FUNCTIONAL ANALYSIS, HOLOMORPHY AND APPROXIMATION THEORY I1
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FUNCTIONAL ANALYSIS, HOLOMORPHY AND APPROXIMATION THEORY I1
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NORTH-HOLLAND MATHEMATICS STUDIES
86
Notas de Matematica (92) Editor: Leopoldo Nachbin Centro Brasileiro de Pesquisas Fisicas and Universitv of Rochester
Functional Analysis, Holomorphy and Approximation Theory I1 Proceedings of the Seminario de Analise Funcional, Holomorfia e Teoria da Aproxima@o, Universidade Federal do Rio de Janeiro, August 3-7,1981 Edited by
Guido 1. ZAPATA lnstituto de Maternatica Universidade Federal do Rio de Janeiro
1984
NORTH-HOLLAND - AMSTERDAM 0 NEW YORK
* OXFORD
@
Elsevier Science Publishers 6.V., I984
All rights reserved. No part of this publication may be reproduced, storedin a rerrievdsystem, or transmitted, in any form or by any meons, electronic, mechanical, photocopying. recording or otherwise, without the prior permission of the copyright owner.
I S B N : 0 444Xhh'45.3
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E I S E V I E K SCIENCE PUBLISHERS 13.V r-'.o.BOX I Y Y I 1000 RZ AMSTERDAM T H E NETH E KLANDS .Sol(, tlr\lrrhiitoi c f o t rlic I/ F A
o i i t l ('tiiiritlti
E I S E V I E K SCIENCE PUBLISHING COMPANY. INC 52VANDERBIL7 AVENUE NEW Y O K K . N Y 10017
Library of Congress Cataloging in Publication Data
Semindrio de An6lise Funcional, Holomorfia e Teoria da AproqimacSo (1981 : Universidade Federal do Rio de Janeiro) Functional analysis, holomorphy, and approximation theory 11. (North-Holland mathematics studies ; 66) (Notas de matemctica ; 92) 1. Functional analysis--Congresses. 2. Holomorphic functions--Congresses. 3. Approximation theory-Congresses. I. Zapata, Guido I. (Guido Ivan), 194011. Series. 111. Series: Notas de matemgtica (Amsterdam, Hollandd ; 92. QA32O.Sb56 1981 515 7 83-25454 ISBN 0-444-66645-3
.
-
PRINTED IN T H E NETHERLANDS
In Memory of
SILVIO MACHADO
Born on September 2, 1932 in Porto Alegre, RS, Brazil Died on July 28, 1981 in Rio de Janeiro
This Page Intentionally Left Blank
vii
FOREWORD
This volume is the Proceedings of the Seminsrio de
Anslise
Funcional, IIolomorfia e Teoria da Aproximaszo, held at the Instituto de M a t e d t i c a , Universidade Federal do Rio de Janeiro (UFRJ) in August 3 - 7 ,
1981.
The participant mathematicians and
contributors
are from Argentina, B r a z i l , Canads, Chile, France, Hungary, Italy
,
Mexico, Spain, Rumania, United States and West Germany. "Functional Analysis, Holomorphy and Approximation
Theory"
includes papers either of a research, or of an advanced expository, nature and is addressed to mathematicians and advanced graduate students in mathelnatics.
Some of the papers could not actually
be
presented at the seminar, and are included here by invitation. The members of the Organizillg Committee
-
J.A. Barroso, M.C.
Matos, L.A. Ploraes, J. Mujica, L. Nachbin, D. Pisarlelli, J.B. Prolla and G.I. Zapata (Coordinator)
-
would like to thank the Conselho de
Ensino p a r a Graduados e Pesquisa (CEPG) of UFRJ, Conselho Nacional de Desenvolvimento Cientifico e Tecrlol6gico (CNPq), and Brasil for direct financial contribution.
We would also
I.B.M.
do
like
to
acknowledge the indirect financial contributions from Universidade Federal do Rio de Janeiro, C o o r d e n a ~ g ode Aperfeisoamento de Pessodl de Nivel Superior (CAPES), Financiadora de Estudos e ProJetos (FTNEP), as well as other universities and agencies. We are happy to thank Professor Sergio Neves Monteiro, president of CEPG of UFRJ for his personal support and understanding; Professor Paulo Emidio de Freitas Barbosa, Dean of the Centro de Ci#
Sncias Matemsticas e da Natureza (CCMN) of UFRJ, in whose facilities the seminar w a s comfortably held; and Professor Leopoldo Nachbin for
viii
FOREWORD
his constant friendly support and understanding.
We also thank Wil-
son Luiz de Gdes for a competent typing job. Finally, let us tell with emotion that one mathematician was sorely missed at the seminar, Silvio Machado, a member of
its
Organizing Committee, who passed away after a heart attack in July
28 of 1 9 8 1 , just a few days before the opening of the meeting.
The
loss caused by his death will surely be long felt by our community, in particular by his friends in which many participants of the seminar are included.
A s a posthumous homage, all of us wish to de-
dicate these Proceedings to the memory of Silvio Machado.
Guido I. Zapata
ix
TABLE OF CONTENTS
Rodrigo Arocena
Richard M. Aron and Carlos Herves
On generalized Toeplitz kernels and their relation with a paper of Adamjan, Arov and Krein
1
Weakly sequentially continuous analytic functions on a Banach space
23
Andreas Defant and Klaus Floret
The precompactness-lemma for sets of operators
39
Zeev Ditzian
On Lipschitz classes and derivative inequalities in various Banach spaces
57
A stratification associated to the copy phenomenon in the space of gauge fields
69
On the angle of dissipativity o f ordinary and partial differential operators
85
Francisco A. Doria
Hector 0. Fattorini
Two equivalent definitions of the density numbers for a plurisubharmonic function in a topological vector space
113
Chebyshev centers of compact sets with respect to Stone-Weierstrass subspaces
133
On the Fourier-Bore1 Transformation and spaces of entire functions in a normed space
139
John McGowan and Horacio Porta
On representations of distance functions in the plane
171
Reinhard Mennicken
Spectral theory for certain operators polynomials
203
Pierre Lelong
Jaros lav Mach
MBrio C. Matos
X
Miklos Mikolgs Petronije S . Milojevi?
Luiza A. Moraes
Vincenzo B. Moscatelli Jorge Mujica
TABLE OF CONTENTS Integro-differential operators and theory of summation
245
Approximation-solvability of some noncoercive nonlinear equations and semilinear problems at resonance with applications
259
Holomorphic functions on holomorphic inductive limits and on the strong duals of strict inductive limits
297
Nuclear Kothe quotients of Fr6chet spaces
31 1
A completeness criterion for inductive limits of Banach spaces
319
Peter Pflug
About the Caratheodory completeness of all Reinhardt domains 331
J o z o B. Prolla
Best simultaneous approximation
Reinaldo Salvitti
Abstract Frobenius theorem - Global formulation. Applications to Lie groups 359
Ivan Singex
Optimization by level set methods. 11: Further duality formulae in the case of essential constraints
383
Gerald0 S. de Souza
Spaces formed by special atoms I 1
413
Harald Upmeier
A holomorphic characterization * of C - algebras
427
A property of Fr6chet spaces
469
Manuel Valdivia
339
Functional Analysis, Holomorphy and Approximation Theory II, G.I. Zaputa (ed.) 0 Elsevier Science Publishers B . V. (North-Holland), 1984
1
ON GENERALIZED TOEPLITZ KERNELS AND THEIR RELATION WITH A PAPER OF ADAMJAN, AROV AND KREIN
Rodrigo Arocena
SUMMARY We consider the relation of the so called generalized Toeplitz kernels with some theorems of Adamjan, Arov and Krein, concerning the unicity of the best uniform approximation of a bounded function, canonical approximating functions and the parametrization of the approximations.
I, INTRODUCTION K
Let
be a kernel in
= K(j+l,n+l), sequence
K
ZxZ.
function in
n E Z)
(c(n):
definite, p.d.,
is said a Toeplitz kernel if
(j,n) E ZxZ,
Y
C
if
the set of integers, that is, a
2,
K(j,n) =
o r , equivalently, if there exists a
such that
K(j,n) = c(j-n ) .
-
K(j,n)s(j)s(n)
P
0
whenever
K
is positive j E Z)
{s(j):
j,n
is a sequence of finite support. int = e e,(t) measure in
T
Let
T
be the unit circle,
where
and integrals are over
T
m
is a complex Radon
unless otherwise specified.
The classical Herglotz theorem says that a Toeplitz kernel p.d. iff
K(j,n) = G(n-j),
Radon measure in
V
(j,n) E ZxZ, where
T, and, moreover,
m
m
K
is
is a positive
is unique.
In this paper we deal with the following extension of the notion of Toeplitz kernel. (1.1) DEFINITION.
K
K(j,n) = K(j+l,n+l),
is a generalized Toeplitz kernel, for every
j,n E Z-(-13.
GTK, if
R. AROCENA
2
This definition is equivalent to saying that there exist four sequences Krs, r , s = 1 , 2 , such that K(j,n) = Krs(j-n), def E Zrs = Zr X Z s , where Z1 = (n E Z: n 2 01, Z2 = 2 - 2
(j,n)
V
1’
For these kernels there is a natural extension of the Herglotz theorem.
A C
As usual,
T,
r,s = 1,2,
T; we write
measures in E Zrs.
M = (mrs),
Let
M
(mrs(A))
be a matrix of complex Radon Krs(j,k) = GrS(k-j),
K - M A if
is said to be positive,
M
2
(j,k)
Y
0, if for every
is a positive definite numerical matrix.
Then
the following theorem holds, a proof of which will be given in next section, (1.2) THEOREM.
Let
K
be a
GTK.
for some positive matrix measure
M = (mr s ) belongs to the class h(K). K
If
and
Then
T; Lp
are as in (1.2) we say that
= (f E Lp: ?(n) = 0, Y n nomials,
P+ =
P
n
H
1
,
dicate a matrix of four that
m12 = rii21.
MA
-K
holds for every
2
03;
P- = P
n H:
and
K
M
2
-
K A , then
we write
M
-
K M‘.
.
= 1,2)
0
is p.d.
iff
is Lebesgue normal-
?
m;
Y
is the Fourier
n
0.
Conversely, if
M’ = (mks)
is such that
Then (1.2) implies the following.
M
5
M’
0
-
M and KA,
*
3
ON GENERALIZED TOEPLITZ m R N E L S
(1.4) THEOREM. M’
such that
If M
If
M
t 0,
there exists a positive matrix measure
- M.
M’
and
(1.4) we say that M‘
M’ are as in
is a posi-
M.
tive lifting of
In [4]
Theorems (1.2) and (1.4) were established in [9].
and [ 5 ] they were applied to uniform approximation by analytic functions and related moment problems, of the type considered by Adamjan, Arov and Krein [l]. the unicity of
M
2
0
These questions led to the problem of
such that
ditions that ensure that
m(K)
M*
-
K,
that is, of finding con-
contains only one element, which is
not always true, contrary to what happens in the classical case.
In [ 6 ] , a different and simpler proof of the generalized Herglotz theorem (1.2) was given, which can be extended to the vectorial case
[7] and leads to a better unicity condition.
That approach happens
to be much closer to the ideas of Adamjan, Arov and Krein.
In par-
ticular, it enables the consideration of the second paper [ 2 ] that those authors dedicated to Hankel operators, generalizing their principal theorems and extending their concepts and methods to obtain new results concerning generalized Toeplitz kernels.
That
is the subject of this paper.
In sections I1 and I11 we review the proofs given in [ 6 ] of the existence of elements in
m(K)
and of the unicity condition,
adding some details that will be needed in the sequel.
In sections
I V to XI some new results are presented, concerning the following subjects and their applications. a)
Conditions that ensure that the class
m(K)
contains only
one element, including an extension of a theorem of [ Z ] concerning the unicity of the best uniform approximation of a bounded function by analytic functions.
4
R. AROCENA
b)
Definition, characterization and properties of the canonical
elements of
h(K),
and, in particular, generalization of some theo-
rems on canonical functions given in [ 2 ] . c) given
f E L"
r
and
(M'2 0: M'
the class
11.
{h E H": IIf-hllm .s r ) ,
A remarkable parametrization of
with
> 0, is established in [ 21 , By means of it
- M),
M > 0, is parametrized.
with given
CONSTRUCTION O F ALL THE POSITIVE MATRIXES ASSOCIATED TO A POSITIVE DEFINITE GENERALIZED TOEPLITZ KERNEL
From now on,
K
GTK.
will always be a p.d.
Let
HK
be
the Hilbert space defined by the linear space P and the metric def = K(n,m). Set H . = the closed linear hull given by (en,em)K J
in
HK
of
{en: nfj] ,
Ven = e
isometry given by class
b(K)
space
HU
3
if
HK
j = 0,-1, and
n+l'
U
V: H-l -I Ho U
We say that
is a unitary extension of
the linear
belongs to the
V
to a Hilbert
.
U E L(K) generates a matrix def M(U) E h(K). F o r every v E HU, J(n,m) = (Unv,Umv) is a p . d . HU ordinary Toeplitz kernel; choosing successively v = eo,e,l,eo+e,l, We shall now see that every
eo+ie-l,
we get four such kernels,
Kll,
K22, F, G ,
which, by the
classical Herglotz theorem, are given respectively by the Fourier transforms of four positive measures,
mil,
m22, u, V.
Set
It is easy dm12 = e-ldm, m21 = G12. def M(U) = (mrS), r,s = 1,2 verifies:
m = 1 (u+iv-(l+i)(mll+m22)), to see that the matrix
If
fl = C a .e ., J J
f2 = C bjej
belong to
P,
the previous equalities
ON GENERALIZED TOEPLITZ KERNELS
5
z
imply that
r, s=l,2
M(U)
Analogously, a straight forward verification shows that
0.
2
K( j,k) = I?I ing
.
rs
(k-j),
(j,k) E Zrs.
Y
(11.2) PROPOSITION.
U E L(K)
Each
so
h(K)
V
gives rise, by means of for-
m(~).
M(U) E
to a matrix mulae (II.~), Obviously,
So we have proved the follow-
always admits a unitary extension to
HK@l 2 ,
is not empty, neither is, by the above proposition, m(K).
So we have proved the generalized Herglotz theorem (1.2). Now we shall consider the reciprocal of (11.2), each
M = (mrs) E h(K)
way that
= M.
M[U(M)]
U(M) E L(K)
we shall associate Since
M
2
the linear space
0,
that is, to in such a
PxP
and
the metric
(11.3) define a Hilbert space
HM
en + ( e , , O )
and
if
isometry from
n 2 0
HK
HM,
to
.
Since
HM
Then
-K
en + (O,en)
the correspondence
if
n < 0
defines an
s o we can identify the former with a
closed subspace of the latter. in
M”
Let
U(M)
be the unitary operator
given by:
U(M)
V,
extends
so
U(M) E I n ( K ) .
Moreover,
= ((en,~),(eo,~))M = ill(-n); from analogous veHU(M) rifications in the other cases, the following result is apparent.
(U(M)neo,e
)
(11.5) PROPOSITIOI~. If M E h(K), fine a unitary operator
formulae (11.3) and
U(M) E L ( K )
such that
M[U(M)I
(11.4) de= M.
Summing up, the construction presented in this section gives A
all the matrices
M
2
0
such that
M
- K.
R. AROCENA
6
111.
THE UNICITY CONDITION
and the theorem of F. and M.
h(K)
The very definition of Riesz imply the following. (111.1) LEMMA.
,
= urr
(mrs) E h(K).
Let
r = 1,2,
u12 = m12
+
Then
h dt,
( u r s ) E h(K)
iff
m rr =
+ fi dt, where
u 21 =
h E HI. Consequently the problem of unicity when
= 1
#[h(K)]
G12(n)
for every
-
-
that is, of knowing
is the problem of knowing when
n
2
0.
K
detrmines
Now, a straight forward calculation
shows that : (111.2)
If eel E H-l if
,
then
eo E Ho, then
Vn+ 1 e,l)K
(V-n-leo,e-l)K
'
= '12(n),
( eo,
=
G12(n),
\d
O;
n
2
0.
Obviously, any of the two conditions in (111.2) implies that determines
m12
and, consequently, there is only one
K
M E h(K).
We shall see, reciprocally, that if neither of those two conditions holds,
h(K)
contains more than one element.
(111.3) NOTATION.
j = -1,0
Let
and
ej
Hj;
let
v
j
be the
the condition
HK, perpendicular to Hj and well determined by def cj = (ej,vj)K > 0; let u be the orthogonal
projection of
e
unit vector in
j
j
on
H
j'
Define the linear operator Vt: HK
-t
HK,
(111.4) It is clear that
Vt E L(K)
and that every unitary extension of t
to
HK
has this type,
Set
(mrs)
= M(Vt);
and the above notation it follows that
(111.5)
from (11.2),
V
HVt = HK
ON GENERALIZED TOEPLITZ KERNELS
#
Since
if
C ~ C - 0 ~,
(111.6) THEOREM. equivalent:
a)
e it f eit'
For every h(K)
,
7
M(v~) # M(v~,),
K, GTK,
so:
the following conditions are
contains more than one element;
b)
Hj,
ej
j = -l,O.
Let us recall (see (1.1)) that every sequences
r , s = 1,2;
Krs,
KI1 = KZ2
if
plications) it is easy to see that V
metry in
,
HK
-
dist e - p H - 1 ) = dist(eo,Ho). (I11 7 ) COROLLARY.
Let
K
(as is usual in the ap-
,
= Hml
B(Ho)
IV.
so
Consequently: be a p.d.
GTK
e - l d Hml; c)
b)
,
defines an antilinear iso-
Kll = K22.
such that
Then the following conditions are equivalent: more than one element;
is given by four
(ej,ek) = (e-j-l,e-k-l)K
B(ae.) = ae j-1 J such that B(e ) = e-l
(j,k)E ZXZ; then
GTK
a)
eo
f$
h(K)
contains
Ho.
EXTENSION O F ADAMJAN, AROV AND KREIN UNICITY THEOREM We shall now
see
how the method employed in section 2 of [2],
for proving a theorem concerning the unicity of the best approximation in
L"
to some
GTK,
by function of
by considering our previous results.
T o each
Ti
,
K , GTK,
we associate two Toeplitz forms,
and one Hankel form,
(IV.l)
j
2
H',
= Kll(j-n)
T;(ej,en) if
can be used to extend that theorem
H",
0,
n < 0;
Ti
and
in the following way: if
j,n 2 0 ;
T;(ej,en)
H'(e.,e-,) J
= K22(J-n)
The method under consideration applies to
if GTK
= K12(jfn) j,n < 0 .
such that
the above forms are bounded, that is, given by bounded operators 2 2 2 2 2 2 T1: H + H , H: H -t H- , T2: H + HNow, we have the follow-
.
ing well known (see for example [lh])
property of Toeplitz operators.
8
R. AROCENA
(IV.2) PROPOSITION. P+ x P+
form in
Let
n E Z) C C
{b,:
T(e. e ) = b . J’ n J -n H2 x H2 iff 3 f E L”
given b y
to a bounded form in
v n 6 Z , and in that case
.
T
be the bilinear can be extended
such that
?(-n) = bn ’
.
IIfll,
IIT/l=
T
and
The corresponding result for Hankel operators is the following. (IV.3) NEHARI’S THEOREM 1131.
Let
(b,]”
c C
be given.
Then
n=1
there exists a bounded operator = b J.+ k
,
V
;(-n) = bn,
j
H: H2
-t
Hf
> 0, iff there exists
2
0, k
Y
n > 0. In that case
(He.,e-k) =
such that f 6 Lm
J
such that
IIHIl = dist(f,H”).
This theorem is a simple consequence of the generalized Herglotz theorem applied to the = /lH/jS(n), V n E Z,
and
GTK
given by
v n > 0.
= bn,
K12(n)
= K22(n) =
Kll(n)
(See f 5 ] ) .
From (IV.2) and (IV.3) we have the following. (IV.4) PROPOSITION.
Let
be a p.d. GTK.
K
The bilinear forms
defined by (IV.l) are given by bounded operators iff M = (w
rs
dt) > 0
When
K
and
K
- MA,
with
wrs E L“.
is as in (IV.4) the unicity condition (111.6) can
be given in terms of the operators
T19 T2, H.
As in [ 21
the proof
rests on the following. (IV.5) KREIN’S LEMMA. in a Hilbert space
Let
E;
let
A
be a bounded non-negative operator
EA
be the Hilbert space obtained by
completing the linear manifold E with respect to the metric def the (g,g’)A = (Ag,g‘). Then in order that, for any h (E E), linear functional
Fh(g)
= (g,h),
is necessary and sufficient that
h E E,
lim ((A
be continuous in
+ sI)-’h,h)
1 lim
. ( 2 ) does.
that
and i t i s e a s y t o s e e t h a t
LL
#[h(K)]
7
to
F
A = ( T ~ + H ) ~ ++ ( T ~ + H * ) ~ -;
1
iff
e. J
H. J'
j
= -l,O,
being continuous i n
j
So from ( i ) and K r e i n l s lemma ( I V . 5 )
-1,O.
m
t h e n (1) h o l d s i f f
respectively.
and t h i s i s e q u i v a l e n t ( I V . 6 )
=
0,
a.e. s o
s+r,
5
m(K)
Consequently,
H".
,
so
= 1
#[h(K)]
For this k e r n e l
iff
K
f
3
s+r
f-hr) 2 0, which s+r def M = (7 s ) > 0.
(FZr
r > 0; therefore
V
has the form
h E H1, which is equivalent to
with
in 2 H-
[f-hrl
r > 0
for every
h E H"
z ( L G f-h) s
0,
/)f-hllm= s
and
([b]).
has a unique best approximation
we have
T1 = s I
in
T2 = sI
H2,
in
from (IV.7) see get the following.
(IV.8) COROLLARY.
Let
Hankel operator given by There is only one
s = dist(f,H")
f E L",
h E H"
(Hej,e-k) = z(-j-k), such that
dist(f,H")
and
+
H
be the
j 2 0,
= IIf-hllm
k > 0.
iff the
following holds: lim ([r21-H*H] ri s
-1 eo,eo) =
m.
This last result constitutes theorem (2.1) of [ 2 ] , proved there by the method the extension of which to
GTK
which is has been
presented in this section.
V.
CHARACTERIZATION OF CANONICAL MATRICES
For
K,
GTK
#[h(K)]
such that
> 1, we shall
use notation
(111.3) and also the following.
(V.2)
G12(0)
Clearly,
= g
+ P(V~,UV-~ for ) ~every ~ U E
I(V~,UV-~)~ s ~ 1, I
so
U E L(K)
L ( K ) and (mrs)=M(U).
can be chosen in such a
ON GENERALIZED TOEPLITZ KERNELS
11
takes any complex value of modulus not bigger HU That implies the following.
(vo,Uv- )
way that than one.
(V.4) DEFINITIONS.
U E L(K)
Let
K
be a p.d.
is a canonical element of
GTK L(K)
such that if
HU = HK
.
M E h(K)
U E L(K)
is a canonical matrix if there exists a canonical
> 1.
#[h(K)]
such
M = M(U).
that
From (111.4) it follows that the set of canonical elements of
L(K)
CVt: t E C 0 , 2 l l ) 3 .
is the same as the set
We have the following characterization of canonical matrices.
(v.5) THEOREM. M = (mrs)
Let
M
and
M = M(U),
be a p.d.
GTK
such that
> 1
#[m(K)]
and
The following condtions are equivalent:
E h(K).
(a)
K
is a canonical matrix;
(b)
lG12(0)-gl
U(H K ) = HK ;
then
(d)
= p ; (c) if U E L(K)
H K = HM
PROOF.
(a) implies (b):
Since
(mrs) = M(Vt)
for s o m e
t,
(b) follows
from (111.5).
In this case (V.2) says that
(b) implies (c): so
U V - ~ is parallel to
= HK
vo;
U(HK) = V(H-l)
then
I = 1, HU (CUV-~: cEC) =
I(vo,Uv- )
+
.
(c) implies (d): and
Since
(ej,O) = U(M)j(eo,O)
every
j
< 0, so
(d) implies (a):
PXP
=
vt/,
=
t = t'.
Then:
(v.6)
= HK
belong to HK
.
HK
(O,en) = U(M)n+l(O,e-l) for every
n
;r
0
and
.
Follows immediately from the definitions.
Note that, since u[M(vt)]
U(M)(HK)
SO
U[M(Vt)]
is canonical, it must be
M(v~) = M ( v ~ / ) and, by (111.5) and (111-4),
U[M(Vt)]
= Vt
,
V
t E [0,2T).
AROCENA
R.
12
VI.
SOME A U X I L I A R Y MATRICES
M = (m ) rs
Let
Radon m e a s u r e s i n
r,s = 1 , 2 ,
T;
set
dm
rs
b e a n h e r m i t e a n m a t r i x o f complex
= w rsd t
+
L = L(M)
W = (w
w i l l denote t h e matrices
vr s
where
rs
)
is
W = W(M)
From now on
s i n g u l a r w i t h r e s p e c t t o Lebesgue measure. and
,
dvrs
L = (v
and
rs
).
The f o l l o w i n g e q u a l i t y i s e v i d e n t ,
v fl,f2 So A
W 2 0
c T
and
L 2 0
M t 0.
imply
Conversely,
B C T
t h e n f o r any
M z 0,
if
v
be a Lebesgue n u l l s e t such t h a t s u p p o r t
E P.
rs
= (mrs(BnA))
( fI w rs d t ) =
(mrs[Bn(T-A)]) a r e p o s i t i v e , so L 2 0 'B ( V I . l ) a l s o shows t h a t W > 0 and L 2 0 i m p l y M > 0 .
and
M'-
l y , if the l a t t e r holds l e t def
=
W'
-
W(M')
L'
and
W
def
=
M
L(M')
= L,
so
W z 0
z 0.
and W
Converse-
M'z 0 ;
be such t h a t
r,s=1,2;
c A,
(vrs(B))
t h e numerical matrices
let
then
L 2 0.
and
Then : ( V I . 2 ) PROPOSITION. W
> 0
iff
and
W(M)
L 2 0;
-
W(b1')
The f o l l o w i n g r e l a t i o n s h o l d : b)
M 2 0
and
L(M)
W 2 0
iff
= L(M');
2
2 Il(f,g)llM = II(f,g)llW + l l ( f , g ) l l ~ 9
"
d)
and if
GTK,
ciated t o
M E h(K) W^.
W = W(M);
and
L 2 0;
M 2 0
denote by
too.
K'
The a b o v e c o n s i d e r a t i o n s show t h a t
d o e s n o t depend of t h e m a t r i x
M
c)
-
iff
M'
then
( f , g ) E pxp*
The l a s t a s s e r t i o n f o l l o w s f r o m ( V I , l ) , p.d.
M > 0
a)
M E h(K)
Let
K
t h e GTK
K'
be a asso-
i s p.d.,
used i n i t s d e f i n i t i o n ,
and, moreover:
(VI.3)
k(K) =
k(K') +
L = L(M).
L,
with
K -
MA,
K'-
WA,
W = W(M)
and
13
ON GENERALIZED TOEPLITZ KERNELS
distH (ej,Pj) = 0, L is the linear space generated by the ek with k f j;
Also, since the
P
where
j
vr s
are singular measures,
dist (ej,P.) = dist (ej,Pj). HK J HK/
then, by (VI.2d),
S o the problems of unicity and of describing all the elements
of
n(K)
can be restricted to the GTK generated by function ma-
trices
(wrS).
M = (m
)
rs
w
h
set
12); for any matrix
N(h)
In fact, if
w = (wll.w22) 1/2
set
0,
5:
111
Wo =
and
mll,m22
.
wll = w22
We can even suppose that
is hermitean and
N = (u
)
;
H 1(M) = (h€H1:M(h)zO).
rs
12
=
U
set
and any function
22
Then : (VI.4) PROPOSITION. Wo > 0
c)
and
9 h E H1(Wo)
Wo(h)
2
Wo
M > 0
0
0
2
iff
r2
0
iff
W
iff
Wo
0
so
L
2
2
3 h E H1(M);
L z 0. Now
and and
M
M
iff
t 0
L z 0;
and
0
2
a)
h E H1(Wo).
iff
Clearly
Obviously
0; b)
t
h E H1(M)
PROOF.
e
L
The following relations hold:
(b) follows from (VI.2). Wo > 0
also,
h E H1(M)
0 u h E H1(Wo)
W(h)
e
and
L
2
2 0;
L z 0
and
L
iff
0
and
so
(a) and (c)
2
0
have been proved. The consideration of canonical matrices can also be restricted to function matrices.
In fact:
(VI.5) PROPOSITION.
K
> 1. Then
#[h(K)] in
Let M
and
K’
be as in (VI.3)
is canonical in
h(K)
iff
W
and is canonical
h(K‘).
PROOF.
Since, for every 2
= dist ((fl,f2), P+xP-), HW the result follows.
(fl,f2) E PXP,
HK
c
HM
iff
2
((fl,f2>, p+Xp-) = HM F r o m (V.5) HK/ = HW dist
R. AROCENA
14
VII.
WPRESENTATION O F THE HILBERT SPACE ASSOCIATED TO A
A
POSITIVE FUNCTION MATRIX
We shall now see how a construction employed in [ 2 ] extended to give a representation of
W = (wrs) =
and
W(y)
p
= wll
2
E = L
Set
B
isometry
from
2
@ L (p dt).
Hw
to
E
E E
63 B ( H W )
2
Set
0.
~ ~ ~ =( 0t . )
I
is easy to see that a linear
is defined by
B;
it is not hard to see that 1 = 0 and w221? = 0,
3
is equivalent to F = 0, w22
dt-a.e., that is, to
M
w121 / w ~ ~ assuming , from now on
We shall now determine the range of (F,G)
, when
2
-
~ ~ ~ ( t )2/ w ~ ~= (0t )whenever
that
HM
can be
dt-a.e. and
G
= 0, p dt-a.e.;
consequently I
( VII .2 )
B ( H ~ )=
x r w22>o~L2
L2(p dt).
Moreover: (VII.3) THEOREM.
P =
-
Wl1
a)
Iw121
Let
M
2
0, W = (wrs) = W(M),
2
X1w22,0~/W22
Xr ~ ~ ~ ’L20€91 L2(p
HW w
.
L = L(M),
Then the following hold.
dt),
where the isometric isomorphism
is given by (VII.l). X(~22>0] L2 €9 L2(p dt) @ HL ,
‘M
b,
If, also,
c) 2
K
-
M A is such that
#[h(K)]
> 1, then
2
HW = L €9 L (P dt)
(a) has been proved already; (b) follows from (a) and (VI.2); as to (c), it stems from:
(VII.4) LEMMA.
Let
M > 0, (wrs) = W(M),
w = (wll.w22)
3.
If M
15
ON GENERALIZED TOEPLITZ KERNELS
has more than one positive lifting, then log w E L 1
or, equivalently,
log wll
log w22 E L1
and
.
This lemma is an obvious consequence of (VI.5) and the following. (VII.5) PROPOSITION.
Let
w
2
0
be an integrable and not trivial
function.
Then the following conditions are equivalent: (i) There w f exists a function f such that ) is weakly positive and has (7 w more than one positive lifting; (ii) l o g w E L1
.
PROOF.
If
(i) implies (ii): such that 2w
w
If+hll
2
and
hl
and
w z
h2
H1
are distinct elements of
If+h21, log w E L1
because
Ihl-h21.
2
3 h E H1(T) such that Ihl = w w h () 2 0; take f 0. h w
(ii) implies (i): not trivial and
VIII.
h
tion equivalent to
(VIII.~)THEOREM.
-
Let
2
we can obtain a condi-
HM
having only one positive lifting.
M 2 0
M 2 0,
I
(wrS) = w ( M ) ,
w = ( w ~ ~ . w ~and ~ )
The following properties are equivalent.
Iw121 /w.
a)
There exists a positive matrix
b)
log w E L1
M'
f M
such that
M'
-
M.
and at least one of the following conditions
holds : i)
log p E L1;
ii)
1
belonging to
H
there exist
2 1 Tl L (w dt)
hl E H1
and a not trivial
IWhl
such that w(lwl
belongs to p(t)
is
ANOTHER FORMULATION OF THE UNICITY CONDITION From the above representation of
p = w
a.e., s o
= 0.
L1
and that
w(t)hl(t)
-
-
2
Wl2h2I
-Iw121
w12(t)h2(t)
2
= 0
2
1
XEP>Ol whenever
h2
16
R.
AROCENA
w
Because o f ( V I . 4 )
PROOF.
Suppose ( a ) h o l d s .
HM
L
N
2
2 @ L (p dt).
trivial
log w E
Then
S i n c e u n i c i t y d o e s n o t h o l d , t h e r e e x i s t s a not
(F,G) E [L2 @ L2(p d t ) ] FW12/w
If
F = 0
-
l o g [ IGI 'p]
a.e.,
+
.
1
and
>
0,
so
3=
Fw
h2,
l o g w 2 log p = loglhll
5
w 12;
G E
since
L2(p d t ) ,
t h e l a s t equal-
( i i ) . So ( a ) i m p l i e s ( b ) .
Now assume t h a t ( b ) h o l d s .
If
H1
function i n
then
(*), b u t n o t t o
not hold.
If
( i i )h o l d s ,
f
h;
= enh2,
set
hl,
h2
r e p a l c e d by
to
E E) B ( H o ) ;
so
e
&'
(VIII.2)
Ho.
r2
so
hl
= en h' l
F = h;/wl/',
so
hi, hi,
,
let
hl
a.e.,
(F,G)
G = lil/p;
B(Ho)
orthogonal t o
E,
eo
be an o u t-
and s e t
q
with
Ho
and u n i c i t y does
h i E H1
such t h a t
G
and
g;(O)f
(*) holds with
i s n o t t r i v i a l and belongs
{ (F,G),B(eo,O))E =
Let
r = ) ) f - h lm )
.
r = dist(f,fl)
E Lm,
f
Then s u c h
h
the following conditions holds:
(f-h)G21 lf-hI2
1
Gi(0)
f
0 and
The r e s u l t f o l l o w s .
hl E
(ii) there e x i s t
Ihl
let
L
lhll
B(eo,O),
on t h e o t h e r h a n d
COROLLARY.
b e such t h a t one of
p =
such t h a t
because of
0;
log p E
i s a n o t t r i v i a l e l e m e n t of
(0,G)
-
2
(*) i t a l s o f o l -
i t y e n s u r e s t h e v a l i d i t y of
er
.
1
is n o t t r i v i a l , ( * ) shows t h a t
F E L2
-
Gpw = h w 1
hl,h2 E H
with
and i s n o t t r i v i a l ; m o r e o v e r , f r o m
(G d t )
lows t h a t
El
If
which i s e q u i v a l e n t t o
0 B(Ho),
Gp =
/Ihllll
E L
2 1
h2 E L
$
12). w21 and, consequently,
(VII.4)
L1
w
M = (
we may assume t h a t
H1
7 0
and
h E H"
i s n o t unique i f f a t l e a s t
1
E L ;
( i ) log[r-lf-hl]
and a n o t t r i v i a l
h2 E H2
such t h a t
2
-
'[r2-If-hl
>0)
E L1
and t h a t
- - -h
h1 =2
whenever
r = If-hi. (VIII.3)
COROLLARY.
Let
There e x i s t s a not t r i v i a l f
= ug/g,
with
u
f
be unimodular and d i s t ( f , H m )
= 1.
IIf-hll m = 1
iff
h E H"
i n t e r i o r and
such t h a t g E H2
and o u t e r .
ON GENERALIZED TOEPLITZ KERNELS
REMARK.
Clearly,
If
EXAMPLE. h E H1
in (VIII.l)
= w,
wrs
such t h a t
may b e assumed t o b e o u t e r .
r , s = 1,2,
w 2
Iw-hl
2 1 k E L (F d t ) ,
such t h a t
h2
17
there exists a not t r i v i a l
iff t h e r e e x i s t s a c o n s t a n t kf0
a.e.
1
;SE
that is, iff
a p p e a r s i n a remarkable theorem of P.
L1.
This condition
Koosis [ 1 2 ] ,
concerning
w e i g h t e d q u a d r a t i c means o f H i l b e r t t r a n s f o r m s ,
t h a t can be obtain-
ed by d e v e l o p i n g t h e a b o v e s k e t c h e d r e a s o n e m e n t
[6].
SOME PROPERTIES OF C A N O N I C A L MATRICES
IX.
Let W = W(M).
So,
M
Then
considering
(VII.l),
s o t h e same h a p p e n s w i t h
be a canonical matrix,
B(Ho)
HW
=
Ho
L
2
2
3 L (p d t )
t
HW
a s a subspace of
p = wll
with
E,
and w i t h
E.
h a s c o d i m e n s i o n one i n
Now,
B
2
= Iw121
/wz.
as in
(F,G) E E 0 B(Ho)
iff
-1 2 Fw12/w22
( I X . 1)
+
A C T,
such t h a t
real in
1 , w = ~h 2 ~
7
a l 2 w 11
2
p 2 a2
Then
i
IG1I2P
Moreover,
F1 E L2
2i,
d t L Ilhll,
IGI2p d t
[3
(F1,G1)
s o t h e dimension of
must b e
p = 0 THEOREM.
Let
.
1
Then t h e r e e x i s t s al,
A.
Proceeding as i n
h E Ha
be such t h a t
(T-A)
+
8
and
:l
k
i s not p a r a l l e l t o
a.e.,
E H
G1
h
a2,
'1-
1
= (LK1-FhGl2/w 2 2) -P
and
trivial,
M = (mrs)
in
hl,h2
and p o s i t i v e c o n s t a n t s
holds i n
F1 = hF
Set
with
does n o t h o ld a . e .
of p o s i t i v e measure,
(T-A).
A.
(IX.2)
F
3 o f [ 2 ] , l e t a non c o n s t a n t
section
in
cl,
p = 0
Suppose t h a t a set
Gp =
E
(3
B(Ho)
I F I 2 d t l , s o (F1,G1)EE
(F,G)
if
0 B(Ho).
the l a s t i s not
would n o t b e o n e ; t h e n i t
that is: K
be a p.d.
a c a n o n i c a l element o f
GTK m(K)
such t h a t and
#[h(K)]
dmrs = w r s d t
> 1,
+
dvrs,
18
R.
with
vrs
wllw22
-
s i n g u l a r w i t h r e s p e c t t o Lebesgue measure.
w12w2, =
F E L~ Q B ( H ~ ) i f f
h
-
-
i n t h e above h y p o t h e s i s
F
;
~
hl = uh,
Set
Then
a.e.
0
Consequently
h 2 E H1
AROCENA
3
~
~
=/
with
G1 W
u
F u G ~ ~ = / w 6,~ ~F u 2Fw =~ u~h 2 ;
~
and ~
F
HW x L
&w = ~h 2~'
i n t e r i o r and
h
2
and
with
hl
outer,
and
so
t h e n t h e same argument c o n c e r n i n g
u
t h e codimension employed i n t h e l a s t p r o o f shows t h a t
1
I
and
leads t o the following.
(IX.3)
THEOREM.
outer
I n t h e same c o n d i t i o n s o f (IX.2) t h e r e e x i s t two
H
functions i n
W
1
hl
and
h2'
such t h a t
W
1 2 - hl -W
-- -w 11 '
22
fi2
lhll
=
21
Note t h a t i f , moreover,
s o there e x i s t s an
w 1 2/ W
outer
= w22 = w ,
wll
h E H1
function
n
then
APPLICATIONS
[a].
=
RELATED TO THE H E L S O N - S Z E G ~ THEOREM
W = (w
Let
wI1
= w
t
rs
)
be an h e r m i t e a n f u n c t i o n m a t r i x w i t h
the kernel
0,
+
s = inf(;(O)
(x.1)
C
-
A
W
+
'
cjEkG12(k-j)
+
2 Re
2 Re
X
It i s not hard t o see t h a t
such c a s e
K
cjG(-j)
t
cjG12(-j))'
(c,}
i s p.d. iff
of f i n i t e j€
s
2
0;
in
s = d i s t ( eoyHo). Consequently:
(1) The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t : with
o,j c
g i v e n by
j 0
1W-h)
= 0
i f f sk
b e s u c h t h a t more t h a n one f u n c t i o n
- z' ( F ( e i x ) ( d x ,
E Hm :
= W(M)
Bk
t h e followi n g remarkable r e s u l t i s s t a t e d .
+
eix
Let
a constant.
AROV AND K R E I N PARAMETRIZATION
121
of
such t h a t , i f
i,
IIvllm
nth
C(C
,
T
n
r o o t s and l e t t i n g
~
~
and f o r a l l
n,
) . s ~ p r I i c p n ( x ) i i: d i s t ( x , ~ )
t a i n e d i n t h e w e a k l y compact s e t
(Hd(Lm
E Hd(Lm)
f
c ( ~ ) ' s u p { I / f ( x ) l l :d i s t ( x , K ) < E } .
cp E C 1 C C L
c
t h e n f o r some a b s o l u t e l y
we would h a v e t h a t for a l l
such t h a t i f
0
xEB(co)}
ll~nllB(co) Taking
Lm,
in
K
is a barrel in
1)
S
i s) n ' t b a r r e l e d .
1161.
H(Lm)
i s b o u n d i n g for
-t
we s e e t h a t
m,
K.
d.
B(co)
i s con-
T h i s c o n t r a d i c t i o n shows t h a t
Hwsc(Lm)
The p r o o f f o r
T h u s f a r our d i s c u s s i o n on b a r r e l e d n e s s o f
i s identical.
Hd(E)
h a s c l o s e l y p a r a l l e l e d t h e analogous s i t u a t i o n f o r
(H(E)
a n d Hwsc(E) An
,TW).
i n t e r e s t i n g d i f f e r e n c e f o l l o w s from t h e f o l l o w i n g simple p r o p o s i t i o n which a l s o h o l d s for Ferrera
reflexive.
(1).
(2).
reflexive or
A = A
(Compare t h i s r e s u l t w i t h
r 91 ) .
PROPOSITION 2 .
PROOF.
Hwsc(E).
(Hd(E)
If
,TOd)
(Hd(E),~wd) i s b a r r e l e d ,
i s convex, b a l a n c e d and a b s o r b i n g .
{ZX
: z E
C,
Therefore
A
i s barreled.
(Hd(E),TOd)
: IIdf(O)ll = s u p ( I d f ( O ) ( x )
( f E Hd(E)
'rod,
then e i t h e r
is
E
is
H ~ ( E )f H ~ ( E ) .
( 1 ) . Suppose t h a t
for
E
i s b a r r e l e d i f and o n l y i f
then f o r a l l unit vectors
IzI
=
13.
Thus,
is a barrel i n
I,
IIxIl
Also, x,
df(O)(x)
f
a
S
if
-+
f
Let
13
5
1.
fa
E
A
Clearly and
f
a -'f
u n i f o r m l y on
=
(Hd(E),~Od)
and t h u s t h e r e i s a n
28
ARON a n d CARLOS HERVES
RICHARD M.
K c E
a b s o l u t e l y convex w e a k l y compact s e t such t h a t
A 3 (f
E
Hd(E)
: IIfllK
i s contained i n c - I K ,
of E , B ( E ) ,
and a c o n s t a n t
E > 0
I t f o l l o w s t h a t t h e u n i t ball
< c}.
and so E is r e f l e x i v e . The c o n v e r s e i s
trivial.
(2).
Assume t h a t
= Hd(E).
IIb(E)
Since
is
(Hd(E),Twd)
b a r r e l e d , a n a p p l i c a t i o n o f t h e open mapping t h e o r e m ( s e e , f o r e x a m p l e , p a g e 299 of [ H b ( E ) -+
(Hd(E),TWd)
141)
shows t h a t t h e i d e n t i t y mapping
i s a t o p o l o g i c a l isomorphism.
a b o v e , we c o n c l u d e t h a t t h e u n i t b a l l o f a weakly compact s u b s e t o f
must b e c o n t a i n e d i n
E
and t h e r e f o r e
E
Arguing a s
must b e r e f l e x i v e .
E
..
Q .E D S i n c e D i n e e n h a s shown t h a t following
Hd(co) = H b ( c o ) ,
we h a v e t h e
. 3.
COROLLARY
(Hd(co),TWd) i s not b a r r e l e d ,
We t u r n now t o a r e v i e w o f
H w ( E ) , Hwsc(E),
t h e spaces
and
t h e q u e s t i o n o f c o m p l e t e n e s s of The b a s i c known r e s u l t s
Hd(E).
a r e summarized i n t h e f o l l o w i n g p r o p o s i t i o n .
L1
The
case i s
discussed l a t e r . PROPOSITION 4 .
i s reflexive or
E
provided a copy of
(1). H w ( E )
E
T~~
and
7
Wd'
i s s e p a r a b l e and d o e s n o t c o n t a i n
Ll.
(2).
Hwsc(E)
(3).
Hd(E)
PROOF.
i s complete f o r b o t h
(1).
i n t h i s case
i s complete f o r b o t h i s complete f o r b o t h
If
E
i s reflexive,
Hw(E) = Hwu(E)
TOd TOd
and and
T 7
wd'
wd'
then the r e s u l t i s t r i v i a l since
and t h e t o p o l o g i e s
TOd
and
T
wd
a r e b o t h 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 o n bounded s u b s e t s o f
E.
If
i s s e p a r a b l e and d o e s n o t c o n t a i n a copy o f
E
Proposition proof
of
3.3 of [ b ] ,
Hwsc(E)
= Hw(E).
(I), i t s u f f i c e s t o p r o v e ( 2 ) .
4,
t h e n by
Thus t o c o m p l e t e t h e
WEAKLY SEQUI?NTIALLY C O N T I N U O U S A N A L Y T I C FUNCTIONS
Let
(2).
E E H(E)
a function
such t h a t
f
a
argument u s i n g t h e c o m p l e t e n e s s of E lIwsc(E).
To see t h i s ,
Since t h e s e t
K = (x} U
f
for
The p r o o f
for the
T
let
u n
and hence f o r
TWd
Hwsc(E).
be a TWd-Cauchy n e t i n
(fa)
T
f
-t
by a r o u t i n e
TWd,
We c l a i m t h a t
(€I(E) , T ~ ) .
c o n v e r g e weakly t o
(x,)
{x,}
for
Then t h e r e i s
(? .I?.
which i s due t o P a r e d e s r 1 8 1 .
The a s s u m p t i o n i n P r o p o s i t i o n
f
-t
a
f ( x n ) -+
c a s e i s o m i t t e d , as i s t h e p r o o f
Od
E.
in
i s weakly compact and s i n c e f
i t i s easy t o see t h a t
Od,
x
f(x).
(3)
of
u. 4 . 1 seems t o be n e c e s s a r y , a s
t h e f o l l o w i n g example shows. EXAMPLE
5.
Let
n xn E
f ( x ) = Cn
H(.Ll)
x = (x,)
where
a',
an e n t i r e f u n c t i o n converge t o t h e f u n c t i o n f o r
IIowever,
f
$?
€IW(L1).
and s u p p o s e t h a t t h e r e i s
Lm
T
W
.
Thus
s i n c e t h e f i n i t e Taylor s e r i e s terms of
IIw,(L1).
such t h a t i f
( f ( x ) ( L 1.
Indeed,
> 0
E
x E D4
i = l , . .. , k .
i
\vj
-
Taking
i
j,
lf(x)
1
I
0
as
j
and
j'
j , j'
,
1
then
we g e t t h a t
I
< c
x
,
E R4
(1sil:k)
can be a r b i t r a r i l y l a r g e .
There a r e two i n t e r e s t i n g
observations
l a t i o n s h i p o f t h e c o m p l e t e n e s s of
Hw(E)
concerning the r e -
t o whether
contains
E
F i r s t , we do n o t know w h e t h e r t h e above argument c a n ed t o t h e c a s e o f a r b i t r a r y Banach s p a c e s show t h a t i n t h i s s i t u a t i o n e i t h e r topology complete when
k)
in
(j,j' E N' )
m
Icpi(x)
1 o
is a semi-norm on the linear hull
span C
such that set of
A.
x E XC}
E
CO,~]
1
mc(x)
0 there are
T1,
H
with
..
{T1,. ,Tn] + cN(A,Bo)
whence n
H(A) Since all
Ti(A)
C
(4) -+ (2a)
+
eBo.
are Bo-precompact (la) follows.
By symmetry the implication
(3)-
u Ti(A) i=l
-t
(lb).
(4) + (2a)
is true and therefore
T o finish the proof it is enough to show
that (2)
-+
(3a)i
By (2b) every
TE H
defines a function
fT; A -+ [BOI] a .rrt T(a)
43
THE PRECOMPACTNE SS-LEMMA
and it is easy to see that
[fT
nuous set of functions from
1
is a uniformly equiconti-
T E H]
(A,%,
(B)~) into
pactness-lemma applied to (2a) implies, that
[BOD.
