 Quantum scattering theory, 1 5 9 Quarter plane problem, 93 Quasielliptic, 64 Quasiequicontinuous group, 3 Quasihyperbolic, 64 Quasinuclear, 169 Radon measure, 1 9 0 , 1 9 4 Radon Nikodym derivative, 200 Range, 1 9 6 Reasonable crossnorm, 1 6 4 Recursion relations, 27 Reduced order, 37 Reduction of order, 117 Reflexive, 193 Regular, 197 Regular (y), 6 3 Related differential equations, viii, 10, 6 9 , 82 Related nonhomogeneous equations, 87 Relatively compact, 1 9 2 Representation, 44 Residual spectrum, 1 9 8 Resolution of the identity, 171, 199 Resolvant, 2 6 , 1 9 6 , 197 Resolvant set, 191 Resolvant strip, 64 Riccati equation, 29 Riemann function, 1 0 0 , 1 4 7 , 150 Riemannian symmetric space, 1 1 4 Riesz operator, vi. 65 Riesz theorem, 1 9 9 Right invariant, 4 4 , 45 Right translation, 44 Scalarly integrable, 1 9 5 Scattering data, 1 5 9 Scattering theory, 5 1 , 157 Schrodinger equation, 92 Schwartz kernal theorem, 189 Seminorm. 1 8 1 Semireflexive, 1 9 3 Separated, 182
243
244
R. W.
Separately continuous, 2, 23 Separation of variables, 98, 163 Sesquilinear form, 5 1 Simple convergence, 1 9 1 Singular problem, 25 Skewadjoint operator, 47 Sobolev equations, 10 Spectral (p), 66 Spectral data, 160 Spectral function, 154 Spectral mapping theorem, 164 Spectral measure, 172 Spectral methods, 10 Spectral resolution, 199 Spectral set, 198 Spectral variables, 65 Spectrum, 1 9 7 Spherical function, 111 Stability, 68 Standard operator, 5 1 Stieltjes transform, 87, 128 Strong operator topology, 12 Strong topology, 1 9 1 Strongly continuous semigroup, 196 Sturm Liouville theory, 146, 156 Submarkov, 115 Subordinate, 57 Sufficiently rich, 37 Support, 183 Symmetric hyperbolic system, 56 Symmetric space, 112 Tensor product, 188 Tichonov theorem, 193 Topology, 1 8 1 Topology ( E ) , 1, 188 Topology (TI), 188 Topological vector space, 181 Trace, 169 Transform (PI, 106 Transform diagram, 206 Transform model, 206 Transform theory, 178, 206 Translation, 139 Translation group, 8 1 Transmutation, 69, 147 Transpose, 193 Tricomi equation, 29 Twisting transform, 205 Type 1 and 2 solution, 58 Ultradistribution, 9 Uniform crossnorm, 164 Uniformly we12 posed, 78 Unitary group, 47, 50 Variable domain, 10, 50 Variational inequalities, 10 Volterra equation, 148
CARROLL
INDEX
Weak topology, 191 Weakly compact, 193 Weighted Banach space, 25 Well posed, 9 2 Weyl Heisenberg algebra, 47 Yosida approximation, 56
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