(A,%'(B)~)
compact semi-metric space; since by (2b) for each
I
{fT(a)
E H) = H(a)
T
The precom-
a E A
is a prethe set
is precompact in the semi-normed space [Bolt
the Arzelh-Ascoli-theorem f o r the vector-valued setting ascertains that
I
{f,
T E H)
is precompact in the space C ( ( A , % ,
with respect to the topology of uniform convergence.
0O ) , [ B o ] ) I t was al-
ready mentioned that the equation sup m
= m
= sup sup I(Ta,b)l
(fT(a))
~ E AB O
aEA M B
(TI
N(A,BO)
holds which implies that precompact with respect to the uniform convergence j u s t means F o r Banach spaces
balls of
and
E
N(A,B')-precornpact. and
E
the sets
F,
A
and
B
being the unit
respectively, this is a well-known result on
F'
sets of collectively compact operators, [ 9 ] ; from this special case it is also clear that the extra-conditions (lb) and (2b) are not superfluous.
3.
SETS OF z-PRECOMPACT OPERATORS
F
Let
E
and
spaces and
C
a cover of
be not necessarily separated locally convex by bounded sets.
E
is precompact in
F
The space
for all
A E
C]
of all C-precompact operators is equipped with the topology of uniform convergence on all
on
PC(E,F)
all of
C
A E
C;
a basis of continuous semi-norms
where A is running through m"A,W) ' through a basis U F ( 0 ) of absolutely convex
is given by all
and
W
closed zero-neighbourhoods of
F.
The symbol
LC(E,F)
stands for
44
A.
t h e space
L(E,F)
DEFANT
K . FLORET
and
of a l l c o n t i n u o u s l i n e a r o p e r a t o r s
E
en-
F
-+
dowed a s w e l l w i t h t h e t o p o l o g y of u n i f o r m c o n v e r g e n c e o n a l l A € C. If
i s the scalar f i e l d
F
w i l l be used.
M
= LC(E,K)
E i := PC(E,IK)
the notation
The f o l l o w i n g p a r t i c u l a r c a s e s f o r
C
a r e important:
b := s e t o f a l l bounded s e t s pc := s e t of a l l p r e c o m p a c t s e t s c o := s e t o f a l l a b s o l u t e l y c o n v e x , r e l a t i v e l y compact s e t s e := s e t of a l l e q u i c o n t i n u o u s s e t s ( i f A € C
J u s t adding q u a n t i f i e r s over a l l
E
i s a dual space).
B = Wo,
and a l l
E
W
UF(0),
t h e ” i n d i v i d u a l ” theorem has t h e f o l l o w i n g ” g l o b a l precompactness theorem” a s a
For
COROLLARY.
(1)-(4)
(In
i s precompact i n
H(A)
(b)
H’(cp)
(a)
H’(w’)
(b)
H(x)
F
F
H
i s precompact i n
P~(E,F)
(4)
H‘
i s precompact i n
P e ( F b c ,E;)
(4)
the topology o f
a
T‘E H’
F’
A
for a l l
for a l l
E C cp E F’
for a l l
E.;:
i s precompact i n
i s precompact i n
for a l l
Ei
i s precompact i n
(3)
topology
all
the following statements
are equivalent:
(1) ( a )
(2)
H C L((E,E’) , ( F , F ’ ) )
w E
~
(la!)
(F’
,a) -+ E i . )
and
For special and W.
I t i s w o r t h w h i l e t o n o t e t h a t b y a simple
( l a ) i s equivalent t o
i s equicontinuous i n
H‘
d e n o t i n g by
6
)
c a n b e r e p l a c e d by any l o c a l l y c o n v e x
a r e weakly c o n t i n u o u s
.
0
*
s u c h t h a t a l l e q u i c o n t i n u o u s s e t s a r e a-bounded
R u e s s [lo] ; s e e a l s o [l]
(
x E E
c a s e s t h i s c o r o l l a r y i s r e l a t e d t o r e s u l t s of A . Geue [5]
argument
~
t h e t o p o l o g y on
E
the statement (2a) i s equivalent t o
L(“bc ,E;:)
i
which i s i n d u c e d by
(Ef)Lc,
45
THE PRECOMPACTNE S S-LEMMA
H
(2al)
L((E,G),F).
is equicontinuous in
If every precompact set in is continuous and
H
Eb
is equicontinuous then
In particular,
is equicontinuous.
L(E,F)
C
this holds true in the following cases C
E c.(E,6)
( F arbitrary):
(i)
E
is barrelled,
(ii)
E
is quasibarrelled,
(iii)
E
has a countable basis of bounded sets and every zeroEL
sequence in
arbitrary
C
3
pc
is equicontinuous (e.g. E
topological = gDF),
C =
a-locally
b.
For Schwartzt e-product of two quasicomplete locally convex separated spaces
G
F
and
G the corollary (with
0
F := Le(GLo,F)
E := G k o ,
= Pe(GLo,F) EL = G
whence
and
(E,6) =
“Lo)
is a refinement of L. Schwartzf characterization of relatively compact sets in
G E F
(see [ 8 ] ,
$44)
which was also obtained by
W. Ruess [lo].
4.
LOCALIZATION
B y definition, a subset
if there is a precompact set pact in
F;
H c L ( ( E , E / ) ,(F,F’)) C-localizes
K C Eh
if the precompact set
equicontinuous
H
such that K C Ei
C-localizes fully.
was initiated by some results of hi.
H(Ko)
is precom-
can be chosen to be
The study of this property
Ruess, e.g. in [lo].
The difference between ”C-localization” and “full C-localization” is of a more technical nature; in most of the applications every precompact set in
Ei
is already equicontinuous (compare the re-
marks after property (2al) in Section 3 . ) .
46
A. DEFANT
and
K. FLORET
PROPOSITION 1.
H c L((E,E‘) , ( F , F ’ ) )
(1) If pact in
H
is precom-
P~(E,F). If
(2)
C-localizes, then
and
G
are quasicomplete, then
F
H
C
Le(GkoyF)= G E F
e-localizes
( = e-localizes fully) if and only if there are compact
sets
and
K C G
Lc F
such that H c (KOOLO)~.
(Here
G‘OF’
PROOF.
C
(G
E
F)’
is the natural embedding).
F o r (1) check conditions (1) (a) and (b) of the global pre-
compactness result, (2) is as easy by taking
L := H ( K o ) .
Our concern is now to find conditions under which all precompact sets in
PC(E,F)
C-localize, i.e. when (1)-(4) of the global pre-
compactness-theorem is equivalent to
(5)
H
C-localizes.
Certainly this includes the problem under which circumstances each b-precompact operator is precompact
( : = there is a zero-neighbour-
hood whose image is precompact) which in general is not true.
In
the setting of e-products the question involves finding out when all compact sets
H
C
in Proposition 1 ( 2 ) .
G
E
F
can be llliftedll in the sense expressed
Note, that this is equivalent to the conti-
nuity of the natural embedding*)
The analysis will be split up in essentially two parts: calizes every precompact set (i.e. there is a
*)
U
E UE(0)
H
C
PC(E,F)
such that
When C-lo-
which is equibounded
H(U)
is bounded)?
When are
There is a close relationship between the notion of e-localization and the duality of E - and lT-topologies on tensorproducts. W e shall deal with this question in another paper.
THE PRECOMPACTNESS-LEMMA
precompact sets in
47
equibounded?
PC(E,F)
For the first question some further notation is helpful: R
of bounded sets of a locally convex space
A family
satisfies the
E
Mackey-condition (resp. the strict Mackey-condition) if f o r every A E R
B E 0
there is an absolutely convex
every (in
precompact subset of
E)
is B-precompact).
such that
A C B
and
is B-precompact (resp.
A
By the precompactness-lemma
strict Mackey-condition if and only if
F'
PC
pc
A
satisfies the
is a Schwartz-space.
One answer to the first question is given by the LEMMA 1
(1)
b = pc
or if H
C
If the family
c
H(U)
(2)
then every equibounded precompact subset
If
is Schwartz or c C pc in E then every precomPC H C P C ( E , F ) which is equiprecompact C-localizes fully.
(1)
If
b = pc
in
U E UE(0)
an equicontinuous set in
F
nothing has to be shown.
such that E'
H(U)
is bounded.
which is chosen to
the Mackey-condition it is enough to show that pact for every C
such
F'
other case take
H
U E UE(0)
there is a
is precompact.
pact subset PROOF.
F,
satisfies the Mackey-condition
Ei
is equiprecompact, i.e.
P (E,F)
that
in
in
e
Pz(E,F))
H(x),
x E E,
But (by the global result)
Uo,
is
according to is W-precom-
are W-precompact it suffices by
the individual theorem to check that
is absorbed by
H(Bo)
If B
Since (by the precompactness of
W E UF(0). all
Uo
In the
H'(Wo)
H'(Wo)
is B-precompact.
is precompact in
the Mackey-condition gives that
Ei
and H'(Wo)
H' (Wo)
is
B-precompact. (2)
Take
U E U,(O)
such that
every equicontinuous set in ness-lemma, whence
H
Eh
H(U)
is precompact.
If C
C
pc
is precompact by the precompact-
C-localizes fully.
48
A.
T h e other case runs as follows: condition of
H(U)
D
C
pc
and
Obviously
in
According to the strict Mackey-
is D-precompact.
i s contained in
H(Ko)
Take
is D-precompact.
compactness of pact.
H
such that
:= H'(Do) c Uo.
K
U
So it
for every
is A'-precompact
to apply the individual theorem note that H(A)
D
whence precompact.
DOo
K = H'(Do)
remains to show that
FLORET
there is a precompact set
F
H(U)
K.
and
DEFANT
absorbs
A
A
E
C:
whence
Again the individual theorem (and the pre-
in
shows that
Pz(E,F))
H'(Do)
is Ao-precom-
U
With rather the same arguments the following lemma can b e shown: LEMMA 2
(1) If in
E
b
in
F
satisifes the Mackey-condition or if
following property ( * ) :
K
in (2)
H
then every equibounded precompact set
If
b = pc
in
has the
There is an equicontinuous precompact set
H(Ko)
such that
E i
C Pz(E,F)
C c pc
F
is bounded.
or if the family Ei
continuous precompact sets in
e
n
pc
of all equi-
satisfies the strict Mackey-
condition then every precompact subset
€I C P C ( E , F )
with ( * )
Z-localizes fully. Collecting the results of the two lemmata gives the PROPOSITION 2.
In each of the following cases (a)-(.)
bounded precompact subset of (a)
C
(b)
Fi
(c)
e
c pc
in
E
and
Pz(E,F)
b = pc
in
every eyui-
C-localizes fully: F.
is a Schwartz-space. in
Ei
satisfies the Mackey-condition and
F'
PC
is
Schwartz, (d)
E
(e)
e fl pc b
is a Schwartz-space.
in
in
F
Ei
satisfies the strict Mackey-condition and
satisfies the Mackey-condition.
49
THE PRECOMPACTNESS-LEMMA
(a) and (c) follows from Lemma 1
PROOF.
ing the following fact: b = pc
Schwartz and
I?;
we11
(b) by ohserv-
as
is Schwartz if and only i f
F’
is PC The statements (e) and (d) come from
F.
in
as
Lemma 2 noticing for the latter that the following holds by the
E
precompactness-lemma: Ei
e fl pc
is Schwartz if and only if C C pc
satisfies the strict Mackey-condition and
in
E.
in
Later on it turns out that the assumption of one of the spaces being Schwartz is not at all artificial,
H C PC(E,F)
H’c Pe(F’ , E i ) PC e-localizes the following result is a corollary of Proposition 2 (d)
Since
and (a)
C-localizes if and only if
(and, of course, the global theorem):
PROPOSITION 3.
Let
F’ PC
every precompact subset
be Schwartz or I1 C PC(E,F)
b = pc
in
Ei.
Then
H‘C L(F‘ ,E&) PC
such that
is
equibounded, C-localizes. Coming back to the original question “When C-localizes every precompact set
H
C
PC(E,F) 7“
(la!) of the global theorem by ( 2 a f ) the set
H
note first that according to condition
H’c L(Fbc,Ek)
is equicontinuous and
itself is equicontinuous in
L((E,6),F)
in most cases implies that it is equicontinuous in
L(E,F).
which
In
view of Propositions 2 and 3 it is therefore reasonable to investigate under which circumstances a given pair
(M,N)
of locally
convex spaces satisfies the following localization principle: Every equicontinuous subset of PROPOSITION
4.
(M,N)
L(M,N)
i s equibounded.
satisfies the localization principle in each
of the following five cases:
N
(a)
M
or
(b)
M
has the countable-neighbourhoods-property (i.e. for
is normed.
every sequence
(Un)
in
which is absorbed by each
UM(0) Un)
there is a and
N
U E UM(0)
is metrizable.
A.
50
(c)
DEFANT
is Baire-like [ll] and
M
K. FLORET
and
N
has a countable basis of
bounded sets. ly
(d)
M
is metrizable and
NL
(or even the completion
NL)
is Baire-like. (e)
M
is metrizable and
N
has a countable basis of bounded
sets.
PROOF.
(b)
(a)
If
is obvious.
(Wn)
is a neighbourhood basis of
N
and
A
C
L(M,N)
n T-l(Wn) and a neighbourhood TE H according to the definition; then H ( U ) is bounded. equicontinuous take
(c)
If
(B,)
Un :=
U
is a basis of closed, absolutely convex, bounded
sets, consider
n
D~ :=
T-'-(B~).
TE H
(d)
H
If
C L(M,N)
continuous; since
Mk
T'
is equicontinuous in
L(N~,M;).
there is in both cases a bounded set
H'(Ao)
is bounded = equicontinuous in
(e)
H
such that
B y dualizing follows
(an alternative p r o o f can be found
4.2.).
There are even pairs
(M,N)
localization principle! type and 209).
Mk.
A c N
is equibounded.
is a special case of (d)
i n [31,
of all
rcI
By ( c )
that
is equi-
5
is a complete (DF)-space the set
rc/
extensions
H'C L(Nk,ML)
is equicontinuous then
N
e.g.
of Frbchet-spaces which satisfy the M
a power sequence space of finite
one of infinite type (see V.P.
Zahariuta n3], p 2 0 8 ,
The localization principle for pairs of Frechet-spaces was
recently charactericed by D. Vogt [12]. I t is not too difficult to see, that for an arbitrary locally convex space
E
and a quasibarrelled space
F
the pair
satisfies the localization principle if and only if on
E
(E,Fk) Q
F
the
THE PRECOMPACTNESS-LEMMA
51
projective and the (b,b)-hypocontinuous topologies coincide.
5.
APPLICATIONS I t was shown
(A)
If
satisfies the localization principle then in any of
(E,F)
the cases of Proposition 2 every equicontinuous precompact subset of
P (E,F) C-localizes fully. C
(For the equicontinuity recall the remarks at the end of 3 . )
(B)
(Fkc,Ei) satisfies the localization principle then in both
If
cases of Proposition 3 every precompact subset of
PC(E,F)
C-localizes.
For the following examples note first, that for metrizable
F
standard manipulations with precompact sets show that
is
F'
PC
Schwartz and has the countable-neighbourhoods-property.
E
for every quasinormable space
the family
e fl pc
Moreover,
in
EL
sa-
tisfies both the Mackey-condition and the strict one. The assumptions of (A) hold true in the following cases: (a)
E
is quasinormable,
F
normed,
I:
= b
(Prop. &(a),
Prop. 2(e)). (b)
E
is Schwartz,
F
normed,
C
arbitrary (Prop. &(a),
Prop. 2(d)). (c)
E
has the countable-neighbourhoods-property (e.g.
0-locally topological),
(d)
is metrizable,
F
b = pc
C
arbitrary (Prop. 4(b),
E
has the countable-neighbourhoods-property and is
Schwartz,
F
metrizable,
E in
is F,
Prop. 2(b)).
C
arbitrary (Prop. 4(b),
Prop. 2(d)). Since 0-locally topological spaces are quasinormable (see [ 71 ,p.260)
52
DEFANT
A.
E
(d) includes the case that
in (e)
E,
E
F
K. FLORET
and
is 0-locally topological,
metrizable and
C
arbitrary.
is metrizable and Schwartz, C
o f bounded sets,
b = pc
F
has the countable basis
4 (b), Prop. 2(d)).
is arbitrary (Prop.
The assumptions of (B) hold true for
(f)
E
has a countable basis o f bounded sets,
C = b (g)
E
(Prop.
Fi
Schwartz,
F
of bounded sets (e.g. (Prop. 4(e),
F
has a countable basis
a n (LS)-space),
3 since
Prop.
is metrizable,
Prop. 3).
4(b),
is metrizable,
F
Fi
C
is arbitrary. F'
Schwartz implies
PC
Schwartz. ) F o r e-products
G E F = Pe(Gbo,F)
PROPOSITION 5.
Let
G
convex spaces such that ciple and: subset
F
be quasicomplete separated locally
(Fbo,G) satisfies the localization prin-
is semi-Monte1 or
H C G E F
(1) H (2)
G
and
the principle (B) gives
Fko
is Schwartz then for every
the following two statements are equivalent:
is relatively compact.
There are compact sets
K c G
and
L
C
F
such that
H c ( K ~ B L ~ ) ~ .
The assumptions of this result are satisfied e.g.
in the following
G
is Banach and
G
and
G
has a countable basis o f bounded sets and
F
Fko
Schwartz.
are FrBchet-spaces. F
is a n
(LS)-space.
G
and
F
both have a countable basis of compact sets
(this implies that p.
266).
G
and
F
are semi-Montel, see [ 7 ] ,
53
THE PRE COMPACTNE SS -LEMMA
$44).
((b) was already treated in [ 8 ] ,
G
i.e. the assumptions o n
and
F
F,
and a n (LS)-space
G
is relatively compact in
F
and
F = F
E
G,
Note that
are semi-Monte1
then for every compact set
H C C ( X ) E F = @ ( X , F ) there is a compact set
(,( fdw
E
For a n illustration of (a) take a
0-locally topological spaces.
X
G
can be interchanged.
(d) comprises the case that both spaces
compact set
Note that
I
f
K C @(X)
such that
5 H , CI E KO}
F.
6. NECESSITY RESULTS By the very nature of C-localization it is clear that once it holds in L((E,6),F)
PC(E,F)
L(Fkc,Eb).
and
assumption of
certain sets have to be equibounded in
E
or
F' PC
However it is surprising that the being Schwartz which appears frequently
is sometimes even necessary. The key for the following results is the external characterization of Schwartz-spaces ( [ 2 ] , 12.4.):
A locally convex space
Schwartz if and only if for every Banach-space
E + G
linear mappings PROPOSITION G
6.
(1)
If
G
E
is
all continuous
are compact.
C
every one-point set in
C pc
in
PC(E,G)
and for every Banach-space
E
E
C-localizes fully, then
is
a Schwartz-space. (2)
G
F
Let
be a semi-Montel-space such that for all Banach-spaces
every one-point set in
Pb(G,F)
b-localizes, then
FL
is
Schwartz. PROOF.
(1)
directly by the external characterization, (2)
simple additional duality argument.
with a
W
Since there is a widely known Frkchet-Montel-space which is not
54
A.
and
DEFANT
K.
FLORET
Schwartz this result shows that there is a Banach-space Montel-(LB)-space
G
and a
such that not every precompact set in P b ( G , F )
F
b-localizes,
In the setting of the c-product (1) implies the following COROLLARY.
a quasicomplete separated locally convex space
If
has the property that for every Banach-space
TE Fc G T E
there are compact sets then
(K0@Lo)O
Fbo
K C F
and
G
F
and for every L C G
with
is a Schwartz-space.
Together with the above-mentioned example the last result shows that the lifting-property of
F a Montel (LB)-space and tion 5(a) and (c)).
G c F
G
may be false in the case
a Banach-space" (compare Proposi-
Since it is immediately clear that in the ex-
ternal characterization of Schwartz-spaces only spaces of the form G = C(X)
(with an arbitrary compact set
this also means, that the property for of 5. does in general not hold if
X
X)
C(X,F)
have to be checked stated at the end
is compact and
F
is only a
Montel (LB)-space. However:
if
H
C F e G
= Pe(Fbo,G)
is precompact it follows ( e . g .
by (2al) of the global precompactness theorem) that tinuous in
L(Fbo ,G)
.
If
(Fbo,G)
H
is equicon-
satisfies the localization prin-
ciple (which is obviously true in the case just mentioned) equibounded which readily means there is a compact bounded
B
C
G
K c F
H
and a
such that
Hc
(KO~BO)O.
BIBLIOGILAPHY 1.
A. DEFANT,
Zur Analysis des Raumes der stetigen linearen
Abbildungen zwischen zwei lokalkonvexen Rtiumen; Dissertation Kiel 1980
is
55
THE PRECOMPACTNESS-LEMMA
2.
K. FLORET, Lokalkonvexe Sequenzen mit kompakten Abbildungen; J. reine angew. Math. 247 (1971) 155-195
3. K. FLORET, Folgenretraktive Sequenzen lokalkonvexer RLume; J. reine angew. Math. 259 (1973) 65-85
4. H.G. GARNIR, M. de WILDE, J. SCHMETS, Analyse fonctionnelle, Tome I; Birkhkuser 1968
5.
A.S. GEUE, Precompact and Collectively Semi-Precompact Sets o f Semi-Precompact Continuous Linear Operators; Pacific J. Math. 52 (1974), 377-401
6. A. GROTHENDIECK,
Sur les applications line'aires faiblement compactes dfespaces du type C(K); Canadian J. Math. 5
(1953) 129-173
7. H. JARCHOW, 8.
G.
KBTHE,
Locally Convex Spaces;
Teubner 1981
Topological Vector Spaces I and 11; Springer 1969
and 1979
9. T.W. PALMER, Totally Bounded Sets of Precompact Linear Operators; Proc. Am. Math. SOC. 20 (1969) 101-106 10.
W. RUESS, Compactness and Collective Compactness in Spaces of Compact Operators; J. Math. Anal. Appl. 84 (1981) 400-417
11.
S.A. SAXON,
Nuclear and Product Spaces, Baire-like Spaces and
the Strongest Locally Convex Topology; Math. Ann. 197 (1972)
87-106 12.
D. VOGT,
Frgchetrlume, zwischen denen jede stetige lineare
Abbilding beschrlnkt ist;
13
V.P. ZAHARIUTA,
preprint
1981
On the Isomorphism of Cartesian Products of
Locally Convex Spaces; Studia Math. 46 (1975), 201-221
Universitat Oldenburg Fachbereich Mathematik
2900
Oldenburg
Fed. Rep. Germany
This Page Intentionally Left Blank
Furrctiowl Analysis. Holomorphy und Approximution Theory rr, G I . Zapata ( e d . ) 0Elsevier Science l'ulrlislirrs fl. V. (North-Holland), 1984
57
ON LIPSCHITZ CLASSES AND DERIVATIVE INEQUALITIES IN VARIOUS BANACH SPACES
2.
Ditzian
1. INTRODUCTION
In a series of papers see [l]
,
[2]
and r 3 ]
it was shown that
certain results on derivative inequalities, best approximation and convolution approximation can be extended from
C
(the space o f
continuous functions) to other Banach spaces for which translation is an isometry or contraction and for which translation is a continuous operator.
In this paper w e shall survey the results o f
those papers and extend them to some Banach spaces for which those This group of Banach spaces will
theorems were not applicable. include
L,,
B.V.
(functions of bounded variation) and duals to
Sobolev or Besov spaces.
2. THE LANDAU-KOLMOGOROV AND SCHOENBERG-CAVARETTA INEQUALITIES
In [4] Kolmogorov has shown that for
where
/Ig((3 sup lg(x)l
and
f E Cn(-m,a)
15 k
5:
n-1,
X
and calculated the best constants
K(n,k).
the result was proved earlier by Landau.
For
n = 2
and
k = 1
In [5] Schoenberg and
Cavaretta developed a method to calculate the best constants of (2.1) for
f E Cn(0,-).
It was shown in [l, p.1503 that if
Supported by NSERC grant A-4816 of Canada.
T(t)
58
2. DITZIAN
is a
Co
contraction semigroup on a Banach space
where
Af = lim t*0+
where
K(n,k)
(The
Tof-f t
in
B
B,
mentioned above are best possible in general, i.e.
for all spaces, but for a particular space
B
and semigroup
it is possible that smaller constants are valid.)
-
< t
l
-c
2
I(AEf(*),g,(*))I =
of norm
gc
-
ilA,'fl/
c ,
l A E F (o ) l
1
in
or
B"
x
and r e c a l l i n g t h a t
r
5
~ ~ ~ h F ~ 4~ Mlha c ( A )f o r a l l
we c o m p l e t e t h e p r o o f .
REMARK.
Our r e s u l t now e x t e n d s i n v e r s e r e s u l t s t o s p a c e s l i k e
B.V.(A),
d u a l of S o b o l e v a n d Besov s p a c e s on
c o u r s e even t h e r e s u l t i n [ Z ] i s a p p l i c a b l e t o
and
A
Lp
L,(A).
spaces,
Of
Sobolev
a n d Besov s p a c e s , O r l i c z and o t h e r s .
5.
AN EQUIVALENT CONDITION ON DERIVATIVES In
[s]
we p r o v e d t h e e q u i v a l e n c e of some a s y m p t o t i c r e l a t i o n s
a n d we r e q u i r e d t h e r e t h a t
IIAhfl/B
= o(1)
for all
f
E B.
This last
c o n d i t i o n c a n b e r e l a x e d i n a way s i m i l a r t o t h a t u s e d i n e a r l i e r s e c t i o n s of t h i s p a p e r . Let such t h a t
B
b e a Banach s p a c e of d i s t r i b u t i o n o v e r
Ilf(-+a)ll =
Ilf(*)l\
and
R
or
T
66
Z.
( I ) IIAhfll
= o(1)
for a l l
(11) ( A h f , g ) = o ( 1 ) (111) ( A h f , g ) = o ( 1 )
Define
Anf
or
for a l l
f
E B
and
g E B*,
+
for a l l
f
E B
and
g E X
0
Gn E L1
and
by
Anf
or (X*=B).
= rf(t+-)Gn(t)dt J
= A n ( A nk - 1 f ) .
k Anf
4 , while
as i n section
E B,
f
h + 0
h
f E B
for
DITZIAN
W e have t h e f o l l o w i n g theorem:
5.1.
THEOREM
fIGn(Y)ldY
M,
I
f E B,
For
Anf
1IB.V.
IIG(r-l)
and
a s above and / G n ( y ) d y = 1,
Gn
f o r some
4 MrUir
f u n c t i o n could b e u n d e r s t o o d i n t h e 0
l y l B IGn(y)ldy
I
4
1 4 On/on+l
IIAEf/lB 5 M h'
[ I ( = d)
rk
rk
that
4
Ma!
for
then f o r
M,
= 0(anrkta)
(derivatives of
s e n s e ) and f o r some
S'
un = o(1) n +
satisfying
a > 0
f o r any i n t e g e r
k An(f,X)IIB
0,
r
m
the following a r e equivalent:
r,
> a.
r
f o r any i n t e g e r s
r, k
such
> a.
11 ( A n - I ) .e f l l B
= O(u:)
such t h a t
.t
f o r any i n t e g e r
.t > a/min(g ,I). Gn
i s even,
RK 5.2. 0
1,
> a/min(@,2) i s s u f f i c i e n t ) .
It is c l e a r t h a t f o r t h e
since (A)
i s much s t r o n g e r .
i s t h a t we do n o t have t o assume B
f
i n q u e s t on
llAhfIIB
= o(1)
The advantage i n t h i s theorem
IIAhfllB
= o(1)
on t h e whole
space
and t h e r e f o r e our theorem a p p l i e s now t o many s p a c e s f o r which
i t was n o t v a l i d b e f o r e .
For example B . V . ,
L,
and d u a l s of
Sobolev and Besov s p a c e s . PROOF.
A combination of
[s]
w i t h t h e simple i d e a used i n s e c t i o n
w i l l c o n s t i t u t e a proof o f o u r theorem.
4
ON LIPSCHITZ CLASSES AND DERIVATIVE INEQUALITIES
67
REFERENCES
1.
Z. DITZIAN,
Some remarks on inequalities of Landau and
Kolmogorov, Aequationes Math., 12, 1975, 145-151. 2.
Z. DITZIAN,
Some remarks on approximation theorems on various
Banach spaces, Jour. of Math. Anal. and Appl., Vo1.77, (2),
19809 567-576. 3.
Z. DITZIAN, Lipschitz classes and convolution approximation processes, Math. Proc. Camb. Phil. SOC., 1981, (go), 51-61.
4. A.N. KOLMOGOROV, On inequalities between the upper bounds of the successive derivatives of an arbitrary function on an infinite interval, 1939, Amer. Math. SOC. transl. 4, 1949, 233-243.
5.
J.J. SCHOENBERG and A. CAVARETTA, Solution of Landau's problem concerning higher derivatives on half line, Proceedings of the international conference on constructive function theory, Varna, May 19-25, 1970, 297-908.
6. E.M. STEIN, Functions of exponential type, Ann. of Math., 65, 1957, 582-592
7. G.I. SUNOUCHI,
Derivatives of trigonometric polynomials of best approximation, in "Abstract spaces and approximation", Proceedings of conference at Oberwolfach, 1968, (P.L. Butzer and B.Sz. Nagy Eds.), 233-241, Birkhtiuser, Base1 und Stuttgart, 1969.
Department of Mathematics The University of Alberta Edmonton, Canada T6G2G1
This Page Intentionally Left Blank
A U I r U H C A T I O N 5 C T 4 S 5 0 C I A T E D T O TIIF C O P Y PlIENOhrCNON I N TIIll s P l C L 01 GAUGE E I1'Ll)i
I-'r a n c i s c o .4n t o n i o D o r i a
We show t h a t g a u g e f i e l d c o p i e s a r e a s s o c i a t e d t o a s t r a t ified bifurcation t o b e t h e l o c u s of
s e t i n gauge f i e l d s p a c e .
Such a s e t i s n o t i c e d
o t h e r b i f u r c a t i o n phenomena i n g a u g e f i e l d t h e -
o r y b e s i d e s t h e copy phenomenon.
1. INTRODUCTION The phenomenon t h a t we a r e g o i n g t o d i s c u s s i n t h e p r e s e n t p a p e r h a s b e e n d i s c o v e r e d by two t h e o r e t i c a l p h y s i c i s t s , T . T . and C . N .
Yang,
Idu
i n s e a r c h for d i f f e r e n c e s b e t w e e n t h e s o - c a l l e d
A b e l i a n gauge t h e o r i e s and t h e i r non-Abelian
counterparts
[7]
.
Gauge f i e l d t h e o r i e s a r e p h y s i c a l i n t e r p r e t a t i o n s f o r t h e u s u a l t h e o r y of
connections
on a p r i n c i p a l f i b e r b u n d l e
u s u a l l y t a k e spacetime manifold (a 4-dimensional
[5].
Physicists
r e a l Hausdorff
smooth m a n i f o l d w i t h a n o w h e r e d e g e n e r a t e q u a d r a t i c f orm, t h e "metric t e n s o r " ) a s base space f o r t h e bundle, while t h e f i b e r i s i d e n t i f i e d with a f i n i t e - d i m e n s i o n a l semi-simple L i e group. general finite-dimensional
More
d i f f e r e n t i a b l e m a n i f o l d s a r e sometimes
used a s b a s e space f o r bundles of p h y s i c a l i n t e r e s t , s o t h a t o u r r e s u l t s w i l l n o t i n g e n e r a l d e p e n d on t h e b a s e m a n i f o l d ' s
dimen-
sion. The a b o v e d e s c r i p t i o n i s t h e m a t h e m a t i c a l s e t t i n g for t h e s o - c a l l e d Wu-Yang a m b i g u i t y or g a u g e f i e l d copy phenomenon.
Let us
b e g i v e n t h e e x p r e s s i o n for a c u r v a t u r e f o r m on s u c h a b u n d l e i n a l o c a l c o o r d i n a t e system:
FRANCISCO ANTONIO DORIA
F = (1/2)F U V (x)dx’
Here the components
F
dxv
A
are Lie-algebra valued objects.
YV
cp
the expression of the bundlels curvature form identity cross-section
U x El],
domain in the base manifold the bundle
P(M,G)
-
(1.1)
where
where
is
at a (local)
is an open trivializing
Curvature and connection forms on
M,
G,
U
F
a semi-simple Lie group as describ-
ed above, is the bundle’s fiber,
-
are related by Cartants structure
equation: cp = du + ( 1 / 2 ) r a A
&I.
(1.2)
(In a local coordinate system, at the identity cross-section,
where the
a cbc
are Lie-algebra structure constants,)
If we are given a curvature form connection form
a 7
is the Abelian group ness of
cp
9,
do we have a unique
The answer is no, in general,
U(1),
If the group
we can immediately check that unique-
can only be a local phenomenon provided that
simply connected.
M
is not
If the group is any non-Abelian semi-simple grcup
it is easy to show that there exists a curvature form ed by a Lie-algebra valued 2-form
F
cp
represent-
which can be obtained out of
an infinite family of connection forms which are
not related any-
where on spacetime by the so-called gauge transformations, that is by the natural action on the bundle
induced by the right action of
G
on the fiber.
M
be the four-plane with Cartesian coordinates and let the fiber
The example is quite simpler
let our spacetime
group be any non-Abelian Lie group (the other assumptions are not essential to the example.)
If
L(G)
is
GIs
Lie algebra, we
71
A BIFURCATION SET
choose as components
for the connection and curvature forms (at
the local identity cross-section),
It is now easy to check that
is also a connection form for
F,
whenever
h(x2)
2
function of the Cartesian coordinate (x ) . and
A
B
9
and
should commute [ 81
8'
gument has a rather more physical flavor:
j
= aVFMV +
#
0
fe,e']
C1
f 0,
cannot be gauge-related, for if it were s o , it is imme-
diate that
j'
Now if
is any
for
[A,,
B.
,FWv]
.
We then check that
.
The Wu and Yang ar-
they formed the current j = 0
for
A
and
As there is no gauge transformation that can make
the current vanish,
A
and
B
cannot be gauge-related.
The Wu-Yang ambiguity has remained a curiosity until now. However some recent work has opened the way for deeper undcrstanding of the phenomenon along physical and mathematical lines [l].
2.
MAIN RESULTS I N THE FIELD COPY PROBLEM Since our goal is to describe the geometry of copied curva-
tures and connections in the space of all curvatures and connectians, we review here the main characterizations for copied curvatures and connections.
FRANCISCO ANTONIO DORIA
72
Let
M
be a differentiable real n-manifold,
simple finite-dimensional Lie group, G-bundle over U C M
If
with projection
TT:
n.
-+ M
P(M,G)
a semi-
a principal
Suppose that over
a nonvoid open set, the curvature form
a'
cp
and
a
Under the above conditions
8
from two different connection forms
G
2
n-'(U),
can be derived
,
a 2 = a' + 8 .
We then conclude: PROPOSITION 2.1.
satisfies:
(2.1)
We now define the auxiliary connection f o r m
[&I.
This implies:
COROLLARY 2.2.
Condition (2.1) is equivalent to d(aO)e
where PROOF.
a 0 = a 1 + (u2) 0
d(ao)
zDef de
+
[ao
A
el
= 0,
denotes the covariant exterior operator w.r.t.
Substitute
a'
= ao
-
(1/2)9
(2.2) ao.
into (2.1).
Condition (2.1) implies also a well-known necessary condition for the existence of gauge copies: COROLLARY 2.3.
PROOF.
One calculates the derivative and then substitutes i n the
result Cartants structure equation and equation (2.1). (2.3) was erroneously considered by the author to be also a
A BIFURCATION SET
73
sufficient condition for the existence of connection ambiguities
f81. A counterexample is given elsewhere [ 121. More on that below. Equation (2.2) can be solved if we suppose that g
@ = Ldg,
where
is a Lie-algebra-valued equivariant function on the bundle.
Covariance considerations indicate that
IJ
= Ad(u),
action of a (possibly local) gauge transformation substitute
8' = Ldg
the adjoint
If we then
u.
into (2.2) we get (2.&a)
(2.4b) Equation (2.4a) can be rewritten as
[a1
A
d@] = -(1/2)[dB
A
dg]
.
(2.5)
If we delete the combined product symbols, we see that solving 2
(dp ) =
(2.1) or (2.2) is equivalent to solving the equation = (al)(dB).
We have two possibilities:
(i)
the infinitesimal,
continuous copies, given by (dp)2 = 0
iff
= 0,
(al)(dB)
and the (ii) discrete, paired copies,
a'
=
1
-
(1/2)dp,
Names are due to the following: EL
0,
5
above iff
A connection form
[a'
[a
A
A p]
a2-a1 =
Cp,
0
> 0,
p l = 0,
a '
is infinitesimally copied as
= 0, p = dp.
Equation (2.1) with
first order). 1
if we put
(2.5b)
we get
LEMMA 2.4.
PROOF.
1
( E )d@ = 0.
@ = ep
implies
d(al)p
= 0
(to the
And this last equation implies and is implied by
P = d@.
Solutions like (2.5b) are said to be discrete because
'8 = dp
is
74
FRANCISCO ANTONIO DORIA
a unique s o l u t i o n f o r ( 2 . 1 ) whenever
6
1
-1 = a
+
dg,
-1
[a
A
dg] = 0.
W e w i l l soon s e e t h a t " i n f i n i t e s i m a l " c o p i e s form a boundary s e t i n t h e s p a c e of a l l copied p o t e n t i a l s . We f i n a l l y n o t i c e t h a t combining
(2.3)
with
( 2 . 5 ) we s e e t h a t
copied curvatures should s a t i s f y
That i s , t h e ( a l g e b r a i c ) o p e r a t o r integrable nullspace.
[cp A
-1
s h o u l d have a n o n t r i v i a l ,
T h i s p r o p e r t y a l l o w s u s t o show t h a t c o p i e d
c u r v a t u r e s form a boundary s e t i n t h e s p a c e of a l l c u r v a t u r e s t h a t [cp A
satisfy
e]
= 0
f o r a nontrivial
on an open s e t i n t h e
8
bundle. A f i n a l r e s u l t w i l l be v e r y u s e f u l i n t h e n e x t s e c t i o n s :
w e w i l l need t h e f a c t t h a t PROPOSITION 2 . 5 .
n-'(U), group
U
H(cp)
cp
has gauge-equivalent d i f f e r e n t p o t e n t i a l s over
and open set i n
M,
i f f i t s Ambrose-Singer holonomy
has a n o n t r i v i a l c e n t r a l i z e r i n
For t h e prof see
[7].
on
G
n-'(U).
T h i s a l l o w s u s t o show t h a t " t r u e t '
c o p i e s a r e d e n s e i n t h e s p a c e of a l l c o p i e s , and t h a t ( l o c a l l y a t l e a s t ) g a u g e - e q u i v a l e n t c o p i e s b e l o n g t o a boundary s e t i n t h e s p a c e of c o p i e d c u r v a t u r e
7.
and c o n n e c t i o n forms.
DEGENERACIES I N C O N N E C T I O N AND CURVATURE SPACE
We a r e g o i n g t o d e s c r i b e some a s p e c t s of
t h e geometry of
c o n n e c t i o n and c u r v a t u r e s p a c e s t h a t have a t l e a s t one of t h e degeneracies l i s t e d i n t h e preceding section.
I n o r d e r t o summarize
t h e s e d e g e n e r a c i e s f o r t h e b e n e f i t of o u r e x p o s i t i o n , we n o t i c e that if
i s a ( p o s s i b l y l o c a l ) 1-form and
covariant derivative operator w . r . t .
a,
d(u)
the exterior
the condition
dz(a)8 = 0
75
A BIFURCATION SET
is equivalent to condition (2.3), or 0
be an ad-type tensorial form.
[cp A
e]
= 0,
provided that
We can thus list:
Covariant cohomology condition: 2
d (a)e = 0
iff
[cp A
e]
= 0,
Necessary condition for copies: [lp A
de] = 0 ,
Existence of copies:
+
d(al
(1/2)f3)9
= 0,
Discrete copies:
[a1
de] = -(1/2)[de
A
A
de],
(3.4)
Infinitesimal copies:
[a1
de] = 0 = -(1/2)[dp
A
A
de]
,
(3.5)
False copies : H(cp)
(3.6)
with nontrivial centralizer.
O u r objects are connections and curvatures for principal
fiber bundles
P(M,G)
with a real n-dimensional smooth manifold
as its base space and a fixed finite-dimensional semi-simple Lie group
G
as its fiber.
The geometry of curvature and connection
space is already a pretty well-known subject [5] and we will sketch here some of its main lines.
Curvature forms can be identified
with cross-sections of the bundle L(G)-valued
2-forms on
M.
all smooth cross-sections M,
nl(M,L(G)).
n2(M,L(G))
of Lie-algebra
Connection forms can be identified with of the bundle of L(G)-valued
1-forms on
Curvatures are ad-type objects; connection forms
will be s o provided that we fix and arbitrary connection (which can be the z e r o , o r vacuum, connection 0) and identify cross-sections of
n'(M,L(G)).
Near any point
xo
a
-
E M,
0
with the any L ( G ) -
76
FRANCISCO A N T O N I O D O R I A
v a l u e d 2-form
f
can b e s e e n a s t h e c u r v a t u r e of a p a r t i c u l a r con-
n e c t i o n form a v i a t h e c o n s t r u c t i o n
[4]
( i n a l o c a l c o o r d i n a t e system)
.
G l o b a l l y , w h i l e t h e map t h a t
sends a c o n n e c t i o n form o v e r i t s c u r v a t u r e i s p r e t t y w e l l - b e h a v e d , t h e i n v e r s e map i s f u l l of p a t h o l o g i e s [13]. W e w i l l consider here connection curvature
3 c C"(n2(M,L(G)))
G
5
C"(n'(M,L(G)))
s p a c e s t o be endowed w i t h a n a t u r a l
F r 6 c h e t s t r u c t u r e i f we endow t h e s e c r o s s - s e c t i o n Cm
topology.
and
spaces with t h e
T h i s r a t h e r weak s t r u c t u r e w i l l b e enough f o r o u r
f i r s t s e r i e s of r e s u l t s . We w i l l f i r s t r e s t r i c t our remarks t o t h e c a s e when a s p a c e t i m e , t h a t i s , a k-dimensional with a nondegenerate m e t r i c t e n s o r . Hodge
+!
operator w . r . t .
M
is
r e a l smooth m a n i f o l d endowed W e t h e n d e f i n e on
the
M
t h e s p a c e t i m e m e t r i c and t h e n check t h a t
(3.1) becomes a s p a c e t i m e - p a r a m e t r i z e d l i n e a r homogeneous system: [cp A
e]
= (Ad+lP)O = 0 .
(3.8)
T h i s system w i l l o b v i o u s l y have n o n t r i v i a l s o l u t i o n s p r o v i d e d t h a t det(Aduep) = 0
p
somewhere i n s p a c e t i m e .
globally s a t i s f i e s a property
t h e whole m a n i f o l d
P
We now s a y t h a t a c u r v a t u r e
iff
i s v e r i f i e d by
P
(or t h e whole bundle
M
P(M,G)).
cp
over
With t h i s
d e f i n i t i o n i n mind we a s s e r t :
P R O P O S I T I O N 3.1.
Curvatures t h a t g l o b a l l y s a t i s f y property
form a c l o s e d and nowhere d e n s e s e t i n PROOF.
Let
det Ad
Y
F
s i o n a l function matrix 2-form
3
i n the
be t h e d e t e r m i n a n t of Ad
*
F,
where
F
topology.
Cm
the finite-dimen-
i s t h e Lie-algebra valued
over s p a c e t i m e a s s o c i a t e d t o a c u r v a t u r e
t h a t globally s a t i s f y property
(3.1)
( 3 . 1 ) w e have
Cp.
d e t Ad
For curvatures
*
F
= 0
on t h e
A BIFURCATION SET
whole of' M.
77
And from the map
+
net: 3 -+ c ~ ( M )
F + * det Ad+Y we see that
t 3
Det-'(O)
3.
is closed and nowhere dense in
This immediately implies: COROLLARY 3 . 2 .
in
5
in the
Globally copied curvatures form a nowhere dense set Cm
topology.
The copy condition (3.3) implies (3.1), as shown i n Corol-
PROOF.
lary 2 . 3 . lu'hat about objects that satisfy one of the properties ( 3 . 1 ) -
(3.6) only locally, that is, over a nonvoid open set in in
F o r property
P(M,G)).
(or
M ?
(3.1) w.r.t. bundles over a spacetime
we can settle that question with: PROPOSITION 3.3.
Curvatures that satisfy property (3.1) locally
over spacetime form a closed and nowhere dense set i n
cm
5
i n the
a9
with the
topology.
PROOF.
We first form
topology induced by the map vanishes over an open set in property (3.1) locally over in
= a9 c Cm(M)
+Det(3)
+Det(-). M
M.
in the induced topology.
h(f)
a
g €
C
Det-'(h)
then Let
and endow f E a9,
F o r any
F E Det-'(f)
h(f)
if
f
satisfies
be a neighborhood of
I t is immediate that there exists
which is nowhere vanishing on
M.
Thus any
will never satisfy property (3.1) on
M.
G E Det-44
The conclusion
follows immediately. This result allows the solution of a question raised by
M. Halpern [Ill.
Halpern suggested that ambiguous curvatures and
connections should give extra contributions to the integrals i n Feynmann quantization techniques.
f
Despite the fact that we don't
78
FRANCISCO ANTONIO D O R I A
have a rigorous measure theoretical construction for a general Feynmann integral, we have several heuristic procedures for such calculations which try to characterize objec s similar to Bore1 sets on connection and curvature spaces.
A s ambiguous curvatures
and connections (which satisfy property (3.1 ) are nowhere dense in 3
and in
in a natural topology like the
G,
Cm
topology, we have
now reason to expect that they should be ignored during Feynmann integral calculations. The next question is:
are copied curvatures dense in the The answer
space of all curvatures that satisfy condition (3.1)? is no: PROPOSITION 3.4.
Copied curvatures are nowhere dense in the space
of all curvatures that satisfy (3.1). PROOF.
If cp
is a copied curvature then it satisfies ( 3 . 2 ) .
5 = [cp
sider all tensorial
e].
A
If
Cm
n'(M,L(G))
Con-
is the space
of all tensorial L(G)-valued 1-forms on the bundle, and 4 the space of all tensorial L(G)-valued 4-forms on Cm 0 (M,L(G)) the bundle, we can form the map d: 3
[CP where
d
nl(M,L(G))
X
A
-t
e]
COD
n4(M,L(G))
d[cpAe]=[CPAde].
is the standard exterior derivative.
differential identity we have that d"(0)
c 3 x Cmnl
dense set and s o is
p,(d-'(O))
c3,
0 1
pl: 5 x C
n
= {[cpAO],
d"(0)
I t is immediate that
Due to the Bianchi 0 a cocycle].
is a closed and nowhere
where
p1
denotes the
I31
= 0 - 1 another closed and nowhere dense subset in 3 x C n ,
projection
-t
3.
Condition
the corresponding restrictions to
d"(0)
rp A
and
p,d-'(O).
defines and so do Thus
globally copied curvatures are nowhere dense in the space of all curvatures that satisfy (3.1).
For locally copied curvatures we
79
A BIFURCATION SET
must follow a reasoning similar to the one used i n Proposition 3.3. The same technique can be applied to prove: PROPOSITION 3 . 5 .
Infinitesimally copied fields are nowhere dense
in the space of all copied fields. PROOF.
Consider the map f: G
X
Cmnl -+ Ci x Cmnl
and apply the same reasoning as i n Proposition 3.4.
We notice that
is the set of all infinitesimally copied curvatures.
f ' ( 0 )
What about "falseffcopies, that is, connection form ambiguities that can be (locally at least) eliminated modulo a gauge transformation?
This question is settled by
PROPOSITION 3 . 6 .
Curvatures with false copies are nowhere dense
in the space of all curvatures with potential ambiguities. PROOF.
Curvatures with false copies are stabilized by gauge trans-
formations that take values in the centralizer of Ambrose-Singer holonomy group generated by
cp.
H(cp),
the
We can thus apply
an adequate slice theorem to get via this symmetry a stratification wherefrom one sees that these symmetric curvatures are nowhere dense in the space of all copied curvatures.
3
the embedding of Lie algebras within
We could also reproduce in L(G)
associated to false-
ly copied curvatures. Propositions 3.4 the dimension of
M.
-
3 . 6 are valid without any restriction on dim M = 4 ,
But if we consider the case when
we can apply the same technique as in Propositions 3.1
-
3.3 to get
a stratification i n the set of all curvatures that obey (3.1) induced by the embedding of ideals in the space of all matrices Ad+F.
FRANC'ISCO ANTONIO D O R I A
4.
DIFFERENTIABLE V E R S I O N S O F O U R RESULTS
I n t h e p r e s e n t s e c t i o n we s u p p o s e t h a t a l l M-defined
objects
h a v e c o m p l e x - v a l u e d componentes t h a t b e l o n g t o a c o n v e n i e n t S o b o l e v space.
More p r e c i s e l y ,
if
U
c M
i s a n a r b i t r a r y open n o n v o i d s u b -
s e t , we w i l l s u p p o s e t h a t o u r o b j e c t s h a v e components i n one o f t h e Hilbert-Sobolev
H m ( U ) = I-12'm(U).
spaces
We t h u s h a v e a d i f f e r e n t i -
a b l e norm f o r o u r o b j e c t s and a v e r y s i m p l e d i f f e r e n t i a b l e s t r u c t u r e i n o u r f u n c t i o n s p a c e s , s o t h a t we c a n nov g i v e a more r e f i n e d v e r s i o n f o r our previous r e s u l t s . When we c o n s i d e r o b j e c t s t h a t are i n a S o b o l e v s p a c e we i m p l i c i t l y a d m i t t h a t o u r smooth ( i . e . t o h a v e compact s u p p o r t s .
C")
o b j e c t s a r e supposed
Such a s u p p o s i t i o n may i n t r o d u c e some
p r o b l e m s when we d e a l w i t h g l o b a l l y d e f i n e d smooth o b j e c t s on a noncompact s p a c e t i m e ; however we n o t i c e t h a t p h y s i c a l c a l c u l a t i o n s a r e a l w a y s done i n a p a r t i c u l a r c o o r d i n a t e domain, a l w a y s r e s t r i c t e d t o a compact r e g i o n , compact c l o s u r e .
A n o t h e r way of
and t h a t d o m a i n c a n b e
or t o a n e i g h b o r h o o d w i t h
looking a t t h i s r e s t r i c t i o n i s t o
s u p p o s e t h a t " p h y s i c a l " o b j e c t s become ( a p p r o x i m a t e l y ,
a t least)
z e r o beyond a c e r t a i n r a n g e ( s u c h a s u p p o s i t i o n i s commonly encount e r e d i n t h e d i s c u s s i o n of some r e s u l t s i n c l a s s i c a l f i e l d t h e o r y ) . Anyway t h e g a u g e f i e l d copy p r o b l e m i s an e s s e n t i a l l y l o c a l phenomenon. With t h o s e r e m a r k s i n mind,
we c a n o b t a i n t h e smooth v e r s i o n s
of our previous r e s u l t s : PROPOSITION
4.1.
Let
U c M
h a v e compact c l o s u r e and c o n s i d e r t h e
c l a s s of a l l c u r v a t u r e s on t h e b u n d l e t h a t do
( 3 . 1 ) anywhere on
U.
not
s a t i s f y condition
T h a t s e t i s a n open s u b m a n i f o l d i n
3.
A BIFURCATION SET
PROOF.
81
Consider the map
and consider its inverse (the norm we use is any finite-dimensional norm composed with the Sobolev norm),
h-l(W+-[O]
smoothness we have here an open submanifold i n
) c 3x0'.
Due to
3.
Appropriate modifications can be made in the other results. A sample is: PROPOSITION
4.2.
Let
U
be as above and consider the class of all
curvatures that have a discrete connection ambiguity all over They form an open, dense submanifold of
so,
U.
the space of all
copied curvatures. PROOF.
Consider the map
and act as i n the preceding proposition.
5. INTERPRETATION AND CONCLUSION Stratified sets first appeared in the study of bifurcation problems in Geometry [lb].
We have here a rather complex stratified
system, which depends i n part on symmetry properties of the systems (the embedding of,Ambrose-Singer holonomy algebras.)
Stratifica-
tions similar to this last one lead in General Relativity to the classification of spacetime geometries that are unstable i n the linear approximation [3]
.
A similar phenomenon leads to the li-
nearization instability of gauge fields uncoupled to any gravita-
82
FRANCISCO ANTONIO DORIA
tional field 1151. Fields that satisfy condition (3.1) ( o u r first stratum) can be shown to be associated to a nonvanishing torsion tensor that satisfies the same set of Bianchi identities.
Such a degeneracy
can be related to well-known "inconsistencies" in higher-spin field theory [2]
.
Fields that possess infinitesimal copies can be shown
to generate a very interesting version of the Higgs mechanism [l] where the gauge field can be shown to generate
,
a field that sa-
tisfies the standard electromagnetic wave equation [lo].
Finally
fields with false copies are shown to imply Nambu's condition for the existence of nontrivial topological effects such as magnetic monopoles and vortices.
This class of fields exhibits also an in-
consistency that appears when one tries to add a gauge-like interaction to spin-0 fields; here this inconsistency is shown to be a symmetry-breaking condition. We do not have a clear interpretation for the coupled sets of nonequivalent potentials that form discrete copies systems, despite the fact that they were one of the first examples of copies
to be found [ 6 )
.
6 . ACKNOWLEDGMENTS The author wishes to thank Professor G . Zapata f o r his kind invitation to expose these ideas at the 1981 Holomorphy and Functional Analysis Symposium in Rio de Janeiro.
He also thanks
Professor Leopoldo Nachbin for his constant interest and encouragement.
A BIFURCATION SET
83
REFERENCES 1.
A.F. AMARAL, F.A. DORIA and M. GLEISER, Higgs fields as Bargmann-Wigner fields and classical symmetry breaking,
J. Math, Phys. 24 (1983), 1888-1890. 2.
A.F. AMARAL,
The Teitler lagrangian and its interactions,
D.Sc. Thesis, Rio de Janeiro (1983) (in Portuguese).
3.
J.M. ARMS,
Linearization instability of gauge fields,
,
443-453.
J. Math, Phys. 20 (1979)
4. C.G. BOLLINI, J.J. GIAMBIAGI and J. TIOMNO, Gauge field Phys. Lett. 83 B (1979), 185-187.
copies,
5.
Y.M.
CHO,
Higher-dimensional unification of gravitational J. Math. Phys. 16 (1975), 2029-2035.
and gauge theories,
6.
S. DESER and F. WILCZEK,
potentials,
'7.
F.A. DORIA,
Non-uniqueness of gauge field
Phys. Lett. 65 B (1976), 391-393.
The geometry of gauge field copies,
Commun.
Math. Phys. 79 (l98l), 435-456.
8. F.A. DORIA, copies:
Quasi-abelian and fully non-abelian gauge field A classification,
J. Math. Phys. 22 (1981),
294'3-2951.
9. F.A. DORIA and A.F.
AMARAL,
gauge field copies,
Linearization instability implies
Preprint , Universidade Federal do
Rio de Janeiro, 1983. 10.
M. GLEISER,
Gauge field copies and the Higgs mechanism,
M.Sc. Thesis, Rio de Janeiro (1982) (in Portuguese). 11.
M.B. HALPERN, Gauge field copies in the temporal gauge, Nucl. Phys. B 139 (1978), 477-489.
12.
M.A. MOSTOW and S. SHNIDER,
Counterexamples to some results
on the existence of field copies,
Preprint, Univ. North
Carolina, 1982,
13.
M.A. MOSTOW and S. SHNIDER,
Does a generic connection depend
continuously on its curvature? Carolina, 1982.
Preprint, Univ. North
84
14.
FRANCISCO ANTONIO DORIA
R. THOM,
L a estabilite topologique des applications poly-
nomiales,
Llenseignement mathematique Vol. 8 ( 1 9 6 0 ) ,
24-33.
Interdisciplinary Graduate Research Program Department o f Theory of Communication Universidade Federal do R i o de Janeiro Av. Pasteur 250 22290
R i o d e Janeiro, RJ, Brazil
1~'unctionalA ndysis, Holoniorphy awd .4pproxitnation Theory 11, C.I. Zopato (cd.) @ Ekevier Scierice P u b l i s h s B. I< (North-Holland), I984
O N THE ANGLE: OF DISSIPATIVITY O F O R D I N A R Y
AND PARTIAL DIFFERENTIAL OPER4TORS"
H.O.
Fattorini
1. INTRODUCTION
Let
A
E.
Banach s p a c e
of
u
where
b e a d e n s e l y d e f i n e d , c l o s e d o p e r a t o r i n a complex
c o n s i s t i n g of a l l
(u*,u)
operator
A
u*
d e n o t e by
@(u)
i n t h e d u a l space
the duality s e t such t h a t
E*
d e n o t e s t h e v a l u e of t h e f u n c t i o n a l
u*
at
u.
The
is dissipative if
Re(u*,Au) If
u E E
F o r each
(u E D ( A ) ,
0
5
u* E @(u)).
(1.2)
we h a v e
(XI f o r some
1 >
0,
then
A
-
A)D(A)
= E
i s called m-dissipative
s t r o n g l y c o n t i n u o u s c o n t r a c t i o n semigroup
The c o n v e r s e i s a s w e l l t r u e . assumed f o r a s i n g l e e l e m e n t replace
(1.2)
(1.3)
{ S(t); t
We n o t e a l s o t h a t
u*
of
and g e n e r a t e s a 2
(1.2)
O]
,
need o n l y be
@ ( u ) ; equivalently,
we may
by Re(8 (u),Au)
0
(u E D ( A ) )
*
(1.5)
T h i s work was s u p p o r t e d i n p a r t by t h e N a t i o n a l S c i e n c e F o u n d a t i o n , U.S.A.
u n d e r g r a n t MCS
79-03163.
. . FATTORINI
86
H 0
8
where with
8:
i s a d u a l i t y map, t h a t i s , an a r b i t r a r y map
e(u) E @(u) (u E D(A)). space, (1.3)
i s a Hilbert
-t
E"
We n o t e a l s o i n p a s s i n g t h a t i f
i s e q u i v a l e n t t o maximality o f
c l a s s of d i s s i p a t i v e o p e r a t o r s . theory see f o r instance
E
A
E
i n the
F o r a l l n e c e s s a r y f a c t s on t h e
[k],
I n c e r t a i n q u e s t i o n s of c o n t r o l t h e o r y ( r e l a t e d t o t h e com-
(I
p u t a t i o n of t h e i n v e r s e
- aS(t))'l)
d e c i d e whether t h e semigroup
I arg 5 I
s cp
(cp
>
More g e n e r a l l y , cp
whether t h e r e e x i s t s
>
and
0
IJJ
= ~ ( c p ) such t h a t
such t h a t ( 1 . 6 )
cp 2 0
a s t h e supremum of a l l
cp
2
0
e*irp(A
8,
the supre-
w = w(cp)
> 0.
may be c h a r a c t e r i z e d
cp(A)
such t h a t
( 1 . 5 ) w i t h r e s p e c t t o some d u a l i t y map cp.
A,
h o l d s f o r some
E q u i v a l e n t l y , t h e a n g l e of d i s s i p a t i v i t y
g e n e r a l of
(1.4)
i t i s o f t e n enough t o i n q u i r e
t h e a n g l e of d i s s i p a t i v i t y o f
cp(A),
mum of a l l t h e
can be e x t e n d e d t o a s e c t o r
i n t h e complex p l a n e i n such a way t h a t
0)
i s preserved there.
We d e n o t e by
S ( * )
i t i s o f importance t o
with
- wI) w
satisfies
depending i n
Some obvious m a n i p u l a t i o n s show t h a t t h i s r e q u i r e m e n t
transl at e s t o Re(e(u),Au) 5 fbIm(8(u),Au)
f o r some d u a l i t y map
8
and
+ wllul12
(1.7)
(u E E )
8 = t g cp.
The o b j e c t of t h e p r e s e n t p a p e r i s t h e c o m p u t a t i o n of t h e a n g l e of d i s s i p a t i v i t y of second o r d e r u n i f o r m l y e l l i p t i c o p e r a t o r s .
m A =
m
m
C ajk(x)DjDk + C b j ( x ) D j + j=1 k = l j=1 C
( a j k ( x ) = a k j ( x ) , x = (xl
R
,...,x m ) ,
of m-dimensional E u c l i d e a n s p a c e
tion
Ap(e)
in
LP(n)
(1 5 p
(1)
(11)
Y(X)U(X),
I' is the boundary of n
...,wm)
Dw
all
u
fi
r).
=EX j a ~k . (x)wj},
r;
the outer normal vector on
when the Dirichlet of all continuous
C(5)
is replaced by the subspace
that vanish on
E
indicates the derivative
boundary condition (11) is used the space functions in
(X
,.
and
in the direction of the conormal vector v = (wl,
o
=
U(X>
consisting of
C,(n')
r.
The results are as follows. assumptions on the coefficients of
Under the standard smoothness A,
on
y
and on
r
the angle
of dissipativity turns out to be independent of the operator and the boundary condition the space 'P(Ap(B))
E = LP(n)
p
(1 < p
0 (depending on 6 )
such that, for every
c,
+
1zI2
We use (2.10)
z =
for
*
a((Rez)2
2
V1zl2.
(2.10)
Gu':
-(p-2) i 6(p-2)
s
b(Rez)(Imz))
2
(
al~l~-~(Re(&~'))~dxi
al~l~-~Re(;u')Im(;u')dx
(2.11)
'0
'0
where
I
f = Iiiu' 2
+ (~-2){(Re(;u'))~ 2
for some
v >
0
(depending on
k 6Re(uu')Im(Gu')]
vliiu' I 2 6 )
2
(2.12)
if
(2.13) We must now estimate the other terms on the right-hand side
91
ON THE ANGLE O F DISSIPATIVITY
of
(2.7).
T o this end, consider a real valued continuously differ-
entiable function
p
+ ppluIp-2 Re(;u/)
0 4 x 4 &.
in
we obtain, f o r any
E
7
= p' lulp +
(pluIp)'
Since
0,
lulPdx
(2.14)
where we have applied the inequality = 2(E
2lul Iu'
-1
IUI)(EIU'I)
We use (2.14) for any
(we may take
p
p
+
E21U'12
4
E
-2 IU
2
(2.15)
such that
linear) to estimate the first two terms on the
right-hand side of (2.7); for the fourth integral we u s e again (2.14), in both cases with the constant of
(2.7)
v
E
in (2.10)).
sufficiently small (in function of We can then bound the right-hand side
by an expression o f the type
JO
for some constants
JO
w = w(6)
and
c = c(6)
7
0.
Upon dividing by
/lulip-2 we obtain we(u),Ap(BO,BL)U)
4
T o extend (2.16) to any imation argument. when
Po,
BL
*-(B
( u ) , A ~ ( B ~ , B ~ ) u+)
u E D(Ap(BO,Bc))
wllull 2
(2.16)
we use an obvious approx-
Inequality (2.16) is obtained in the same way
(or both) are of type (11).
In the case
1< p c 2
the function
e(u)
nay not be contin-
u o u s l y differentiable; however a simple argument based on the Taylor
formula shows that if
u
is a polynomial (or, m o r e generally,
an
92
. FATTORINI
H .O
e(u)
analytic function) then
is absolutely continuous and the com-
putations can be justified in the same way.
Details are omitted.
We have completed half of the proof of the following result: THEOREM 2.1. Ap(BO,BL)
1< p
v,.
We sketch the argument for
boundary conditions of type (I).
Assume that (2.18)
Then we can find a complex number 1Zl2
Let
q
z
(say, of modulus 1) such that
-
+ (~-2)((Rez)~ 6(Rez)(Imz))
be a smooth real valued function in
= -p < 0 . 0
S
x
5
L.
Then the
function u(x) = ezq(x) belongs to
D(Ap(eo,B,)) q’(0) =
We have
Yo/%
satisfies the boundary conditions q‘(L) =
f
(2.20)
YL/Z.
= z ~ ’ ( x ) e ~ ( ~ ~ ~ ) q=( ~z$(x) )
u(x)u‘(x)
Accordingly, if
q
if
(2.19)
with
JI
real.
is the function in (2.11)’ f = -p$
2
= -plul
2
lu’
I2
.
(2.21)
Making use of this equality and estimating the rest of the terms i n ( 2 . 7 ) in a way similar to that used i n Theorem 2.1 we obtain an
ON THE ANGLE OF DISSIPATIVITY
93
inequality of the form
- 6 I m ( e (u),Ap(BO,BC)U)
Re(0 (u) ,Ap(BO,BL)U) 2
CII
UII 2-p
r,“
I2dx
-
CIl~11~.
(2.22)
Assume that (2.16) holds as well for the same value of
c
6.
Then
we obtain from (2.22) that
lulp-21u’I2dx
C’
5
’,I
lulPdx
q
for all functions of the form (2.19) where boundary condition (2.20). f o r instance taking
rl
(2.23)
satisfies the
But (2.23) is easily seen to be false,
to be rapidly oscillating function.
This
completes the proof of Theorem 2.1.
If .A
REMARK 2.2.
AOu(x)
is written in variational form,
+
b(x)u’
a(.),
b(*)
= (a(x)u’ (x))‘
we only need to require that
(x)
3.
0
(2.24)
C(X)U(X)
(resp.
uously differentiable (resp. continuous) i n observation will apply i n Section
+
Z
be contin-
.(a))
x
5
4,.
The same
4.
AN APPLICATION: COMPUTATION OF THE NORM OF CERTAIN MULTIPLIER OPERATORS
We limit ourselves to the following example. A 0 u = u”
operator ditions
in the interval
u ’ ( 0 ) = u ‘ ( r ) = 0.
0
is the multiplier operator
(for
-c
defined for
Re
ancosnx)
5 >
0;
x s rr
Then the semigroup
Ap(Bo,B,)
u(x)
L
in the space
LP(O,n).
the alternative formula
Consider the
with boundary conS(C)
generated by
Note that
S(C)
is
H. 0 . FATTORINI
94
can be used, where
u
is extended 2n-periodically to
i n such a way that
u
is even about
x = 0
-m
x = rr.
and
-
< x
1
(3.4)
(Rec > 0)
A far more precise estimate can be obtained from Theorem 2.1 o r ,
Noting that in this
rather, from a close examination of (2.7). a = 1, b =
case
c = y
0
= 0
= YI,
of ( 2 . 7 ) is non-positive for
6
5
we see that the righ hand side tg (pp
(~p,
given by (2.17)).
A ccord ingly,
Ils(c)II, = in the sector
(arg 5 1
(that
5 (pp
(3.5)
1 z 1
IIS(C)llp
is obvious).
On
the other hand, it follows from the necessity part of Theorem 2.1 that ( 3 . 5 ) does not extend to any sector
in other words there exists
15,1
1< p
5
> 1). P 2
5 =
5
I
5
(in fact, a sequence
(lcpl
with
Cp
5,
rp > c p P ;
with
such that IlS(c)II P > 1 The same results can be achieved in the range
+ 0) in the ray
(llS(cn)ll
5
larg
arg
(p
> ep,)
(for instance, by using duality).
ON THE ANGLE OF DISSIPATIVITY
4. ORDINARY DIFFERENTIAL OPERATORS I N
L1
AND
95
C
We consider again the formal differential operator A 0u(x)
+ b(x)u’(x)
= a(x)u”(x)
+
(4 1)
C(X)U(X)
under the assumptions on the coefficients used i n $ 2 ) . Al(Po,B,,)
in
has already been defined there for boundary
L1(O,C)
conditions of any type. i n the space
C[O,L]
The definition of the operator
of continuous functions i n
ed with its usual supremum norm) is D(A)
The operator
A(po,pL)u
u
consisting of all functions
0
L
x
A(B0,B,) 1,
S
(endow-
with domain
= A 0u
twice continuously differ-
entiable satisfying the boundary condition at each end.
Note,
however, that if the boundary condition at zero is of type (11), D(A)
will not be dense in
E = C[O,L] u ( 0 ) = 0.
by its subspace
E;
this is remedied replacing consisting of all
Co[O,&]
When the boundary condition at
L
u
with
is of type (11)
(resp. when both conditions are of type (11)) the corresponding subspace is defined by
CLIO,L]
u(L) = 0
defined by
u ( 0 ) = u(L)
(resp.
Co,LIO,&]
= 0).
The first difficulty we encounter here is that will not be dissipative for any
w
THEOFEM
4.1. (a)
WI
unless the boundary conditions
(if of type (I)) are adequately restricted. with the operator
A1(BO,p,)-
The same problem exists
A(pO,pI,). Assume the boundary condition at
0
is of type
(I). Then the inequality yoa(0)
-
a’(0)
is necessary for dissipativity i n
+
b(0)
L1(O,l,)
5
0
(4.2)
of any operator
A ( p ,p ) using the boundary condition p o at x = 0. If the 1 0 L at x = 9 is of type (I) the corresponding boundary condition p,
96
FATTORINI
H.O.
inequality is
-
Y!,a(&)
+ b(b)
a'(&)
5;
(4.3)
0
If (4.2) and (4.3) hold (or if the corresponding boundary conditions are of type (11)) then
A1(pO,BL)
for sufficiently large
w.
(b)
-
WI is m-dissipative in
L1(O,l,)
Assume the boundary condition at 0
is of type (I). Then the inequality
Yo
2
is necessary for dissipativity i n LIO,&])
cO
(1
(4.4)
PROOF.
Let
(L1)* = L"
A(Bo,BL)
at
where
5;
1
u E L
,
u(x)
- wI
is m-dissipative in
u f 0.
5
x
a.
and we have
the definition of
Then any
u* E @ ( u ) equals
IIuII1
ON THE ANGLE: O F DISSIPATIVITY
[ -
+r
+
u(0) = 1
and
= -(Yoa(0)
+
((au’)’
a’(0)
97
(b-a‘)u’ + cu)dx
b(O))u(O)
+
(4.7)
(a”-b/+c)udx
-
If yoa(0)
a‘(0) + b(0) < 0, the right hand side of
made positive taking
shows the necessity of (4.2);
C[O,l]
space
sufficiently small.
the argument for
This
(4.3) is identical.
(4.4). Recall that the dual space
We prove the necessity of of
a
(4.7) can be
can be identified linearly and metrically with the
C[O,l]
of all finite Bore1 measures defined in
0 5
x
5
1
endowed with the total variation norm, application of a functional p E C
u E C
to an element
given by
If the boundary condition at space is
~ ( ( 0 ) =) 0.
Co[0,4,]
of
4,
is
= (x;
C[O,&]
u(&) = 0
u ( 0 ) = u(G) = 0.
u E C
w
with
or where the two boundary conditions
The duality set
@ ( u )E
c
of an element
consists of all measures supported by the set
I u(x) I
yo < 0
= IIull]
such that
I+
m(u)
=
is a positive measure and
II UII ‘
w e can obviously construct a real element of
D(A(@,,@,)) that
then the relevant
consisting of all
llr-111 =
If
u(0) = 0
Similar comments apply to the case where the boundary
condition at are
is
0
(4.8)
udll
whose dual can be identified through (4.8) to
Co[O,G],
the subspace
[
=
(Ll,u>
having a single positive maximum at
u ( 0 ) = 1,
fixed later.
u”(0) =
Then
a
where
O(u) = ( 6 } ,
6
a
x = 0
and such
is arbitrary and will be the Dirac delta and we have
98
H . 0 . FATTORINI
a.
which c a n b e made p o s i t i v e by j u d i c i o u s c h o i c e of
The s t a t e m e n t s c o n c e r n i n g m - d i s s i p a t i v i t y of t h e o p e r a t o r s
-
A1(Bo,pZ,)
w I
and
A(BO,~{,)
- wI
can be r ead o f f t h e f o llo win g
(4.3),
two more g e n e r a l r e s u l t s where we show t h a t c o n d i t i o n s ( 4 . 2 ) ,
( 4 . 4 ) , ( 4 . 5 ) c a n i n f a c t b e d i s c a r d e d if one a r e n o r m i n g of t h e s p a c e s Let
1 5 p
0 5 x 5 L.
i s w i l l i n g t o perform
P
8
P
: Lp
Lp‘
given
t h e d u a l i t y map c o r r e s p o n d i n g t o t h e c a s e For
p = 1
t h e d u a l i t y s e t of a n e l e m e n t
up
c o i n c i d e s w i t h t h e d u a l i t y s e t of (see
-t
(4.6)).
We t a k e now
u
smooth and p e r f o r m t h e
c u s t o m a r y i n t e g r a t i o n s by p a r t s , a s s u m i n g t h a t tinuously d i f f e r e n t i a b l e a s well:
a s an element
p
i s t w i c e con-
ON THE ANGLE OF DISSIPATIVITY
{ (apP-’p‘)‘
+
It is obvious that
1 (a”-b’+pc)PP P
+ (a’-b)pP-lp’] [ulpdx.
(4.11)
~ ‘ ( 4 , ) can be chosen at will, hence
and
p’(0)
99
we may do s o in such a way that the quantities between curly brackets in the first two terms on the right-hand side of (4.11) are nonpositive, say, for
1< p
2.
5
Since the first two integrals
together contribute a nonpositive amount, we can bound (4.11) by an W‘Ilullp 5 cullu\l~where
expression of the form p.
Consider now the space
L1(O,L)p.
tion (4.10) the duality set u* E L m ( O , ~ ) with
all where
U(X)
f 0 and
limits in (4.11) as
Op(u)
u* x) =
Iu*(x p + 1
I
h
w
Again under the identificaof an element
II ulIp I u(x)
IIuI(
P
does not depend on
I -%x)
u
=
elsewhere.
consists of
II UPll I 4 x ) l - 1 w
We can then take
and obtain an inequality of the form (4.12)
in
L1,
The inequality is extended to arbitrary
by means of the usual approximation argument.
u E D(A1(BO,Bl,))
Now that A1(BO,~l,)-wI
has been shown to be dissipative, m-dissipativity is established by
100
H.O.
FATTORINI
The c a s e where one ( o r b o t h )
u s i n g Green f u n c t i o n s a s i n S e c t i o n 2 .
of t h e boundary c o n d i t i o n s a r e of t y p e (11) i s t r e a t e d i n a n e n t i r e l y s i m i l a r way; n a t u r a l l y ,
t h e u s e of t h e weight f u n c t i o n i s un-
necessary i n the l a s t case. We have completed t h e proof THEOREM 4 . 2 .
Let
Then t h e o p e r a t o r
s(.)
Here
in
IIS(t)\lp
be a p o s i t i v e t w i c e c o n t i n u o u s l y d i f f e r e n t i -
0 5 x 4 t,
able function i n
group
p
such t h a t
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 semi-
A1(BO,BL)
such t h a t , f o r some
L1(O,L)p
i n d i c a t e s t h e norm o f
Assumption (4.13) ( r e s p .
L1(0,4,)p.
of
boundary c o n d i t i o n a t
0
(resp. a t
> 0,
W
a s an o p e r a t o r i n
S(t)
( 4 . 1 4 ) ) does not apply i f t h e 4,)
i s of t y p e (11).
To prove a s i m i l a r r e s u l t f o r t h e o p e r a t o r space
C
A
we renorm t h e
o r t h e c o r r e s p o n d i n g s u b s p a c e by means of
(4.15) where
in
p
is a positive,
0 5 x 5 4,.
twice continuously d i f f e r e n t i a b l e f u n c t i o n
The u s e of t h e weight f u n c t i o n
p
i s a g a i n un-
n e c e s s a r y when b o t h boundary c o n d i t i o n s a r e of t y p e (11): below i n d e t a i l t h e c a s e where
Po
and
lfmixedfl c a s e b e i n g e s s e n t i a l l y s i m i l a r .
B4,
we t r e a t
a r e of t y p e ( I ) , t h e
Choose
p
i n s u c h a way
that ~ ' ( 0 +)
if
yo
0 ) .
( r e s p . ~ ' ( 4 , )+ Y ~ P ( G ) 4 0 ) The d u a l o f
C[O,G]
(4.16)
equipped with
ON THE ANGLE OF DISSIPATIVITY
11 . / I p
C[O,L],
can again be identified with
u E C[O,L]
acting on functions
(Ll,u) =
QI
lip))
=
Op(u)
sets
i,“
U(X)P
up)
I
mp(u)
p E Z[O,L]
(4.17)
(X)P(dX) *
p E
c
C*
as a n element of
is
and the identification of the duality
Op(u)
is the same as before;
with support i n
(or
Ip (dx)
an element
through the formula
Accordingly, the norm of a measure still
101
= ( x ; lu(x)p(x)l
is a positive measure i n
= Ilull,]
mp(u)
p E Z
consists of all
with
same comments apply of course to the spaces
Co,
and such that upp lip11 =
I)UI/~.
C L , C0,&
the corresponding measures are required to vanish at
The
where
0, 4 ,
and
0
4 . We now show that large enough.
u(t) - ,!Y
then
A(oo,p,)
- WI
u’ (0) = y o u ( G ) ,
Observe first that if up
is m-dissipative for u’(C)
w =
satisfies the boundary conditions
where
Using elementary calculus we show that for any u E D ( A ( B O y D )),
L
u
#
0
the set
Y 0 , P ’’4 YP
> 0,
mp(u)
does not contain either endpoint if both
s o that
(4.20) On the other hand, if either
Y0,P
Or
Y&,p
vanish,
m,(u)
may
contain the corresponding endpoint but we can prove again that (4.20) holds. Writing q = p 2 we have ( l u p 1 2 ) ’ = 2 ( u1u 1 ’ + u2 u 2 ’)q + 2 2 2 1 + 2(u; +u;Z)q + + (u1+u2)q’, ( I u P 1 2 ) ” = 2 ( u 1u”+u2u’;)q 2 2 + 4(u1u;+u2u;)q‘ + (u1+u2)q”. Hence, it follows from (4.20) that
102
. FATTORINI
H .O
(4.21)
for some constant pative.
(XI
That
which shows that
W,
-
A(po,bt))u
= v
A(Po,Pg)
- wI
has a solution
u
is dissifor all
v
is once again shown by means of Green functions. The following result, that settles completely the question of angles of dissipativity in
LL
and
C
is a simple consequence
of the identification of angles of dissipativity in
L2
in Theorem
2 . 1 and of the theory of interpolation of operators between
L2
and between
3 . (a)
- L~I p,,~,)) in
C
L2
and
C.
Let ( 4 . 2 ) and ( 4 . 3 ) be satisfied. is m-dissipative in
= 0.
L1 and
(b)
(Co,Cg,C,,t)
L~
for
co
Then
sufficiently large
The same conclusion h o l d s for if
(4.4) and (4.5) hold.
ON THE ANGLE OF DISSIPATIVITY
5.
ELLIPTIC PARTIAL DIFFERENTIAL OPERATORS IN
1
W(x,f)]
E
i s a F r 6 c h e t space o r i f
or E
g:
= E.
i s a B a i r e space and
122
P I E D LELONG
i s a continuous plurisubharmonic function (it i s
f
tinuous) then
#
gk
i s a p l u r i p o l a r cone i n
E
h(y) = -v(x,y,f)
Let us w r i t e
X
f o r fixed
X #
h(Xy) = h ( y )
Then
i s a plurisubharmonic f u n c t i o n i n
h*(y)
C,
€
i s con-
E.
p o s i t i o n 1,
for
ef
x;
and
0
by t h e Proh(y)
0.
5
because i t i s
E
t h e upper r e g u l a r i z a t i o n of t h e u p p e r e n v e l o p e i n ( 1 0 ) of $(x,y,r) h*(y) h"
h*
0,
5
which i s a p l u r i s u b h a r m o n i c f u n c t i o n s of
0,
5
i s u p p e r bounded i n
y € E
for
and f i x e d
x.
y.
By
and t h e n , i t i s a c o n s t a n t
E,
W e have
h(y)
h* = V ( x , f ) ,
G
which
proves ( 2 0 ) ;
( 2 1 ) i s a consequence of t h e d e f i n i t i o n of t h e r e g u -
larization.
If
f(x)
#
V ( x , f ) = 0.
and t h e r e f o r e
f(x+uy)
4
f(x) =
by
G
-m.
has a f i n i t e value; there e x i s t s
IuI < 1 ;
for
-CU
= inf v(x,y,f)
y
v(x,y,f)
[ x E G; v ( x , f )
The s e t
tained i n the s e t defined i n v(x,f) 2 0
V(x,y,f) = 0
we have
-m,
i s con-
03
x € G,
For e a c h
#
0
such v(x,f) =
and
m
y;
has a f i n i t e value.
Y
To prove (11): g:
by
~ ( x , y , X f )= w ( x , y , f )
f(x)
#
i n which and
For
-m.
f
v(x,y,f)
y E gx,
0 E gk
and
f(x) =
if
y
#
To end t h e proof
By ( I ) , we have
then
-m
Therefore
-OD.
y E g:,
gx c gk
and
S(Y)
and
s,(y)
= sup SJY)
0
5
4
0.
n V(x,f),
we have i n s*(y)
E
0.
E
and
0,
By t h e
0.
8
gk =
if D
X1Y
v(x,y,f) =
+a,,
i s proved.
of (11) and (111),we w r i t e f o r
sn(y) E P(E)
By t h e d e f i n i t i o n of
and
X #
C,
there e x i s t s a disk
0,
has the constant value
> v(x,f);
1 E
i s a cone of v e r t e x
= [ y E E ; w(x,y,f) > y(x,f)]
Remark 1, we have seen:
for a l l
rn + 0:
TWO EQUIVALENT DEFINITIONS O F THE DENSITY NUMBERS
Now we have t w o p o s s i b l e s i t u a t i o n s : a)
Yo
Suppose t h e r e e x i s t s
such
S(Y0)
= sup s n ( y o ) = 0.
Then
p a s s i n g t o a subsequence i f n e c e s s a r y , we may P suppose C l s n ( y o ) l < m . Then Vp(y) = C s n ( y ) < 0 i s a d e c r e a s n 1 Then i n g sequence o f p l u r i s u b h a r m o n i c w i t h f i n i t e l i m i t i n y o .
by
sn(yo)
V(y) = s(y)
0, j E
is plurisubharmonic in u E 6,
such that
E
v(O,y,f)
N.
if we take
we obtain at the origin
will be the
0
Then c
7
0.
For
= inf cj,
for
j E supp y
j E IN.
j
loglxjl
fy(u) = f(yu),
x = 0:
= v(O,f Y ) = inf c j v(O,f)
j
f(x) = sup c
PIERRE LELONG
130
If s
c
-
2-j,
j -
6.
for
y E E
and
.
= Csup j ; j E SUPP Y]
REMARK
> v(0,f) = 0
v ( O , y , f ) = 2-'
then
C a l c u l u s of
w(x,y,f)
and
v(x,f)
E = @ En
if
More g e n e r a l l y , l e t u s c o n s i d e r a s e q u e n c e l o c a l l y convex complete s p a c e s ; i n
M = nE n ny
En
.
of complex
E
we d e f i n e
by
E = $ E n C M and t a k e on denote
E
E + En
p,:
E
d e f i n e d by
We s u p p o s e on e a c h monic f u n c t i o n
Let u s d e f i n e Given
xo E E ,
jn: En
+ E;
= 0,
p,(x)
i s the
T
i s t h e c l o s e d sub-
jn(En)
m f n
and
p,(x)
= x
= [ x n E E n ; Un(xn) = --]
En.
C
we d e f i n e a p o s i t i v e i n t e g e r s u c h p n ( x o ) = 0 for n
Now we s u p p o s e t h a t t h e o r i g i n b e l o n g s t o L
G i v e n a complex l i n e
in
qn
0
> s(x )]. En
in
f o r each
E:
x = x o + u y ,
u E C
P j b ) = Pj(XO) + U P j ( Y ) . x E L,
t h e number s(x)
s(x)
i s bounded
s s u p ~ s ( x O ) , s ( y ) ]= s ( L ) .
L e t us d e f i n e l i k e i n t h e p r e c e d e n t example f (x)
(33) Then
pn(x) = 0
i f m=n.
and we d e n o t e :
s ( x o ) = [ i n f n , n E IN,
For
We
we h a v e d e f i n e d a c o n t i n u o u s p l u r i s u b h a r -
En
Un(xn),
:n
xn = p n ( x ) ;
t h e p r o j e c t i o n s and
t o p o l o g y of t h e mappings space o f
T.
t h e l o c a l l y convex d i r e c t s u m t o p o l o g y
and
= s u p Unopn(x) = sup v n ( x ) n n Unopn(x) =
-m
for
n
.
> s(L)
and
n.
TWO EQUIVALENT DEFINITIONS OF THE DENSITY NUMBERS
f I L = sup U j o p j ( x o + u y ) f o r 1 5 j 5 s(L) j tion ( o r z -m) d e f i n e d by t h e formula
i s a subharmonic f u n c -
N o w we suppose t h a t t h e p o i n t
xo
belongs t o
xz = p n ( x o ) E :n
or
Un(xz)
Then i f
L
f o r each
n
i s a complex l i n e t h r o u g h
v(xo7y,f) = v(xo,flL)
(33
131
-
nrln
n.
f o r each
m
xo,
it is
of d i r e c t i o n
y = [yj}.
= inf v[x;,pj(y),ujl J
and we have t o t a k e i n
1 s j
s(y).
(33) the
inf
The d e n s i t y number
f o r given
v(xo7y,f)
y E $En
and
is
(34) for
y E E = $En,
I n the
En
and
1 5
s(y).
a r e closed subspaces o f
t h e space
E,
E
i s not a
i s a continuous plurisubharmonic function.
Baire space;
f(x)
From ( 3 3 ) and
( 3 4 ) we g e t
(35)
j S
v ( x o , f ) = inf v ( x o , y , f ) = inf v[pj(xo),uj1. Y
The t a n g e n t i a l d e n s i t y number
j (xo,y,f),
g i v e n by
(33) i s the
inf
o f a f i n i t e s e t of p o s i t i v e numbers; we can choose i n ( 3 5 ) t h e continuous plurisubharmonic functions v(xo,f) = c
> 0
and
v(xo,y,f) > c
Uj €
P(Ej),
for a l l
i n order t o obtain
y E E.
PIERRE LELONG
132
BIBLIOGRAPHY
1.
AVANISSIAN, V . , Fonctions plurisousharmoniques et fonctions doublement sousharmoniques, Ann. E .N. S , , t. 78, p. 101-161.
2.
C O E d , G., Fonctions plurisousharmoniques sur les espaces vectoriels topologiques, Ann. Inst. Fourier, 1970, p.361-432.
3. KISELMAN, Ch.,
a
Stabilite du nombre de Lelong par restriction une sous-variet6, Lecture Notes Springer ne 919,
P. 324-9379 (1980)
4. L E M N G , P.,
a/ Plurisubharmonic functions in topological vector spaces, Polar sets and problems of measure. Lecture Notes, no 364, 1973, p. 58-69. b/ Fonctions plurisousharmoniques et ensembles polaires sur
une aog8bre de fonctions holomorphes,
Lecture Notes, no 116,
1969, pa 1-20. c/ Calcul du nombre densit6 v(x,f)
et lemme de Schwarz pour
les fonctions plurisousharmoniques dans un espace vectoriel topologique,
.
Lecture Notes Springer nP 919, p. 167-177,
(1980)
d/ Integration sur un ensemble analytique complexe,
Bull.
SOC. Math. de France, t. 85, p. 239-262, 1957.
5.
RAMIS, J.-P., Sous-ensembles analytiques dcune variet6 banachique complexe, Ergebnisse der Math., t. 53, Springer,
1970
6.
SIU, Y.T., Analyticity of sets associated to Lelong numbers, Inv. Math., 6. 27, p. 53-156, 1974.
7 - NOVERRAZ, Ph.,
Pseudo-convexit6, convexit6 polynomiale et domaines dlholomorphie en dimension infinie, North Holland,
Math. Studies, vol. 3 (1973).
Dgpartement de Mathematiques Universite de Paris VI
4 Place Jussieu 75230
Paris
CEDEX 0 5
Functionnl Anolysis, Holoniorplry orid Appruxiniotion Theory 11,G.I. Zupata ( E d . ) 0 Elsevier Science Publislrers B. V. (Nurtli-Holland), 1984
133
CHEBYSHEV CENTERS OF COMPACT SETS WITH RESPECT T O STONE-WEIERSTRASS SUBSPACES
Jaroslav Mach
Let C(S,X)
S
be a compact Hausdorff space,
X
a Banach space,
S
the Banach space of all continuous functions on
X
values in
with
In this note two
equipped with the supremum norm.
results concerning Chebyshev centers of compact subsets of with respect to a Stone-Weierstrass subspace of
C(S,X)
C(S,X)
are es-
In particular, a formula for the relative Chebyshev
tablished.
radius in terms of the Chebyshev radius of the corresponding set valued map is given.
I t is shown further that the proximinality
of all Stone-Weierstrass subspaces implies the existence of relative Chebyshev centers for all compact subsets of
C(S,X).
The proximinality of Stone-Weierstrass subspaces has been studied by many authors.
Mazur (unpublished, c.f.,
e.g., [ 6 ] )
proved that any Stone-Weierstrass subspace is proximinal if the real line (a subspace led proximinal if every ximation
x
in
G,
G
of a normed linear space
yE Y
X
i.e., if there is an x € G).
xo € G
is cal-
such that
The question for which
every Stone-Weierstrass subspace of
proximinal is due to Pelczynski
C(S,X)
X
and an L1-predual space, respectively.
In
is uniformly convex [2]
those Banach spaces
for which any Stone-Weierstrass subspace is proximinal were
characterized.
is
[4] and Olech [ 3 ] . Olech [ 3 ] and
Blatter [I] showed that this is true if
X
is
possesses a n element of best appro-
/(y-xo)S [/y-x((holds for every Banach spaces
Y
X
134
JAROSLAV MACH
We will employ the following notations and definitions.
Let
E X,
x
center
r > 0.
B(x,r)
and radius
x
pact subsets of
X.
C(X)
r.
T
V
g E C(T,X).
G
of
V
C(S,X)
C(S,X)
is said to be a
if there is a compact
and a continuous surjection
is the set of all functions
some
Let
f
S
'p:
T
-t
such that
f = goCp
having the form
be a set-valued mapping from
@
171 ) if for every
borhood
U
of
E S
so
and every
sup
E
for into 2
S
is said to be upper Hausdorff semicontinuous (u.H.s.c.)
and
with
will denote the class of all com-
A subspace
Stone-Weierstrass subspace of Hausdorff space
X
will denote the closed ball in
we have
E .
xE@( s ) is lower semicontinuous if the set
@
for any open set [ s :
@ ( s )fl H
f € C(S,X)
#
iP
G.
61
( s :
n
G(s)
#
G
@) is open
is upper semicontinuous if the set
H.
is closed for any closed set
is said to be a best approximation of
A function
@
in
C(S,X)
if the number dist(f,G) = sup SES
is equal to
inf dist(g,C)
g E C(S,X).
Let
F
sup
IIx-f(s)l/
X€@(S)
where the infimum is taken over all
be a bounded subset of
X,
G
a subspace of X.
The number
rG(F) = inf sup IIx-yll xEG yEF is called the Chebyshev radius of x
E G
if
F
with respect to
is said to be a Chebyshev center of
IIx-yI(
denoted by
2
rG(F) cG(F).
note the number
for all
y E F.
F
G.
A point
with respect to
G
The set of all such x will be X For a set-valued map I: S -t 2 , r@ will de-
sup sEs
.)(a,
135
CHEBYSHEV CENTERS OF COMPACT SETS
THEOREM 1.
Let
cp
fined by
V
be a Stone-Weierstrass subspace of
T.
and
F
Let
be a compact subset of
r (F) =
V
where
@:
PROOF.
T
n
tu E {t: @(t)
E cp
iP
-1
H f 0}
n
(t,)
U.S.C.
fa E F s
U
s
-+
and
@(t) fl H f 0 .
g E
C(T,X)
sup
we show that the set
tu
sup
-+
t.
f
U
H.
Let
Then there are
fu(sa) E H.
such that
so
f(s) E H , For any
is
be such that
generality assume that and
Then
r@
is closed for any closed set
H f 0)
and
C(S,X).
de-
is the set-valued map
To prove that
(t € T: @(t)
S&
C(X)
-+
C(S,X)
-+ f.
Without l o s s of
Then clearly s E cp”(t)
we have
IIx-g(t))) = dist(g,@).
tET x€@ (t) It was proved in
[2]
that dist(g,@) = r@.
inf
& c (T,X) It follows
THEOREM 2 .
Let
of
is proximinal.
C(S,X)
X
be such that every Stone-Weierstrass subspace Then
cV(F)
#
0
for every compact sub-
set .F
of
PROOF.
By Theorem 2 of [ 2 ] , the proximinality of every Stone-
C(S,X)
and every Stone-Weierstrass subspace
Weierstrass subspace of Hausdorff space
C(S,X)
T, any u.H.s.c.
V.
implies that for any compact map
@:
T
-+
C(X)
has a best
136
JAR0SLAV MACH
approximation
g
c(T,x).
in
Let
Then inf (h,@) = sup Ilf-go(pll = dist(g,$) = f€F h€ C (T,X ) inf hEC(T,X)
sup
Ilf-hocpll
= rV(F).
f€F
I t follows The following corollary is a consequence o f Blatterls result and Theorem 2. COROLLARY 1.
Let
be an L1-predual space.
X
for any compact subset subspace
F
of
C(S,X)
Then
cV(F) f Q
and any Stone-Weierstrass
V.
In [8], a bounded subset of an L1-predual space has been constructed whose set of Chebyshev centers is empty.
This shows that
Corollary 1 does not hold if compact subsets are replaced by bounded subsets.
It was shown in [2] that if
X
is a locally uniformly convex dual
Banach space then every Stone-Weierstrass subspace of proximinal.
is
The next corollary follows from this and Theorem 2.
COROLLARY 2.
Let
space.
cV(F)
Then
C(S,X)
X
be a dual locally uniformly convex Banach
#
Q
for any compact subset
any Stone-Weierstrass subspace
V.
F
of
C(S,X)
and
CHEBYSHEV CENTERS OF COMPACT SETS
137
REFERENCES
1.
J. BLATTER, Grothendieck spaces in approximation theory,
Mem.
Amer. Math. SOC. 120 (1972). 2.
J. MACH,
On the proximinality of Stone-Weierstrass subspaces,
Pacific J. Math. 99 (1982), 97-104. 3.
C. OLECH, Approximation of set-valued functions by continuous functions, Colloq. Math. 19 (1968), 285-293.
4. A. PELCZYNSKI,
Linear extensions, linear averagings and their
applications to linear topological classification of spaces of continuous functions, Dissert. Math. (Rozprawy Math.)
58, Warszawa 1968.
5.
W.
POLLUL, Topologien auf Mengen von Teilmengen und Stetigkeit von mengenwertigen metrischen Projektionen, Diplomarbeit, Bonn 1967.
6.
Z.
SEMADENI,
Banach spaces of continuous functions,
Monografje Matematyczne 55, Warszawa 1971.
7. I. SINGER, The theory of best approximation and functional analysis, Reg. conference ser. appl. math. 13, SIAM, Philadelphia 1974. 8.
D.
AMIR, J. MACH, K. SAATKAMP,
Existence of Chebyshev centers,
best n-nets and best compact approximants, Trans. Amer. Math. SOC. 2 7 1 (1982), 513-524.
Institut fiir Angewandte Mathematik der Universitgt Bonn Wegelerstr. 6 5300 Bonn
This work was done while the author was visiting the Texas A&M University at College Station.
This Page Intentionally Left Blank
O N THE: FOURIER-BOREL TRANSFORMATION AND SPACES O F ENTIRE FUNCTIONS I N A N O W , n SPACE
Mdrio C . Matos ( D e d i c a t e d t o t h e memory o f S i l v i o Machado)
1. INTRODUCTION We i n t r o d u c e h e r e t h e s p a c e s o f e n t i r e f u n c t i o n s i n a normed s p a c e which a r e of o r d e r ( r e s p e c t i v e l y , n u c l e a r o r d e r ) ( r e s p e c t i v e l y , nuclear t y p e )
k E [l,+m]
Here
and
s t r i c t l y l e s s than
A E
(O,+-].
k E [l,+m]
and
A E [O,+m).
and t y p e
A.
The c o r r e s p o n d i n g s p a c e s
i n w h i c h t h e t y p e i s a l l o w e d t o be a l s o e q u a l t o when
k
A a r e introduced
These spaces have n a t u r a l t o -
p o l o g i e s and t h e y a r e t h e i n f i n i t e d i m e n s i o n a l a n a l o g o u s of t h e spaces considered i n Martineau [l]. Fourier-Bore1
I n t h i s p a p e r we s t u d y t h e
t r a n s f o r m a t i o n i n t h e s e s p a c e s a n d we a r e a b l e t o show
t h a t t h e s e t r a n s f o r m a t i o n s i d e n t i f y 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 t h e s t r o n g d u a l s of t h e a b o v e s p a c e s w i t h o t h e r s p a c e s of t h e same
I n a s e c o n d p a p e r , t o a p p e a r e l s e w h e r e , we p r o v e e x i s t e n c e
kind.
and a p p r o x i m a t i o n t h e o r e m s for c o n v o l u t i o n e q u a t i o n s i n t h e s e s p a c e s . The n o t a t i o n s we u s e d a r e t h o s e u s e d by N a c h b i n [l] a n d Gupta
[ 11,
Hence,
E
if
i s a complex normed s p a c e ,
s p a c e of a l l e n t i r e f u n c t i o n s i n a l l j-homogeneous norm
I/ *I/
and
E,
P(%)
continuous polynomials i n
PN('E)
#(E)
is the vector
t h e Banach s p a c e o f
E
with the n a t u r a l
t h e Banach s p a c e of a l l j-homogeneous
t i n u o u s p o l y n o m i a l s of n u c l e a r t y p e w i t h t h e n u c l e a r norm for a l l
j E N.
II.IIN
con-
MARIO C .
140
MATOS
S P A C E S O F ENTIRE FUNCTIONS I N NORMED SPACES
2.
I n t h i s section 2 . 1 DEFINITION.
p > 0
If
f E B(E)
s p a c e of a l l
d e n o t e s a complex normed s p a c e .
E
we d e n o t e
Bp(E)
t h e complex v e c t o r
such t h a t m
11 * ] I p .
normed by all
2.2
f E H(E)
>
0
t h e complex v e c t o r space of
znf(0) E pN(%)
p
For each
n E N
f o r each
> 0 , t h e normed s p a c e s
Bp(E)
and
and
a r e complete. m
(fn)n,l
If
PROOF.
a
such t h a t
PROPOSITION.
nN,p(E)
BN,p(E)
We d e n o t e
there i s
i s a Cauchy sequence i n
n
a
E IN,
(E)
m
element
P j E P (’E)
for a l l
rn
f o r every
such t h a t
.
for a l l
,
and n 2 n It f o l l o w s t h a t (djf,(O))OD is a a E n= 1 and i t c o n v e r g e s t o an Cauchy sequence i n t h e Banach s p a c e P ( j E ) 2 n
2
n
E
.
.
Hence, u s i n g ( 3 ) , w e have:
If w e prove t h a t m
f(x) =
C
1 3 Pj(x)
j=O
d e f i n e s an element of
#(E),
then w e get:
(X
E E)
ON THE FOURIER-BOREL TRANSFORMATION
Therefore
to
f.
f
141
and ( 4 ) i m p l i e s t h e c o n v e r g e n c e of
E Wp (E)
f E #(E)
I n o r d e r t o prove t h a t
(fn)m
n= 1
we n o t e t h a t
Hence
f E 1(E).
and
2 . 3 DEFINITION. v e c t o r space topology.
aN,p
A s i m i l a r p r o o f may be u s e d f o r A E
If
u
Bp(E)
we d e n o t e
(O,+m)
Exp;(E)
*
t h e complex
w i t h t h e l o c a l l y convex i n d u c t i v e l i m i t
P 0
E (l,+m)
pTj E P ( j E ’ ) ,
and
A E [O,+m).
Thus t h e r e a r e
p > A
such t h a t
P E pN(jE),
for all
k
T E [ E x p N , O , A ( E ) ] ’ be g i v e n .
Let and
161
j
E
By G u p t a l s r e s u l t , mentioned e a r l i e r ,
N.
BTj((p) = T j ( ( p J )
and
IIPTjll = llTjll.
T h u s , by
(45),
and
Hence
FT
E
Exp
k’
(E’).
A 1(k)
Now l e t
H E Exp
k’
(E‘)
ah(k) and
c(p)
> 0
such t h a t
be g i v e n .
Hence t h e r e a r e p
E
A
162
MARIO
MATOS
C.
By a G u p t a t s r e s u l t m e n t i o n e d e a r l i e r t h e r e i s that
T . = s''(djH(0)) J
f o r every
f
E
/ITjll = IldJH(0)ll.
and
k (E). Exp N,O,A
We u s e ( 4 6 ) and
T j E [PN('E)]'
such
Now, we d e f i n e
I/Tjll = IldJH(0)ll
and
we o b t a i n
(47)
= c
If e > 0
i s such t h a t
p>
l+S
ITH(f)I
for all
f
NOW
E
k (E). Exp N,O,A
A
' '(P)'(') Hence
-
w e get
TH
IlfllN,k E
' l+C
[ E x P N , O , A( E ) ] '
t h e r e i s only t h e c a s e ( 2 ) w i t h
k = +a
and FTH = H. and A E c 0 , + O D ) .
I n t h i s case (El = HNb(B1(0))' A The p r o o f s w e r e d o n e i n G u p t a E l ]
a n d Matos [l] and [ 2 ] .
Q.E .D.
ON THE FOURIER-BOREL TRANSFORMATION
4.4
REMARK.
cp E E'
and
A s we saw i n P r o p o s i t i o n
llcpll
d e f i n i t i o n f o r i t s Fourier-Bore1
= T(e')
cp E E '
for a l l
t h a t we c a n d e f i n e
FT
with
eV E Exp N,O ,A(E)
T E [ E x p N , O , A( E ) ]
Hence if
A.
6
2.17,
transform
FT
'
,
if
the natural FT(cp) =
would be
However i t c a n b e p r o v e d
IIrp/( L A.
for a l l
163
cp E E '
with
)IcpII
s PT
,
PT > A ,
i n such a v e r y t h a t i t a g r e e s w i t h t h e p r ev io u s d e f i n i t i o n f o r
Ilcpll
cp E E ' t
E3
A.
4 . 5 DEFINITION.
If
T E [ExpN,O,A(E)]',
Bore1 tra nsf orm
FT
by
for a l l
cp E E'
T . = TIpN(jE) J
such t h a t and
PROOF.
If
(49) converges a b s o l u t e l y .
BTj E F ' ( j E ' )
A s we w r o t e p r e v i o u s l y
Thus
i s g i v e n by
IIBTjll = llBTjll
f E E X ~ ~ , ~ , ~ ( Hence E ) .
Here
BTj(cp) = Tj(cpn).
by a G u p t a l s r e s u l t .
T E [ExpN,O,A(~)]'there are
such t h a t
f o r all
we d e f i n e i t s F o u r i e r -
p 7
o
and
C(P)
> 0
164
MARIO C .
It f o l l o w s t h a t c o n v e r g e n c e of
s i n c e ( 5 0 ) s a y s t h a t t h e r a d i u s of
FT E H b ( B p ( 0 ) ) FT
is
2
MATOS
p. Q.E.D.
4.7
THEOREM.
The F o u r i e r - B o r e 1 t r a n s f o r m a t i o n
isomorphism between
(E)]’
and
Exp;(E‘)
-
rExpN, o , A PROOF.
F
for A E [O,+m),
A
4 . 6 and t h e d e f i n i t i o n o f
By P r o p o s i t i o n
i s a v e c t o r space
Expy(E’)
-
A FT E Exp;(E’)
clear that
T E [Exp
for all
-
A
T E [ExpN,O,A(E)]’ i s such that all 0
cp E E ’ ,
IlcPll
= IIBTj/l = llTjll
and Hence
< p
f o r some
for a l l
P E PN(jE),
Thus
Therefore, f o r every
j E N.
T j E [ P N ( ’E)]’
H E Exp;(E’)
-A
H E ab(Bp(0)).
such t h a t
T h u s , by
j E IN.
(see
C
> 0,
If
(FT)(cp) = 0
4.6).
T(P) = 0
for
Hence for a l l
j E N
T = 0.
= Ub(BA(0)).
Therefore
there i s
c(e)
such t h a t
such that
pT
.
= ajH(0)
J
and
I(Tjll = I I a J H ( O ) I l ,
( 5 1 ) , we h a v e
a
C
T j ( b :jf(O))
j=O f
Hence t h e r e i s
By a G u p t a f s r e s u l t m e n t i o n e d e a r l i e r t h e r e i s
TH(f) =
for a l l
.
i t f o l l o w s from P r o p o s i t i o n 2.15 t h a t
Now we c o n s i d e r
for all
> A
p
then
(E)]’
i s an i n j e c t i o n .
F
p > A
FT = 0 ,
N,O,A
it is
E Exp
N,O,A
(E).
We use (52) t o o b t a i n
ON THE FOURIER-BOREL TRANSFORMATION
for a l l
E
f
p> A. l+e
p
E X ~ ~ , ~ , * ( E )a ,l l
Thus
TH
E
>
[ E x p N , O , A( E ) ] ’
and a l l
A
.
165
e > 0
such t h a t
It i s e a s y t o s e e t h a t FTH = H. Q.E.D.
4 . 8 THEOREM.
The F o u r i e r - B o r e 1
transformation
i s a topological
F‘
isomorphism between:
and
E
A
(O,+m].
k [ExpN,A(E)]
Here
k CEXPN,*(E)I‘ a n d PROOF.
k rExpN,o,A(E)];
and
CExPN,O,A( E ) ] ’
k ExpN,,(E)
Since
denote the duals
with the strong topologies.
i s a DF s p a c e ,
k’ Exp
(E’)
is a
oq$T i s a n a l g e b r a i c isomorphism, i n o r d e r t o
F
F r 6 c h e t s p a c e and
p r o v e ( a ) i t i s enough t o show t h a t
F-’
t h e Open Mapping Theorem i m p l i e s t h a t
Now w e prove t h a t A E
Let
(O,+=],
By 3.2
j
E
N.
E > 0
p
In fact,
i s continuous.
i s continuous f o r
b e a bounded
there i s
Hence, f o r e v e r y
for all
bd
F-’
F
i s continuous.
s u b s e t of
E (0,A)
such t h a t
there i s
c(c) 2 0
k
E
(l,+m)
k Exp (E). N ,A
such t h a t
and
166
i4ARIO C . MATOS
W e have:
Since
that
W e know
Hence there is
d(e)
z 0
such that
Theref ore
N o w we choose 1
X ( k ) ( p t$ ) ( 1+e 7
e > 0 1 7
such that
--
the other values of
and k
for every
that
p
< A.
is continuous.
The proofs for all
to,+-).
and
A E
be a continuous seminorm in
Exp
p < A,
Hence
for case (a) f o l l o w the same pattern.
N o w we prove case (b) for
Let
F-l
(p+e)(l+e)
we have
k E (1,+-)
1
k'
6,
(E').
Hence,
X(k)A
>
and there is
c(p)
such
ON THE FOURIER-BOREL TRANSFORMATION
k T E [ExpN,O,A(E)]‘.
f o r every
We c o n s i d e r
m
If
03 =
for all
and
8
u j=O
$3
j E IN.
w e have
Hence
i s a b o u n d e d subset o f Now w e w r i t e
Since
k
by P r o p o s i t i o n 3 . 4 . E X P ~ ,( E~) , ~
167
MARIO C .
168
c(c) 2 0
there i s
j E N.
or a l l
MATOS
such t h a t
Hence
where
o > 0
f o r all
such t h a t
p ( l + e ) < A. F.
T h i s proves t h e c o n t i n u i t y of
I n o r d e r t o p r o v e t h e c o n t i n u i t y of bounded i n
by P r o p o s i t i o n
3.4.
Now, for e v e r y
such t h a t
for a l l
F-’
we c o n s i d e r
63
E X kP ~ , ~ , ~ ( EHence )
j E N.
We h a v e
p
>
A,
there i s
c(p) 2 0
ON T m FOURIER-BOREL TRANSFORMATION
for a l l
e
for a l l
r
A.
Hence
-j--$jA-.
T h i s proves t h a t
F-l
i s continuous.
The p r o o f s f o r t h e o t h e r v a l u e s of
k
f o l l o w t h e same
0
pattern.
REFERENCES
1.
C.P.
GUPTA,
Malgrange theorem f o r n u c l e a r l y e n t i r e f u n c t i o n s
of bounded t y p e neiro,
2.
C.P.
-
Notas de Matemitica
3 7 , IMPA, R i o d e Ja-
1968.
GUPTA,
C o n v o l u t i o n o p e r a t o r s a n d h o l o m o r p h i c m a p p i n g s on
a Banach s p a c e ,
S e m i n a i r e d c A n a l y s e Moderne, n2 2 , U n i v e r -
s i t 6 d e S h e r b r o o k e , S h e r b r o o k e , 1969. 1.
L. N A C H B I N ,
T o p o l o g y on s p a c e s o f h o l o m o r p h i c m a p p i n g s ,
E r g e b n i s s e d e r M a t h e m a t i k , 47
1.
A.
MARTINEAU, Bull.
1.
M.C.
Equations d i f f b r e n t i e l l e s d f o r d r e i n f i n i ,
S O C . Math. F r a n c e ,
MATOS,
(1969)~ Springer-Verlag.
95 ( 1 9 6 7 ) , p . 109-154.
O n M a l g r a n g e Theorem for n u c l e a r h o l o m o r p h i c
F u n c t i o n s i n open b a l l s of a Banach s p a c e ,
Math.
Z.
162
( 1 9 7 8 ) , 113-123. 2.
M.C.
MATOS,
C o r r e c t i o n t o "On M a l g r a n g e Theorem N u c l e a r h o l o -
m o r p h i c F u n c t i o n s i n open b a l l s of a Banach S p a c e " ,
Z.,
1 7 1 ( 1 9 8 0 ) , 289-290.
D e p a r t a m e n t o d e MatemAtica IMECC
- UNICAMP
Caixa P o s t a l 13100
-
6155
Campinas, S . P . ,
Brasil
Math.
This Page Intentionally Left Blank
Functional Analysis, Holomorpliy and Approximation Theory 14 G.I.Zapata (ed.) @ Ekeuirr Science Publishers B. V. (North-Holland), 1984
ON REPRESENTATIONS OF DISTANCE FUNCTIONS I N THE PLANE
John McGowan
and
Horacio Porta
INTRODUCTION
ly symmetric curve assume that
r
8,sin 8 )
P(8) = r(e)(cos
Suppose that C
describes a centralr(8+rr) = r ( 8 ) ) .
in the plane ( s o that
is a continuous function and that
r(8) > 0
We on
c 0,2lTI The distance function (sometimes called the Minkowski functional
of
is the real valued non-negative function
X = IXl(cos a,sin a )
for
is convex,
Lc
C
Lc
in
by
iR2
Lc(X)
is the norm associated to
consisting of the union of
C
Lc
= IXl/r(a).
C,
defined
When
C
the unit ball for
and the region enclosed by
C.
The objective of this paper is to find integral and differ-
Lc. More specifically, for a class of
ential representations for
curves (essentially those having finite angular variation and satisfying an interior cone condition) w e obtain in $4 the formula LC(X) = where
x
dM C
Ip(e)xxl
dM(8)
denotes cross-product of vectors and where
appropriate Borei measure. tion
6'
of plane norms (where is unique, and
dM Z 0
is convex in the section
dM
is an
This generalizes the Levy representadM Z
0;
see 7 . 8 below).
on an interval
(a,p)
The measure
if and only if
a < 8 < p.
As usual, this representation for
Lc
gives when
convex the classical result that the normed space
(W2,Lc)
C
is can be
172
JOHN MCGOWAN
embedded isometrically in
and
HORACIO PORTA
L1[O,l]
(as shown in $ 2 below using
ultraproducts). I n view of the uniqueness result, it is reasonable to expect a simple expression for
dM
P(f3).
in terms of
We prove in $ 5
that dM =
where
R(8)
= l/r(e)
1 2
+
R(R
d2R T )d8 df3-
and where the derivative
interpreted in the sense of distributions. turn be used for curves of class ternative expressions of
dM
C2
d2/de2
is to be
This formula can in
(convex o r not) to find al-
in terms of geometric notions.
Among
other results, we prove in 96 that
where
n
is the curvature of
the ray through
0
and
P(0)
C
a
and
is the angle formed by
and the tangent to
C
at
P(0).
Finally, in $7 we consider several notions related to including its moments.
dM,
Among several inequalities and identities
we prove, for example, that [r2 dM
is invariant under inversion
through the unit circle. We want to thank R.P. Kaufman, H.P. Lotz and T. Morley for their valuable comments on parts of this work.
6 1.
POLYOGONAL EQUATIONS Let us consider a polygon in the
i = l,Z,...,n.
..,PZn
plane, symmetric
Pn+i = -Pi Here and in the following we do not distinguish
about the origin with vertices for
x, y
P1,P2,.
where
between points in the plane, and their position vectors with origin at
0 = (0,O).
Suppose that the
...,Pzn
P1,P2,
are all non zero
and that they form a radial sequence, i.e., they are totally ordered
ON REPRESENTATIONS OF DISTANCE FUNCTIONS
by their central angle (varying counterclockwise).
173
We assume
further that this order is strict s o that no two distinct vertices lie on the same ray from the origin.
For convenience we set P o = - P n
a .
denote the component of the cross product iJ Pi X P . in the positive z-direction, f o r 1 (= i, j 5 n (here J and in the following we consider the x,y-plane as the set in Let now
x,y,z-space characterized by 1.1 PROPOSITION.
The matrix with entries
with inverse having entries
for
1 5 i 5 n-1,
z = 0).
and
Bij
pij
= 0
laijl
is invertible
given by
for all other
i,j.
The proof of this proposition along with various observations on the matrices
([uijl) and
(Bij)
will be found in the
.
174
JOHN MCGOWAN
Appendix, $ 8 . of
(Pij)
and
HORACIO PORTA
We remark that by definition, the diagonal entries
are non-positive while the off diagonal entries are po-
sitive. Consider now the expression
n
c
L(P) =
mjlPjxPl
j=1
where
P = (x,y)
is an arbitrary point in the plane and the
m Is
are arbitrary real numbers not all zero.
It is clear that for
between the rays from the origin through
Pi
tion
L(P) = 1
P
Pi+l, the equa-
and
is equivalent to P .
-
m.P.xk (j$i
where
i
J
J
C j> i
m.P.Xk) J J
= 1
k = (0,0,1) is the unit vector in the z-direction; whence,
it is the equation of a straight line. Therefore,
L(P) = 1
represents a polygonal line with
vertices on the rays from the origin through the
P.ls,
J
ly symmetric about the origin
I /”
\ ‘
and clear-
175
ON REPRESENTATIONS OF DIST4NCE FUNCTIONS
1.2 PROPOSITION.
Given
P1,P2,...,Pn,Pn+l = -P1,...,P2n
radial sequence, there exist unique
L(P) = C m.lPxPil = 1 that order) mi
P1,P2,
ml,m2, ...,m
n
= -Pn
in
such that
is the equation of the polygon joining (in
...,Pn, -P1,...,-P n,P1.
Further, for each
is non-negative if and only if the quadrilateral
i,
OPi-lPiPi+l
is convex (hence the polygon is convex if and only if all
"'j
are
non-negative).
mn
But since the area of the triangle
-
-1 2 &i-l,i
'
1
OPi-lPi
is
1 T
IPi-lXPil =
and similarly for all the other values of
i,
from the last equation
and therefore,
mi t 0
if and only if
area(OPi-lPi+l) z
area(^^^,^^^)
+ area(OPiPi+l),
we get
J O H N MCGOWAN
and
that is, if the quadrilateral
H O R A C I O PORTA
i s convex.
OPi-lPiPi+l
$ 2 . ISOMETRIC EMBEDDINGS C o n s i d e r a s y m m e t r i c onvex p o l y g o n
r
n m.IP.XPI. J J
it is clear that
= lalJ+(P),
t o t h e convex body bounded by ed space
= 1 on
L
Since
r
with equation
r
Lr(aP) =
and
Lr
i s t h e n o r m on
r.
We s h a l l d e n o t e by
R2
associated
B,
t h e norm-
2
(IR , L r ) .
n
m =
Let
C
j=1
m. J ’
= m./m
p
J
d i s j o i n t union of i n t e r v a l s
E(3) =
[ul
+
u2,
p1 + b 2 +
and decompose
E ( 1 ) = [O,F,),
w3),
etc.
as the
[O,l)
+
E ( 2 ) = [bl,l-ll
with lengths
p2),
k(E(j)) = pjY
1 s j s n . Suppose now t h a t t h e c o o r d i n a t e s of t h e and d e f i n e f u n c t i o n s
E( j ) ,
TP = m(xf Then
j = 1,2,.
for
+
yg)
f,
g
. ., n .
where
on
r0,l)
by:
Finally define
P = (x,y)
P . J
f = b j ,
T: R 2
-t
are
P . = ( a .,b.) J J J
g = -a
j
L1(O,l)
i n an a r b i t r a r y v e c t o r i n
on by 2
IR
.
ON REPRESENTATIONS OF DISTANCE FUNCTIONS
n = m
-
C (mj/m)lxbj j=1
177
ya.1 = J
n
Thus
T
B~
is an isometry from
into
L'(O,I).
This is a particular case of the classical result that all 2-dimensional normed spaces can be isometrically embedded in 1 L1 = L (0,l)
(see for example
[4]). The following argument shows
that the general case follows from the case of polygons very simply. Suppose that convex curve polygons
)I 11
IIPI/ = 1.
r(n)
is a norm on
We can approximate
inscribed in
= lim L ~ ( ~ ) ( P ) . Let now
n
(T,)
C
of countably many copies of
we have
T:
a separable subspace of
L1/b,
so
that
LL/b,
C
(see
T
C
be the
by a sequence of IlPIl =
be isometries.
R2 + L1/k
([&], Proposition 11
IITpllL1/ll = IIPII
sublattice of
L1
and let
in the sense that
+ L1 T ~ B : r(n)
defines a linear operator
self an Ll-space
R2
The family
into the ultraproduct
r4], p.
121ff),
2.10).
Since
is an isometry.
which is it-
But
T(R2) is
and thus contained in a separable
which, by the classical Kakutani representa-
tion theorem can be identified with
L'(0,l).
It follows that all
2-dimensional normed spaces can be isometrically embedded in
L1( 0,l).
178
$3.
and
J O H N MCGOWAN
HORACIO PORTA
SOME ESTIMATES
Consider again the vertices = -p,,
,...,Pn,Pn+l
P1,P2
= -P1
,...,P2n
=
(ordered counterclockwise in radial sequence) of a symmetric
r.
polygon chosen
Denote by 0
0 and where
where
dN = C wn 6
with
dM.
en
.
IX'YnI.
n.
Write
t i7
IIX((= 1.
is the equation of
P(e)
Then
By uniqueness, this measure must coincide
Thus:
5 . 5 COROLLARY.
The normed space
t1
space of
if and only if
(R2,1/
11)
is isometric to a sub-
In particular,
dM
is purely atomic.
+m
is not isometric to a subspace
n
IRL
with the p-norm,
of
t?
2
s p
q-1
then
and t h e
then T
yh,j
= 0
for
(m+ j + l )( p ) = 0
j
for
...,q-11 [ q , ...,h]
E {h+l, j
E
and, i f whence
SPECTRAL THEORY F O R OPERATOR POLYNOMIALS
The e x p a n s i o n
and t h e r e l a t i o n s h i p
(3.18)
lead t o
Because q-1-j
rn
c
c
m=O
r=O
=
q-1-j
q-1-j
r=O
m=r
c
c
we o b t a i n
q-1 q-1-j
= j =cO
c r=O
q-1
q-1-j
j=O
r=O
= c
c
q-1-j-r
c
m=O
q-1-j-r
c
m=O
9
.L=m+r+j+l
e ) X r p L - r - j-1 m+r+ j+l TLyh- j
q-1-j-r
c
t -m
.L =m
where we u s e d t h e i d e n t i t y
Since, according t o
and because
(3.16),
'
the foregoing equation (3.22)
implies that
221
222
REINHARD MENNICKEN
s o t h a t , i n c o n s i d e r a t i o n of ( 3 . 2 1 ) ,
t h e proof of t h e r e l a t i o n s h i p
(3.20) f i n a l l y i s complete.
E u(T)
( 3 . 2 3 ) THEOREM.
i) If
i s equal t o
and t h u s i n d e p e n d e n t of
Eo
@
then
I m ( P ( 1 ) ) c D(P((1))
which
p.
The b i o r t h o g o n a l r e l a t i o n s h i p
ii)
(3.24 holds f o r a l l
u1,b2 E u ( T ) .
PROOF
E u(T).
Let
(1
Obviously
Res (RV) = Res (HT(R,p)V)
1
cc
s o t h a t , by ( 2 . 6 ) ,
Hence, if
where
y E D(P(p))
V(j)(w)
then
d e n o t e s t h e j - t h d e r i v a t i v e of
V
a t the point
1.
A change of t h e o r d e r of summation l e a d s t o
whence, t a k i n g i n t o c o n s i d e r a t i o n t h e r e l a t i o n s h i p ( 3 . 9 ) ,
the
a s s e r t i o n i) i s a l r e a d y p r o v e d .
Now we a r e g o i n g t o p r o v e t h e s t a t e m e n t i i ) : A c c o r d i n g t o (1.5),
for a l l
( 2 . 8 ) and ( 3 . 2 0 ) we h a v e
h E [0,1,2,.
and b e c a u s e of ( 2 . 7 ) ,
.,
,%(p)}
and a r b i t r a r y
(3.25) we o b t a i n
?, E C .
Therefore
SPECTRAL THEORY FOR OPERATOR POLYNOMIALS
for
s mk(bl),
0P h
0 5 k S nu1 T(pl),
Thus, again in view of
(3.25), the proof of ii) is complete.
4. BIORTHOGONAL EXPANSIONS In the sequel we adopt the assumptions made in the preceding sections. spectrum
We denumerate the elements
up(T)
p E
Let p
...
of the (point)
in such a way that
We call the operator bundle
[N.
T
regular of order
if and only if the following properties are fulfilled: i)
there exist a sequence
sequence
all ii)
("j j E N
of natural numbers and a
of simply closed Jordan curves in
(rj)jcN
so that exactly the eigenvalues
9
pl,p2,
pl,p2,.. .,p
nj
a:
around 0
are inside
. for J
j E R;
dj := max{
1x1
:
E
rj}
tends to infinity and there is a
> 0 so that 6 d . S dist(0,r .) J J iii)
there exists a real number
r,
IR(X)l
( j E N); c
> 0 such that
Idxi s c dp
Obviously the property iii) is positive constants
(j
E
N).
j
cl, c2
so that
fulfilled if there exist
REINHARD M E N N I C K E N
224
l e n g t h ( r .) J
z c
1
d
j
and
j E IN.
for a l l
( 4 . 1 ) THEOREM. Let
f E
Let t h e o p e r a t o r bundle
J-l(D(MP+¶))
and assume t h a t
T f
b e r e g u l a r of o r d e r f u l f i l l s t h e "boundary
conditions
(4.2)
j=O
J
Then we a s s e r t t h a t
(4.3) PROOF.
Accorrting t o (22) and (3.3), the d e f i n i t i o n of t h e TD,ff&-1I+.;JP'
= 0
whence t h e b o u n d a r y c o n d i t i o n s ( 4 . 2 )
,
f' j1
we have
(& = 1 , 2 , . . . , p ) a r e e q u i v a l e n t 60 t h e condi-
tions
Let
for
X
E p(T)
& E [O,l,
Using
...,PI
and u n e q u a l t o
and show t h a t
(3.1) and ( 4 . 4 ) w e i n f e r
0,
We s e t
S4+l(X) = S,(X):
Obviously
p.
SPECTRAL THEORY FOR OPERATOR POLYNOMIALS
1
[ fC q"]
p+C+1
-
R(X)
-
1 -.-pi R(X)
'il
3
hkTk f[q+']
=xL+~R(X) 1
225
Tq fCq+"
k= 0
- -
q+d-l m=&
q-1 T, Tm m= 0
1 h~+1 R(X)
C,
m+tI
Tm-t.
C,
ml
which completes the proof. According to (3.4), (3.5) and (4.5) we obtain R(X)V(X)f
= R(X)
q-1 1 C m+l (T(X) m=O ?,
1 = ' c -'
,C~I
-
lm+l
m=O
q-1 m=O
-
m C
kkTk)frm'
k= 0
m
'
1 R( ?,)Tkfr m1 m+l-k
k=O
= S0(X)f whence
which, by (4.5), immediately leads to
I f we integrate both sides of the preceding equation along
rj
we
obtain
f r o m the Definition ( 3 . 6 ) .
that all
d . 2 1. J
We assume without loss o f generality
We estimate
c 1 p+l-k d
5 - -
-
for all
0 2 k 2 m-p,
p z m
j
s q+p-1 whence the assertion of the
226
REINHARD MENNICKEN
expansion theorem is proved. We would like to add the remark that according to (3.11) the eigenvectors and the associated vectors fulfill the conditions and thus the boundary conditions (4.2). conditions
(4.2) are also necessary:
frjl
(j = 0 , 1 , 2 ,. . . , q )
then
f
(4.4)
Therefore, in a sense, the
if we know that all functions
are expandable into a series of type
has to fulfill the conditions (4.2).
(4.3)
This fact is well-
known for functions with (pointwise convergent) Fourier expansions: they have to satisfy the same periodicity properties as the sine and cosine functions.
It is not difficult and therefore left to the reader to state
(4.3) is absolutely convergent
conditions under which the series
(in parantheses) which means that the sums
are bounded if
j
tends to infinity.
We would like to point out that, on the contrary to almost all other authors, we do not require that all eigenvalues with the exception of only finitely many are normal,
5 . APPLICATIONS TO DIFFERENTIAL EQUATIONS If
s E Z
and
u
is an open subset of
denotes the Sobolev space of order viated by
Hs.
If Ns
s E IN
s
over
:= ( U E Hs
: supp u
c
Hs[a,b]
:= Hs/Ns
IR\(a,b))
;
then
Hs(R)
we set
and define
(5.1)
w;
IR
Hs(w)
is abbre-
SPECTRAL THEORY FOR OPERATOR POLYNOMIALS
the corresponding quotient mapping is denoted by to show that
s E N
:
E IIs
UJ
U=W
we define := (v E I I m S : supp v
IIc [a,b]
(5.2)
It is easy
can be identified with the vector space
lIs[a,b]
(u E L2[a,b] Again for
‘p,.
227
-S
Obvious ly Lm[a,b]
(5.7)
c L2[a,b]
= Hi[a,b]
C
F o r later use we state
(Hs/Ns)’ = HC [a,b].
(5.4)
-S
It is well-known that the dual space o f the quotient space I (Hs’H:s) , i.e. the orthogonal complement Hs/Ns is isomorphic to N S of
Ns
with respect to the dual pair
(Hs,H-s). Therefore we only
have to prove that
The inclusion ” C ” immediately follows from the definition o f the (R\[ a,b]
support of a distribution because
C :
the proof o f the inverse inclusion
”1”:
Let
for
i
v E H : s [
E (1,2,
a,b]
...,s ]
.
Since
v E H-s
)
C
Ns.
We sketch
we can choose
vi E L2
such that v =
c
,(i)*
i=O
It is not difficult to show that the functions even f r o m
L2[a,b].
Thus, if
i=O
vi
can be chosen
u E Ns,
i=O
where the derivatives are taken in the sense of distribution theory.
REINHARD MENNICKEN
228
We would like to point out that in this paper
is
H:s[a,b]
understood to be the Banach space dual, i.e. the space of continuous linear functionals, and not to be the Hilbert space dual, i.e. the space of continuous conjugate linear functionals.
Consequently we
have e.g.
for
u,v E L2. If
is a vector space,
G
nxn-matrices with elements in
Mn(G)
G.
n E w\[O},
In the following
m E N,
a = a < a2
TR(X)y
(Y E H?Ca,bI)
from which the assertion ii) is immediate. iii)
Let
X E
be fixed,
Q:
The relationship
T(X) E P(H?[a,bl,L:Ca,b1xCn)
(5.11)
remains to be proved.
We have codim Im(TR(X))
1
with the period
s
of a func-
in the form
1
and
an(.)
= (an-ao)cos 2nrrx
bn(x)
=
(an-a )sin Pnnx
+ p,,sin
-
Znnx,
pncos 2nnx,
@ n denoting the corresponding Fourier coefficients of f. If we apply the most effective of the above-mentioned summation methods, namely a suitable extension of the so-called Mittag-Leffler summation* (due to M. Riesz) to the series in
-mined
(4), then it can be deter-
completely the characterizing Mittag-Leffler star for both of
these series, so that we get an explicit expression of
fC
s ] (XI 9
holding everywhere in the common part of the mentioned domains. This means that the method yields about the maximal infornation on the holomorphy domain and the singularities of as a function of
s
f[s] (x)
which can be hoped in full generality,
By the
way, we have the integral representation:
(6)
fcs,(X)
=
f(x-6) [s,(t)
-
,S,(x)ldt
(Re
5
where
*This
process is defined for an arbitrary series m
(ML)
C
m= 0
m
um = lim 6++0
C m=O
r(1+6m)''
urn.
Cum by
> 1)
INTEGRO-DIFFEmNTIAL OPERATORS AND THEORY OF SUMMATION
249
Thus the result depends also upon the properties of the generalized (Hurwitz) zeta-function
C(s,u),
defined by analytic continuation
m
C (u+m)-'. m= 0 tion ( 7 ) the formula:
of the series
It holds namely for the kernel func-
The other side of the exposed connexion will be discussed more in detail, i.e. that certain function series can be handled very effectively by means of integro-differential operators.
It is
about a qew summation method which was introduced in its simplest form in the works [4]-[6]
of the author and has been developed
further since the sixties in several directions, e . g . in [9]-[11]. Let
'pn(x)
(n=1,2,...)
a sequence o f functions bounded and
Lebesgue integrable in an interval
c
series
: I (pn
(xo,xl)
converges at a point
x E
and suppose that the (xo,xl)
for any v > 0.
0
Then the limit
(9) is called the (W)-= existence of ( 9 ) ,
of the series
C cpn
C (pn;
and in case of the
is said to be (W)-summable at
x.
This (W)-method leads to sharp results e.g. for trigonometric series and ordinary Dirichlet series. x
=
In particular, putting
and using certain properties of the Hurwitz zeta-function,
-m
a simple
-
necessary and sufficient
-
summability condition for
trigonometric Fourier series can be deduced.
Moreover we find that
the local "strength" of the (W)-method is beyond that of any classical summation process. By formal grounds, it is reasonable for these applications to define two variants of the method: We say that the series
M.
250
. a
is
+
MIKOLAS
C (an cos nx + bn sin nx) n=1
(W-)-summable at a point
(W+)-or
x,
if there exists,a > 0
being chosen sufficiently small, the limit for
8
-t
+O
of the sum
of the series
or
co
. a respectively.
+
n-9[an cos(nx
n=1
T8 - Tr) +
bn sin(nx
- Trrs) ]
,
These limits are called the (W+)- resp. (W-)-sum of
(10)
111. MAIN RESULTS
On the above mentioned lines, the following theorems can be obtained.
--1. f
The trigonometric Fourier series of a bounded function
at a point
x
is (W+)-summable if and only if the limit 5 f(x+O)
(11)
exists, where
6
=
is an arbitrary positive number (but a fixed one).
does not depend on
f(x+O)
f(x+t)t'-ldt]
6
and in case of its existence, it
yields also the (W+)-sum o f the Fourier series.
-2 .
We have especially
f(x+O)
= f(x+O)
the function has a limit from the right.
at every point where
Furthermore, the (W+)-sum-
mability holds uniformly in each closed interval where tinuous.
--3 .
f
is con-
(Two-side continuity at the end-points being assumed.) Analogous statements hold also for the (W-)-method.
have only to put
x-0
instead o f
x+O.
We
INTEGRO-DIFFERENTIAL OPERATORS AND THEORY OF SUMMATION
4. -
There exist such trigonometrical series which are not
summable neither by any ( C , r ) -
-
( W i ) summable 5. -
251
nor by the (A)-method, yet are
.
In case of a related summation method, defined for (10)
m
. a
+
all our theorems are valid with of
f(xit)
cos nx + bn sin nx),
C n-'(an W + O n=l
lim
rpx(t)
= 1 [f(x+t)+f(x-t)]
instead
and with
[
b
(13)
f((x))
instead of tion
f
f(xf0).
= '++o lim
['
rpx(t)tO-ldtl
In addition, we obtain for any bounded func-
the result, that the set of points at which
f((x))
exists
is wider than that of the points where the so-called Lebesgue condition holds i.e.
the limit /
lim
exists.
x+e
-
f(t)dt
As it is well-known, this last one is the most general
summability condition of practical use. We remark that these results are based essentially on the connection between the closed form of the series occuring in the definition of the (W,)-method
and the
Hurwitz zeta-function.
By
certain propositions of Tauberian type it can be shown still:
the
effectiveness of the (W*)-methods in case of bounded functions exceeds not only the effectiveness of the Abel-Poisson method but also that of a more general class of processes, namely the so-called Abel-Cartwright methods. The latters are defined for an arbitrary series
c
Un
by
252
M. MIKOLAS
where
is a fixed positive number.
q
Recent investigations indicate wide application possibili-
ties to boundary
asymptotics of power series in Hadamardls sense,
namely concerning improvement and localization of the results in question.
IV. PROOF OF A TYPICAL SUMMATION THEOREM For illustration, we will consider the above-mentioned theorem
2. -
which can be formulated in detail as follows:
The trigonometric Fourier series of a bounded function is summable in the sense (12) at a point
x
if and only if the
limit (13) (with an arbitrarily small but fixed
In particular we have
f( (x))
f
6 > 0)
= [ f (x+O)+f (x-0)] /2
exists.
whenever f (x*
exist, and the summability is uniform in each closed continuity interval of the function.
The domain of effectiveness of our sum-
mation process is greater than that of any CesAro method o r of the Abel-Poisson summation. PROOF,
We start with some elementary lemmas which can be deduced
easily from the classical theory of the Hurewitz zeta-function Ccf.
(811 *
LEMMA 1.
0 then
n
i.e., are such {yn]
is
For o u r study of Eq. (2.1) we need introduce a multivalued
bounded mapping
G: X -+ K(Y)
such that
0 $f G x
for
)IxII
large
and which satisfies the following conditions: (2.2)
F o r each large
large
n,
where
r > 0, deg(WnGVn,Bn(O,r),O) Bn = Vil(B(0,r))
#
0
for each
and the degree is that
.
defined in [ 17,181
(2.4)
There exists
Kn: Vn(En)
-I
K:
2
for each
(WnY,u) = (Y,v) f o r each u E Knx,
n
v E Kx,
such that
x E VnEn, y E Y.
262
MILOJEVIC
P.S.
Y = X
When
Y = Y"
or
there are natural choices f o r
t h a t s a t i s f y (2.2)-(2.4).
Y = X
If
i s a lll-Banach
G = I,
Kn = K I X n
p i n g , and
x E Xn,
each
K = J: X
we
n z 1.
(2.2)-(2.4)
X
-t
possible t o find a
mapping. if
X
K:
K = JG,
J: Y
r
Then (2.4) h o l d s f o r
Q ~ K X= K X
x E Xn.
for
"'2
Let
J:
a semi-inner (=min ( @ ( x )
X
G:
-t
i t i s always
Y
= {Xn,Vn,Yn,Qn)
with K n = K'Xn I n applications i t i s often possible
K, K n l G
r
and a scheme
2.2)-(2.4). X
"'2
b e t h e n o r m a l i z e d d u a l i t y mapping a n d d e f i n e
product
I
Kn = I l x n we s e e
and
i s t h e normalized d u a l i t y
-I
t o c o n s t r u c t o h e r t y p e s of mappings which s a t i s f y
and
s u c h t h a t (2.3) h o l d s ; f o r e x a m p l e ,
"'2
-t
where
K = I
, " ' 2
for
JX
c o m p l e t e scheme f o r
F o r a given
t h a t t h e y s a t i s f y (2.2)-(2.4).
IIPn(I = 1,
P ~ J Xc
since
Y = X*
i s reflexive,
G = J: X
X,
t h e n o r m a l i z e d d u a l i t y map-
i s a projectionally
then taking
we c a n t a k e
, * ' 2
have
If
T o = {XnlPn;R(P:),P:] (X,X*),
-t
Kn
s p a c e and
T o = {XnlPn) i s a p r o j e c t i o n a l l y c o m p l e t e scheme f o r t h e n choosing
G, K,
@
( a , * ) - :
XXX
-t
R
(x,y)- = inf{@(x)l@EJy}
by
We a r e i n a p r o p o s i t i o n t o g i v e o u r b a s i c
E Jy)).
approximation-solvability
r e s u l t f o r Eq.
(2.1) w i t h
T
being
A - p r o p e r which was announced f i r s t i n [ 281.
T H E O R E M 2.1.
T: X
Let
+
-$
'2
G,T
Suppose t h a t
(2.2)-(2.4) h o l d a n d t h a t f o r e a c h
e x i s t s an
Then E q .
rf > 0
A - p r o p e r and A - c l o s e d
w.r.t.
bounded a n d
pG
K = JG
be K - q u a s i b o u n d e d ,
f
r in
and
G
1
2
for
Y
0.
there
such t h a t
(2.1) i s f e e b l y a p p r o x i m a t i o n - s o l v a b l e
(i.e,,
there exists a solution
large
n
and a s u b s e q u e n c e
V
un E E n "k
u
"k
-t
x
of with
f o r each
Wnf
E WnTVnu
f E Tx).
f
in
Y
f o r each
263
A P P R O X I MAT1ON- S O LVABI LITY
PROOF.
Let
e r n e s s of
Y
in
f
and ( 2 . 5 )
T
I t i s e a s y t o s e e t h a t t h e A-prop-
be f i x e d .
imply t h a t t h e r e e x i s t s a n
n
1
2
such
that
Therefore,
deg(lJnTVn-\Jnf,
Bn,
0 ) = deg(WnTVn, B n ,
0)
To show t h a t t h i s d e g r e e i s n o n z e r o , we d e f i n e on H n ( t , u ) = tldnTVnu
homotopy
n znl
each
w i t h some
n1
+ (1-t)WnGVnu.
E $Bn
u
and
a
tk E r O , l ] ,
f o r each
tk + t0
k.
)
Suppose
(0,l)
and l e t
+ (1-tk)W
tkWnkvk
vk w
E
u and wk E GV u nk nk "k "k = 0, k 2 1. Then f o r e a c h TV
nk
k
w
(l-t0)tilw
+
v nk
( 'nkUnk )
= rf.
u E Tx
3
+ (l-to)t;l w
w
-
1
( l - t k ) t ; ]wnwk
+
and A - c l o s e d n e s s
of
{Vmum)+ x
with
0 €
o
k +
as
T
+
toTx
+
pG,
v E Gx
1
(l-to)ti,
then
a r e such t h a t
+
tou
T h e r e f o r e , i t remains t o c o n s i d e r t h e c a s e
Then wnkVk = e r n e s s of T,
- ( lmtk)
ti
'nkWk
+ 0
some s u b s e q u e n c e
2
U
and if
and
X(v,v)- = 0 .
to = 0 , l .
k + m
as
-
some s u b -
since,
(1-to)v = 0
))uI/IIvII + ( u , v ) - = XIIvll
1
=
a,
(1-to)Gx
This leads t o a contradiction with (2.6)
and
w nk
nk
a n d , by t h e A - p r o p e r n e s s sequence
w
= -(l-tk)t;l
w nk
C ( l - t o ) t-1 o
=
=
)
0 C Hnk(tk,u
such t h a t
to E
be s u c h t h a t
X
gn
nk
first that
IIxll
X
[O,l]
no.
2
fixed
nk
large
n
W e n e e d show t h a t for
( 2 . 7 ) d i d n o t h o l d , t h e n t h e r e would e x i s t
If
for
t o = 1.
Let
a n d , by t h e A-prop+ x
with
0
E
Tx
vnk(i) "k(i) and
IIxII = r f ,
i n contradiction with ( 2 . 5 ) . and
Then f o r
xk E KV
and ( 2 . 4 )
and n o t h i n g t h a t
yk
E
K
nkUnk
tk f 0
V u nk nk "k
Finally,
let
to = 0.
we o b t a i n u s i n g ( 2 . 3 )
f o r i n f i n i t e l y many
k
264
P.
which i m p l i e s t h a t
= ( 1-tk)
T h i s completes t h e proof
of
ri 2 n1
Wnf E
tha
osed,
A-C
(b)
WnTVnun
(a)
tkll Wnkwk((
n
for
(2.7).
2
V
nl
.
u k n nk
(2.5)
Conditions
Analysing t h e proof
of T.
a contradiction.
-t a ,
deg(WnTVn
-
T
Since -Ix
u n E Bn
i s A-proper
and ( 2 . 6 )
of Theorem 2 . 1 ,
0
such
and
0
f € Tx.
with
#
Wnf,Bn,O)
and c o n s e q u e n t l y t h e r e e x i s t s a
some s u b s e q u e n c e
REMARK 2 . 1 .
MILOJEVI~
(2.7) t h a t
Now, i t f o l l o w s f r o m f o r each
.
i s bounded by t h e K - q u a s i b o u n d e d n e s s
{v,}
-1
(1 Wnkvk((
Hence,
s
a r e i m p l i e d by
we s e e t h a t c o n d i t i o n
( 2 . 6 ) c a n be r e p l a c e d by (2.9)
f E Y
For e a c h
x
L e t u s now e x t e n d Theorem 2 . 1 t o t h e c a s e when
T
Whenever and
{x,)
f E Y,
THEOREM 2 . 2 .
C X
for
x E aB(o,rf)
i s bounded and
then there e x i s t s an
0
0
$
or s a t i s f i e s t h e f o l l o w i n g c o n d i t i o n :
s t r o n g l y A-closed
#
such t h a t and
~ ( x n) X G ( X ) =
(*)
rf > 0
t h e r e e x i s t s an
T
of
T
r e p l a c e d by e i t h e r
or t h e s t r o n g A - c l o s e d n e s s
l a r g e and e a c h
f o r each l a r g e
remains v a l i d i f
and A - c l o s e d n e s s
> 0
n. Then T(X) = Y .
(2.6)
of
T
and t h a t
s m a l l , deg(pWnGVn,Bn(Qr),O) Moreover, t h e c o n c l u s i o n
i s r e p l a c e d by ( 2 . 9 ) .
#
265
APPROXIMATION-SOLVABILITY
PROOF.
Let
po > 0
s u c h t h a t for e a c h
p E
Let
in
f
(O,yo)
-
Wnf,Bn,O) = deg(l\rnTVn
d e f i n e on
x fin
[O,l]
+
and
nl(pl)
exist
s n2(p2)
t k E [O,l],
for e a c h
Let
k.
t k l J n k v k + pWnkwk
W
v
+
z no
H
E aB
"k
u nk nk
and by t h e A - p r o p e r n e s s 0 E
toTx
+ pGx
i 1,.
-1 = p(to
of
and
T
.t
-
yG
no.
;r
+
0
-1
tk )Wn
-
Next,
FWnGVnU
-
such t h a t f o r each
If n o t ,
u
Hpnk(tk,unk )
be s u c h t h a t
nk nk
< to
then there
0 E
such t h a t
nk
Suppose f i r s t t h a t
-1 p t o WnkwX
n
( t , u ) = tWnTVnu
wk E GV
and
"k
with
Pn
for
deg(WnTVn+pWnGVn
f o r each
p 2 < p1
t k -+ t o , u
= 0.
Hence,
and A - c l o s e d ,
+ pWnGVnu
WnTVnu
nl = n l ( p ) z no
whenever
vk E TV
d
.
i-(WnGVn,Bn,O)
a hornotopy
i s A-proper
pG
tWnf
n
We claim t h a t t h e r e e x i s t s a n
and
+
T
Since
t € [ 0,1]
there exists a
(O,po)
z 1 such t h a t
n
u 5 a B n ( 0 ,rf),
E
i-(
be f i x e d .
there e x i s t s an
By condition (2.5)
be f i x e d .
Y
1.
I
wk + 0
as
Then
k
-t
m
k some s u b s e q u e n c e
I(xI/ = r f .
V
+ x nk(i)unk(i)
A s i n Theorem 2 . 1 ,
this
leads
t o a contradiction with (2.6).
t o = 0.
Suppose now t h a t many
k,
we h a v e t h a t
xk E KV
any
1J v nk
Since
-1
= -btk T
for i n f i n i t e l y
f o r such
and f o r
k,
u nk nk
-1 (vkSxk) = ( W n vkSYk) = - p t k k
Since
o
tk f
-1 = -btk Wnkwk
\Iwkl( ((xkll +
i s K-quasibounded,
-1
= -btk -a
we g e t t h a t
as
k +
{vk]
(Wk'xk)
=
-. i s bounded and con-
MILOJEVI~
P.S.
266
-1
ness of
( / W v 1) = ptk llWnkwkl1 -+ m as k -+ m by the A-propernk G. This contradicts the boundedness of {v,] and there-
fore
#
sequently
to
0. Hence, ( 2 . 1 0 )
is valid.
Now, it follows from ( 2 . 1 0 )
= deg(kWnGVn,Bn,O) f E Tx
+ pGx
)Ik -+ 0
and
for each
is solvable in xk E n(O,rf)
dition (*) we obtain an
n
e
g(O,rf).
e
( 0 , ~ ~be )
such that
fixed, condition
X
T
theref ore
deg(WnkTVn
k
un nk k
be such that as
k -+
a,
+
WnkVk
holds for each
-
W
f,B
nk
"k'
0)
#
n
2
nk
and
0.
lkWnkwk = W x E X
-
f. Since ))Wnkvk Wnkfll -+ 0 nk such that f E T x by the strong
0
T.
(2.1) with
ing ( 2 . 5 ) - ( 2 . 6 )
H
k
u
there exists
A-closedness of Eq.
'
F o r each
decreasingly.
E B such that W f E W T V u + nk nk nk nk nk nk for each k . Let vk E T V u and wk E GV u nk nk nk nk
Hence, there exists a
+ pkWnkGV
and using con-
is strongly A-closed and let
kn k W GV nk nk
+
with
f € Tx.
such that
p k -+ 0
for
(2.10)
pk E ( O , k o )
Taking
f E Txk + bkGxk
Let us now suppose that pk
deg(WnTVn+lWnGV,-W,f,BnsO)
n l , and therefore, the equation
2
with x
that
T
A-proper o r strongly A-closed and satisfy-
or (2.8) has been earlier studied by the author
under other additional conditions on
T.
So,
i n [ 2 0 ] have announced
the following results (see also Note added i n proof). THEOREM 2 . 3 .
(a)
Eq.
(2.1)
T: X
Let
H(t,x)
closed homotopy on T(X) = Y
K = JG
2',
and
G
be bounded and
is feebly approximation-solvable f o r each
if, in addition,
(b)
3
[O,l]
= tTx
+ (1-t)Gx
f
in
Y
is a n A-proper and A-
x X.
if, in addition,
e r and A-closed homotopy at
0
HW(t,x)
on
= tTx
+ (IGX is a n A-prop-
[O,l]xX\B(O,R)
f o r some large
APPROXIMATION-SOLVABILITY
R
p E
and a l l
(0,p)
6 > 0,
f o r some
T
267
e i t h e r s a t i s f i e s con-
( * ) or i s s t r o n g l y A - c l o s e d a n d f o r e a c h
dition
f
deg(pWnGVn,Bn(O,r),O)
0
r > R
and p
e
(O,p),
r.
f o r each large
The f o l l o w i n g s p e c i a l c a s e i s u s e f u l i n a p p l i c a t i o n s
(C
20-221
)
COROLLARY 2 . 1 .
y
(p > 0 ,
2 0
for
If
u E Tx,
H(t,x)
X
and
resp.) v E
Kx
(Hp(t,x),
x
[O,l]
G
a r e A-proper
yG
and e i t h e r
a n d some
i n a n A-proper
t h e c o n c l u s i o n s of Theorem 2 . 3
(a)-(b)
c
> 0,
( u , v ) 2 -cllvli then
and A - c l o s e d homotopy on
R 2 Ro,
[O,l]XX\B(O,R),
and-A-closed f o r
i s bounded or
T
IlxII 2 R o
with
resp.)
( a t 0 on
+
T
resp.).
Therefore,
a r e v a l i d provided t h e o t h e r
i t s hypotheses hold. We n o t e t h a t i n t h e above r e s u l t s we h a v e n o t assumed ( 2 . 4 ) . D e t a i l e d p r o o f s of Theorem 2 . 3
and C o r o l l a r y 2 . 1 a n d t h e i r a p p l i c a -
t i o n s t o v a r i o u s s p e c i a l c l a s s e s of n o n l i n e a r m a p p i n g s a n d BVP f o 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 s c a n be f o u n d i n t h e a u t h o r ' s p a p e r s
[21,22]
( c f . also [28]).
C o r o l l a r y 2 . 1 we s e e t h a t
A n a l y s i n g t h e p r o o f s of Theorem 2 . 3 a n d
(2.6)
c a n be r e p l a c e d by ( 2 . 9 ) .
We h a v e
a l s o proven t h e r e t h e f o l l o w i n g PROPOSITION 2 . 1 .
for
v E Knu
Mn: E n -+ F n .
and
(2.3),
Suppose t h a t 0 f u E En
f 0
(Mnu,v) ' 0
a n d some l i n e a r i s o m o r p h i s m
Then, f o r e a c h
deg(pWnGVn,Bn(O,r),O)
( 2 . 4 ) h o l d and t h a t
r
for
and l a r g e
n
p > 0,
large.
The a b o v e r e s u l t s a r e a p p l i c a b l e t o s t u d y i n g p e r t u r b e d equations
f
(2.11) with
F
such t h a t
E
Tx
+ Fx
P
268
( u , v ) - 2 -allvll
(2.12)
.s . M I L O J E V I ~ u E F x , v E Gx, IIxIl
for
T
i s K-coercive,
next r e s u l t ,
We r e c a l l t h a t
If
if
(x,}
in
Y.
then (2.12)
([28]).
T: X
Let
( u , v ) - b c ( ~ ~ x I I ) ~ \ vfIoI r
u E Tx,
such t h a t
r
2
c(r)
as
m
-b
- c ( ~ ~ x ~ /f )o r~ ~u vE ~ F~x ,
Suppose t h a t
G,
i s A-proper and A-closed
> 0.
8,
Then, i f
+
T
c o n d i t i o n s ( + ) and
(*),
F:
-+
X
2'
x E X\B(O,R)
Let
f E Y
be g i v e n .
By c o n d i t i o n
rf
and
y > 0
such t h a t
I ( u + v - t f ) )2 y
((~11=
rf
t E [O,l].
and
0, E ( p o , l )
6 E
Therefore, f o r each
t E [O,l] T
+
BF
w e have t h a t s a t i s f i e s (2.5) u E Tx,
over, f o r each
B E (61,1)
Since
( O , , 1) ,
we obtain t h a t
and t h e r e f o r e
T + pF
x E E(O,rf)
(u+OVtW)- 2
s a t i s f i e s (2.6)
such t h a t Since
F
exists a
8,
-t
1
such t h a t and
E
v
v E Fx,
/\xll = r f .
11 x(I
Fx,
= rf
E
on
B(O,rf)
such t h a t
+
Fx.
0
More-
(fj,,l).
-
C(IIXII)llWII
and
BC(IlXli)IIWII
> 0,
f o r each
E (Bl,l) Let
and i.e.
\lxll = rf
with
f E Tx + BFx.
f E Tx
e
f o r a given
xk E f 3 ( 0 , r f )
such t h a t
(T+F)(X) = Y .
v E Fx,
i s bounded, i t f o l l o w s f r o m c o n d i t i o n
x E X
satisfies
u E Tx,
for
f o r each
w E Gx,
B E ( ~ ~ ~Hence, 1 ) . by Theorem 2 . 2 , exists
w i t h some
= Ilu+v-tf-(l-B)v// t y / 2 ,
BB(O,rf)
v € Fx,
F
pG
(+), there e x i s t s an
for
u E Tx,
IIu+Bv-tfll on
+
+
BF
i s bounded, t h e r e e x i s t s a
F
( l - ~ l ) l l v l \ 5 y/2
such t h a t
T
is surjective, i.e.
PROOF. t R
p E (po,l)
and
R > 0.
+
T
Rf
-t
(u,v)- 2
and some
i s K-quasibounded and
T + F
i.e.
b e bounded a n d
p > 0
f
and some c:R+
a r e a s i n Theorem 2 . 2 and
f o r each
BF
x E X,
and
I< = J G ,
be K - c o e r c i v e ,
v E Gx,
Kn
and
K
y n E Axn
f o r some
v EGx,
m ,
-t
(+)
s a t i s f i e s condition
f
2'
-b
c a n b e weaken a s i n t h e
2'
-t
y n -+
i s bounded whenever
THEOREM 2 . 4
X
A:
R and some a > 0 .
b
Bk
E
f E Txk
there
(p,,l) +
be
BkFxk.
(*) t h a t t h e r e
2 69
APPROXIMATION-SOLVABILITY
We c o n t i n u e our e x p o s i t i o n by d e r i v i n g s o l v a b i l i t y r e s u l t s f o r Eq
(2.1)
pings.
Recall that X + 2
K:
f o r some
i n v o l v i n g v a r i o u s s p e c i a l c l a s s e s o f n o n l i n e a r map-
z E K(x-y)
X
A:
Y"
if
and some
ro
I
allx/I
for
u
E
Kx
2
c l l x - y ~ ~f ~ or
c = 0,
If
0.
W e s h a l l a l w a y s assume t h a t and a g i v e n scheme
i s s a i d t o be c - s t r o n g l y K-monotone
(AX-AY,~)
>
c
Y
-t
i s c a l l e d K-monotone.
A
= { X ~ , P , ; Y ~ , Q ~f ~o r
>
and some
0.
When
f o r x E Xn
QEKx c Kx
i s such t h a t
K
y E X,
x,
and
Y = X"
(Y = X,
s u c h mappings a r e c a l l e d c - s t r o n g l y monotone ( c - s t r o n g l y with
K = J,
t h e normalized d u a l i t y mapping, r e s p . ) .
m e a s u r e of n o n - c o m p a c t n e s s by
= inf(r > 0
x(D)
-t
5 kX(Q)
f o r each
BK(Y)
Q
c D
EXAMPLE 2 . 1 and
F:
X
-t
(a)
of a bounded s u b s e t
D C X
Q
I
D C
c
i t i s ball-condesing i f
D;
f 0.
The b a l l i s defined
([19,25])
Let
X + Y
A:
k-ball-contractive
with
F
of p a r t
(u.s.c.)
upper semicontinuous
6 = supllQnII.
k6 < c ,
w . r . t . a p r o j e c t i o n a l l y c o m p l e t e scheme
+
x(Q)
be c - s t r o n g l y K-monotone
i s A-proper
where
x(T(Q))
0;
f o r some
X\B(O,R)
X
reflexive.
s a t i s f i e s e i t h e r one of c o n d i t i o n s ( 2 . 1 3 ) - ( 2 . 1 5 )
i s K-quasibounded
and s a t i s f i e s ( 2 . 1 6 ) .
Theorem 2 . 5 w i t h
F = 0
a r e s u l t of F , Browder [ 61. e n t i a l e q u a t i o n s , our p r o o f
Then
( T + F ) ( X ) = Y.
y i e l d s t h e f o l l o w i n g e x t e n s i o n of
Unlike h i s approach based on d i f f e r i s much s i m p l e r .
271
APPROXIMATION -SOLVABILITY
Let
COROLLARY 2.,?
be a n l - s p a c e
X
and
A:
+
X
i.e.
a c c r e t i v e and
X
-+
X = X**.
e i t h e r c o n t i n u o u s or d e m i c o n t i n u o u s w i t h m-accretive,
X
i s s u r j e c t ve € o r e a c h
A
A
Then ),
0.
7
The f o l l o w i n g r e s u l t i s u s e f u l i n a p p l i c a t i o n s
is
of Theorem
2.6 ( c f . c91). LEMMA 2 . 1
Let
be r e f l e x i v e ,
X
s t r i c t l y c o n v e x and condition
(*).
whenever
x
then
n
+ F
A
-
be m - a c c r e t i v e .
X -+ X
A:
have normal s t r u c t u r e ,
I f , i n addition, x
Axn -+ f
and
Then
and
X
satihfies
A
i s s t r o n g l y demiclosed,
A
then
Ax =
r,
and
X*
i.e.
i s compact,
F'
(*).
satisfies
L e t u s n o w d i s c u s s more g e n e r a l t h a n c - s t r o n g l y K-monotone mappings.
Let
Cb(D,Y)
d e n o t e t h e normed l i n e a r s p a c e w i t h t h e
supremum norm of a l l c o n t i n u o u s bounded f u n c t i o n s f r o m t h e t o p o l o g i c a l space
DEFINITION 2 . 2 a-stable for
([28]).
if
A mapping
U:
there exists a
-I B K ( Y )
x
XXX
ij
(Xt.1
-+
-+
BK(Y)
2
s
= (Xn,Pn;Yn,Qn}
such t h a t
Tx = U ( x , + ,
x E
(ii) F o r each
In particular,
U:
XXX
>
c
for some
x, 0
-t
i s a-stable
U(X,*
and e a c h l a r g e
T: X -+ B K ( Y )
BK(Y)
is
and
K:
&
U
> 0
As before,
and
GKx
X -+ ZY*
we assume t h a t C
into
Kx,
x
E
w.r.t.
To,
cJIT,Y) i.e.,
n
c - s t r o n g l y K-monotone
( i ) of D e f i n i t i o n 2 . 1 h o l d s and for e a c h l y K-monotone.
6
i s compact f r o m
D c X.
f o r e a c h bounded s u b s e t
and some
i s s a i d t o be
and
( i ) t h e mapp n g
exists a
T: X
Y.
a p r o j e c t i o n a l l y c o m p l e t e scheme
w.r.t.
(X,Y)
x E X,
i n t o t h e normed l i n e a r s p a c e
U
Xn.
such t h a t x E X,
there
Tx = U ( x , x ) ,
U(x,.)
)IuII < U//xll
if
for
i s c-strong-
u E Kx
It i s c l e a r t h a t s u c h map-
P.S. MILOJEVI~
272
of semi K-monotone m a p p i n g s .
f 25,281
To.
w.r.t.
p i n g s a r e semi a - s t a b l e
c = 0,
If
we h a v e t h e c l a s s
F o r s u c h mappings we h a v e p r o v e n i n
t h e f o l l o w i n g r e s u l t whose p a r t ( a ) a n s w e r s p o s i t i v e l y t h e
q u e s t i o n r a i s e d by Browder
[ 7 ] and e x t e n d s some of h i s r e s u l t s ( c f .
c 6,71) . THEOREM 2.7
(a)
To.
w.r.t.
A-proper
If
( + ) and i s e i t h e r i s a l s o A-closed
T: X + B K ( Y )
then
I f , a l s o , QnT i s i n j e c t i v e i n X n ,
U.S.C.
or d e m i c o n t i n u o u s w i t h
and t h e e q u a t i o n
s o l v a b l e f o r each
i s semi a - s t a b l e ,
f
Y.
in
If
f
E
is
T satisfies
Y = Y**,
then i t
i s f e e b l y approximation-.
Tx
i s continuous a - s t a b l e
T
T
and
s i n g l e v a l u e d , i t i s a n A - p r o p e r and A - c l o s e d homeomorphism. (b)
If
i s a s i n ( a ) and
T
contractive with T + F
kb
i s A-proper
< c
F: X + BK(Y)
is
or b a l l - c o n d e n s i n g if
and A - c l o s e d
Now, i n v i e w of Theorem
w.r.t.
and k - b a l l
U.S.C.
b = c = 1,
then
r o o
2.7 ( b ) , o u r g e n e r a l r e s u l t s y i e l d
(r 281 ) THEOREM 2 . 8
The c o n c l u s i o n s of Theorem 2 . 5 r e m a i n v a l i d i f we
(a)
assume i n i t t h a t (b)
that
A
i s s e m i c - s t r o n g l y K-monotone w i t h
The c o n c l u s i o n of Theorem 2 . 6
r e m a i n s v a l i d i f we assume i n i t
i s s e m i K-monotone.
A
REMARK 2 . 2
I n Theorem 2 . 8
k-ball-contractive
with
( a ) one c a n assume t h a t
kb < c
(i.e.
t i s f y i n g ( i ) of D e f i n i t i o n 2 . 1 and f o r each
x),
x
+ Y
F
F ( x ) = U(x,x)
u(x,.)
or semi b a l l - c o n d e n s i n g i f
We n o t e t h a t ( i ) of D e f i n i t i o n 2 . 1 h o l d s i f
U( * ,x):
K = JG.
i s semi with
U
sa-
i s k-ball-contractive
6 = c = 1 X
( c f . [ 281).
i s r e f l e x i v e and
i s completely continuous uniformly f o r
x
in a
bounded s e t , I n [ 311
,
Pohogaev s t u d i e d a c l a s s of A - p r o p e r mappings T:X
* X"
APPROXIMATION-SOLVABILITY
such t h a t
0: X
R
-t
c: R+
(Tx-TY, x - y )
c(llx-yII)
2
-
c(0) = 0
i s continuous,
-I R +
x,y E X ,
~(x-Y),
i s weakly u p p e r s e m i c o n t i n u o u s a t
0
-t
where
@(O) = 0
and
0
r
and
273
and
whenever c ( r )
0.
-t
A s l i g h t l y more g e n e r a l c l a s s i s g i v e n by
PROPOSITION 2 . 2
q(r)
constant
Let 0
7
(2.17) Then
=
and
T
i s of t y p e I
0,
Let
PROOF.
x
there exists
7
and 0
with
x),
r,
2
-
q(r)c(llx-yll)
2
(i.e., -t
Ilx-yll
whenever
xn
Q(x-Y).
-
x
and
l i m sup
and i s t h e r e f o r e A-proper w . r . t .
l i m sup (Txn,xn-x)
such t h a t
(/xn-x/I z
r
2
0.
If
xn
f x,
then
n.
f o r i n f i n i t e l y many
n
(2.17) holds f o r such n l s l e a d i n g t o a c o n t r a d i c t i o n .
Hence,
some
To.
or
n -x r
xn
r > 0,
be s u c h t h a t for e a c h
X*
x-Y)
(S+)
then
(xn,vn;xn’v;l *
-t
x,y
(TX-TY,
(Txn,xn-x)
ra
T: X
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 s s t u d i e d r e c e n t l y by H e t z e r
[15],
i n a much more c o m p l i c a t e d way u s i n g g e n e r a l i z e d d e g r e e t h e o r y
o f Browder [ 6 ] , s a t i s f y ( 2 . 1 7 ) modulo a compact mapping. be a bounded r e g i o n .
L e t Q c Rn
We a r e i n t e r e s t e d i n a g e n e r a l i z e d s o l u t i o n
of
u E V
(2.18)
with Let
c
f E L2(Q),
sm = # ( a
> 0
and
I
where
lalsm].
m V c W2
om
i s a clo s ed subspace with
Suppose t h a t
aaB
E L ~ ( Q ) for
W2
la1 ,
c V.
l ~ 5:l
m,
P.
274
la1
For e a c h
(2.20)
5
(2.21)
E
1 E R,
.
MILOJEVI~
A,:
m,
dory c o n d t i o n s and and some
s
QxRSm
IAa(x,y)I
-t
R
6
L(yI
s a t i s f i e s t h e CarathBo-
a(.)
+
for
E
x
Q(a.e.),
L2(Q).
p: R'\(O]
There e x i s t s a f u n c t i o n
[O,c)
4
([15])
such t h a t
s -s for
x
E
Q(a.e.),
Iz-z'
I
2
r.
y
r > 0,
E Rsm-',
z,z'E
and
D e f i n e c o n t i n u o u s and bounded mappings
(N2u,v) =
and
C
(Aa(x,u,Du
l a I-
i s a w e l l known f a c t t h a t
A2
i n Hetzer [15]
t h a t , for e a c h
(N1~-N1vy u-v)
2
T(r)
-s(r)c(llu-vll)
,...,Dmu),Dav) and
+
A1
N1
i s A-proper
and A-closed
satisfies
- o(u-v)
ro.
veloped i n [ 24,26-28] In particular,
[27]
V
-t
It
I t was shown
E
w i t h IIu-vI/ 5 r ,
V
where q ( r ) = ( c 0 + 3 r ( s ( r ) ) ) / 4 ,
Since
T
@(u-v) =
i s applicable t o (2.18)
t h e r e s u l t of H e t z e r [ l 5 ]
+
A
5
+
A = A
s o l v a b i l i t y of
f o r a more g e n e r a l c a s e when
V
u , v € V.
for
u,v
( 2 . 1 7 ) and s o
holm of i n d e x z e r o , t h e t h e o r y of
(see
and
with
L2
a r e compact.
N2
r > 0,
w.r.t.
m-l
A1,A2,N1,N2:
= r / 4 1 ~ 1 ~ / ~c , a s i n P r o p o s i t i o n 2 . 1 and
Therefore,
R
Ax
A2
N1
+
Nx = f
de-
assuming ( 2 . 2 0 ) , ( 2 . 2 1 ) .
i s d e d u c i b l e by t h i s t h e o r y A = A*).
When
N(A)
= (01,
w e have: THEOREM 2 . 9 (a)
If
Suppose t h a t T
(2.19)-(2.21)
s a t i s f i e s condition
hold.
( + ) and e i t h e r
=
A,(X,-Y)
S
= -A,(x,y)
for
x
E
Q(a.e.)
and
y
E
N2
i s Fred-
A2
+
+
R r n , la1
5
m,
or
T
sa-
APPROXIMATION -SOLVABILITY
t i s f i e s (2.9),
then Eq.
If
(b) s i o n of
N(A)
= {O}
i s feeby approximation-solvable
(2.18)
V
the v a r i a t i o n a l sense i n
f o r each
and
27 5
f E L2
.
i s s u f f i c i e n t l y small, t h e conclu-
2
(a) holds.
(c)
N(A)
If
the value
c
= (03,
and
T
PROOF.
Part
Remalk 2.1,
p
i s sufficiently small,
s a t i s f i e s condition
a generalized solution
u
E
(*),
A1
+
Now
A1+N1
+ a1
N1
Part
s a t i s f i e s (2.17)
for
a. > 0 .
r > 0
Hence,
+ a1
{Xn,Pn]
for
Theorems 2 . 1 - 2 . 4
V
is
f o r each i n [28]
a. >
(or Co-
127,281.
and s t r o n g l y A-closed mappings d i s c u s s e d
We c o n l c u d e t h e s e c t i o n w i t h a c o u p l e of a l s o C o r o l l a r y 3.1N in [ S O ] ) .
more a p p l i c a t i o n s p r o v e n i n [ 2 8 ] ( c f .
x
=
ro
x**,
= {x,,P,},
))P,J) = 1,
s: x
-+
c(x)
a g e n e r a l i z e d c o n t r a c t i o n ( i n t h e s e n s e of B e l l u c e a n d K i r k ) a n d C:
v
X
C(X)
-+
E Jx,
u.s.c.,
/ / x / / 2- R ,
0
and C o r o l l a r y 2 . 1 a r e a p p l i c a b l e t o many
o t h e r c l a s s e s of A - p r o p e r
Let
T
and
n
r o l l a r y 1 i n [24] ) .
THEOREM 2.10
Let us prove
for e a c h
V\B(O,r)
To =
w.r.t.
Theorem 2 . 1 and
1231.
a n d t h e c o n c l u s i o n now f o l l o w s from Theorem 4.5.2
in detail in
(2.18) has
( b ) f o l l o w s from t h e g e n e r a l i z e d
i s monotone on
A-proper and A-closed
t a k e s on a l s o
V.
(a) follows f r o m a r e s u l t i n [30,31],
respectively.
p
then Eq.
f i r s t F r e d h o l m t h e o r e m for A - p r o p e r mappings i n (c).
in
compact a n d c
> 0.
( u , v ) 2 -cllx))* f o r
Then t h e e q u a t i o n
f e e b l y approximation-solvable
f o r each
p i n g s we h a v e
([ 281 ) :
THEOREM 2 . 1 1
Let
X
be r e f l e x i v e ,
T: X
e i t h e r p s e u d o monotone a n d d e m i c l o s e d ,
x
-
Sx
-
Cx
is
f E X.
2.4.
The p r o o f f o l l o w s f r o m Theorem
f E
u E Cx,
F o r monotone l i k e map-
+ 2’”
quasibounded and
or g e n e r a l i z e d p s e u d o mono-
276
MILOJEVIC
P.S.
t o n e , or q u a s i - m o n o t o n e ,
and
F: X
-+
such t h a t
2"'
Ta = [ X , V n ; X n x, V z }
s t r o n g l y A-closed w . r . t .
T t F
(e.g.,
c o u l d be
F
c o m p l e t e l y c o n t i n u o u s , or, f o r t h e f i r s t two t y p e s o f
(u,x)
bounded and g e n e r a l i z e d pseudo monotone w i t h
f o r some c o n t i n u o u s Then, i f
T
+
DEFINITION 2 . 3 R > 0
v
E Ty,
and
where
c(R,r) 2 0
t
-+
0'
~
c : R + -+ R',
x
5
11 - 1 1 '
COROLLARY 2 . 2
Let
yz
~R , ~ ( u , - v~, x - y~) i s a norm on
i s continuous i n
f o r fixed
T
and s a t i s f y ( 2 . 5 ) - ( 2 . 9 ) .
( T + F ) ( X ) = X".
R
-~ c ( R~, l / x - y l l ' )
for
u
E
\l*ll
compact r e l a t i v e t o
X
r
and
and
c(R,tr)/t
Tx,
-+
0
and
as
r.
and
R
-c(IIxII)II xi1
i s of semibounded v a r i a t i o n i f f o r
T: X -+ BK(X") ~
quasi-
T,
or t h e sum of two s u c h m a p p i n g s ) .
satisfies (2.5)-(2.9),
F
3
is
be h e m i c o n t i n u o u s , Then
of semibounded v a r i a t i o n
T(X) = X".
Theorem 2 . 1 1 f o l l o w s f r o m Theorem 2 . 2 and e x t e n d s t h e known s u r -
It i n c l u d e s t h e s u r -
j e c t i v i t y r e s u l t s f o r monotone l i k e n x p p i n g s . j e c t i v i t y r e s u l t s of Brezis [ Z ]
f o r bounded c o e r c i v e pseudo mono-
t o n e mappings, of Browder and H e s s [ S ]
f o r g e n e r a l i z e d p s e u d o mono-
t o n e mappings, of H e s s [ 1 4 1 , C a l v e r t and Webb [ 101 and F i t z p a t r i c k
[ 131 f o r quasimonotone mappings, a n d of W i l l e [ 341 and Browder [ 51 f o r maximal monotone and bounded g e n e r a l i z e d pseudo monotone mappings t h a t s a t i s f y (2.8)
and
(Tx,x)
z -1IxII z R ,
l a r y 2 . 2 f o l l o w s from Theorem 2 . 1 1 s i n c e
respectively. T
i s pseudo monotone,
It extends the
d e m i c l o s e d and q u a s i b o u n d e d by a r e s u l t i n [ 2 9 ] . e a r l i e r r e s u l t s of Browder [ b ]
and D u b i n s k i i [ 121.
r e s u l t s a r e v a l i d f o r mappings between
X
Corol-
and
Y
B o t h of t h e s e under s u i t a b l e
r e s t r i c t i o n s on t h e s p a c e s . F i n a l l y w e s h a l l l o o k a t i n t e r t w i n e d p e r t u r b a t i o n s of mappings of semibounded v a r i a t i o n . Banach s p a c e s ,
X
Let
X
and
Xo
be s e p a r a b l e r e f l e x i v e
c o n t i n u o u s l y a n d d e n s i l y embedded i n
X
and l e t
-
APPROXIMATI ON SOLVABI LITY
I: X
the injection
xo
of
-t
( C 331 ) .
T: X
-t
X*
mappin5 i f
Tx = U ( x , x )
such t h a t
for
and
(i) F o r each
y E X,
U(y
(ii) F o r each
x E X,
U(.
R 7 0
(iii) F o r each
and
c : R+XR+
where
r > 0,
each
R
-t
R+
>
0.
x
+
X*
i s c o m p l e t e ~ yc o n t i n u o u s ;
x): X
-+
X"
i s hemicontinuous;
a ) :
IIXII
*
,x-Y)
(U(X,X)-U(Y,X
2
dition
(*),
COROLLARY 2 . 3
holds,
l i m c(r,tR)/t t+0+
(see
= 0
for
[ 3 3 ] ) and s a t i s f y con-
gives the following extension of t h e r e s u l t
.
C33l
T
R,
-C(R,Ilx-Yllo)
i s c o n t i n u o u s and
Theorem 2 . 2
*
IIYII
R,
S i n c e s u c h m a p p i n g s a r e p s e u d o monotone
if
the n o r m
i s c a l l e d a G:rding
X X X -+ X*
U:
t h e r e e x i s t s a mapping
Oden
1/'110
D e n o t e by
.
DEFINITION 2 . 4
x E X
he compact.
Xo
277
Let
T
be a G i r d i n g mappings and
(2.5) hold.
i s q u a s i b o u n d e d and e i t h e r o n e o f c o n d i t i o n s ( 2 . 6 )
Then,
and ( 2 . 9 )
= X*.
T(X)
We s h a l l now a p p l y C o r o l l a r y 2 . 3 t o f i n d i n g a g e n e r a l i z e d s o l u t i o n u E V
of
C
(2.22)
( - l ) l a lD a ( x , u , D u
,...,D m u )
= f,
f
E Lq(Q),
la I s m where
Q
c Rn
i s - a hounded domain w i t h t h e smooth b o u n d a r y a n d
i s c l o s e d s u b s p a c e of
with
WF(Q)
im C V P
and
p E
V
(-1,m).
Assume (2.23)
F o r each
la1
5
m,
Aa:
QXRSm
c o n d i t i o n s and t h e r e e x i s t that
-t
R
K > 0
s a t i s f i e s t h e CarathAodory and
k ( x ) E Lq(Q)
such
P.
278
IA,(x,y)l
s
. MILOJ-EVI~ +
S
k(x))
x E Q a.e.,
for
A s b e f o r e , a g e n e r a l i z e d form a s s o c i a t e d w i t h ( 2 . 2 2 ) T: V + V".
bounded and c o n t i n u o u s mappings
(wr,v) = (Qfvdx
that
s o l u t i o n s of
TU = w f
Y
D u = { (Dau)
induces a
E V"
wf
be such
Then f i n d i n g g e n e r a l i z e d
i s equivalent t o solving the operator equation
(2.22)
(2.24) Let
v E V.
f o r each
Let
y E RSm.
1
)a1 g m - 1 1 ,
,
U E V .
W e n e e d r e q u i r e t h e f o l l o w i n g con-
ditions. (2.25) c:
Let
R+XR+
Rf
+
(2.26)
v E V,
R > 0
i n e q u a l i t y i n (2.25)
a n d for e a c h
.$
0
if
c,(r)
v E V,
R > 0
I I w ~ ~ S ~ R, ~ t h e
s R,
and
c1
(2.27)
and ( 2 . 2 6 ) ,
lyil
+
lzil
each S
R,
(2.25)
The f o l l o w i n g a l g e b r a i c c o n d i t i o n s i m p l y
respectively.
There a r e c o n s t a n t
y E Rsm-l,
integral
0.
3
Some a l g e b r a i c c o n d i t i o n s t h a t i m p l y a s t r o n g e r v e r s i o n o f
(2.25)
and
b e a s i n D e f i n i t i o n 2 . 4 and s u p p o s e t h a t
holds with
c a n be f o u n d i n [ 1 2 1 .
+ 0,
k S m-1
we h a v e f o r some
c : R + x R + + R+
Let
f o r each
R
I
R
1"
i s weakly u p p e r - s e m i c o n t i n u o u s
c(R,*)
= 0 f o r each
Ilwllm,p
R,
b e c o n t i n u o u s and
be s u c h t h a t
a t 0 and c ( R , O ) IIullmrp
R+ + R+
cl:
R
>
0
and
i = 1,2,
c1
>
0,
(yi,zi) and
c
5:
0
s u c h t h a t for e a c h
E Rsm-l
x E Q
a.e.
x R
s -s
we h a v e
m-l
with
APPROXIMATION-SOLVABI LITY
(2.28)
y E R Sm-l ,
F o r each
z1 z2
279
Rsm-sm-l
x E Q
and
a.e.
we have
c
THEOREM 2.11
-
[Aa(x,y,zl)
la I =m
Aa(x,y,z2)](z1-z2) a a
Let (2.23) hold and
T
0.
2
satisfy condition(2.5).
Sup-
pose that either one of the following conditions holds:
T
(2.29)
is odd on
= -Aa(x,I)
la1
V\B(O,r) for
for some
E Q
x
r
> 0, i.e.
I
15
a.e., and all
Aa(x,-5)
=
large and
m.
(2.30)
h E V"
For each Tu
where
J: V
(a)
-t
V"
#
XJu
there exists an for
rh > 0
X
C
f
E H)
i s a l i n e a r d e n s i l y d e f i n e d c l o s e d map-
such t h a t i t s n u l l space
t h e range
(x E D(A),
H.
Let
= X
h a s f i n i t e dimension, l.
Here A1
= A
and
N:
is a
V -+ H
denotes t h e orthogonal
X i
restricted t o
V
n
R(A)
i s c l o s e d , i t i s c o n t i n u o u s by t h e c l o s e d 0
be s u c h t h a t
llA~l~lL l
( A x , x ) 2 -IIAxl(llxll 2 - ~ " l l A x ( ( ~ for
b e t h e supremum of a l l s u c h
C.
Then
a E
[O,m]
1 ; IIxII
for
x E V. and
Let
-
A P P R O X I M A T I O N S O LVAB ILITY
(3.1) w i t h
Equation R(A)
A;':
i s compact
R(A)
-+
N i r e n b e r g [ 31
,
X = H
and
a s above and s u c h t h a t
A
has
been s t u d i e d by B r k z i s and
f 11
B e r e s t y c k i and de F i g u e i r e d o
u n d e r v a r i o u s c o n d i t i o n s on t h e p e r t u r b a t i o n b i b l i o g r a p h y on t h e s e p r o b l e m s ) .
s t r o n g l y A-closed.
and many o t h e r s ( c f . [1,3] for t h e
N
Our r e s e a r c h h a s f e e n m o t i v a t e d
( 3 . 1 ) such t h a t
by [ 13 and d e a l s w i t h E q .
281
+
A
N
i s A-proper
or
U s i n g t h e d e g r e e t h e o r y f o r m u l t i v a l u e d map-
p i n g s i n s t e a d of t h e B r o u w e r f s d e g r e e , we s e e t h a t t h e r e s u l t s of t h i s section are also v a l i d
f o r m u l t i v a l u e d n o n l i n e a r i t i e s of
the
same t y p e . We b e g i n w i t h t h e f o l l o w i n g r e s u l t proven i n [ l ] w i t h and
R ( A ) -+ R ( A )
A;':
compact.
s e e s t h a t t h e compactness of
X = H
However, a n a l y s i n g i t s p r o o f one
i s n o t needed and we g i v e i t s
A;'
proof for t h e s a k e o f c o m p l e t e n e s s . LEMMA A;'
+
(Ai'x
3.1
Let
1
+ a-
x
> 0).
( A X ~ , X ~ =)
+
-a
2
+
x
E
and some
i.e.
and
(3.3) holds. x1 E R ( A ) . x1
f
setting
Then
= 0,
or
t h e con-
and t h e n ( 3 . 3 ) becomes
0
u =
Ax1
x = x o + x1
x1 = 0 ,
If
AX^,
we g e t
The s t r o n g m o n o t o n i c i t y of a-lu
R(A)
H.
in
Suppose t h a t
= 0. A;u '
for e a c h
a-lI)xIl
l l ~ ~ ~, l or, l
a-'u,u)
implies t h a t
2
+
be such t h a t
-1
and
(0,m)
be s t r o n g l y monotone ( i . e . ,
xo 6 N ( A )
clusion follows.
a E
be such t h a t
I f e q u a l i t y holds i n (3.2),
x E- V
u n i q u e l y , where
1 (A; u
(A;'
H
-+
N ( A ) CB N ( A + a I )
Let
PROOF.
cII
2
X,X)
c
E
V c X
R ( A ) -+ R ( A )
U-%:
constant
then
A:
+
a x l = 0.
A;'
+ 0
U'lI
P.
282
s. MILOJEVIC
Lemma 3.1 i n c l u d e s t h e c a s e when
REMARK 3 . 1 selfadjoint.
Assuming a d d i t i o n a l y i n Lemma
i s compact t h e n a s i n r l ]
X = H
and
A
3.1 t h a t
AY1:
R(A)
-a
one o b t a i n s t h a t
is -t
R(A)
i s a n e i g e n v a l u e of
A. Introduce i n
11 ' \ I O .
a new norm
V
Then t h r o u g h o u t t h e
s e c t i o n we s h a l l assume t h a t t h e n o n l i n e a r mapping quasibounded,
N: V
-I
H
is
i.e.
(3.4) We s a y t h a t
i t has the p r o p e r t i e s discussed
has Property I i f
A
a t t h e b e g i n i n g of t h e s e c t i o n and i f Let
ra
= (Xn,Vn;Yn,Qn)
ped w i t h t h e norm
3.1
THECREM
Let
II.IIO. A:
G
a(u,v)
Whenever
IIxnllo
i s compact.
and
N(A)
(V,H)
with
equip-
V
We a r e r e a d y now t o p r o v e v a r i o u s s u r -
(3.1).
V c X
s t r o n g l y monotone,
( i i i ) (2w,v)
onto
a n a d m i s s i b l e scheme f o r
j e c t i v i t y r e s u l t s f o r Eq.
+ a1
H
b e t h e o r t h o g o n a l p r o j e c t i o n of
Q
R ( A ) -+ R ( A )
A;':
-t
H
O u r first result is
have P r o p e r t y I ,
and
C,N:
V
-t
H
0 < a < m;
A;'
+
quasibounded and such t h a t
i n ( i )and ( i i )
a n d one of t h e i n e q u a l i t i e s
is strict.
(3.6)
uo
E
N(A),
-t
m ,
> o
A-closed
w.r.t.
xn
uo
u1
H
in
Nxn/\\xnllo -L v ,
(v,ul)
< a\\ul\12 i f
+
+
with
then u1
#
v
0
#
uul;
and
u1 = 0 .
if
Suppose t h a t
mr0++
and
u1 E N ( A + d I )
i n p a r t i c u l a r , t h i s i s s o when
(v,uo)
=
Un
A
and
ra
H ( t , x ) = Ax for
(V,H)
(1-t)Nx
tCx
and f o r each l a r g e
are A - p r o p e r and R,
-
283
APPROXIMATION SOLVABILITY
deg(QnH1,B(O,R) r l X n , O )
#
0
f o r each l a r g e
f e e b l y approximation-solvable PROOF.
f E H
Let
IIx[lo < R
that
f o r each
be f i x e d .
t E [O,l].
I f not,
such t h a t
t n -+ t o
Then t h e r e e x i s t s a n
/Ixn/(
Ex,]
R = R(f)
C V
such
and
and t n E [ O , l ]
and
m
-+
(3.1) i s
x E V
for some
t h e n t h e r e would e x i s t
,
Then E q .
f E H.
H(t,x) = (1-t)f
whenever
n.
0
(3.7)
H ( t n , x n ) = (1-tn)f
Set
un = xn/l1Xnllo.
and
uln E R ( A ) ,
u n = uon
Since
( 3 . 7 ) by
dividing
NXn + (l-tn)
AUln
f o r each
+
uln
IIxnlI
u n i q u e l y w i t h uon E N ( A ) we o b t a i n
0
wiC xn
+
tn
n o
n.
f
(l-tn)
1 3=
0,
or
(3.8)
+
(l-tn)Al
-1 f (I-Q)
= 0
n
(3.9)
Since
{Nxn/llxnll
t h e q u a s i b o u n d e d n e s s of
and
N
{Cx,/IIx,II
and
C,
v e r g e weakly t o
v
t i n u i t y of
we o b t a i n f r o m ( 3 . 8 )
u
In
-+
u1
AY1 in
H,
and
w,
u1 E R ( A ) ,
(3.10)
a r e bounded i n
H
by
we may assume t h a t t h e y c o n -
respectively,
and by t h e c o m p l e t e c o n -
passing t o the l i m i t t h a t
and
u1 + ( 1 - t o ) A y 1 ( I - Q ) v
or
o]
+
t o A ; 1( 1 - Q ) w
= 0,
P,
284
dim N ( A )
Now, s i n c e
2
+ to/lw/I
2
N(A+aI)
+ au) = 0 ,
-
Au = Aul = - a u l
and
(3.6).
0
be a such one.
N
and
(1-t)gfll 2 y
and a
for
Y >
( 3 . 5 ) - ( i ) and
and t h e r e f o r e and by ( 3 . 1 1 )
= -v
Therefore, our claim i s
a r e bounded on
C
no 2 1
t h a t t h e r e e x i s t an
-
u =
ato(w,u).
+ u1
u = u
Since
aB(0,R)
and
A
and
it i s easy t o see
t,
a r e A-proper and A-closed f o r each
II&,H(t,x)
-t
= 0.
W
(Au,Au
i n contraction t o
R
un
( 3 . 9 ) we o b t a i n
rIJvll 2 - a ( u , v ) I
we g e t t h a t
u1
v = uu1
H(t,x)
and s o
i t f o l l o w s f r o m ( 3 . 2 ) and (3.11) t h a t
N(A) @ N(A+aI)
Hence,
u
t h i s l e a d s t o a c o n t r a d i c t i o n i n v i e w of
0,
(ii). I f
-t
it follows t h a t
a n d by ( 3 . 1 0 )
toQw = 0 ,
(AU,AU
u on
Passing t o the l i m i t i n
(3.11) Using
. MILOJEVI~
w e may assume t h a t
m ,
H.
in
s
such t h a t f o r each n 2 no
0
x E aB(O,R)
n
Xn,
t E [O,l].
Con-
s e q u e n t l y , b y t h e homotopy t h e o r e m f o r t h e B r o u w e r l s d e g r e e we n 2 n
o b t a i n t h a t f o r each
Hence, t h e r e e x i s t s a that
&,Axn -+ x
x
+
and
xn E B ( 0 , R )
S N x n = Q,f, Ax
+
n
Xn
f o r each
and t h e r e f o r e ,
n 2 no
such
some s u b s e q u e n c e
Nx = f .
nk The f o l l o w i n g s p e c i a l c a s e i s s u i t a b l e i n many a p p l i c a t i o n s . COROLLARY
)IxIIo < R
such t h a t
0
= (1-t)f
+
un = uon
uln
Then, a s i n Theorem
= 0.
t
tn
= (1-tn)f
-I
to
and
x
3 . 1 we s e e t h a t
n
n.
(Axn,uo) = 0.
COROLLARY 3.2
on
+ a1
(V,(I*llo)
in
H
H(tn,xn) =
,Uo)
,
R
>
0
r7
exists.
i s compactly embedded i n
s t r o n g l y monotone and
ra
A-closed
E R(A).
3.1, w e o b t a i n
A-proper and A-closed w . r . t .
I n c a s e when
In
uln -+ 0
T h e r e f o r e , such a n
Suppose t h a t
f
u
and
and
Therefore
Suppose t h a t
A;"
=-
,
-+ u
and s a t i s f i e s ( 3 . 1 2 ) - ( 3 . 1 3 ) .
Nx
un = xn/lIxnIlo
Set
uon E N ( A )
u
-+
I/xnll 0
+ t n ( c x n , u o ) = ( 1-tn)( f
A s i n t h e c a s e of C o r o l l a r y
+
with
u o , we o b t a i n for e a c h
i n contradiction t o (3.13).
Ax
E V
Taking t h e i n n e r p r o d u c t of t h e e q u a t i o n
with
a E (o,m),
+ ( 1 - t ) N x + tCx =
Ax
I
Again, a r g u i n g by c o n t r a d i c t i o n ,
uniquely with
( l - t n )( N X n , U o ) since
H(t,x)
H(tn,xn) = ( l - t n ) f f o r each
such t h a t
and
.
t E [O,l]
for some
suppose t h a t t h e r e e x i s t
write
whenever
for
V + H
N: A
(V,H).
+
tN,
H,
quasibounded
t E [O,l], i s
Then t h e e q u a t i o n
i s f e e b l y aporxirnation-solvable.
Ho = A
+
N
i s n o t A-proper b u t j u s t s t r o n g l y
i n s t e a d , we have t h e f o l l o w i n g e x t e n s i o n s of t h e above
results.
THEOREM 3.3
Let
and A-closed w . r . t .
A
and
ra
H ( t , x ) = Ax for
(V,H)
+ ( 1 - t ) N x + tCx for e a c h
be A-proper
t E (0,1] and A
+
N
APPROXIMATION -SOLVABI LITY
N
also that
and
Suppose
Then
= H.
(a)
If a l l o t h e r c o n d i t i o n s of Theorem 3 . 1 h o l d ,
(b)
If a l l o t h e r c o n d i t i o n s of Theorem 3 . 2 h o l d , t h e e q u a t i o n
PROOF.
Let
f
3 . 1 and 3 . 2 ,
E E
Let
on
E
aB(0,R)
n
B(0,R)
ck E (0,l)
-
ek(Nxk
R = R(f) > 0
Since
condition
such t h a t
i s A-proper
H(t,*)
t E (0,1], t h e r e e x i s t s an
f (l-t)Qnf
for
x
such t h a t
Ax
c
be such t h a t
Cxk) + f
E
+
(l-e)Nxe
ek + 0
n
Q(O,R)
el < e2
whenever
H(ek,xk) = ( l - a k ) f .
such t h a t
Then, a s i n t h e p r o o f s of Theorems
and A-closed
n(c) 2 1
such
n(c)
2
2 n(e2)
"(El)
E
be g i v e n .
f o r each
QnH(t,x) and
H
be f i x e d .
(0,l)
R(A+N)
i s solvable.
t h e r e e x i s t s an
t h a t for e a c h
+
(*).
and s a t i s f y c o n d i t i o n
a r e bounded.
C
Ax + Nx = f
x E
Ta
w.r.t.
be s t r o n g l y A-closed
287
.
Therefore,
+
there e x i s t s a
E
d e c r e a s i n g l y and
Then
,1]
= (1-e)f.
cCx
xk
Let
E
B(0,R)
Axk + Nxk = ( l - e k ) f
by t h e boundedness of
(*) there e x i s t s a
t E
Xn,
x E G(O,R)
N
and
+
F i n a l l y , by
C.
Ax + Nx = f .
such t h a t
0 COROLLARY 3 . 3 0
0
such t h a t
ro
i s A-proper w . r . t .
(3.25)
,
> 0,
c
for
can be r e l a x e d t o
p: R+\{O]
-I
[O,c)
such t h a t
S'
r > 0,
and
z,z'E
R 2m w i t h
Iz-z'Izr.
P
292
A + t N r V + L2
Then
A + N
and, i f
A
+ N
Po
p
A
of
+
t N
can be found i n [ 2 4 , 2 7 , 2 8 ] .
3.5
we s e e t h a t Theorems
and
for
t E [O,l]
by a
t a k e s on a l s o t h e v a l u e
i s j u s t s t r o n g l y A-closed.
t h a t imply t h e A-properness of
MILO J E V I ~
i s A-proper w . r . t .
v a r i a n t of P r o p o s i t i o n 2 . 2 then
.s .
O t h e r c o n d i t i o n s on
c,
N
or t h e s t r o n g A-closedness I n view of t h i s d i s c u s s i o n ,
3.6 e x t e n d some r e s u l t s o f Dancer [ l l ]
and B e r e s t y c k i and F i g u e i r e d o [ 13 i n v o l v i n g n o n l i n e a r i t i e s t h a t depend o n l y on
x
and
u,
a r e a l s o c o n s t r u c t i v e when
and,
i n c o n t r a s t t o t h e i r s , our r e s u l t s
A+N
i s A-proper.
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Arch.
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Functional Analysis, Hu lo mo rp h y and Approximation Theory II, GI. Zapata ( e d . ) 0 Ekevier Science Publizlrers B. C< (Nurtlr-HuNmrd), 1984
297
HOLOMORPHIC FUNCTIONS ON HOLOMORPHIC INDUCTIVE LIMITS AND ON THE STRONG DUALS O F STRICT INDUCTIVE LIMITS
L u i z a Amdlia Moraes
INTRODUCTION
The c o n c e p t s of ho l o m o r p h i c a l l y b o r n o l o g i c a l ( h b o ) p h i c a l l y b a r r e l e d (hba
,
,
holomor-
h o l o m o r p h i c a l y i n f r a b a r r e l e d ( h i b ) and
h o l o m o r p h i c a l l y Mackey (hM) s p a c e s h a v e b e e n i n t r o d u c e d by B a r r o s o , Matos a n d N a c h b i n i n [ l ] .
I n t h i s note
we w i l l s u r v e y r e s u l t s con-
c e r n i n g t h i s h o l o m o r p h i c c l a s s i f i c a t i o n of t h e s p a c e s .
More e x p l i c -
i t l y , we w i l l b e c o n c e r n e d w i t h t h e f o l owing s i t u a t i o n s : be a i n d u c t i v e l i m i t o f
E
Let
l o c a l l y convex s p a c e s
Ei
i E I.
1.
2.
E'
is a
F i n d s u f f i c i e n t c o n d i t i o n s on
i E I
Ei
-
(a)
Ei
hbo
f o r every
i E
(b)
Ei
hba
f o r every
i E I
(c)
Ei
hib
f o r every
i E I
(d)
Ei
hM
f o r every
i E I - E
F i n d s u f f i c i e n t c o n d i t i o n s on hbo
I
E
hbo
=
E
hba
3
E
hib
Ei
a n d on
E
so that
hM i E
I
and
E
so that
space.
PRELIMINARIES r e v i e w o f what w i l l b e n e e d e d h e r e .
L e t u s make a b r i e f Unless s t a t e d otherwise, spaces,
U
E
and
F
d e n o t e complex l o c a l l y c o n v e x
i s a non v o i d open s u b s e t of
a l l m a p p i n g s of
U
into
F.
If
I
E
and
i s a s e t and
Fu
F
i s t h e s e t of is a seminord
298
L U I Z A AMALIA MORAES
s p a c e , we d e n o t e by mappings of
into
I
morphic mappings of of a l l mappings of
t h e seminormed s p a c e of a l l bounded
lm(I;F) F.
#(U;F) into
U
and
F;
into
U
i s t h e v e c t o r space of a l l holo-
F
H(U;F)
i s t h e v e c t o r space
which a r e h o l o m o r p h i c when c o n s i d 4
e r e d a s mappings of
i n t o a f i x e d completion
U
F
of
F.
We w i l l
s a y t h a t a mapping
f: U
-t
F
i s holomorphic i f f
#(U;F).
f: U
-t
F
i s a l g e b r a i c a l l y holomorphic ( e q u i -
A mapping
v a l e n t l y G-holomorphic) i f t h e r e s t r i c t i o n
flu
fl S
S
of
f o r every f i n i t e dimensional v e c t o r subspace where
c a r r i e s i t s n a t u r a l topology.
S
P,(%;F)
PHy(%;F),
and
k-homogeneous
o f a l l k-homogeneous p o l y n o m i a l s bounded s u b s e t s of polynomials of
aM(U;F)
E;
-+
P: E
meeting
E
F,
-t
F
-t
P(%;F),
t h e v e c t o r space
t h a t a r e bounded on
E
and t h e v e c t o r s p a c e o f a l l k-homogeneous
F
t h a t a r e c o n t i n u o u s on t h e compact s u b s e t s
(respectively:
#=(U;F))
s p a c e of a l l G-holomorphic mappings of e d on t h e compact s u b s e t s of
on t h e compact s u b s e t s o f
U
U
into
t h a t a r e bound-
F
( r e s p e c t i v e l y : t h a t a r e continuous F = 6,
When
U).
w i l l denote t h e v e c t o r
it i s not included
i n t h e n o t a t i o n f o r f u n c t i o n s p a c e s ; s o , we w i l l w r i t e P(%;C),
for
#(U)
l m ( I >f o r
#(U;C),
s e t of a l l c o n t i n u o u s seminorms on mapping every U
f: U
+ F
0 E SC(F);
$nto
F
E
p o f
the collection
p 6 CS(F).
z
A given
E
A
U
of mappings o f
=
p o x
A given
morphically b o r n o l o g i c a l space i f , f o r every
CS(E).
The
i s l o c a l l y bounded f o r
more g e n e r a l l y , a c o l l e c t i o n
i s l o c a l l y bounded f o r e v e r y
for
P(%)
and s o on.
lQ(I;6),
i s d e n o t e d by
i s amply bounded i f
i s amply bounded i f
#(U;F) = gM(U;F).
U,
t h e v e c t o r s p a c e of a l l
P: E
P: E
i s holomorphic,
We d e n o t e by
respectively,
continuous polynomials
belongs t o
f
E
and
f p o f
I
f E Z]
i s a holoF,
we h a v e
i s a holomorphically b a r r e l e d space
( r e s p e c t i v e l y : a holomorphically i n f r a b a r r e l e d space) i f f o r every U
and e v e r y
F,
we h a v e t h a t e a c h c o l l e c t i o n
bounded i f , and a l w a y s o n l y i f ,
Z
Z c H(U;F)
i s amply
i s bounded on e v e r y f i n i t e d i -
299
HOLOMORPHIC FUNCTIONS ON HOLOMORPHIC INDUCTIVE LIMITS
m e n s i o n a l compact s u b s e t of s u b s e t of
U).
f o r every
U
belongs t o
A given and e v e r y
H(U;F)
lo the
s e n t by
s e t s and by
lof
iff
E F,
(respectively:
on e v e r y compact
i s a h o l o m o r p h i c a l l y Mackey s p a c e i f f: U
we have t h a t e a c h mapping
$of E W(U)
$ E F’.
f o r every
of s c a l a r s .
A net
( ha
(Xa)a€A
la E A
of e l e m e n t s i n
-t
F
An i n d u c t i v e l i m i t o f ( E . ) 1
we have
SECTION 1:
f E #(U;F)
iff
E
is
c o n v e r g e s t o z e r o f o r any n e t by
iEI
i s s a i d t o b e a holomorphic i n d u c t i v e l i m i t i f f o r e v e r y U
We r e p r e -
t h e t o p o l o g y of u n i f o r m convergence on t h e f i n i t e
very strongly convergent i f
f:
F
-t
t o p o l o g y of u n i f o r m convergence on t h e compact sub-
d i m e n s i o n a l compact s u b s e t s .
(‘a )a€ A
U
fapi E #(Ui;F)
(Pi)iEI U, F ,
f o r every i E I.
STRONG DUALS O F STRICT INDUCTIVE LIMITS O F FmCHET-
MONTEL SPACES
B a r r o s o , Matos and Nachbin prove i n [ l ] t h a t S i l v a s p a c e s a r e holomorphically bornological spaces. of Dineen
r4]
W e i n f e r from t h e r e s u l t s
t h a t t h e s t r o n g d u a l s of Fr6chet-Monte1 s p a c e s (DFM-
s p a c e s ) a r e a l s o h o l o m o r p h i c a l l y b o r n o l o g i c a l s p a c e s and t h i s i m p r o v e s t h e r e s u l t of B a r r o s o , Matos and Nachbin a s e v e r y S i l v a s p a c e i s a DFM-space and t h e r e a r e DFM-spaces spaces.
t h a t a r e not S i l v a
Boland and Dineen g i v e i n [ 21 an example of a s t r i c t i n -
d u c t i v e l i m i t of Fr6chet-Monte1 s p a c e s (FM-spaces)
t h a t i s n o t ho-
l o m o r p h i c a l l y b o r n o l o g i c a l ( s e e P r o p o s i t i o n 1 4 ( a ) of going t o prove i n t h i s s e c t i o n t h a t i f
E
t 21 ) .
i s a s t r i c t inductive
l i m i t of FM-spaces and t h e r e e x i s t s a c o n t i n u o u s norm on E’
i s a holomorphically b o r n o l o g i c a l space.
Proposition quently Q‘
3 of [ 81 :
#=(U;F)
w e have now
= #(U;F))
U
E,
then
T h i s r e s u l t improves
HM(U;F) = # ( U ; F )
f o r every
We a r e
( a n d conse-
and for e v e r y
s a t i s f i e s o u r c o n d i t i o n s , we have i n p a r t i c u l a r
F.
As
XM(U;F) = # ( U ; F )
300
LUIZA A d L I A MORAES
U c i9’
f o r every of
f
and f o r e v e r y
F.
T h i s improves P r o p o s i t i o n 1 4
21.
The r e s u l t s c o n t a i n e d i n Lemmas 1 & 3 a r e well-known and we h a v e s t a t e d t h e n h e r e o n l y f o r t h e s a k e of c o m p l e t e n e s s . LEMMA 1. E‘
i s t h e s t r i c t i n d u c t i v e l i m i t of FM-spaces
E
If
PROOF.
T: E ‘ + F
Let
compact s u b s e t of E’
= 12m E n ,
“n f o r every
n E N,
nn: E’ -+ E L
i s the canonical surjection.
t h e r e e x i s t s a l i n e a r mapping Let
By h y p o t h e s i s ,
bounded on
Kn.
Eh
Tn
i s a DFM-space,
i s continuous.
So,
T
i s bounded on
E‘
such
such t h a t
and s o ,
K
So,
By Example
EA.
of
K
+ F
Tn: E’ n
be a compact s u b s e t of
Kn
K = nn(K).
So,
of [ 5 ] t h a t
We know f r o m P r o p o s i t i o n 2 . 8
of [ 5 ] , t h e r e e x i s t s a compact s u b s e t
2.10
E’
9
be a l i n e a r mapping t h a t i s bounded on e v e r y
E’.
where
T = Tnonn.
that
as
En
i s a bornological space.
Tn
is
i s bounded on e v e r y compact s u b s e t o f E h ; we i n f e r from C o r o l l a r y 11 of r k ]
T = T on n n
that
Tn
i s c o n t i n u o u s and t h i s p r o v e s t h a t
i s a bornological space.
DEFINITION 2. and
flX
LEMMA
n
3.
A set
U = 0
Let
E
X
imply
f: U + G
i s determining f o r F
if
X
n
U
$6
0.
I
be a Monte1 s p a c e s u c h t h a t
E’
i s bornological.
The f o l l o w i n g a r e e q u i v a l e n t : (a)
T h e r e e x i s t s a c o n t i n u o u s norm
p
on
= (cp E E : p(cp) 4 11 f o r e a c h P t h e r e e x i s t s a t l e a s t a p E SC(E) such t h a t (b)
If
f o r every of
V
f E HM(W),
where
W
There e x i s t s a s u b s e t
i n g for e v e r y
E E = (E’ ) ‘
.
p E SC(E),
then
Vo
i s determining P i s any convex b a l a n c e d open s u b s e t
E‘ ; t h i s i s t r u e f o r e v e r y c o n t i n u o u s norm (c)
E.
K
of
E’
p
such t h a t
on K
E.
i s determin-
301
HOLOMORPHIC FUNCTIONS ON HOLOMORPHIC INDUCTIVE LIMITS
PROOF.
(a)
(b):
=)
Let
p
i s t h e Minkowski f u n c t i o n a l o f E'
K = Vo P
i s Montel,
E.
be a c o n t i n u o u s norm on
V p = (rp E E
11.
: p(tp) sz
p
Since
a n d we c l a i m i t
E'
i s a compact s u b s e t of
Then
satisfies (b). f E HM(W)
We show t h a t if and
f7 W =
flK
0,
it is clear that = 0
fl
W
=
f
f Q
0.
i s convex b a l a n c e d o p e n i n E '
W
Since
a n d if
flK
n
W = 0
f
I
0
H e n c e , t o show
= E
(E')'
= V tc 1 E Q:
tc T E P M ( % ' ) ; p ( h T ) 1 1 %' X E C
3
for a l l e l e m e n t s of be s u c h t h a t
P/K
PM(%')
= 0,
form c o r r e s p o n d i n g t o
f a c t o r s ) a n d v a n i s h e s on
Lx:
E'+
d e f i n e d by
Q:
PM(n-%')
= L(z,y, ...,y)
every
y E E'
induction every
K x...x
Then
K, Ly:
and
Ly/K
= 0.
But t h e n i n p a r t i c u l a r
E'.
K
k E IN.
=)
X T E Voo = T
3
0.
i s determining
K
Let
P
E
P,(%')
K.
z)
and hence E'+
C
x E K.
Now f i x
...,
= E'xE'x
(E')n
...x E '
Then
i s a n e l e m e n t of Lx
I
0.
d e f i n e d by
Next l e t Ly(z) =
i s a l i n e a r f o r m t h a t i s c o n t i n u o u s on t h e bounded
E'
s u b s e t s of
E'
d e n o t e t h e symmetric n - l i n e a r
Lx(z) = L ( x , z ,
w h i c h v a n i s h e s on
be a r b i t r a r y .
as
From t h e p o l a r i z a t i o n f o r m u l a we s e e
(n
on
L
n = 1
p(T) = 0
3
k < n.
i s bounded on t h e bounded s u b s e t s of
L
y E E'
f o r every
and l e t
P.
E K
V x
1 p ( T ) 4 T~;TV X E C
3
for e v e r y
i s a c o n t i n u o u s norm on E :
p
n > 1 and s u p p o s e we h a v e shown t h a t
Now l e t
that
i t s u f f i c e s t o show PM(%')
T(x) = 0
so
V l K r 7 W =
then
I t i s c l e a r for
We p r o v e t h i s by i n d u c t i o n .
i s a b o r n o l o g i c a l s p a c e (Lemma 1) a n d T E
i s convex a n d b a l a n c e d ,
K
i s d e t e r m i n i n g f o r t h e e l e m e n t s of
K
n E N.
K
n E N.
f o r each
that
then
where
.
T h i s shows t h a t
By t h e i n d u c t i o n h y p o t h e s i s Ly(y) = L( y , . . . , y )
P
P
0
on
E'
(b)
3
(c):
obvious.
(c)
3
(a):
L e t K be a compact s u b s e t of
(a)
E'
5
= P(y) = 0
0
for
and t h e r e f o r e by
i s d e t e r m i n i n g f o r t h e e l e m e n t s of T h i s c o m p l e t e s t h e p r o v e of
Ly
3
PM(kE')
for
(b).
such t h a t i f
cp E E =
302
L U I Z A AMALIA MORAES
= (E')'
= 0,
cp/K
and
cp s 0.
then
W e claim t h a t
u n i t a r y b a l l of a c o n t i n u o u s norm on
E
cp
exists
=?
#
cp
hypothesis
#
a E d:
E
acp
such t h a t
0,
true, there
f o r e v e r y a E C.
KO
and c o n s e q u e n t l y
Contradiction.
0.
I f t h i s i s not
E.
vp/K = 0.
By
T h i s completes t h e prove of
(a).
4.
PROPOSITION If
cp
IIacplIK d 1 f o r e v e r y
So
(c)
= E,
(E')'
i s the
KO
Let
be a s t r i c t i n d u c t i v e l i m i t of FM-spaces En.
E
h a s a c o n t i n u o u s norm,
E
E'
i s a holomorphically b o r n o l o g i c a l
space. From P r o p o s i t i o n 54 of [ l ] i t i s enough t o show t h a t
PROOF.
= H(U)
%(U)
U c E'
f o r every
and
i s a holomorphically
E'
in-
f r a b a r r e l e d space.
then
E,
Since
WM(U) = # ( U ) :
Let u s show
1)
p
Lemma 3
i s t h e Minkoswki f u n c t i o n a l of K = Vo
i s Montel,
E'
convex b a l a n c e d open s u b s e t of f IK
n
W = 0
then
As
f e 0.
V = [ c p E E : p(cp) c
i s a compact s u b s e t of
i s determining f o r every
K
i s a c o n t i n u o u s norm on
p
If
= l i m EL
f
and f r o m
E'
where
WM(W)
t h a t i s , if
E', E'
E
f
11.
is a
W
E WM(W)
and
i s an open and compact
nn s u r j e c t i v e l i m i t ( s e e Example 2.10 [ 5 ] ) we may assume w i t h o u t l o s s U = T Tm- ~ ( W )
of g e n e r a l i t y t h a t Eh).
s u b s e t of
Now l e t
f E HM(U).
s u r j e c t i v e r e p r e s e n t a t i o n of
Eh
i.e.
x,
x+y E U
some
n E IN
t h e r e e x i s t s an
n,
whenever
yn
#
0
and
n
of
n
2
aM(V)
r).
For f i x e d
where
V
E'
i s an open and compact
If
f(zn+yn)
#
i n o r d e r t o show
f(x+y) = f(x)
for a l l
does n o t f a c t o r t h r o u g h
f
zn,
f(zn).
E'
i s an open
W
f a c t o r s t h r o u g h some
f
there e x i s t s
n,
(where
such t h a t
v e r y s t r o n g l y convergent t o z e r o i n ever
m
by DFM-spaces,
rrn(y) = 0 .
t h e n f o r each
nn(yn) = 0 ,
E'
As
i t s u f f i c e s t o show t h a t
f E w(U)
that
f o r some
Note t h a t
(since
z cf(z+yn)
-
zn+yn E U
rrr(yn)
f(z)
where
(Y,)
= 0
is
when-
d e f i n e s an element
i s some convex b a l a n c e d neighbourhood of z e r o
303
HOLOMORPHIC FUNCTIONS ON HOLOMORPHIC INDUCTrVE LIMITS
E'
in
,
and hence t h e r e e x i s t s
f(xn+yn)
#
f(xn).
n
For
c o n s t a n t e n t i r e f u n c t i o n on such t h a t ever
C,
closure i s contained i n
and
U
in
i.e.
EA,
E'
Suppose
.
Hence
E'
(
We want t o show
1, E C
> n. U
How-
whose
(Xn+'nyn)nzm
must f a c t o r
f
i s a holomorphically
i s a T -bounded
(fU)a€A
i s a non
E #(U).
f
Let u s prove t h a t
space:
If(xn+knyn)I
i s unbounded on
f
f E aM(U).
contradicting the f a c t that
2)
i.e.
such t h a t
f(xn)
i s a r e l a t i v e l y compact s u b s e t of
(xn+'nYn)n2m
t h r o u g h some
-
and hence t h e r e e x i s t s
> n + If(xn)l,
lgn(Xn)l
( a < 1)
(1-a)U
gn(X) = f ( x n + X y n )
m,
2
n
xn E K
fU)aEA
infrabarreled
s u b s e t of
#(U),
i s l o c a l l y bounded.
U
open
A s i n part
1) of t h i s p r o o f , we may assume w i t h o u t l o s s of g e n e r a l i t y t h a t U = n-l(W)
f o r some
m
claim t h a t
m
where
W
i s a n o p e n s u b s e t of
f a c t o r s uniformly through
(fa)aEA
t h e r e e x i s t s nEN such t h a t f
= f
o n
EL
EL.
f o r some
We
n ie.,
f o r a l l U E A where FU€#(nn(U)).
a a n If n o t , we c a n , a s i n p a r t 1) of t h i s p r o o f , f i n d a s e q u e n c e (1-a)U
(xn+Xnyn) C
which i s r e l a t i v e l y compact a n d a s e q u e n c e ( f
)
%nm such t h a t
( f a a€A
(fan(xn+Xnyn)I 2 n. i s ro-bounded,
t h r o u g h some
(F
)
a aEA
[4]
EL.
Now
(fa)(*EA limit).
and h e n c e
E'
i s lo-bounded i n
we c o n c l u d e t h a t
This contradicts the f a c t that
(fa)aEA
i s a compact s u r j e c t i v e l i m i t a n d h e n c e
a(nn(U)).
('a)aEA
i s l o c a l l y bounded on
U
(since of
We would l i k e t o p r o v e t h a t i f
l o g i c a l space, then
En E
under such h y p o t h e s i s
T h e r e f o r e from P r o p o s i t i o n
such t h a t
E'
E'
i s an open s u r j e c t i v e
Proposition E
4.
i s a s t r i c t inductive
i s a h o l o m o r p h i c a l l y borno-
h a s a c o n t i n u o u s norm.
En
6
i s l o c a l l y bounded a n d h e n c e
T h i s completes t h e proof
l i m i t of FM-spaces
f a c t o r s uniformly
A s we know t h a t
h a s a c o n t i n u o u s norm f o r e v e r y
n
(see
P r o p o s i t i o n 2 [ 8 ] ) , t h e f i r s t i d e a t o d o t h i s was t o p r o v e t h a t i f
304
E En
L U I Z A AMALIA MORAES
i s a s t r i c t i n d u c t i v e l i m i t of h a s a c o n t i n u o u s norm, t h e n
FM-spaces E
En
h a s a c o n t i n u o u s norm.
t u n a t e l l y F l o r e t proved, giving a counterexample,
a r e FN-spaces.
that this state-
These counterexamples can be found i n
The o r i g i n a l p r o b l e m r e m a i n s a n open p r o b l e m . that if
[6].
We know now
i s a s t r i c t i n d u c t i v e l i m i t of FM-spaces
E
Unfor-
A c t u a l l y i t i s f a l s e e v e n i n t h e c a s e when t h e
ment i s n o t t r u e .
En
such t h a t every
En,
at least
one of t h e f o l l o w i n g two s t a t e m e n t s must be f a l s e :
i s a holomorphically b o r n o l o g i c a l space
(1) E '
E
3
has a
c o n t i n u o u s norm. PM(%') = P(%')
(2)
k
f o r every
=,
HM(U) = H ( U )
for e v e r y
U C E'. ( 2 ) i s correct,
If
the following assertions are equivalent:
n.
(a)
En
h a s a c o n t i n u o u s norm f o r e v e r y
(b)
E'
i s a holomorphically bornological space.
I n [ 3 ] D i e r o l f and F l o r e t prove t h e f o l l o w i n g : LEMMA
5.
Let
F
r e s p e c t t o a c o n t i n u o u s norm norm
q
on
F
p
E.
on
can be extended t o
E
which i s c l o s e d w i t h
E
be a l i n e a r s u b s p a c e of
Then e v e r y c o n t i n u o u s a s a c o n t i n u o u s norm.
T h i s lemma a l l o w s u s t o s t a t e t h e PROPOSITION
6.
Let
such t h a t e a c h
En
be a s t r i c t i n d u c t i v e l i m i t of FM-spaces E
E
i s norm-closed
in
Then
En+la
E'
n
i s a holo-
morphically bornological space.
I t i s a c o n s e q u e n c e of Lemma 5 a n d P r o p o s i t i o n 3 f 8 ] .
PROOF.
LEMMA
7.
If
E
such t h a t each
i s t h e s t r i c t inductive l i m i t of Frbchet-spaces
En
h a s a c o n t i n u o u s norm.
If
E
h as a n uncondi-
t i o n a l b a s i s t h e n t h e r e e x i s t s a c o n t i n u o u s norm on PROOF.
See F l o r e t [ 61
.
En
E.
305
HOLOMORPHIC FUNCTIONS ON HOLOMORPHIC INDUCTIVE LIMITS
PROPOSITION 8.
En
E
and
If
i s t h e s t r i c t i n d u c t i v e l i m i t of FM-spaces
E
has an unconditional b a s i s ,
then the following are
equivalent: (a)
En
(b)
E’
PROOF.
3
(a)
h a s a c o n t i n u o u s norm f o r e v e r y
n.
i s a holomorphically bornological space.
It i s a c o n s e q u e n c e of Lemma 7 a n d P r o p o s i t i o n
(b):
3
mi. (b)
It i s a c o n s e q u e n c e of P r o p o s i t i o n 2 [8].
(a):
3
SECTION 2 .
HOLOMORPHIC INDUCTIVE LIMITS
PROPOSITION
9.
be t h e h o l o m o r p h i c i n d u c t i v e l i m i t of
E
Let
(pi)
by
l o c a l l y convex s p a c e s
iEI
(Ei)iEI
.
the
The f o l l o w i n g s t a t e -
ments a r e t r u e : If
i s a holomorphically bornological space f o r every
Ei
i E I,
If
i s a holomorphically b a r r e l e d space.
E
then
i s a holomorphically
Ei
then
If
i s a h o l o m o r p h i c a l l y Mackey s p a c e f o r e v e r y i E I ,
Ei
(a) such t h a t
E Let
f
arbitrary.
f: U
i s continuous.
therefore
i n f r a b a r r e l e d space.
-+ F
be a a l g e b r a i c a l l y holomorphic mapping
i s bounded on e v e r y compact s u b s e t of
fopi
fopi
i s a holomorphically
i s a h o l o m o r p h i c a l l y Mackey s p a c e .
Then
l i n e a r , and
space,
E
i n f r a b a r r e l e d space f o r every
i E I,
then
pi
i s a holomorphically b a r r e l e d space f o r every i E I,
Ei
If
i s a holomorphically bornological space.
E
then
fapi
U.
i s a l g e b r a i c a l l y holomorphic a s
Let
pi
i s bounded on e v e r y compact s u b s e t o f Since
E #(Ui;F)
f E g(U;F)
Ei
T h i s completes t h e proof
of
E
is
as
i s a holomorphically bornological
f o r every (as
Ui
i E I
i E I
a n d for e v e r y
F
and
i s a holomorphic i n d u c t i v e l i m i t ) .
(a).
LUIZA AMALIA MORAES
30 6
From P r o p o s i t i o n 35 of r l ] i t f o l l o w s t h a t i t s u f f i c e s t o
(b)
U C E
show t h a t f o r e v e r y
X
i s l o c a l l y bounded i f
U,
compact s u b s e t of
on e v e r y f i n i t e d i m e n s i o n a l
i s bounded
X
i.e.,
pi:
As
Tof-bounded.
Ei
i s l i n e a r and c o n t i n u o u s ,
E
-t
U.
f i n i t e d i m e n s i o n a l compact s u b s e t of
= [fopi
: f
E X] c
i E I).
every
x E U
for
and
f E
lm(X)
g: U +
X.
E
H(Ui;lm(X)):
If
S
We c l a i m t h a t
gopi:
Ui
lm(X)
-t
hence
s i n c e X i C W(Ui)
p h i c . On t h e o t h e r h a n d ,
i s bounded
gopi(K)
S o , gopi
i s G-holomor-
i s l o c a l l y bounded, i t i s i E #(Ui;lm(X))
f o r every i E I . A s E i s a holomorphic i n d u c t i v e l i m i t , g
t h e image of e v e r y compact s u b s e t of U
and c o n s e q u e n t l y ,
i s To-bounded;
C W(Ui)
Xi
space,
X
f E
and
bounded, (d) that
as
g: U
-t
lm(X)
belongs t o
Let
U
Ui
d e f i n e d by
W(U;l”(X))
a n d c o n t i n u o u s and
f o r every
6 E
$ o f E #(U)
Ei
pi:
+ E
Xi
i € I
i s continuous,
i s a compact s u b -
= pil(U)
: f €
= [fopi
i E I.
X)
Now we p r o v e a s
= f(x)
g(x)(f)
X
and t h e r e f o r e
for
x E U
is locally
of ( c ) .
b e an open s u b s e t of
E W(U)
is locally
i s a holomorphically i n f r a b a r r e l e d
T h i s completes t h e proof
g o f
As
f o r every Ei
gEH(U;lm(X))
(a).
i s l o c a l l y bounded f o r e v e r y
i n (b) t h a t
X
i s l o c a l l y bounded, t h a t i s ,
bounded. T h i s c o m p l e t e s t h e p r o o f of (c) Let X C H ( U ) be l o - b o u n d e d .
s e t of
is al-
pi E
go
i s l o c a l l y b o u n d e d . S o , we h a v e g o p
and c o n s e q u e n t l y
is
i s a f i n i t e d i m e n s i o n a l v e c t o r s u b s p a c e of E
a n d K i s a compact s u b s e t of S fl U i ,
c l e a r t h a t gopi
Xi
g ( x ) ( f ) = f(x)
d e f i n e d by
i s a f i n i t e d i m e n s i o n a l s u b s e t of U.
as Pi(K)
is a
i E I
So, f o r every
g e b r a i c a l l y h o l o m o r p h i c and l o c a l l y bounded;
m e e t i n g Ui
t h e image
-1 = p i (U)
Ui
be
i s a holomorphically b a r r e l e d space f o r
Ei
Consider
C W(U)
i s Tof-bounded a n d t h e r e f o r e
#(Ui)
l o c a l l y bounded ( s i n c e
X
is ~~~-bounded L .e t
of e v e r y f i n i t e d i m e n s i o n a l compact s u b s e t o f
Xi
X c #(U)
we h a v e t h a t e a c h c o l l e c t i o n
F‘.
E.
Consider
Since
w e have t h a t
pi:
f: U + F
Ei + E
Jlofopi
such
is linear
E W(Ui)
where
307
HOLOMORPHIC FUNCTIONS ON HOLOMORPHIC INDUCTIVE LIMITS
Ui = pil(U)
is a n open subset of
ly Mackey space for every
i E I
Q E F ’ , we conclude that
fapi
is,
fopi
E FUi n W(Ui;?)
Ei
.
Ei
As
and
E W(Ui)
$ofopi
for every
i E I.
Now,
i E I
E
f E Fu
n
that
is a holo-
W(U;G) =
This completes the proof of (d).
If
REMARK.
for every
E H(Ui;F) for every
morphic inductive limit and so we infer that
= H(U;F).
is a holomorphical-
E
E
is a DFM-space, m
has a fundamental sequence of
compact sets,
(Bn)n,l,
and increasing.
denote the vector space Bn and endowed with the norm generated by the Minkowski
En
spaned by
functional of
which we may suppose are convex, balanced
For each
Bn.
n
For every
let
n E
E
(N,
E
is a Banach space and
Bn s o , it is a holomorphically barreled space. morphic to the inductive limit
The space
E
is iso-
lim E
in the category of locally n Bn convex spaces and continuous linear mappings (see $ 2 of [hi). We
E = lim E is a holomorphic inductive limit. This n Bn conjecture is equivalent to that of Dineen in the p. 163 of [4]. conjecture if
Matos introduced, in bornological space.
[7], an other notion of holomorphically
We will call S-holomorphically bornological
the spaces that are holomorphically bornological in the sense of Matos
[7] and holomorphically bornological the spaces that are ho-
lomorphically bornological in the sense of Barroso, Matos and Nachbin.
Every S-holomorphically bornological space is holomorphi-
cally bornological.
In [4] Dineen proves that DFM-spaces are ho-
lomorphically bornological and asks if they are S-holomorphically bornological (see conjecture, p . 163 of
[4] )
.
We are going to
prove that the holomorphic inductive limit of S-holomorphically bornological spaces is a S-holomorphically bornological space. Let
BE
denote the family of all closed absolutely convex
bounded subsets of
E.
For each
B E OE,
let
EB
denote the
308
AMALIA
LUIZA
B
v e c t o r s p a c e s p a n n e d by
and endowed w i t h t h e norm g e n e r a t e d by
t h e Minkowski f u n c t i o n a l of DEFINITION 10. morphic i n
B E EE
U
A mapping if
MORAES
B.
U
from
f
i s c a l l e d S-holo-
F
into
i s f i n i t e l y holomorphic i n
f
flu
t h e mapping
n
i s continuous
EB
U
and for e v e r y
(or, e q u i v a l e n t l y ,
h o l o m o r p h i c ) r e l a t i v e t o t h e normed t o p o l o g y .
Let
HS(U;F)
U
mappings f r o m DEFINITION 11.
d e n o t e t h e v e c t o r s p a c e of a l l S - h o l o m o r p h i c
into
F.
The s p a c e
U
space i f f o r every
i s a S-holomorphically b o r n o l o g i c a l
E
and
H(U;F) = H S ( U ; F ) .
it i s true that
F
For b a s i c p r o p e r t i e s , s e e [ 7 ] . PROPOSITION 1 2 .
Let
b e t h e h o l o m o r p h i c i n d u c t i v e l i m i t of
E
l o c a l l y convex s p a c e s
(Pi)iEI.
by
( E ii ) EI
morphically b o r n o l o g i c a l space f o r ever y
If
Ei
i E I,
the
i s a S-holo-
E
then
is a
S-holomorphically b o r n o l o g i c a l space. PROOF.
Let
f: U
-I
F
be a G-holomorphic
on t h e s t r i c t compact s u b s e t s of
1)
If
E = 1 2 Ei
mapping which i s bounded
U.
i s a i n d u c t i v e l i m i t and
is a strict
Ki
p i compact s u b s e t of E:
Ei,
by d e f i n i t i o n ,
exists Since
Bi E BE
pi
Ki
then
pi(Ki)
i s a s t r i c t compact s u b s e t of
such t h a t
i
K i c (Ei)Bi
i s l i n e a r and c o n t i n u o u s ,
t o prove t h a t
pi(Ki)
t h e norm.
E P~
As
e v e r y sequence
(x,)
i s compact i n
n E IN.
pi(Bi) E
c pi(Ki)
Then
(y,)
Ei
i s compact i n
E
aE
there
iff (Ei)Bi.
and we a r e g o i n g
i n t h e t o p o l o g y of
Pl(B1)
i s a normed s p a c e ,
Pi(Ki)
i s compact i f f
a d m i t s subsequence which converges
( i n t h e norm t o p o l o g y ) t o a p o i n t o f for e v e r y
i s a s t r i c t compact s u b s e t o f
pi(Ki).
i s a sequence i n
Let
Ki
y,
-1 = P i (x,)
and,
as
Ki
is
309
HOLOMORPHIC FUNCTIONS ON HOLOMORPHIC INDUC'IIVE LIMITS
(y ) = subsequence of nk which converges in (Ei)Bi to y o E K i , i.e., l/Y -YollBi+ 0 "k k -+ m . It follows that given c > 0, there exists No such
a compact subset of (y,) as
k > No
that f o r every
> 0
inf(X
-yo) E
pi(y
llYnk-YollBi < c
.
such that
'c ,k
But this implies
we have
XBi) < E
: ynk-y0 E
there exists
there exists
(Ei)Bi,
E ,k
9
c > 0, f o r each
So, given 0< h
i.e.
< e
and
k > No
E h E ,kBi *
Yn -Yo k
X c ,kpi(Bi)
and consequently
"k
P~(Y
nk
-
)
inf{X > 0 : pi(y k > No.
every
)
nk So,
pi(Ki)
C E
pi(yo) E
f: U + F
E,
such that
(as
by 1). Ki
is continuous).
pi
If
E i , Pi(Ki) c U fi = fopi
Ei
P q q -
*
is a strict
(u) = ui
f E U(U;F).
is an open
Ki C pil(U)
is a
is a strict compact sub-
is bounded on every K i c pyl(U)
U,
Ei
(as
by hypothesis).
f
is bounded
fiE
So,
is a S-holomorphically bornological space
i s continuous f o r every
fi
ACKNOWLEDGEMENTS.
E
in
pi(Ki)
i s a strict compact subset of
tive limit is holomorphic and s o , implies
yo E Ki)
(x,)
G-holomorphic and bounded on the
pi
So,
and, as
by hypothesis,
i s a subsequence of
It i s clear that
on the strict compact subsets of E HS(Ui;F)
))
U.
strict compact subset of set of
and this is true f o r
< c
E.
compact subsets of
Ei
It follows
*
and is compact, i.e.,
We consider now
subset of
hPi(Bi)]
(xnk) = (pi(y
P1(B1)
compact subset of 2)
-
,kPi(Bi)
nk pi(yo) E pi(Ki) (as
which converges to So,
X,
Pi(y0) E X c ,kPi(Bi) c
f
i E I.
E U(Ui;F)
But the induc-
for every
i
E I
This completes the proof of Proposition 8.
I would like to express my thanks to Professor
Leopoldo Nachbin and to Professor Mario Matos f o r some useful discussions concerning this paper.
This research was supported in
part by FINEP, to which I express may gratitude.
LUIZA
310
AMALIA
MORAES
REFERENCES
1.
J.A. BARROSO, M.C. MATOS and L. NACHBIN,
O n holomorphy versus
linearity in classifying locally convex spaces. Infinite Dimensional Holomorphy and Applications, Ed. M.C. Matos, North Holland Math. Studies, 12, 1977, p. 31-74. 2.
P.J. BOLAND and S. DINEEN, Duality theory for spaces of germs and holomorphic functions on nuclear spaces. Advances in Holomorphy.
Ed, J.A. Barroso, North Holland Math. Studies,
34, 1979, P. 179-207. 3.
S. DIEROLF and K. FLORET,
Normen.
4.
S. DINEEN,
ober die Fortsetzbarkeit stetiger
Archiv. der Math., 35, 1980, p. 149-154. Holomorphic functions on strong duals of FrBchet-
Monte1 spaces. cations,
Infinite Dimensional Holomorphy and Appli-
Ed. M.C. Matos, North Holland Math. Studies, 12,
1977, p. 147-166. 5.
S. DINEEN,
Surjective limits of locally convex spaces and their
application to infinite dimensional holomorphy.
Bull. SOC.
Math. France, 103, 1975, p. 441-509.
6. K. FLORET,
Continuous norms on locally convex strict inductive
limit spaces.
7.
Preprint.
M. MATOS, Holomorphically bornological spaces and infinite dimensional versions of Hartogsf theorem. J. London Math. SOC., 2,
17, 1978, P. 363-368.
8. L.A. MORAES,
Holomorphic functions on strict inductive limits. Resultate der Math., 4, 1981, p. 201-212.
Universidade Federal do Rio de Janeiro Instituto de MatemAtica Caixa Postal 68.530 21.944 Rio de Janeiro, RJ, Brasil
-
Functional Analysis, Hobmorphy and Approximation Theory 11, G I . Zapata (ed.) @Elsevier Science Publishers B. K (North-Holland), 1984
NUCLEAR KOTHE QUOTIENTS OF FF&CHET
SPACES
V.B. Moscatelli ( * )
The structure theory of Frechet spaces is, at present, the object of an intensive study not only because of its intrinsic interest, but also because of its applications to approximation theory and to concrete function spaces.
Within this framework, one
is led to problems concerned with the determination of what kinds of subspaces and quotients can be found in arbitrary Fr6chet spaces, and here I shall attempt to sketch briefly the history of one of these problems up to its present state.
In order to introduce the
problem, let us first explain the title. will be infinite-dimensional.
t9l
Background references are [4], [ 8 ] ,
and [ 111. We recall that a Fr6chet space
operators
uk: Ek+l + Ek
(k E N )
is the set of all sequences with the product topology. choose Banach spaces that each
("1
Of course, all our spaces
uk
Ek
(xk)
E
E
is a projective limit of
on Banach spaces, that is, such that
xk =
E
U ~ ( X ~ + ~( L)E N )
is said to be nuclear if we can
and linking maps
uk: Ek+l
3
Ek
such
can be represented as
~
The author gratefully acknowledges partial support from the Italian CNR through a travel grant.
V.B.
P
Given a set K8the space
h(P)
x(p) =
MOSCATELLI
of non-negative sequences
a = (an),
the
is defined as
mn) nE :
an15,
0.
in a topological vector space
E E
m E N
is a
E
there exists a unique scalar sequence
x = C snxn in E. A basic sequence is a sequence n which is a basis for the closed subspace it generates. such that
Now let
E
be a nuclear Frechet space with a basis
(x,).
Then, by the fundamental Basis Theorem o f Dynin-Mitiagin [ 6 ] , (x,)
is an absolute basis in the sense that the above series con-
verges absolutely for each of the semi-norms defining the topology k that E is of E. From this it follows, putting an = pk (xn) , k isomorphic to the K8the space X(P), where P = ((a,)). The matrix
P
is a representation of the basis
be taken to satisfy
0 5
:a
5
ak+l n
(x,)
and it can always
and the following condition,
known as the Grothendiek-Pietsch criterion:
(*I
for each
k
there is a
j
such that
ak n n an J
C -
0
for all n.
NUCLEAR
KBTHE
QUOTIENTS
OF F ~ C H E TSPACES
313
With a slight abuse of language, from now on by a nuclear K6the space I will mean a nuclear Fr6chet space which has a basis and admits a continuous norm, and the problem under consideration is : Which Fr6chet spaces have nuclear K6the quotients?
REMARK 1.
The continuous norm business is crucial hare.
Already
in 1936 Eidelheit [7] showed that any non-normable Fr6chet space has a quotient isomorphic to
w
(the topological product of
countably many copies of the real line) and, of course, not have a continuous norm. dual
E'
The proof is simple:
as the union of an increasing sequence
w
does
represent the (EL)
of Banach
EA 4 EA+l. Pick elements x' E E;+l-EA; the required quotient is then E/r span(xL)] a spaces, where we can assume
.
REMARK 2.
Nuclearity is also crucial in the sense that the problem
is likely to be much more difficult without it.
Indeed, the answer
is unknown even in the Banach space case and it is a celebrated open problem to know whether every Banach space has a separable quotient
.
Thus, nuclearity rules out Banach spaces, but the above remark points at the difficulty that might lie at the heart of the problem and, indeed, this has been solved s o far only for separable Fr6chet spaces (see (4) below). The problem may be raised, of course, for subspaces as well as quotients and it is instructive to look at the subspace case. Again, nuclearity rules out Banach spaces (but it is an old and classical result that every Banach space has a subspace with a basis) and the subspace problem for Fr6chet spaces was ultimately solved about twenty years ago by Bessaga, PeZczyfiski and Rolewicz
[ 21
, [ 31, who
showed that
314
V.B. MOSCATELLI
(1) a non-normable Frgchet space
if and only if X x w,
with
X
E
E
has a nuclear Kbthe subspace
is not isomorphic to a product of the form
a Banach space (possibly
Note that all closed subspaces of
{O]).
X x w
(X
w.
either of the same form, or Banach or isomorphic to of proof can quickly be summarized as follows.
Banach) are
If
A
(p,)
method is an
increasing sequence of semi-norms defining the topology of first one finds a separable subspace norms
pn
F C E
on which the semi-
are mutually non-equivalent norms.
inductively basic sequences
: k E N)
(x:
(F,P,+~)" such that, denoting by (F,P,+~)",
all embeddings
Xn
Xn + Xn-l
E,
Next, one chooses
in each Banach space
their closed linear spans in are nuclear.
Finally, one
takes suitable linear combinations of elements from the set n (xk : n,k E N) to construct a basic sequence (xk) in F (hence in
E)
whose closed linear span is the required nuclear K6the
subspace. Now let us go back to our problem.
To work directly with
quotients is generally difficult and s o one is tempted to work with subspaces in the dual clude by duality.
E' of a Frbchet space E
and then con-
What I mean is that one is led to represent
as the union of an increasing sequence of Banach spaces to look for a subspace
F
of
E'
EL,
E'
then
on which the EL-norms form a de-
creasing sequence of mutually non-equivalent norms and, finally, to try to construct a basic sequence
(xk)
in
F
this approach does not work because the sequence ed is basic i n each Banach space (strong topology).
E;
as above. (xk)
Well,
thus obtain-
but might not be basic in E'
Indeed, there are examples to the contrary due
to Dubinsky and we refer to [ 5 ] for this as well as for related pathologies.
Of course, this is not surprising, for it occurs all
the time when one deals with inductive limits.
We note however
NUCLEAR K~THE QUOTIENTS
OF
FRFCHET
SPACES
715
that the approach through the dual space works, but with entirely different methods, if the original space is already nuclear (cf
.[5 3 ,
leading to the positive result that (2) Every nuclear Fr6chet space not isomorphic to
w
has a nuclear
Kbthe quotient. Now let us see what can be said on the negative side.
There
is a class of Fr6chet spaces which can be ruled out without any it is the class of those Fr6chet spaces that
assumption whatsoever:
are now called quojections. class of Fr6chet spaces
E
I introduced this class in [lo] as the that are projective limits of a sequence
of surjective operators on Banach spaces.
Obviously, a countable
product of Banach spaces is a quojection, but there are a lot of quojections which are not products (these were called "twisted" in [lo]).
Quojections fail to have nuclear Kbthe quotients
in a very
strong way, for it is not difficult to show that
(3)
If
E
is a quojection and
F
is a quotient of
E,
then:
(a)
F
is nuclear if and only if it is isomorphic to
(b)
F
admits a continuous norm if and only if it is Banach.
By (3)(a),
w
W;
is the only nuclear quojection, so in the
light of (2) and ( 3 ) we can ask: Are quojections the only Fr6chet spaces without nuclear Kbthe quotients? The answer is unknown and in general no more can be said at present (remember Remark 2).
However, there is an important posi-
tive result obtained only recently by Bellenot and Dubinsky [l] under the assumption of separability. generalizes (2):
I t is the following, which
316
V.B.
(4)
A separable Frechet space
and only if Banach spaces
E
has a nuclear Kbthe quotient if
is not the union of an increasing sequence of
E'
EL
MOSCATELLI
with each
being a closed subspace of E' n+l*
EL
F
What we are saying here is that there is a subspace E'
and an increasing sequence
the dual norms
(ph)
(p,)
of semi-norms on
E
of
such that
form a (clearly decreasing) sequence of
F.
mutually non-equivalent norms on
Unfortunately, the proof of
(4) is quite technical and its heavy use of separability points at the difficulty that may be encountered in trying to solve our problem in general.
In proving ( b ) , first one goes over to a
quotient with a continuous norm stated in (4). quence
(d,)
p
and whose dual has the property
Then separability comes in, and we may choose a se-
which is dense in this quotient.
linear span of
(d,),
Calling
the
Eo
we then use the B e s s a g a - P e $ c z y f i s k i - R o l e w i c z
method mentioned above to construct a biorthogonal system
(xn,fn)
such that:
Condition (ii) enables us to extract a subsequence (fn ) j
(fA) such that, if quotient map, then
N =
n
fnl(0)
j
J
)
6(xn
and
6 : Eo
-t
Eo/N
C
is the
is a basis in the completion
(Eo/N)
j
By construction, the latter space is a Kbthe quotient of condition (iii), when reflected on the
),
@(xn
E
-.
and
ensuresnuclearity.
j
Let us remark that the condition of (4) is on the dual of
E.
E'
We know that the dual of every quojection satisfies the
condition.
I s the converse true?
This is at present unknown and
an answer to it would settle our original problem in the separable
NUCLEAR K~~THEQUOTIENTS
317
What we can say is that if the condition of (4) holds, then
case. E’x
OF F ~ C H E TSPACES
(the space of bounded linear functionals on
E’)
is a quojec-
tion (easy to prove) and therefore we can conclude that
(5) Within the class of separable, reflexive Frechet spaces, quojections are exactly those spaces without nuclear KOthe quotients,
REFERENCES 1.
S.F. BELLENOT and E. DUBINSKY, Fr6chet spaces with nuclear KOthe quotients, Trans. Amer. Math. SOC. (to appear).
2.
C. BESSAGA and A. PEgCZYiSKI, On a class of B -spaces, Bull. Acad. Polon. Sci., V . 4 (1957) 375-377.
3.
C. BESSAGA, A . P E g C Z d S K I and S . ROLEWICZ, On diametral approximative dimension and linear homogeneity of F-spaces, Bull. Acad. Polon. Sci., IX, 9 (1961) 677-683.
4.
E. DUBINSKY, The structure of nuclear Fr6chet spaces, Lecture Notes in Mathematics 720, Springer 1979.
5. E. DUBINSKY, On (LB)-spaces and quotients of Frechet spaces, Proc. Sem. Funct. Anal., Holomorphy and Approx. Theory, Rio de Janeiro 1979, Marcel Dekker Lecture Notes (to appear).
6. A.S. DYNIN and B.S. MITIAGIN,
Criterion for nuclearity in
terms of approximative dimension, Bull, Acad. Polon. Sci., 111, 8 (1960)
535-540.
7. M. EIDELHEIT, Zur Theorie der systeme linearer Gleichungen, Studia Math., 6 (1936) 139-148. 8.
H. JARCHOW,
9. G. 10. V.B.
KtlTHE,
Locally convex spaces, Teubner 1981.
Topological vector spaces I, Springer 1969.
MOSCATELLI,
Fr6chet spaces without continuous norms and
without bases,
Bull. London Math. SOC., 12 (1980) 63-66.
318
11.
V.B.
A. PIETSCH,
Nuclear locally convex spaces, Springer 1969.
Dipartimento di Matematica Universita
73100
-
Lecce
C.P.
MOSCATELLI
193
- Italy
Functional Analysis, Holomorphy and Approximation Theory 11, G.I.Zapata ( e d . ) 0 Elsevier Science Publishers B. K (North-Holland), f 984
A COMPLETENESS CRITERION F O R INDUCTIVE LIMITS OF BANACH SPACES
Jorge Mujica(*)
INTRODUCTION By an (LB)-space
X = lim X +
j
we mean the locally convex in-
ductive limit of an increasing sequence of Banach spaces X =
X
*where j
0
U X
.ko
and where each inclusion mapping
Xj
-t
Xj+l
is contin-
j
It is often of crucial importance to know whether a given
UOUS.
(LB)-space is complete o r not, but there are very few criteria to establish completeness for (LB)-spaces, and in many situations these criteria do not apply.
In Theorem 1 we show that if there exists a
Hausdorff locally convex topology
T
on the (LB)-space
with the property that the closed unit ball of each then
X
is complete.
that the space
H(K)
compact subset
K
X = lim X +
j
X j is ‘T-compact,
As an easy consequence of Theorem 1 we prove of all germs of holomorphic functions on a
of a complex Frechet space is always complete.
This result had already been established by Dineen more complicated way.
[4] in a much
In Theorem 1 we also give a sufficient con-
dition f o r an (LB)-space to be the strong dual of a quasi-normable Fr6chet space.
As an application of this criterion we show that if
K is a compact subset o f a complex quasi-normable Frechet space, then H(K)
is the strong dual of a quasi-normable Fr6chet space.
This improves a previous result of Aviles and the author El].
(*)
This research, partially supported by FAPESP, Brazil, was performed when the author was a visiting lecturer at the University College Dublin, Ireland, during the academic year 1980-1981.
JORGE M U J I C A
I would like to thank Richard Aron and S6an Dineen for many helpful discussions that we had during the preparation of this I would also like t o thank Klaus-Dieter Bierstedt, for a
paper.
problem we had discussed some time ago was one of the principal motivations for this research,
That problem, that up to my knowledge
Is every regular (LB)-space
still remains open, is the following: comple te7
1. T m M A I N RESULT
A basic tool in this paper is Berezanskiifs inductive topo-
convex space then we will denote by
Yf
Y
If
logy on the dual of a locally convex space.
is a locally Y‘
the dual
Y, endow-
of
Y! =
ed with the locally convex inductive topology defined by
,
= l$m(Y’)
where
varies among all neighborhoods of zero in Y.
V
VO
This inductive topology is stronger than the strong topology. Berezanskii
f2] o r Floret
ductive topology on
.
f7]
See
KCIthe has also studied this in-
Y‘ in the case where
Y
is a metrizable local-
l y convex space, and he has proved that in that case the space is always complete.
See KCIthe f 9 , p.4001.
used without further reference. topology
T~
We should remark also that the
introduced by Nachbin on the space
continuous m-homogeneous polynomials on Berezanskii topology in the case
THEOREM 1. (a)
X
Let
X = lim X +
j
rn = 1.
Y
of all
reduces to the See Dineen [ 5 , p.511.
be an (LB)-space.
with the property that the closed unit ball
ticular
P(?)
If there exists a Hausdorff locally convex toplogy
T-compact, then
X
X =
Yi
is complete.
Yf
This result will be
K
j
of each
for a suitable Fr6chet space
Y.
on
7
X
j
is
In par-
A COMPLETENESS CRITERION
If in addition
(b)
X
has a base of 7-closed, convex, balanced
neighborhood of zero, then
Y
and
321
X
is actually the strong dual of
Y,
is a distinguished Frechet space. The statement and proof of part (a) are nothing but an
PROOF.
adaptation of a characterization of dual Banach spaces, due to N g [ll, Th.11.
Ng's result is, on the other hand, a variant of an
old result of Dixmier [ 6 , Th. 191.
X + (X,T)
First of all we observe that the identity mapping is continuous. forms on
X
denote the vector space of all linear
whose restrictions to each set
Y
If we endow sets
Y
Now,let
are 7-continuous.
j
with the topology of uniform convergence on the
Y
then it is clear that
Kj,
K
is a Frechet space (it is
actually a closed subspace of the strong dual J: X
-+ Y*
ping.
(Y* = algebraic dual of
Since
Y
3
( x , ~ ) 'and
separates the points of Let
'
and
a
(X,Y)
and
every
j,
X
since
Y) r
x).
of
X;
Let
denote the evaluation mapis Hausdorff, we see that Y
and hence the mapping
is injective.
J
denote the polars with respect to the dual pairs
(Y,Y*), respectively. we see that
J
maps
X
Since clearly
J(K.) J
C KO*
j
for
Yf. Now,the
continuously into
mapping J:
(K.,T) J
-t
(Y',U(Y',Y)) Y.
is clearly continuous, by the definition of Hence dense in J(Kj)
J(Kj)
(J(Kj))**,
= (J(K.))*'. J
is
u ( Y ' ,Y)-compact.
Since
J(Kj)
is U(Y',Y)-
by the Bipolar Theorem, we conclude that Thus,since clearly
(J(Kj))*
= KQ
,
we conclude
that J(K.) J
and hence
J
= (J(K.)*' J
= K
0 .
j
is a topological isomorphism between
T o show (b) let
U
X
and
Y i .
be a 7-closed, convex, balanced neigh-
722
JORGE MUJICA
borhood of zero in Now, since
I c Y
J
,
0 0
X.
Then
U = U
X
onto
Y' we see that
maps
by the Bipolar Theorem. = @*
J(4)'
for each
and hence J(U) = J(Uoo) = Uo*.
Since
Uo
YL
0-neighborhood in tinuous.
Y
is clearly bounded in
we conclude that
and hence the mapping
5-l: Y;
Yf + YL
Since the identity mapping
J(U)
+ X
is a is con-
is always continuous,
the proof is complete. T o verify the second condition in Theorem 1, the following
lemma will be useful. LEMMA 1.
X = lim X
Let
+
j
be an (LB)-space, and assume that there
exists a Hausdorff locally convex topology
7
on
X
with the fol-
lowing properties: (i) (ii)
The clsed unit ball
K
j
for each 0-neighborhood
of each
U
X
X
in
is
j
there exists a sequence
of 7-closed, convex, balanced 0-neighborhoods
vj
n
K. c J
compact;
Vj
in
X
such that
u.
Then
X
has a base of 7-closed, convex, balanced neighbor-
hoods of zero. PROOF.
Let
U
X.
be a 0-neighborhood in
We choose a sequence
m
(ej)
of positive numbers with
C
Cj
4
1
such that
j=O
m
where
C
j=o
m
E .K
denotes the set
~j
u n=O
n
C
j=o
By (ii) we can find a sequence of 0-ne ighb orhood s
Vj
in
X
EjKj
closed, convex, balanced
such that
c -1 2u .
A COMPLETENESS CRITERION
323
Define n
m
(3)
V =
fl
( c
n=O
V
Then
Since
i s a 0-neighborhood
X.
show t h a t
V t U.
in
3'
i s b a r r e l l e d we c o n c l u d e t h a t
X
and choose
n
V
is suffices t o
To c o n c l u d e t h e proof
z E V
Let
Vn).
i t i s convex and b a l a n c e d , and
K .
absorbs every
by ( i ) i t i s r - c l o s e d .
+
EjKj
j=O
such t h a t
z
E
nKn.
BY ( 3 ) we can w r i t e
n z = x+y,
(4)
with
x E
C
e .K.
j=O
and
J J
y
E
Vn
.
Hence
E
y = z-x
(5)
nKn
n
+
C j=O
c .K. J J
C
(n+l)Kn
m
C c 1 *: 1 and s i n c e we may assume, w i t h o u t l o s s of g e n e r J j=O a l i t y , t h a t t h e sequence (Kj) i s increasing. Thus from ( b ) , ( 5 ) since
and ( 2 ) we conclude t h a t
Y
E vn n
(n+l)Kn c
1 2 u
and t h e r e f o r e n
by
(4)
,
( 6)
and (1). Thus
V c U
and t h e proof
i s complete.
2. APPLICATIONS TO COMPLEX A N A L Y S I S If
i s a compact s u b s e t o f a complex F d c h e t s p a c e
K
then t h e space
#(K)
of a l l germs of holomorphic f u n c t i o n s on
E K
i s d e f i n e d a s t h e l o c a l l y convex i n d u c t i v e l i m i t
where
(Uj)
borhoods of
i s a d e c r e a s i n g fundamental sequence o f open n e i g h K
and where
#"(Uj)
d e n o t e s t h e Banach s p a c e of a l l
324
JORGE MUJICA
U
bounded holomorphic functions on
j'
with the norm of the supremum.
This (LEI)-space has received a good deal of attention in recent years and we refer to the survey article of Bierstedt and Meise [ 3 ] or to the recent book of Dineen c5] for background information and open problems concerning #(K)
The problem of completeness of
#(K).
remained open for several years until it was finally solved
by Dineen [4, Th. 81, who proved that the space complete.
is always
#(K)
Dineen's proof is quite complicated, but we can now
obtain Dineen's result as an easy consequence of Theorem 1. THEOREM 2. E. #(K)
Then
Let
#(K)
be a compact subset of a complex Fr6chet space
K
= Yi
Y.
for a suitable Frechet space
In particular,
is complete.
Let
Before proving Theorem 2 we fix some notation.
#(U)
U
denote the space of all holomorphic functions on an open subset of a complex locally convex space then we let expansion of set
f(n)(x) f
11 f(n) "A,B =
denote the
If
nth
x.
sup
I f(n) ( x ) ( s ) I
f E #(U)
x E U
and
term in the Taylor series
f E #(U),
at
xEA
If
E.
A C
U
and
B
C
E
then we
.
sE B
PROOF OF THEOREM 2.
Let
of open neighborhoods of pology on
W(Uj),
(Uj) be a decreasing fundamental sequence Let
K.
denote the compact-open to-
T~
and by abuse of notation let
locally convex inductive topology on
As Nicodemi [12] has remarked, for the seminorms uous on
(#(K),T~)
f +
I f(n)(x)
for all
#(K)
(#(K),T~) (s)
I
n E N,
T~
also denote the
which is defined by
is a Hausdorff space,
are well-defined and continx E K
and
s E E.
other hand, by Ascoli theorem, the closed unit ball of compact in
(#(Uj),To),
and hence in
(#(K),T~).
On the Wa(Uj)
is
An application
A COMPLETENESS CRITERION
325
of Theorem 1 completes the proof. REMARK.
(#(K),T~)
The locally convex space
has recently been
studied by the author [lo] in great detail and it turns out that
Y
the Frechet space of
(H(K)
,To)
that appears in Theorem 2 is the strong dual
-
Avil6s and the author [l, Th.21 have shown that
#(K)
sa-
K
is a
tisfies the strict Mackey convergence condition whenever compact subset of a complex quasi-normable Frechet space.
This
result can be improved as follows: THEOREM 3.
Frechet space
be a compact subset of a complex quasi-normable
K
Let E.
Then
#(K)
is the strong dual of a quasi-
normable Fr6chet space. We refer to Grothendieck f 81 for information concerning quasi-normable spaces and the strict Mackey convergence condition. T o prove Theorem 3 we need the following lemma, which is essentially a reformulation of the proof of [l, Th.21. Let
LEMMA 2.
Frgchet space
K
be a compact subset of
E.
a
complex quasi-normable
Then there exists a decreasing fundamental se-
quence of open, convex, balanced 0-neighborhoods that, if we let
Xj
U
denote the closed unit ball of
for each 0-neighborhood
L
in
convex, balanced 0-neighborhoods
#(K)
k
j
in
E
such
Hm(K+Uj),
then
there exists a sequence of in
#(I()
with the following
properties: (i) (ii) PROOF.
each IJ
k
n X J. c
Since
E
is closed in
(#(K) , T ~ ) ;
L
j.
for every
is metrizable and quasi-normable, we can induct-
ively find a fundamental sequence of open, convex, balanced O-neighborhoods
U . in J
E
such that:
326
JORGE MUJICA
~ c uj ~
(a)
2
(b)
for every
set
B
E
in
IJ
j;
6
and for every
j
there exists a bounded
0
7
2Uj+l C B + 6U
such that
xj
Let
+ ~ for every
j ’
denote the closed unit ball of
is a 0-neighborhood in
Since
we can find a sequence of positive
#(K)
3 c j$t
Hm(K+2Uj).
c IJ
numbers
E
f E X j .
Then using (a) and the Cauchy integral formulas, we can
j
such that
Fix
j.
and fix
j
N E N:
write, for each
Now, since
for every
fE X j
the Cauchy integral formulas imply that
If(n)(x)(s)\
L
x E K
for all
1
and
s
E 2U j
and hence that (2)
I(f(n)(x))(k)(s)(t)I
for all
1
4
Next we note that by (b), given ed such that each
s E
2U
j+l
s = b + 6t,
Hence, for each
x E K,
x E K
s,t E U j ’
and
6 > 0 there exists B
C E
bound-
can be written in the form
with
b E B
t E Uj.
and
(2) implies that
and we conclude that
m
First we choose
N E N
such that
C 2’n n=N _. .
m
choose
0< 6 < 1
such that
C bn k=1
5
N
4
0 j+l
.
If
and next we
B
is the bounded
A
set associated with
COMPLETENESS CRITERION
327
in (b) then from (1) and ( 3 ) we conclude
6
that N-1
II f11K+2U j+l
(4)
J+1
n=O
*
If we define
then Ir .
is a convex, balanced 0-neighborhood in
J
closed in
#(K),
is
l.rj
(#(K),'T~) and by (4)
L
. n x J. c
J
30 j+lx j+l c b.
The proof of Lemma 2 is now complete. PROOF OF THEOREM
3.
Let
neighborhoods of zero in
(Uj) be the fundamental sequence of E
given by Lemma 2.
know that the closed unit ball of each
gm(K+Uj)
Since we already is compact in
(~(K),T~), then from Lemma 1 and Lemma 2 we conclude that
g(K)
has a base of convex, balanced neighborhoods of zero, each of which is closed in #(K)
(#(K),T.).
Then we conclude from Theorem 1 that
is the strong dual of a Fr6chet space
[l, Th.21
#(K)
we conclude that complete
.
Y.
But since by
satisfies the strict Mackey convergence condition, Y
must be quasi-normable.
The proof is now
JORGE MUJICA
REFERENCES
1.
P. AVILXS and J. MUJICA, Holomorphic germs and homogeneous polynomials on quasi-normable metrizable spaces, Rend. Mat. 10
2.
(1977)s 117-127.
J.A. BEREZANSKII,
Inductively reflexive locally convex spaces,
Soviet Math. Dokl. 9 (1968), 1080-1082.
3.
K.-D. BIERSTEDT and R . MEISE,
Aspects of inductive limits in
spaces of germs of holomorphic functions on locally convex spaces and applications to a study of
(H(U),T~), in
Advances in Holomorphy (J.A. Barroso, ed.),
North-Holland,
Amsterdam, 1979, p. 111-178.
4.
S. DINEEN,
Holomorphic germs on compact subsets of locally
convex spaces, in Functional Analysis, Holomorphy and Approximation Theory (S. Machado, ed.),
Lecture Notes in Math. 843,
Springer, Berlin, 1981, p . 247-263.
5.
S. DINEEN,
Complex Analysis in Locally Convex Spaces,
North-
Holland, Amsterdam, 1981.
6. J. DIXMIER, S u r un Theoreme de Banach, Duke Math. J. 15 (1948), 1057-1071.
7. K. FLORET,
h e r den Dualraum eines lokalkonvexen Unterraumes,
Arch. Math. (Basel) 25 (1974), 646-648.
8.
A.
GROTHENDIECK,
Sur les espaces (F) et (DF), Summa Brasil.
Math. 3 (1954), 57-123.
9. G. KdTHE, Topological Vector Spaces I, Springer, Berlin, 1969. 10.
J. MUJICA,
A new topolo#w
on the space of germs of holomorphic
functions (preprint). 11.
On a theorem of Dixmier,
K.F. NG, 279-280
0
Math. Scand. 29 (1971),
A COMPLETENESS CRITERION
12.
0. NICODEMI,
329
Homomorphisms of algebras of germs of holomorphic
functions, in Functional Analysis, Holomorphy and Approximation Theory (S. Machado, ed.), Lecture Notes i n Math. 843, Springer, Berlin, 1981, p . 534-546.
Department of Mathematics University College Dublin Belfield, Dublin 4 Ireland and Instituto de Matemgtica Universidade Estadual de Campinas Caixa Postal 6155
13100 Campinas, SP (current address)
-
Brazil
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Functional Analysis, Ho lo rno rp hy and Approximation Theory II, G I . Zapata ( e d . ) 0 Ekevier Science Publishers B. V. (North-Holland), 1984
ABOUT THE CARATHEODORY COMPLETENESS OF ALL REINHARDT DOMAINS Peter Pflug
It is well known that in the theory of complex analysis there are different notions of distances on a bounded domain
G
in
Gn,
for example, the Caratheodory-distance dealing with bounded holon
morphic functions, the Bergmann-metric measuring how many Lc-holomorphic functions do exist or the Kobayashi-distance describing the sizes of analytic discs in
G.
A survey on these notions, also ge-
neralized to infinite dimensional holomorphy, can be found in the book of Franzoni-Vesentini [ 31
.
The main problem working with these distances is to decide which domain
G
is complete w.r.t.
one of these distances.
There
is a fairly general result for the Bergmann-metric due to T. Ohsawa and P. Pflug [ 6 , 7 ]
which states that any pseudoconvex domain with
C1-boundary is complete w.r. t. the Bergmann-metric.
O n the other
hand it is well known that the Caratheodory-distance can be compared with the other two, in fact, it is the smallest one, but there is n o relation between the Bergmann-metric and the Kobayashi-metric [Z]. Thus the question remains which domains are complete w.r.t. Caratheodory-distance or, at least, w.r.t.
the
the Kobayashi-distance.
In this short note it will be shown that any bounded complete Reinhardt domain
G
which is pseudoconvex is complete in the sense
of the Caratheodory-distance; in fact, it will be proved that any Caratheodory ball is a relatively compact subset of
G.
Using the
above remark on the comparability of the distances it is clear that
332
PETER PFLUG
those domains are also complete w.r.t.
the two other distances.
First, some definitions should be repeated. DEFINITION 1. A domain
zo E G
domain if for any
for
1 4 i
L
n)
G C Cn
is called a complete Reinhardt {z E Cn:
the polycylinder
has to be contained in the domain
lzil
G
lzil
G.
It is well known that a complete Reinhardt domain pseudoconvex iff
S
G
is
is logarithmically convex which means the set
loglGl := {x E Rn: for
,...,loglznl)]
x 3 z E G: x = (loglzll
is convex in the u s u a l sense, DEFINITION 2. in
Let
G
be a domain in
Cn
then, for points
z’, z”
G, CG(Z’ ,z”) := sup { 1 F log 1 1
+
-
If(Z”)I
.
lf(z/’)l
*
f:G+E holomorphic with f(z‘)=O] is called the Caratheodory distance between in the future
E
It is easy to check that
on a bounded
E =
denotes the unit disc
G; hence
(G, CG(
,
CG(
,
))
)
z’
and
{ I E C:
z ” ; here and
1x1
< 13.
is, in fact, a distance
is a metric space.
Asking
whether this space is complete it suffices to establish that any Caratheodory ball
{z E G: CG(z,zo)
latively compact subset of dory-completeness of
G;
< M]
around
zo E G
is a re-
this is called the strong Caratheo-
G.
It is well known that a Caratheodory-complete domain (i.e.
(G, CG(
,
))
is complete) has to be Hm(G)-convex
and a domain of
bounded holomorphy [ 81 ; the converse, in general, is false.
In fact
m
there exists a H (G)-convex domain of bounded holomorphy which is not Caratheodory-complete.
On the other hand it should be repeated
ABOUT THE CARATHEODORY COMPLETENESS OF ALL REINHARDT DOMAINS 333
m
that any pseudoconvex domain with a smooth boundary is H (G)-convex and also a domain of holomorphy fl]. following problem:
This remark may induce the
is any pseudoconvex domain with smooth boundary
Caratheodory-complete o r , at least, complete w.r.t. distance? THEOREM.
the Kobayashi-
Here only a simple partial result can be presented, G , . which is
Any bounded complete Reinhardt domain
pseudoconvex, is strongly Caratheodory-complete. Without loss of generality we can assume that
PROOF.
tained in the unit-polycylinder.
is con-
Then assuming the proposition is
(z"] C G
false there exists a sequence
G
with
zv
-t
zo
E G
such
n
that
CG(zv,O)
4
M
0, consider the
6
A % E
b = 1
and is equal to
f
-1 TT (y): IIg(x)-f(x)/l
ping
5.5, [ 61 , each
and the constant function
belongs to
is
A = {ban; b E C ( Y ; F ) ] ,
the closure of
in
W
Then
from
Let X
X
Z
and
= 0
0,
is a closed subspace. rp
Let us define a carrier closed subsets of
E.
For each
rp(Y) =
cs
E E;
y E Y,
SUP XE ll
It is clear that pact subset of
X
cp(y)
-
define Ilf(x)-sl/
B y hypothesis, there exists some
63.
(Y)
is closed.
and therefore
into the non-empty
Now
K = (f(x); so E E
n-’(y)
x E n”(y))
such that
is a comis compact.
BE ST SIMULTANEOUS APPROXIMATION
3 47
We claim that
(""1
rad(K)
Indeed, let
Since
E W
g
be given.
8 ,
I;
Then
was arbitrary,
g
rad(K)
inf I\f-g//, &W
S
and (**) is true. Now, from ( * ) and ( * * ) it follows that Hence,
~ ( y ) f
We claim that
is open in Let
#,
yo E
Y
E ep(y).
y E Y.
for all
is lower semi-continuous, i.e. that
Q
for each open subset
Y
so
be such that
G C E.
cp(yo)
n
G f
@.Let
so
E
epbo)n
G.
Then SUP
llf(x)-~oll
5
6
-1
XEl-f This means that /ls-soll
81.
c fe1(B(s0;6)). n
is closed.
with
c B(so;8),
f(i7-'(y0))
Notice that Since
(Yo)
X
B(so;S)
where
B(so;8)
is open, and that
is compact and
Y
= ( s E E;
-1 IT
(yo) c
is Hausdorff, the map
Hence there is some saturated open set
V
in
X
J O I O B. PROLLA
348
TT
U = n(V)
Then y E U,
all
rr-'(y)
U;
y
C
y E U,
cp,
(because
V C f-'(B(s0;6)).
TT
Hence
cp
and
and, for any
Hence
11 f-wll
W
for all
x E X
y E Y.
y = TT(X).
let
and therefore
for
E cp(y)
for
Let
g E C(Y;E)
w =
gow.
Then
Then
w E Pw(f).
This ends
(E,ll=\\) admits Chebyshev centers, a
better result can be proved, namely that
X
not only a
centw(B) f $
B = (f}, but for equicontinuous bounded sets Let
so
is proximinal.
When the Banach space
B
C
C(X;E).
be a 0-dimensional compact T1-space and let
be a non-archimedean Banach space over a non-trivially
(E,l/.ll)
valued division ring
(F,I
-1).
If
E
admits Chebyshev centers,
and
W C C(X;E)
f $
for every non-empty bounded subset
is a Weierstrass-Stone subspace, then
continuous at every point of PROOF.
Let
TT:
X -+ Y
C
C(X;E)
B c C(X;E)
Y
centw(B) f
which is equi-
X.
be a continuous surjection of
compact Hausdorff space
B
B(so;G)
C
is lower semi-continuous.
< dist(f ;W) ,
the proof that
Let
f(n-'(y))
But this means that
g(y) E c p ( y )
i.e.,
THEOREM 16.
and for any
14, there is a continuous selection
w E W
for
-1 (U) = V)
that is
By Theorem for
Y
is open in
t E v -1(y), y E U .
for all all
-1(yo) c V c f-'(B(s,;*)).
X
onto a
such that
be a non-empty bounded subset which is equicontin-
B E S T SIMULTANEOUS A P P R O X I M A T I O N
uous a t e v e r y p o i n t of Let CASE I :
X.
6 = radli(B).
6 > 0. Define a c a r r i e r
s e t s of
349
E
cp
i n t o t h e non-empty
Y
from
c l o s e d sub-
by p(y) =
( S
E E ; sup
sup
61.
Ilf(x)-sll
f E B xEll-l(y)
It is clear that B C C(X;E)
so
for e a c h
E E
E,
and by h y p o t h e s i s
cent(B(y))
# #,
~ ~ f ( x ) - s=o r~a ~ d(B(y)).
SUP
M B xEft-l(y) We claim t h a t
("1
rad(B(y)) I n d e e d , f o r any
=
g E
W,
SUP
IIf-gll
MB
g
Y.
such t h a t SUP
Now
y E
Since
i s bounded,
i s bounded i n exists
i s closed,
p(y)
was a r b i t r a r y ,
so
5
we h a v e
.
6.
i.e.,
there
350
J O X O B. PROLLA
and s o ( * ) is true, as claimed. Therefore
E cp(y),
so
cp
We claim that
Y,
that
n
rp(y0)
q(y)
Choose
so
n -1(yo)
finite open covering
# $,
1
i
J;
4
n,
f E B.
for all
f E
G C E.
Vl,V2,
...,Vn
of
yo E Y
Let
be such
Then
s 6
. X,
n-'(yo),
there exists a with
Vi
n n-1(yo)#
such that
vi
=)
llf(x)-f(x')ll
This is possible
Vo = V1 U V2 U...U
x E Vo x E Vi
such that
$3
11 f(x)-so]l
and
,
< 8
because the set
C C(X;E)
B
is
X.
equicontinuous at every point of Let
f
is a compact subset of
x,x' E for all
G
E ~(y,) n G.
sup sup fEB xErr-l (Yo) Since
n
Y; V ( Y )
for each open subset
G f $.
is non-empty.
is lower semicontinuous, i.e., that
EYE is open in
and
Vn.
We claim that
\lf(x)-soll
6
J:
Indeed, given x E Vo choose Vi -1 and choose t E Vi n TT (yo). Then, for all f E B.
B
llf(x)-soll
max(llf(X)-f(t)il,
6
llf(t)-soll)
*
Theref ore TI
c Vo
-1(Y,)
C
n
f-1(B(so;6)).
fEB Choose a saturated open set (This is possible because an open neighborhood of
E V
C
f-l(B(s0;8))
x
E ~"(y)
yo
for all
and
in
X
with
is a closed map), in
Y,
f E B.
llf(x)-~oll for all
V
f E B.
'IT
-1
Then
and for every
(yo) C
V
C
U = n(V)
y E U,
=6 so
is
rr-l(y) E
Hence
This means that
Vo.
E cp(y)
for
BEST SIMULTANEOUS APPROXIMATION
all
y E U,
and s o
351
is lower semicontinuous.
cp
By Theorem 14 there exists a continuous selection g(y) E q(y)
such that
and for any
x
E X,
sup IIf-wll < 6 fEB
Hence
CASE 11:
Let
w =
gon.
.
and s o
Then
fE B
y = ~ ( x ) Then for any
let
,
y E Y.
Y
-I
E
w E W
we have
w E centW(B).
6 = 0.
Now
radW(B) = 0
= 0. Therefore REMARK.
for all
gr
f
E W
implies
B = [f}
and
dist(f;W) = radW(B)=
and there is nothing to prove.
In Olech [ 5 ] the formula dist(f;W) = sup rad(f(n-'(y))) YEY
was proved for Weierstrass-Stone subspaces W C C(X;E), is compact and
E
where
is a uniformly convex Banach space (over
X
R or
C),
We will show that (*) is a consequence of the Stone-Weierstrass Theorem. THEOREM 17.
Let
X
be a compact Hausdorff space, and
( E , l l * \ \ ) be
a normed space over a non-archimedean non-trivially valued division ring
(F,I*I).
W
C
C(X;E)
v: X
-I
Y
subspace
where
For every
Hausdorff space
f E C(X;E)
and every Weierstrass-Stone
we have
is the continuous surjection of
Y
such that W = { P n ; g E C(Y;E)}.
X
onto a compact
JOXO B. PROLLA
352
Let
PROOF.
y E Y.
w E W
Then, for every inf
sup
we have
Ilf(x)-zll
z E E xEl?(y)
SUP Ilf(x)-w(x)ll xEn -I (Y) because
w
is constant on
rl
-1 (y).
rad(f(r-l(y)))
IIf-wll
5
Since 5
w
was arbitrary,
dist(f;W)
and then sup rad(f(n YE y Conversely , by Theorem 6.4,
-1
(y)))
S
dist(f;W).
f 61 , we have
dist(f;W) = sup inf Ilf(X)-w(x)ll. YEy wEW xEn-I(y) Let
y E Y.
x E X,
For each
belongs to
z E E,
W.
the constant function
Hence, for each
inf
sup
z
E E
Ilf(x)-w(x)ll
wEw xEn-l(y)
* Since
z
SUP llf(x)-zIl xE rl- Y)
was arbitrary, we have inf
sup
WEW
xErr-l(y and from this it clearly follows that dist(f
REMARK.
of
In the proof given above we used the following properties
w; (1) every (2)
w E W
for each
is constant on each
y E Y,
and
z
E E,
TT
-1
(y),
y E Y;
there is some
w E W
such
BEST SIMULTANEOUS APPROXIMATION
w(x) = z
that
(3) W
such t h a t modulo
x E fl-l(y);
for a l l
i s an A-module,
where
n"(n(x))
i s a s u b a l g e b r a of
A
C(X;F)
i s t h e e q u i v a l e n c e c l a s s of
x
x E X.
f o r each
X/A,
353
Hence t h e f o l l o w i n g r e s u l t i s t r u e : THEOREM 18.
Let
b e a compact Hausdorff s p a c e and- l e t
X
(E,\\*ll)
b e a normed s p a c e o v e r a non-archimedean n o n - t r i v i a l l y v a l u e d d i v i sion r i n g
n: X
3
Y
(F,I
01).
A c C(X;F)
Let
be t h e q u o t i e n t map of
onto t h e q u o t i e n t space
X
o f a l l e q u i v a l e n c e c l a s s e s modulo
be a s u b a l g e b r a and l e t
Let
X/A.
W c C(X;E)
Y
be an
A-module such t h a t w E W
(1) e v e r y
i s c o n s t a n t on e a c h e q u i v a l e n c e c l a s s ~ - ' ( y ) ,
Y E y;
f o r each
(2)
that
y E Y
w(x) = z
and
t h e r e i s some
E
w E W
x E n-'(y).
for a l l
t o ask t h e f o l l o w i n g q u e s t i o n :
i s an A-module, where i s such t h a t
f o r each
x E X,
A
C
g i v e n a subspace
C(X;F)
W(x) = ( w ( x ) ; w €
W
c C(W;E) which
i s a s e p a r a t i n g s u b a l g e b r a , and W)
does i t f o l l o w t h a t
C E
W
i s proximinal i n
,I] *I[ )
X
and
s a t i s f y t h e h y p o t h e s i s of t h e s e l e c t i o n Theorem 1 4 .
THEOREM 1 9 . (E
E,
i s proximinal i n C(W;E)
O u r n e x t r e s u l t shows t h a t t h e answer i s y e s i f
(E,ll-ll)
such
Under t h e h y p o t h e s i s of Theorems 17 and 18 i t i s n a t u r a l
REMARK.
W
z €
Let
X
b e a 0 - d i m e n s i o n a l compact T1-space
and l e t
be a non-archimedean Banach s p a c e o v e r a n o n - t r i v i a l l y
valued d i v i s i o n r i n g
(F,
I . I ).
Let
A c C(X;F)
be a separating
?
354
JOXO B. PROLLA
subalgebra and let is proximinal in Then
W
Let
PROOF.
W
x E X.
for every
E,
C(X;E).
is proximinal in
f E C(X;E)
> 0, because
be a closed A-module such that W(x)
C(X;E)
C
be given with
is closed.
W
x E X,
F o r each
n
there is
( S
cp(x) f $.
so
Clearly,
lower semicontinuous. Choose and
so
n
E cp(xo)
G.
x E X,
F o r each
E.
w € W
G c E
Let
Then
E E ; Ilf(x)-s11
some
Ilw(x)-f(x)ll and
W.
L - t us define a carrier
into the non-empty closed subsets of
cp(x) = W(X)
f
f $
E,
E
but has the
a better result can be proved,
for every equicontinuous bounded subset
9
BEST SIMULTANEOUS APPROXIMATION
B
C(X;E).
C
Let
THEOREM 2 0 . A
C
355
C(X;F)
X
and
(E,\l-ll) be as in Theorem 19.
closed A-module such that property in
for every
E,
2,
centw(B) f
Then
bounded subset Let
PROOF.
B
B C
W(x)
B
C(X;E)
for every non-empty equicontinuous
be a non-empty bounded subset which is
Define a carrier E
with
(f]
cp
X.
b
from
X
Let f E W
>
6 = radW(B).
If b =
0,
and there is nothing to
0.
into the non-empty closed sub-
by
x E X,
F o r each
be a
x E X.
Hence we may assume that
sets of
C(X;E)
C(X;E).
C
is a singleton
prove.
C
has the relative Chebyshev center
equicontinuous at every point of then
W
be a separating subalgebra and let
Let
there is some
B(x)
w E W
= (f(x);
f E B}
is bounded in
E,
and
such that
Now
= inf SUP Ilf(x)-w(x)II
radW(x)(~(x))
wEW fEB inf sup
g
1) f-wll
= 6.
W E W fEB
Hence
cp(x) f
2.
Clearly,
lower semicontinuous, i.e. for every open subset
f
2
and choose
s o = w(xo)
and
so
cp(x) that
G C E.
is closed. [x E X; cp(x)
Let
E rp(xo) fl G.
SUP Ilf(xo~-w(xo)II
xo E X
We claim that
n
G
f 23 w E W
is
is open,
be such that
There is some
cp
cp(xo)
n
Gf
such that
6
f€ B Hence B(0;b) =
[ S
xo E (f-w)-l(B(O;b))
€ E ; IIs(I
L
b ] .
f E B,
for every
By continuity of
w
where
and equicontinuity
356
of
JOXO B. PROLLA
{f-w; f E B),
such that
x E U.
E G
W(X)
Then
there is some neighborhood
W(X)
x E
and
E cp(x)
n
U
(f-w)-l(B(O;6))
G,
in
xo
X
f E B
for all
x E U,
for all
of
and
and the carrier
is lower semicontinuous. By Theorem 14 there is a continuous selection for the carrier
Then
by [ 6 ] , 6.4.
g E W,
x E X,
for all
COROLLARY 2 1 . that
rp,
Let
X
and
W = { g E
is such that bounded subset W
space,
C(X;F) x E X,
COROLLARY 2 2 . space
( E , \ \ * \ l ) be as in Theorem
E
centW(B)
W
C(X;E);
# #
Z C X,
given by x E Z],
g(x) = 0,
for every non-empty equicontinuous
B C C(X;E).
is separating over W(x) = 0 ,
Let
X
and
being a 0-dimensional T 1X. On the other hand, f o r X
x E Z;
if (E
,I1 *I1 )
and
non-empty equicontinuous bounded subset PROOF.
W = C(X;E)
x E X.
Since
is a C ( X ; F ) - m o d u l e ,
C(X;F)
W(x) = E
if
x
be as in Theorem 19.
admits Chebyshev centers, then
Theorem 2 0 .
19. Assume
F o r each closed subset
c C(X;E),
is a C(X;F)-module, and
PROOF.
and s o
x E X,
and therefore
admits Chebyshev centers,
E
for a l l
On the other hand
the closed vector subspace
every
g ( x ) E W(x)
g E C(X;E)
cent(B) B
C
and
# #
Z.
If the
for every
C(X;E). W(x) = E
for every
is separating, the result follows from
rp
BE ST SIMULTANEOUS APPROXIMATION
357
REFERENCES
1.
BLATTER, J., Grothendieck spaces in approximation theory, Memoirs Amer. Math. SOC. 120 ( 1 9 7 2 ) .
2.
M I C H m L , E.,
Selected selection theorems,
Amer. Math. Monthly
63 ( 1 9 5 6 ) ~ 233-238. 3.
MONNA, A . F . ,
I.
4.
S u r les espaces lineaires normes non-archim&diens,
Indagationes Mathematicae 18 ( 1 9 5 6 ) , 475-483.
MONNA, A.F.,
Remarks on some problems in linear topological
spaces over fields with non-archimedean valuation,
Inda-
gationes Mathematicae 30 ( 1 9 6 8 ) , 484-496.
5.
OLECH, C.,
Approximation of set-valued functions by contin-
uous functions,
6.
PROLLA, J.B.,
Colloquium Mathematicum 19 ( 1 9 6 8 ) , 285-293.
Topics in Functional Analysis over Valued Divi-
sion Rings, North-Holland Publ. Co., Amsterdam, 1982.
7.
SMITH, P.W. of
8.
and J.D. WARD,
C(X),
WARD, J.D., tions,
Chebyshev centers in spaces of continuous funcPacific Journal of Mathematics 52 ( 1 9 7 4 ) , 283-287.
Departamento de Matemitica UNICAMP
-
IME'cc
Campinas, SP
Restricted centers in subalgebras
Journal of Approximation Theory 1 5 ( 1 9 7 5 ) , 54-59.
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Functional Analysis, Holomorphy and Approximation Theory Il, G I . Zapata (ed.) 0 Elsevier Science Publishers B. V. (North-Holland), 1984
359
ABSTRACT FROBENIUS T H E O W M
-
GLOBAL FORMULATION
APPLICATIONS TO LIE GROUPS
Reinaldo Salvitti
The main goal of this work is to give the Global Formulation of the Abstract Frobenius Theorem in the context of Scales of Banach Spaces and to applie it in the construction of Lie Subgroups.
The
motivation of this work was the study of germs of analytic transformations of
Cn
that vanish at the origin,
studied by Pisanelli in [l]. scale
gh(n,C),
gh(n,C),
as it was
The applications of this work, in the
will appear in another paper.
1. INTRODUCTION 1.1 DEFINITION.
A Scale
X
of Complex Banach Spaces is a topolo-
gical vector space, obtained from the union of a family of Complex Banach Spaces (a)
,
V(P(.,>)
in a neighborhood of
neighborhood of 2.7
the map
is a neighborhood of p(zo)
COROLLARY 1.
for
is a neighbor-
is an integrable distribution with the (P) property, where V(p(zo))
2.6
W(zo)
.(.,a)
z(x,a):
fi
where
If the system ( S )
has analytic solution
then
h € H
z o , and for each
is LF-analytic.
E W(zo),
z
z,
f(zo)H
+
We take
V(zo),
nerality, like a product of neighborhoods of
then z o .
we define two
without lost of gep(zo) in
f(eo)H
and
ABSTRACT FROBENIUS THEOREM
in
q(zo)
4
X
F(a) = p(a)
G: V(zo)
3
X
G(a) = z(p(A),
F(zo) = z
and
G(zo)
= a
(GoF)(a)
2.10 REMARK.
Let
X
= z
A-p(A) 0
P(zo)).
f
. we have
z
be a complex Banach space. = Ix
P(zo)
(F~G)(A) = A
(ii)
F’ (G(A))G’(A) A = a = z
-
+ Z(P(z,),a)
In convenient neighborhoods of
2.9 THEOREM.
If
We define
F: V(zo)
We see that
(i)
q = I-p.
S o ,
367
Then
G’ (F(a))F’ (a) = Ix
and
.
we have
0
F’ (zo)G’ ( z o ) = Ix
G ’ (zo)F‘ ( z o ) = Ix
and
.
By the Inverse Function Theorem there is a neighborhood of where
F
and
G
In a convenient neighborhood of
2.11 THEOREM.
the bracket
[
,]
for all
f‘(z)h
fixed
.
2.13 THEOREM.
is involutive if
h,k E H
= f’(z)(f(z)k)h
is differential of
Let
we have
z E
and for all
U , where
is
[f(z)h,f(z)k] and
B
A distribution
E f(z)H
[f(z)h,f(z)k]
z
= f(zo)H.
F’(a)f(a)H 2.12 DEFINITION.
zo
are inverses to each other.
B
f
-
f’(z)(f(z)h)k
with respect to
be a distribution and for all
have a continuous projection
p
onto
f(zo)H.
h
for
z
zo
E U
If for all
we
zo
the map pof(z): is invertible, where
z
H
-P
f(zo)H
belongs to a neighborhood
V(zo)
and for
368
each
REINALDO SALVITTI
h E f(zo)H
is analytic then bracket where
we have that the map
69
is an involutive distribution if only if the
[@(z)h,$(z)k]
belongs to
f(z)H
for all
h,k E f(zo)H,
= f(z)[pof(z)]-'h.
g(z)h
2.14 THEOREM.
Let
of Theorem (2.4).
69
be a distribution with the same hypotheses
Then the second term of = f(z)[pof(z)]-lh
z'(x,a)h
satisfies the integrability condition. 2.15 COROLLARY.
On integrable distribution with the property (P)
is involutive.
2.16 THEOREM. such that
Let
a9
be an involutive distribution and
is a continuous projection onto
p
f(zo)H.
z E U Suppose
as well that H -+ f(zo)H
paf(z): is invertible for each h E f(zo)H
z
belonging to
V(zo)
and for each
fixed, the map Vbo) z
is analytic,
+
H
+
[pof(z)]-'h
Then the second term of the equation z'(x,a)h
= f(~)[pof(z)]-~h
satisfies the integrability condition for all 2.17 LOCAL FROBENIUS THEOREM
z E V(zo).
FOR A DISTRIBUTION IN A SCALE OF
BANACH SCALES
We can now apply the Theorem (1.7) to an involutive distri-
369
ABSTRACT FROBENIUS THEOmM
bution. and
u
X =
lie assume that if
Xs
is a Scale of Banach Spaces
wss 1 H
is closed vectorial subspace then
H
is also a Scale of
Banach Spaces, in fact
H = lim (H n
xS).
-4
THEOREM. and
r9
Let
u
X =
be an involutive distribution in
Xs
a Scale of Banach Spaces.
U c X
U,
open
We suppose that for
Oiss1
each
z
E U
there is a continuous projection
p
onto
f(zo)H
such that the map
H
pof(z): is invertible for
E V(zo)
z
=
u O<Sd
h E f(zo)H
Bs(zo,R)
and for each
1
the map
is LF-analytic.
Consider the system
(
PROOF.
f(zo)H
-4
z'(x,a)h
= f(z)
[pof(z)]-lh
B y (2.15) we have the integrability condition for z'(x,a) Applying (1.7) and (2.5) we have that
= f(z)[pof(z)]-'h.
19
=
is a
inte gr ab le di s tributi on. 2.18 THEOREM.
Let
Q,
S c X , be a Lie Group and
algebra and as well as a closed subset of distribution in the group
Q.
S
If 5
X.
Let
H
a Lei Sub-
L(z)H
be the
where
L
is an infinitesimal transformation of
and
q
are analytic maps
370
REINALDO SALVITTI
2.19 COROLLARY 1.
The distribution
2.20 COROLLARY 2.
We suppose that for each
continuous projection onto
f(zo)H
z E
is involutive. zo E tJ
there is a
such that the map
H + L(zo)H
poL(z): is invertible for each
L(z)H
Then the second term of the
V(zo).
equation
z' (x,a)h = L(z)[poL(z)]
-lh
satisfies the integrability condition.
3 . FROBENIUS THEOREM
- GLOBAL FORMULATION
First of all we suppose that we are always with the (2.5) hypotheses.
Then we have an integrable distribution
the (P) property.
Then f o r a neighborhood of
zo,
8
zo
in
E U,
U
with
the in-
tegrable manifolds are solutions of = f(z)[pof(z)~-~
z'(x,a)
= a
z(p(a),a)
zo E U
We will prove that for each maximal.integrable manifold
3 ZO
manifold of
19
,
passing through
there is a connected
i.e., every connected integrable
zo
.
is contained in g Z 0
3.1
A
NEW TOPOLOGY We define for
U
a new topology
of neighborhoods at each point A neighborhood of
zo
z
E U
TN
by a complete system
that will be denote by
is a subset of
U
b,.
that contains the
371
ABSTRACT FROBENIUS THEOREM
z = z(x,z )
image by
of a neighborhood of
p(zo)
in
f(zo)H,
being the solution of the system (S) with the initial
z(x,zo)
condition
= zo
z(p(zo),zo)
zo
(1) For each
,
.
we have
#
b,
because the system (S)
@
0
has solution. For each
(2)
and
z
V
E IJ
zo E V
we have
because
20
= zo
Z(P(ZO),ZO)
(3) If
V E b
*
and
then
V’2 V
V‘E b z
zO
(4) I f
and
V
0
W
I I ~ then
belongs to
.
V
n
W
belongs to
0
because the solution
z(x,zo)
is injective.
I’ZO
(5)
We will prove that if
V E IJ
there is
0E b
that
and
0t V
0 E bz
such ZO
zO
z E 0.
for all
The proof of
this will require two lemmas. 3.2 LEMMA.
Since
z1
we have that
in MZo when x
belongs to a convenient neighborhood of
-
(Fo@~)(x) F ( z l )
is taken in a convenient neighborhood of
f(zl)H.
of
is the map defined in (2.8),
p1
is a continuous projection onto
@l
is the solution of (S) replacing
z1
f(zl)H, p
and
zo
by
p1
and
respectively.
We denote
C $ ~ ( X , Z ~=)
@l(x)
and
z(x,zo) = z(x).
By the Theorem (2.11) we have F‘ (a)f(a)H
where
pl(zl)
f(zo)H
Here
F
PROOF.
has values in
z
5
= f(zo)H
belongs to a convenient neighborhood
Let
V(p(z,))
V(z,).
be a convenient neighborhood in
f(zo)h
such
372
REINALDO SALVITTI
Applying
q = I-p
to the two terms of the last equation we
have
Since
(qOFo@l)(xl)
-
(qoF)
is connected, we have, replacing
(2,)
x1
= constant, because by
w(P,(Z,))
P1(Z1),
and therefore
3 . 3 LEMMA. for all
There is a neighborhood
zl,
z l E z(V(p(zo))),
V(p(zo))
in
f(eo)H
such that
there exists a neighborhood
1=
W(P1(Z1)
1
PROOF.
By the Theorems ( 2 . 9 ) and (2.11) we have a neighborhood
V(zo)
in
f(zl)H
such that for
such that
a E V(zo) F(a)f(a)H
and
@l(w(P1(zl)
= f(zo)H
z(v(P(zo))
1.
ABSTRACT FROBENIUS THEOREM
373
(GoF)(a) = a.
(3.2) l e t
As in
and
z1 E
V(p(z,))
z(V(p(zo))).
Let
+
The maps neighborhood
and
V(0) c V ( p ( z o ) ) @1
and
f(zo)H
+
F(zl)
such t h a t
V(0) C V ( z o ) .
a r e continuous hence t h e r e i s a
Fool
W = W(pl(zl))
z ( V ( p ( z o ) ) ) c V(zo)
x1 = p ( z l ) .
W e denote
be a n e i g h b o r h o o d i n
V(0)
x1
such t h a t
in
f(zl)H
such t h a t
@,(W)
c V(zo)
and
-
( F o @ ~ ) ( W ) F ( z l ) C V(0). Then
+
C)F(zl) (Fo@~)(W
Applying
G
t o b o t h t e r m s we h a v e
L e t u s s e e what i s
Taking
V(0).
t E V(0)
G(F(zl)
+
V(0))
we h a v e
G ( F ( z l ) + t ) = G(X1+Zo-P(zo)+t) = = z ( p ( xl+zo-p(
2 o)
+ t ) ,xl+z .-P ( z o) + t -P ( X1+ZO'P ( zo)+ t1-P ( z o ) 1 = = z(xl+t,z
= z(xl+t,xl+zo-p(zo)+t-xl-t+P(zo)) Then
G(F(zl)
+
V(0) = z ( x l
=
+
0
) = z(xl+t).
V(0),zo) = z(xl + V(0))
hence
Z(Xl+V(0)) c Z W P ( Z 0 ) ) ) .
Now i t i s s i m p l e t o p r o v e t h e p r o p e r t y 5 . Let
V E bz0,
such t h a t f o r each
m,(w)
c Z(V(P(Z,))),
by t h e ( 3 . 3 ) t h e r e i s
zlE
0
there i s
(recall
w
b
0 = z(V(p(z,)))
W = W(p1(zl))
I*
z1
with
c
V
374
REINALDO SALVITTI
With those 5 properties it is defined a new topology in that we denote
3.4 THEOREM. PROOF.
IN.
(U,TN)-t U
The identity
Let
V(zo)
is continuous,
be a neighborhood of
The solution of the
zO.
system (S) is continuous hence there is a neighborhood such that
z(V(p(zo)))
borhood of
3 . 5 THEOREM. $4
in
U.
Let
c V(zo).
The set
z(V(p(zo)))
V(p(zo)) is a neigh-
(U,IN) hence the inclusion above is continuous.
in
zo
U
M
be an integrable manifold of the distribution
Then the inclusion
is continuous
(',IN)
.
T o prove this theorem we u s e two lemmas like ( 3 . 2 ) and (3.3).
3.6 COROLLARY. system (S).
For each
The map
E U
z
let
z
be the solution of the
restrict to a neighborhood of
z,
continuous one taking values in
p(zo),
is
(U,TN).
As a consequence of the last corollary there is a fundamental system of connected neighborhood in ly convex space.
(U,TN)
because
(U,TN)
is a local-
(U,IN) are open.
Then the connected components of
Now we consider in
H
the connected component 3
for zO
each
zO.
5,
It is easy to prove that
is an integrable manifold 0
of the distribution
$4
in
U.
is a
From the last theorem if MZO
connected integrable manifold which pass through connected in
(U,TN),
so
c 3,
M 20
0
.
zo
then
M,
is 0
Then 3
integrable manifold of the distribution
is the maximal zO
Q.
3 . 7 THE EXISTENCE OF A MAXIMAL INTEGRABLF MANIFOLD IN SCALE OF BANACH SCALES. The existence of a maximal integrable manifold came from the solution of the system (S) in the neighborhood of each point z o
375
ABSTRACT FROBENIUS THEOREM
of an open set
Then under, the hypotheses of (2.17)) an invo-
U.
B
lutive distribution
in a Scale of Banach Spaces is integrable
z E U
and for each point
manifold which contains
3.8 REMARK.
1.
there is a connected maximal integral
0
z
0
.
We constructed the maximal integral manifold based
on the existence of solutions of the system (S) for each where the distribution is defined. of a family of projections
,
pz
z C U 0
I t was necessary the existence
z E U.
The Theorem (3.5) shows
that the existence of a connected maximal integral manifold does not depend on the family of projections used, i.e., and
p;
,
zE U
if
pz
,
zE V
are two families of projections satisfying the
hypothesis of Theorem ( 2 . 5 ) then they produce the same connected maximal integral manifolds. 2.
If
X
is a complex Banach Space any family of projections
satisfies the hypothesis of Theorem ( 2 . 5 ) .
We will see this proof
in another paper.
4. APPLICATION TO A LIE GROUP 4.1 THEOREM.
Let
X
be a complex, Hausdorff, sequentially com-
plete, locally convex sapce and
S,
S c X, a Lie Group.
be a Lie subalgebra of
X
with topological supplementary.
sider the distribution
&
of
s
giving by
L(z)H,
S.
the infinitesimal transformation of the group that for each that
zo E S
there is a projection
p
where
Let
H
We con-
L
is
We also suppose onto
L(zo)H
such
376
Rl3INALDO S A L V I T T I
H
poL(z): is invertible for all
( @(x,a)h
z E W(zo)
where
V(p(zo))
and the system
= L(d)CPOL(@)I
has holomorphic solution
h E L(zo)H
-lh
@(.,a)
is a neighborhood in
S
z E
Then for each point
belongs t o is a Lie subgroup
of
L(zo)H.
there is a connected maximal in-
2,
tegral manifold and that one which
PROOF.
L(zo)H
-t
the unit o f the group,
q.
We have to prove only that the connected maximal integral
manifold who passes through
2
is a Lie Subgroup of
Q.
We need
two lemmas.
4.2 LEMMA. in (3.1).
Let
Q
be endowed with the new topology
Then for each
9
is continuous, where PROOF.
a E 9,
the translation
is the operation o f the group
First of all we will see that if
o f the distribution
In fact, if x E Y,
a,
Y
z(x)
19
s N defined
then
aN
is a integral manifold
N
is also a integral manifold o f
is a parametrization of points of
parameter space.
Consider
N,
8.
where
@(a,z(x)),
From the Lie equations of the group is an integral manifold we have
S.
Q
and the fact that
N
ABSTRACT FROBENIUS THEOREM
Then the tangent space at an integral manifold of Let in
(S,lN).
L($(a,z)H)
and
aN
is
S
and
be a neighborhood of a
W
zo
From ( 3 . 5 ) we know .that the inclusion
is continuous. such that
is
a9.
belong to
z
az
377
Hence there is a neighborhood
in L(zo)H
c W.
$(a,z(V(p(zo)),zo))
Since
V(p(zo))
z(V(p(zo)),zo)
is a neighborhood of
in
z
(B,SN)
the translation (S,'N) a
*
(S,'N)
+
az
is continuous.
4.3 LEMMA.
The connected maximal integral manifold
Let
x, y
belong to
Me.
From (4.2)
integral manifold which passes through and
x
xMe
X.
Since
so
x'he
is a connected x E Me,
xMe C M e
xy E Me. Analogously
-1
is a group
S.
with the operation of PROOF.
Me
x'he
2
contains
C
Me
therefore
E Me.
PROOF OF THE (4.1). e = 0.
If p
We denote
n
We may, without lost of generality, suppose
is a projection onto = p-'(V(p(e)))
L(e)H
= H
and we define two functions
inverse to each other,
f(z) = z
we have
- @(P(.))
g ( w ) = @(P(W)
+
w
-
P(Z>
P(W))
p(e) = e. f
and
g,
FEINALDO S A L V I T T I
where
@(x)
= a)(x,O). = I,
f o g
Because
and
f
g
= I
g o f
n '
are continuous there a r e
a l o c a l group [ 2 ]
(fll,U,V,W)
and
around
0,
U,V,W,W
=
n n
G,
where
= f[$(g(z),g(w))l*
$,(Z,W)
W e have
z E
but f o r
V(p(e))
Taking
g(z) = @ ( z ) .
w e have
h E H
g'(0)h = ,$'(O)h
= L(@(O))(poL(a)(O)))"h
= h
theref o r e L1(Z)h
As
(fog)(z) = z ,
=
f'
z E
(g(z))L(g(z))h.
n,
w e have
f ' ( g ( z ) ) g ' ( z ) H = H.
and s o
Then
and s o where
L1(z)H
= H
C1(z)H
= H,
z E V(p(e)). So i f
z
and
w
b e l o n g t o a neighborhood o f
2
in
H
we
ABSTRACT FROBENIUS THEOREM
have
L1(z)dil(w)H Take
C
H (*).
and
z
379
in a neighborhood of
w
2
in
H,
h E H
and
consider the systems
where
TI
is the projection,
and
[6
(k’fk)rrl
(x,~)
The solution of
z
-1
(S2
= el(z,e) =
From (*) each term of the series above is in H
i s
closed
zw
and
z
-1
belong to
We then have a local group G1 =
Also the map
g
H
and since
H.
G1 c H,
($l,ul,vl,wl).
is a local homomorphism between
Consider the local group in
S,
[6],
(Z,g(U1) ,g(vl) ,g(w1)
1
S1
and
S.
380
REINALDO SALVITTI
2
and the topological group that passes through
u
N =
We can take so
N
U1
u
=
[g(u1)ln
n2 1
n21
U
conncected then follows that
is connected too.
Since the set
2,
tegral manifold which passes through manifold which contains The set open in
N
(S,lN).
closed in
ze,
ag(U1)
a€[ g(ul)l n-l
e.
g(U,) N
Hence
N C 3,
3,
because
is open in
g(U,)
is connected
is a connected inis a connected integral
. g(U,)
= $(V(p(e)))
is
u aN is open therefore N is aE3 e/N Ze is connected N = 3,. So 3, is a to-
The set
since
pological group. Now we prove that the operations
3,x3, (Z,W)
3e z
-t
3,
-t
zw
+
+
e' z-1
are analytics. Let with
z
E 3,,
wo E
z ( t ) = z o , w(To)
and
~ ( t ) , w(T),
parametrizations
= w0
The map (tSl.1 + PO(8(Z(t),W(t)) is analytic, where
po
is a projection onto
Therefore the map Analogously the map
( z , ~ )+ zw z + z-l
L(zowo)H.
is analytic. is analytic.
ABSTRACT FROBENIUS THEOREM
381
REFERENCES
1.
PISANELLI, D.,
An example of a infinitive Lie group.
Am. Math. SOC. 6 2 (1976) no 1, 156-160 2.
PISANELLI, D.,
Grupos Analiticos Finitos de TransformagEes,
Publicaggo da Sociedade Brasileira de Matemdtica An6lise de
3.
Proc.
(1977).
-
Escola de
1977.
PISANELLI, D.,
Theoreme dtOvcyannicov, Frobenius et groupes
de Lie locaux dans une Qchelle d'espaces de Banacli,
C.R.A.
S. Paris, 277 (1973).
4.
LANG, S.,
5.
PISANELLI, D.,
Differential Manifolds,
Addison, Wesley D.C.,
1972.
Linear Connected Subgroup of a Lie Group i n a
locally convex space.
Anais da Academia Brasileira de Cicn-
cias (1979), 51(4).
6.
PISANELLI, D.,
Sulltintegrazione di un sistema di differen-
ziali totali in uno spazio d i Banach.
Academia Nationale
dei Lincei, Rendiconti della Classe di Scienze Fisiche, Matematiche e Naturali Serie VII, Vol. XLVI, fase
Instituto de Matemgtica e Estatistica Universidade de S g o Paulo
CX 20570
-
Aggncia Iguatemi
S g o Paulo, SP
-
Brasil
6, giugno 1969.
This Page Intentionally Left Blank
Functional .4nulysis, Holomorphy and Approximation Theory 11, G I . Zupata ( e d . ) 0 Ekevier Science Publishers B. K (North-Holland), 1984
383
OPTIMIZATION B Y LEVEL SET METHODS.
11: FURTHER DUALITY
FORMULAE I N THE CASE OF ESSENTIAL CONSTRAINTS
Ivan Singer
ABSTRACT Let
F
h: F
and
for some
F
W e show that if
(this happens e.g.
for
I
GS = ( $ E F"
I
a functional.
y' by the elements of
141
( $ E GS
E
-I
y'E
mation of lae of
be a non-empty subset of a real locally convex space
G
inf h(G),
h(y') < inf' h(G)
in the theory of best approxi-
G),
then, in the duality formu-
one can replace the set
$ f 0, sup $(G)
.
iii) and iv) follow from [4], Theorem 1.5 and ii) above. the proof of v) is similar to that of Theorem 1.3 v).
Finally,
This com-
pletes the proof of Theorem 1.5. REMARK
1.4. From Theorems 1.2 and 1.5 there follows
milar to Theorem 1.4. in [ 3 ]
, Theorem
a result si-
Such a result has been proved, essentially,
2.3 and its proof.
396
IVAN SINGER
1.4
RESULTS OF WEAK DUALITY IN TERMS OF SUPPORT HYPERPLANES OF G
sup $(GI E
If we have (1.24), then, since
a(c),
there holds
also
Thus, it is natural to ask whether the opposite inequality holds (similarly to the obvious inequalities
2
in the preceding
formulae of weak duality), and whether one can replace in Theorem 1.5 the closed strips
B -$,G
the support hyperplanes
I
= EYE
H
$ ,G
$(Y)
(see
c41)
by
defined by (1.12) or by the support
hyperplanes
";I
,G =
Since the set hyperplanes (1.12), Indeed, even when
I
{Y E F
GS
$(Y) = inf
JI(G)I.
(1.121)
is too large and since for
$ E GS
the
(1.129) are too Ifthinff, the answer is negative. G
is a closed convex set and
h
is a finite
continuous convex functional on a finite-dimensional space
F
(so
(1.24) holds), the inequality in (1.27) may be strict, as shown by EXAMPLE 1.1.
so
Let
F = R2 ,
a = inf h(G) = 1.
the euclidean plane, and let
F o r each
Jlc(Y) = 'rll
c
with
0< c < a = 1
(Y = (Tl1,q2) E
let
(1.30)
F).
Then sup
qC(c)
= -1 < inf
qC(sc) =
inf
YE F
(-ql) = -c,
II Y1I so
(1.5) holds.
However, for
Q0
=
-6,
we have
sup
q0(G)
= 3
OPTIMIZATION BY LEVEL SET METHODS
q0 E
(so
397
and
Gs)
and hence the inequality in (1.27) is strict. Nevertheless, we shall show now that for the subset
( $ E GS
I
sup $(G) < $(y’)]
of
GS,
occurring in (1.26),
the si-
tuation is different, under certain additional assumptions.
To
this end, let us first give the following generalization of [2], Lemma 2.1: PROPOSITION 1.1. of IJJ
with
F
E GS,
GS
Let
f
@,
F
be a locally convex space,
fi
h: F +
such that either i)
Aa
a functional with
G
a subset
f 0,
and
is an interval (n=1,2,...),
$(A
a + :
$(s
1) is an interval (n=1,2,...) a+n $(Aa) is an interval, or iv) G fl Sa
or ii)
and
If there exists
interval.
y’E Aa
or iii)
#
0
and
Gn
f 0
$(Sa)
is an
such that
then a = inf h(G)
PROOF, i)
Take
E $(A by
l). a+n $(gn)
By
gn E G
2
inf h(Y). YE F 1 (Y)=suP $ (G)
such that
y’E Aa C A
h(gn) < a +
we have
sup $(G)
1 n,
$ (g,)
E
and (1.31), and since
1), whence, a+n $(A 1) is an interval, a+;; ‘I
we obtain SUP $(GI E t$(gn),$(Y’)1
Thus, for each such that
so
$(Y‘) E $(A
a+n C
(1.32)
n
there exists
$(yn) = sup $(G),
ii) is similar.
=
$(A
(1.33)
1)’ a+n
yn E F
with
which proves (1.32).
h(yn) < a
1
+ 5,
The proof of
398
IVAN SINGER
iv)
Take
we have
6 n sa,
yo E
$(Y' ) E $(Sa),
and (1.31), and since
SO
whence, by $(Sa)
Thus, there exists
n o(sa).
$(yo) s SUP
BY
Y'E
$(6) =
c
sa
SUP $(G)
is an interval, we obtain
ylE F
with
h(yl)
which proves (1.32).
= sup $(G),
$(yl)
$(yo) E a ( 6 )
5
a,
such that
The proof of iii) is si-
milar, which completes the proof of Proposition 1.1. We shall also need the following proposition and corollary, corresponding to
[&I,
Proposition 1.1 and Corollary 1.1 respective-
ly : PROPOSITION 1.2. of
F, h: F + i-it)
F
Let
r?
be a locally convex space,
a functional,
If either (1.4) or
c E R
with
Ac f
a subset
G
a,
$,
and
E
F".
1.41) holds, then
I inf
h(Y
).EF
oc(Y)=suP JlC(G) ii)
Conversely, if
y'E Ac
Q ~ ( A ~ )is an interval and there exists
such that SUP
s OC(Y'
1,
(1.36)
and if we have
then
(1.4) holds.
i i t ) If
qc(Ac)
is an interval and there exists
y'E
Ac
such
that
and if we have
(1.37')
OPTIMIZATION BY LEVEL SET METHODS
399
then (1.4f)holds. PROOF.
Similarly to
[&I,
proof of Proposition 1.1, it is immediate
that we have (1.37) if and only if
i-if) If (1.4) or (1.41) holds,then obviously, we have (1.38). Alternatively, i-if) follows also from and ii)
SUP Jlc(G), inf $,(GI
-
E ?Jc(G)
[4], Proposition 1.4 i). By
y'E
and (1.36), (l.38),
Ac
and since
Jlc(Ac)
is an in-
terval, we obtain (1.4) (whence, in particular, sup Qc(G) < Qc(y' Finally, iil) is equivalent to ii), considering
QL =
>.
-Jlc
This completes the proof of Proposition 1.2.
F, G
Even when
and
h
have "nice" properties, as in
Example 1.1, one cannot omit the assumptions (1.36) and (1.361) in Proposition 1.2 ii) and i i f ) respectively, as shown by EXAMPLE 1.2.
Let
F, G
and
h
0 < c < 1 = a = inf h(G)
be as in Example 1.1 and for each let
c
with
so
we have (1.35), but neither (1.4), nor (1.41).
The same example
motivates also the assumptions (1.36) and (1.369) in the following corollary of Proposition 1.2: COROLLARY 1.1. of
F,
h: F +
Let
F
be a locally convex space,
a functional,
c E R
with
Sc
#
0,
G
a subset and
$, E GS.
400
IVAN SINGER
If either (1.5) o r (1.51) holds, then we have (1.35).
i-it) ii) y'E
Conversely, if
$,(SC)
is an interval and there exists
satisfying (1.36) and if we have
Sc
(1.40)
then (1.5) holds.
If
ii')
is an interval and there exists
$,(SC)
y'E
sa-
Sc
tisfying ( 1 . 9 6 1 ) and if we have
then (1.5') holds. PROOF.
Parts i), i f ) follow from
Ac
C
Sc
and Proposition 1.2 i)
and it) respectively. The proof o f part ii) is similar to that o f Proposition 1.2 ii),
observing that if (1.40) holds, then
SUP fc(G)
f
(Y E s c ) .
$,(Y)
Finally, iil) is equivalent to ii), considering
Jr/c = -$,.
This completes the proof of Corollary 1.1. Now we are ready to prove THEOREM 1.6. h: F
and a)
be a locally convex space,
a functional with
-t
is F*-connected
A
a+n (n=1,2
,...) ,
d)
F
Let
G' fl
or
Sa f @
c)
and
G
and
a subset of F
6 , such that either
(n=l,2,...), o r b)
rl Aa f d,
Sa
Aa f
G
Aa
is 8'"-connected.
S a+n
is F*-connected
is F*-connected, Let
or
y'E
Aa
and con-
8, E
GS
satisfying
sider the following statements: lo.
(1.4).
F o r each
c E
(h(y'),a)
there exists
401
OPTIMIZATION B Y LEVEL SET METHODS
.'2
c E [h(y'),a)
For each
6 , E GS
there exists
satisfying
(1.5). . ' 3
There holds inf h(G) =
SUP inf $EGS YEF SUP 6 ( G k Q(Y' ) Q(Y)=suP
(1.41)
h(Y) '
6 (GI
11-3f, obtained from lo-3O similarly to the corresponding p r o cedure of Theorem 1.3. i)
We have the implications 2'
If the sets
ii) then
' 1
then lo
PROOF. a
2O
t)
Ac
with
c E
Sc
with
c E [h(y'),a)
nc
(n=1,2,3).
o
i) The implication '2
.'3
3O.
(h(y/),a)
are F*-connected,
are F*-connected,
3O.
We have no
iv)
' 1
t)
a
3O.
e
If the sets
iii)
a ' 1
By a), b),
sup Q(G) E
lo is obvious (since
Ac
C
Sc).
c) or d), y'E Aa, lo, Proposition 1.1 and
we have
inf h(G)
inf
;r
h(Y)
(1.42)
9
whence w e obtain 3 O (by Theorem 1.5). ii)
Assume that the sets
and that 3' Q c E GS
since
Ac
holds.
Ac
with
c E
(h(y'),a)
are F*-connected
Then, by (1.41), for any such c < a there exists
satisfying (1.11) (whence (1.36)) and (1.37).
Hence,
is F*-connected, by Proposition 1.2 ii) we obtain (1.4).
402
iii)
IVAN SINGER
If the sets
Sc
with
c E [h(y'),a)
are F*-connected and if
3' holds, then, similarly to the above proof of ii), using now corollary 1.1 ii), we obtain (1.5). Finally, the proof of iv) is similar to that of Theorem 1.3 v). This completes the proof of Theorem 1.6. REMARK 1.5.
Geometrically, formula (1.41) of Theorem 1.6 means
that inf h(G) =
sup
(1.44)
inf h(H),
HExG ,y'
where
#G,yl
denotes the collection of all hyperplanes
which support G Y'
H
in
and have a translate separating strictly G
F from
. Finally, let us make some complementary observations to
Proposition 1.1, collected in REMARK 1.6.
Under the assumptions of Proposition 1.1, but replacing
(1.31) by the stronger condition
(1.45)
Moreover, replacing (1.45) by the stronger condition
we have even
Indeed, if (1.45) holds, then f o r any
y'E Aa
and hence, by Proposition 1.1, we obtain (1.32). the inequality 2
we have (1.31)
Furthermore, if
in (1.43) is strict, then there exists
yo E F
OPTIMIZATION BY LEVEL SET METHODS
JI(G),
$(yo) E
with
403
such that
inf yEF $ (Y)=suP
h(Y)
’ h(Yo)*
(1.49)
Q (GI
But, by (1.32) and (1.49), we have
yo E Aa
and hence, by
(1.451,
SUP @(G)
inf @(Aa)
On the other hand, by 5
sup $(G),
5
whence, by (1.49), we obtain
contradiction with (1.50).
(1.50)
$(Yo).
$(yo) E $(G) we have
5
$(yo) < sup $(G),
This proves (1.46).
holds, then we have (1.45), whence also (1.46). by (1.47) and Proposition 1.2 i) (with
$(yo)
c=a)
in
(1.47)
Finally, if
On the other hand,
we have the opposite
inequality to (1.46) and hence the equality (1.48) (alternatively,
(1.48) also follows from (1.32), (1.43), (1.47) and [ 4 ] , Proposition
1.4 i) with
c=a).
Moreover, under some additional assumptions
(see Proposition 1.2 ii) and [4], Proposition
1.4 ii)), one can also
give results of converse type.
$2.
RESULTS OF STRONG DUALITY
2.1
RESULTS OF STRONG DUALITY IN TERMS OF CLOSED HALF-SPACES CONTAINING G
THEOREM 2.1. and
h: F -+
Let
0.
+
424
GERALD0 S O A R E S DE SOUZA
PO 1
- I(Tfl1; P
-I P-Po
1
p
MI
u
,
- P'P1 (7
B
where
is a
We get PO
(5.4)
III
PI-P
u = B 1I-F
is arbitrary we may take
constant to be determined.
Since
P-PI
P1
- P-Po
1
u
As
P-Po
MO
1
Mo
P IlTflIpP
P-Po
P1 M1
=P-P0
+ - P1-P B
B > 0 is arbitrary, replace B
P'P1
by the value that makes the
expression minimal, namely take
Substitutin in (5.4)
. p1-p
Po
5.P-Po
P1-Po Mo
Po
Observe that P 1-t =
and
PI
P1-P P1 .+ -.-P1-Po P
P-Po
.-
-
P
P1-Po'
( 5 . 5 ) becomes
IITfllp
k
h(t) =
if
Z ' cnbn(t) n=1
p-atoms, that is,
bnls
then
P-Po P1'PO -= p
- 1.
- + - lmt Po
p1
. M1
Taking
,
P1-P .-P1-Po
Po t =-.-
P
0 < t < 1,
.
(p-p,
.
Pl'PO
and thus
Consequently,
is a finite linear combination of special are equal p-atoms of type
f, we have
and thus
(5.6)
IIThllp
t 1-t m O M l ( ~ ~ h ~ l C p ) l where ~P
p
K = (p )P-Po , + P1 P
That is, the theorem is proved for a dense subspace of Cg
,
so
it can be extended to all
( 5 . 6 ) in
Cp.
Then,if
f
E Cp,
Cp,
U P
Cp,
'
namely
preserving the inequality
( 5 . 6 ) implies that
SPACES FORMED B Y SPECIAL ATOMS I1
IITfllp
l-t(l fl
KMOMl
. ' / ' )
425
The p r o o f is completed.
CP
The interested reader can find similar types of interpolation theorem in [ 81 and 191
.
REFERENCES 1.
R.R. COIFMAN,
A real variable characterization of
Hp,
Studia Math., 51 (1974) 269-274. 2.
P.L. DUREN, B.W. ROMBERG and A.L. SHIELDS, Linear functionals on Hp with 0 < p-1, J. Reine Angin Math., 238 (1969) 32-60.
3.
GERALD0 SOARES DE SOULA,
Spaced formed by special atoms,
Ph.D. dissertation, SUNY at Albany, 1980.
4.
.......................
5.
-----------------------, Spaces
and RICHARD O"EIL, Spaces formed with special atoms, Proceedings Conference on Harmonic Anal y s i s , 1980, Italy. Rendiconti del Circolo Matematico di Palermo, 139-144, Serie 11, #l. 1981. formed by special atoms I,
to appear, Rocky Mountain Journal o f Mathematics.
6.
.......................
and GARY SAMPSON,
A real character-
ization of the pre-dual of the Bloch functions, to appear, London Journal o f Mathematics.
7.
....................... acterization o f
and
------------, An
cP, In preparation.
8.
-----------------------, Two theorems
9.
-----------------------, An interpolation
10
11.
analytic char-
on interpolation of operators, Journal of Functional Analysis, 46, 149-157, (1982).
o f operators, to appear, Anais la Academia Brasileira de Cigncias 54, # 3 (1982).
-----------------------, The dyadic
special atom space, Proceeding Conference on Harmonic Analysis, Minneapolis, April 1981, Lectures Notes in Mathematics, #908, Springer-Verlag, 1981. A. ZYGMUND, Trigonometric Series, 2nd red. ed., Vols. I,II, Cambridge University Press, New York, 1959.
Department of Mathematics Syracuse University Syracuse, New York 13210
This Page Intentionally Left Blank
Fui~ctionul~4rralysis, Holornorphy and Approximation Theory 11, C.I. Zapata (ed.) @ Ekevier Sciencc Publishers R. V. (North-Holland), 1984
427
A HOLOMORPHIC CHARACTERIZATION OF c*-ALGEBRAS
Harald Upmeier
$1. INTRODUCTION
The theory of operator algebras (i.e. C*-algebras and von Neumann algebras on complex Hilbert spaces) is of increasing importance to many branches of mathematics, e.g.
integration theory,
operator theory, algebraic topology and in particular mathematical physics and quantum mechanics.
Since C*-algebras provide a natural
framework for the foundations of quantum mechanics and quantum field theory it is an important problem to characterize the class of C*algebras by certain properties, for instance motivated by physical experiments.
So far two characterizations of operator algebras in
different categories have been obtained.
The first is A. Connest
characterization of von Neumann algebras in terms of self-dual homogeneous Hilbert cones [ 81,
the second is the work of Alfsen and
Shultz [2,1] characterizing the state spaces of C*-algebras using the geometry of compact convex sets and their affine function spaces. Although the methods of these papers are quite different, approaches have a common feature:
both
the characterization of C*-al&ras
in the respective category can be divided into two steps, the first being the characterization of the larger class of Jordan C*-algebras (JB*-algebras) and the second being the characterization of (associative) C*-algebras among all (non-associative) Jordan C*-algebras. ( F o r the case of self-dual homogeneous Hilbert cones this point of
view has been adopted by Bellissard and Iochum
[S]).
The second step
428
HARALD UPMEIER
i n v o l v e s some concept of " o r i e n t a t i o n " of The g e o m e t r i c
2
Torden o p e r a t o r a l g e b r a .
o b j e c t s a s s o c i a t e d w i t h an o p e r a t o r a l g e b r a A
i n t h e p a p e r s mentioned above a r e t h e H i l b e r t cone a s s o c i a t e d w i t h A
v i a Tomita-Takesaki t h e o r y ( i f
s p a c e of ometry. A
A,
A
h a s a p r e d u a l ) and t h e s t a t e
r e s p e c t i v e l y , b o t h endowed w i t h a n a t u r a l a f f i n e ge-
On t h e o t h e r hand,
D
t h e open u n i t b a l l
h a s an i n t e r e s t i n g holomorphic s t r u c t u r e :
D
of a C*-algebra
i s homogeneous with
r e s p e c t t o holomorphic automorphisms and i s t h e r e f o r e a bounded symm e t r i c domain.
The aim of t h i s p a p e r i s t o g i v e a holomorphic
c h a r a c t e r i z a t i o n of C*-algebras
i n terms of t h e holomorphic s t r u c -
t u r e of t h e a s s o c i a t e d open u n i t b a l l . f o r J o r d a n C*-algebras
The c o r r e s p o n d i n g r e s u l t
i s t h e main theorem i n [ 6 ] .
a c h a r a c t e r i z a t i o n of C*-algebras
Here we o b t a i n
among a l l J o r d a n C*-algebras
u s e s t h e s t r u c t u r e of t h e L i e a l g e b r a of a l l d e r i v a t i o n s .
which
A similar
i d e a i s u n d e r l y i n g i n [ B ] ; t h e c o n c e p t of H i l b e r t cones however i s o n l y a p p r o p r i a t e f o r o p e r a t o r a l g e b r a s h a v i n g a Banach p r e d u a l . I n $ 2 we g i v e a s h o r t i n t r o d u c t i o n i n t o t h e t h e o r y of J o r d a n C*-alg e b r a s and t h e i r holomorphic c h a r a c t e r i z a t i o n i n c l u d i n g t h e cons t r u c t i o n of t h e J o r d a n t r i p l e p r o d u c t f o r bounded symmetric domains i n complex Banach s p a c e s .
In
$3 some p r o p e r t i e s o f t h e L i e a l g e b r a
of a l l d e r i v a t i o n s of a J o r d a n C*-algebra needed i n t h e s e q u e l .
The c o n c e p t of " o r i e n t a t i o n " o f a J o r d a n C*-
algebra i s introduced i n result,
a r e s t u d i e d which w i l l be
$4,
and
$ 5 c o n t a i n s t h e p r o o f of t h e main
s a y i n g t h a t o r i e n t a b l e J o r d a n C*-algebras
$ 2 . JORDAN c*-ALGEBRAS
AND THEIR
HOLOMORPHIC
a r e C*-algebras.
CHARACTERIZATION
J o r d a n a l g e b r a s made t h e i r f i r s t a p p e a r a n c e i n m a t h e m a t i c a l p h y s i c s and quantum t h e o r y ,
( P . J o r d a n 1932):
Let
H
s t a r t i n g from t h e f o l l o w i n g o b s e r v a t i o n be a complex H i l b e r t s p a c e .
Then t h e
A HOLOMORPHIC CHARACTERIZATION
Banach space
#(H)
OF c*-ALGEBRAS
429
H (which
of all bounded hermitian operators on
can be interpreted as (bounded) observables of a quantum mechanical system) is not closed under the associative operator product
xy,
but with respect to the anti-commutator product
#(H)
becomes a non-associative algebra.
As a consequence the anti-
commutator product of two observables has a physical interpretation whereas, i n general, the operator product does not.
The product
(2.1) satisfies the following identities: (J1)
xay = yox
(52)
x o(x0y) = xo(x
(Commutativity)
2
2.2 DEFINITION.
Let
A
2
(Jordan-Identity).
oy)
be a (not necessarily associative) algebra
over the real o r complex numbers with product denoted by all
x,y E A.
Then
A
xoy
for
is called a Jordan algebra i f the identities
(Jl) and (J2) are satisfied. Since the algebras appearing i n quantum mechanics are in general infinite-dimensional, it is desirable to consider Banach Jordan algebras.
A particularly important class of Banach Jordan
algebras are the so-called Jordan C*-algebras (i.e. JB*-algebras and JB-algebras) which have been introduced and thoroughly studied by Alfsen, Shultz and St$rmer
2 . 3 DEFINITION. and unit element
Let e.
X
131.
be a real Jordan algebra with product
Then
X
is called a JB-algebra i f
Banach space with respect to a norm x,y 6 X
X
is a
x ~ 1 x 1 such that f o r all
the following properties hold:
xoy
430
HARALD UPMEIER
I*I
I n t h i s c a s e t h e JB-norm
on
i s u n i q u e l y d e t e r m i n e d by ( i )
X
and ( i i ) . Note t h a t J B - a l g e b r a s a r e " f o r m a l l y r e a l " , i.e. 2 2 x = o f o r a l l x1 xn E X . x1 +...+ xn = o i m p l i e s x1 =...= n
,...,
The complex a n a l o g u e of J B - a l g e b r a s
2.4 DEFINITION. ZOW,
Let
u n i t element
a JB*-algebra
u,v E Z
such t h a t f o r a l l luovl
(i) (ii)
and i n v o l u t i o n
Then
Z I - Z * .
i s a Banach s p a c e w i t h r e s p e c t
Z
if
be a complex J o r d a n a l g e b r a w i t h p r o d u c t
Z
e
a r e t h e s o - c a l l e d JB*-algebras:
i s called
Z
t o a norm z - l z l
t h e following p r o p e r t i e s hold:
IUI*IVI
I;
=
IEUU*U31
IUI
3
9
where
(2.5)
{Uv*w)
:=
uo
-
(v*ow)
v*o
denotes t h e Jordan t r i p l e product o f Wright and Youngson [29,30] i n t r o d u c e d above a r e e q u i v a l e n t : Z := X'
plexification
+
(w.u)
wo
(uav*)
u,v,w E 2 .
have shown t h a t t h e c o n c e p t s Given a J B - a l g e b r a
X,
becomes a JB*-algebra
= X @ i X
t h e com-
with respect
t o a unique ftJB*-normff; c o n v e r s e l y t h e s e l f - a d j o i n t p a r t
x of a JB*-algebra JB*-algebras
:= [ x E Z : x* = x)
i s a J B - a l g e b r a under t h e r e s t r i c t e d norm.
Z
and J B - a l g e b r a s a r e o f t e n c a l l e d " J o r d a n C*-algebras"
b e c a u s e o f t h e f o l l o w i n g example.
2 . 6 EXAMPLE.
Let
be a u n i t a l C*-algebra,
Z
i.e.
Z
i s a complex
a s s o c i a t i v e Banach * - a l g e b r a w i t h u n i t s u c h t h a t product, i n v o l u t i o n and norm a r e r e l a t e d by t h e c o n d i t i o n
for all
z E 2.
Then
Z
becomes a JB*-algebra
commutator p r o d u c t ( 2 . 1 ) and t h e JB*-norm
under t h e a n t i -
c o i n c i d e s w i t h t h e C*-norm.
To s e e t h i s , n o t e t h a t f o r a s s o c i a t i v e * - a l g e b r a s product (2.5)
1 (uv*w + [ uv*w} = 2 I n particular,
t h e Jordan t r i p l e
reduces t o
[zz*z)
= zz*z.
wv*u)
.
This implies, v i a the s p e c t r a l
theorem for h e r m i t i a n o p e r a t o r s and t h e C*-condition:
A HOLOMORPHIC CHARACTERIZATION
OF c*-ALGEBRAS
431
In a fundamental paper [12] Jordan, von Neumann and Wigner classified all formally-real (simple) Jordan algebras of finite dimension.
A natural extension of this classification is the follow-
ing list of all JB-factors of type 1 [ 2 2 , 3 ] :
2.7 EXAMPLE. (i)
Let
M
denote the field
R
of real numbers, the field
of complex numbers o r the skew-field respectively. H(E)
Let
E
IH
be a Hilbert space over
C
of quaternions,
M.
Then the set
of all bounded K-linear self-adjoint operators on
E
is a
JB-algebra under the product (2.1) and the operator norm. (ii)
Let
Y
be a real Hilbert space of dimension
scalar product
(xly).
Then the direct sum
r2
with
X := R
@
Y
becomes a JB-algebra called spin factor under the product
(a1
Y,)
0
(a2 CB Y,)
:=
(ala2
+ (y11y2))
CB (aly2 +
a2y1).
The name "spin-factor" stems from the fact, that in quantum mechanics spin systems obeying the canonical anti-commutation relations are in close connection with Jordan algebra representations of a suitable spin factor as defined above. (iii)
The set
Z3(CD)
of all self-adjoint
9x3 octonion ma-
trices is a JB-algebra under the product (2.1) which cannot be embedded into an associative algebra and is therefore called the exceptional Jordan algebra.
The octonion matrices of higher
rank do not form a Jordan algebra, since
0
is not associative
[ 7 ; Ch. VIII, Lemma 8 . 2 1 . Jordan algebras and JB-algebras in particular have found several applications to quantum mechanics, for instance using the affine geometry of the state space of a JB-algebra or the projective geometry associated with the exceptional Jordan algebra.
How-
ever, even more promising seems to be the relationship between J o r d a n C*-algebras and complex analysis, more precisely the Jordan algebraic
432
HARALD UPMEIER
description of bounded symmetric domains i n complex Banach spaces. F o r the sake of completeness we give a short survey about this relationship. a mapping
Z
Given a complex Banach space f: D
a E D
each point
is said to be holomorphic if locally around
Z
-+
D C Z,
and a domain
there exists a convergent expansion m
Pm: Z -+ Z.
into a series of m-homogeneous continuous polynomials The polynomials
where
f(m)(a)
Pm
are uniquely determined by
is the m-th derivative of
ed as a multilinear mapping [18; $51.
in
f
and
f
a E D,
all biholomorphic automorphisms of a domain
D
matter of notation, denote by
Z(E)
namely
consider-
A bijective mapping which is
holomorphic in both directions is called biholomorphic.
led the autornorphism group of
a,
D
The set of
forms a group, cal-
and denoted by
Aut(D).
As a
the algebra of all bounded
linear operators on a real or complex Banach space
E.
Generalizing a well-known theorem of H. Cartan for domains in
Cn,
the author [ 2 4 ]
and independently J.P. Vigu6 [ 2 8 ] have s h m
that for a bounded domain automorphism group
D
in a complex Banach space
G = Aut(D)
The essential idea behind G
this result is the construction of the Lie algebra of set of all complete holomorphic vector fields on
5
on a domain
DC
2
6 = h(z) where
h: D -+ Z
cates that
6
the
carries in a natural way the
structure of a real Banach Lie group.
vector field
Z,
D.
as the
A holomorphic
can be written as
a , az
is a holomorphic mapping and the symbol
a az
indi-
is viewed as a holomorphic differential operator,
associating to each holomorphic mapping
f: D -+ Z
the holomorphic
A HOLOMORPHIC CHARACTERIZATION OF c*-ALGEBRAS
mapping
5f: D
-I
Z
f’(z) E X ( Z )
where
433
defined by
is the derivative of
f
in
z
E D.
Given
another holomorphic vector field
the commutator vector field
has the characteristic property
for all holomorphic mappings vector space
3(D)
f: D
-t
2.
It follows that the complex
of all holomorphic vector fields on
a Lie algebra under the commutator product.
D
Regarded as an ordina-
ry differential equation, each holomorphic vector field on nerates a local flow on
D.
A
becomes
ge-
D
vector field is called complete if
D.
this local flow can be enlarged to a global flow on
An equiva-
lent formulation is given in
2.8 DEFINITION.
A holomorphic vector field
5
= h(z)
aZ
on
D
is
called complete if there exists an analytic real 1-parameter group tl-gt
E Aut(D)
for all
z E D.
satisfying the differential equation
In this case the transformations
determined for all
since locally
gt
t E R
aut(D).
are uniquely
and are denoted by
is given by an exponential series.
The set of
D
is denoted
all complete holomorphic vector fields on a domain by
gt
For domains i n general,
aut(D)
is not closed under
434
HARALD UPMEIER
In case
formation of sums and commutators. aut(D)
Q:=
D
is bounded, however,
Satz 2 . 6 1 .
9
Moreover
nential mapping
r24;
3(D)
turns out to be a real Lie subalgebra of
is a Banach Lie algebra and via the expo-
5-exp(la5),
this Banach Lie algebra induces on
the structure of a real Banach Lie group with Lie algebra j.
Aut(D)
As a consequence of Liouvillels theorem, we have
9 n is, =
0.
A crucial property of bounded domains is the following result.
2 . 9 CARTANIS UNIQUENESS THEOREM. g E Aut(D)
Suppose some
PROOF.
a E D.
f
of
g
a =
g f idD.
Assume,
0.
o
o
of order
for
Choose
k
2
5
there is an expansion
h
By induction, the n-th iterate
>k.
gn
has the expansion
= z
gn(z) where
g’(a) = idZ
is a k-homogeneous continuous polynomial and
0
vanishes in
and
be a bounded domain.
g = idD ’
Then
We may assume
pk
g(a) = a
satisfies
minimal, such that around
where
Dc Z
Let
>k.
h’ has order
bounded domain
D
+
+
nPk(z)
h‘(z),
Since the transformations
gn
leave the
invariant, it follows from Cauchyfs inequalities
[18; 5 6 , Prop, 31 that geneous polynomials.
{nPk : n
2
This implies
O}
is a bounded set of k-homo-
Pk =
0,
a contradiction. Q.E.D.
By differentiation, Theorem 2 . 9 implies that each vector field
5 =
f o r some
h(z)
aE a Z
a E D
aut(D)
satisfying
h(a)
= o
and
h’(a)
= o
must vanish identically.
The possibility of describing a bounded domain ically (e.g. in terms of
Aut(D)
or
aut(D))
D
algebra-
can only be expected
HOLOMORPHIC CHARACTERIZATION OF c*-ALGEBRAS
A
if
D
435
admits sufficiently many holomorphic automorphisms. The class
of bounded symmetric domains defined below fulfills this requirement in an ideal manner. 2.10 DEFINITION.
D
A bounded domain
a E D
is called symmetric if for each point s E Aut(D) a
~'(a) = -idZ ' a
By Theorem 2.9 the automorphism a,
is uniquely determined.
bounded symmetric domain
D,
transitive on
D
s a , called the symmetry
Moreover,
D
is homogeneous, i.e.
and
a
is an
Aut(D)
is
and is biholomorphically equivalent to a bounded
satisfying
o E D
and
e
it
D = D
mains of this kind are called circular).
D
homogeneous domain
o E D
2 a = idD
s
s a . It can be shown [28,14] that each
isolated fixed point of
domain
there is a mapping
with the following properties: s (a) = a, a
in
Z
in a complex Banach space
for all
t E R
(do-
Conversely, every circular
is symmetric, since the symmetry
can be transported to any other point in
D.
zu-z
at
In particular,
is a bounded symmetric domain. It turns out that there is a natural way of associating with each bounded symmetric domain a Jordan triple product generalizing the triple product ( 2 . 5 ) .
The following Lemmas are the crucial
steps towards this algebraic construction. 2.11 LEMMA.
D
Let
field
5 = h(z)
h = h
+ hl + h 2 ,
polynomials.
a=
E
be a bounded circular domain.
9
:= aut(D)
where
Then each vector
is polynomial of degree
hk: Z + Z
4 2 , i.e.
are k-homogeneous continuous
Moreover, the vector fields
436
HARALD UPMEIER
belong to
3
and there is a Cartan decomposition
y = 4 e p
(2.12) where
4
PROOF.
[s,
:=
Since
:
D
be the expansion of vector fields
]
,b :=
and
5,.
since
:
5 E y].
a E 9 az
.
Therefore m
is a Lie algebra.
$4
5 =
Let
5,
C m= o
3
5 E
(5-,
6 := iz
is circular,
:= ad(6) E C( J ) ,
8
y
5 E
I
around
into m-homogeneous polynomial
o
By Euler's relation
This implies for any polynomial
p E El[@],
m
2 p(8) = 8 ( e +l).
Choose
~(8)s = o for all
Then
p(i(m-1))
= o
as a consequence of Theorem 2.9.
m > 2, which implies
as asserted.
Further,
5,
=
2
5-,
= -0 5 E
0.
y
for
m
But
Hence
2.13 LEMMA.
Let
D
Ad(s),
f o
5 = 5 , + 5, + g 2
5,
and hence
where
hence
p(i(m-1))
I t is clear that (2.12) is a Cartan decomposition of the involutive automorphism
2,
5;
:=
s(z)
- z-,
= 5
ZXZXZ + Z
denoted by
E f.
relative to
j
Q.E.D.
-z.
be a bounded symmetric circular domain.
there is a unique mapping
,
Then
( u , a , v ) c {ua*v]
having the following properties: (i)
{ua*v]
is complex bilinear symmetric in the outer
variables and conjugate linear in the inner variable a E Z . (ii)
The subspace
p
p =
c j [(U
defined in 2.11 has the form
-
{ZU".])
Moreover, defining the operator I=
u
a v*
: u E z].
on
Z
by
(u
V*)Z
{ u v " ~ ] , the following Jordan triple identity is valid:
:=
A HOLOMORPHIC CHARACTERIZATION OF c*-ALGEBRAS
u,v,x,y E Z ,
for all
[x,~]
where
-
:= ? q ~ p?,
437
denotes the com-
mutator of linear operators. PROOF.
D
Since
is homogeneous, it follows from (2.12) that the
evaluation mapping I ) -+ Z , ive.
defined by
h(z)
By definition, the vector fields in
(u-qu(z))
=, a
where
Z + Z
9 , :
u
u
?
c p rp,p11
c
(since (since
Then (i) and (ii) are satisfied. sequence of the fact that
/J
is surject-
have the form
is a 2-homogeneous polynomial,
which is uniquely determined by depends conjugate linearly on
a. a -h(o),
D
3 n
is bounded) and iJ =
Define
0).
Property (iii) is a direct con-
is a Lie triple system, i.e.,
p.
A Banach space
Z
with a composition
{ua*v}
satisfying
properties (i) and (iii) is called a Banach Jordan triple system. Using Jordan triple systems, W. Kaup [ l ' r ] has obtained an algebraic
In particular,
characterization of all symmetric Banach manifolds.
the Banach Jordan triple system associated to each bounded symmetric domain
D
D
via 2.13 characterizes
uniquely.
A particularly
important class of Jordan triple systems are the JB*-algebras with Jordan triple product ( 2 . 5 ) .
Therefore the problem arises which
bounded symmetric domains correspond to JB*-algebras.
It turns out
that the appropriate holomorphic condition relates to the notion of tube domain. 2.14 DEFINITION. tion
Z := X
6:
.
Let Let
X
be a real Banach space with complexifica-
R C X D~
:=
be an open convex cone.
iz
E
z
:
z-z*
2i E n ?
is called the tube domain with the base the conjugation of
Z
Then the domain
with respect to
R. X.
Here
z
HZ*
denotes
HARALD UPMEIER
438
X := R
I n case
n
and
:= ( x E R : x > o } ,
i s t h e f a m i l i a r upper h a l f - p l a n e
in
( x 2 : x E X}
n.
terior
of
,
the
i s a convex cone h a v i n g non-empty
in-
i s c a l l e d t h e upper
Dn
The holomorbhic c h a r a c t e r i z a t i o n
Z.
i s g i v e n i n t h e f o l l o w i n g theorem, t h e main r e s u l t
which g e n e r a l i z e s t h e p i o n e e r i n g work by M .
2.15 THEOmM.
A bounded symmetric domain
a l g e b r a i f and only i f domain.
D
More p r e c i s e l y ,
o f a JB*-algebra
n
X,
o f t h e JB*-algebra
JB*-algebras
o f [ 61
For any JB-algebra
The a s s o c i a t e d t u b e domain
half-plane
D
T h e r e f o r e t u b e domains a r e
C.
o f t e n c a l l e d ‘!generalized h a l f - p l a n e s “ . s e t of s q u a r e s
t h e domain
D C Z
Koecher [ 171
.
b e l o n g s t o a JB*-
i s biholomorphically equivalent t o a t u b D :=
t h e open u n i t b a l l
( z E Z
: IzI
1.
Supposing that we have obtained
such that Iuj(xj)l
let
H
and
L
*,...,un}
(ul,u
> 1,
n
in
E' TO
in
E'
and
E'
F
[x1,x2,
..., n}
and
x
,...,un]
respectively.
Since [u1,u2
which is a topological complement of
X
these is a
X >
1
such that
n
H)
2 TO.
TO X(X
On the other hand,
F
i,j = 1,2,...,n,
ifj,
be the orthogonal subspaces of
generate a linear space
H
= 0,
uj(xi)
and therefore
L
+
TO
(1)
is a finite codimensional closed subspace of
T flL
is not a neighbourhood of the origin in
L
and thus there is 1
Yn+1
~22(n+l) Ln
B y using (1) we can find
u n+ 1 E To
Un+l
9
n H
and
T*
X
E K,
j = 1,2,.
..,n,
A PROPERTY OF F ~ C H E TSPACES
so that
By setting
XY,+~
= x
~
+ it ~ follows that
1
Xn+l E
~22(n+l)un+l
9
Un+l E
Xn+l @
U ~ ( X ~ +=~ U) ~ + ~ ( X=~ 0),
TO
9
j=1,2,...,ny
n
Then, sequences
c
AjUj
+
(up)
in
F
= IX(
Iun+l(xn+l)l (x,)
and
> 1.
Un+l)(Yn+l)l
j=1 and
E' can be select
such that x € - 1 U P 22P P
For every positive integer
,
xp@T,
n
we find a sequence
u € T o , P
...,k(n), ...
l(n),2(n),
of even positive integer numbers which are pairwise differente such that if
k(n)
#
h, k, m
h(m).
and
Let
n
V
m f n
then
be the linear hull of
EU2n-1
and let
are positive integers with
+
1 m\(n) n,k = 1,2,..*1 :
be its closure in
E'[u(E',F)].
The closure of
V
in
contains
E' TO
b1'U3,
* * *
lu 2n-19
*.I
and therefore 2n-1 E
U
Let
A
7,
n = 1,2,...
.
be the closed absolutely convex hull in
{ 22xl, 24x2,. , Since the sequence
(22nxn)
E
of
.,22nxn,. . .} .
converges to the origin in
E
the set
472
A
MANUEL VALDIVIA
is compact.
P
Let
be the linear hull : n = 1,2
{U2n-1
P
We take a non-zero element of
Then there is a positive integer
p,
Given a positive number
2e
of
v E V
with
E
,...] .
1
%
p
n,
6
< lapl,
so
that
a
P
f 0.
suppose the existence
such that : x E A] < E .
supfI(v-u)(x)l We can write m
c c j=1 kEP
v = with and in
m
;?
n, P j
@ j,k E K ,
u-v
1 @
j,k ('2j-1
+
"k)
j
a finite subset of even positive integer numbers
k E
,
P
.
j = 1,2,. .,m.
The coefficient of
u
2p-1
coincides with
a
-
C kEPp P p 9 k
e
+
and therefore
from where we get
lapl In u-v,
%,
the coefficient of
take an element
ek
of ' k
we have that, since
I kEP c
'p,kI '
k E P p , is
*
If we
of modulus one such that
K
k E Pp
Pp,k = l@p,kl'
Pi n P j
c kEP
e > sup(I(u-v)(x)l
1 k Bp,k
= 0,
ifj,
i,j=l,2,,..,m,
and
e k zk xk E A,
: x E A) 2
I(u-v)(
c kEP
k ek2 xk)l
2
A PROPERTY OF FRESCHET
SPACES
473
tion.
< 2 e , according to ( 2 ) , which is a contradiclap We can conc ude that P f l M = (01 if M is the closure o f
V
E’[y(E‘ , E ) ] .
and therefore
in
M
be the orthogonal subspace of
M
gonal subspace of
in
5
it follows that
E
7,
is U(E’ ,F)-dense in
M
closed in
and of infinite codimension in that space.
u(E’,E)
for
Then
F.
in
S
If
E
and let
S
be an algebraic complement of
in
denotes the closure of
R.
R
be the ortho-
R.
is o f infinite codimension in
S
Let
?
in
S
Y
Let
We take linearly inde-
pendent vectors
in Y.
denotes the closed linear hull of ( 3 ) in
Z
If
5 n
is separable and
of infinite codimension in
Z
to a) we obtain a quasicomplement subspace Then
=
G
is closed in
G
n ( 5 n z)
=
E,
G
Z.
5 n
of
E
Z
it has infinite dimension and
(03.
Z
According in G
n
Z.
F =
q.e.d. G
In order to prove the following lemma let subsets of a space
then
E
be a family of
which are bounded closed absolutely convex
and satisfying the following conditions: a.
If
is a finite part of
(2
that
A
3
If
A1,A2 E G.
C.
If
A
We denote by
G
there is
and if
E’[-i]
A3 E G.
with
then
LA E G .
A > 0
the space
Let
di tions :
F
such
be a subspace of
A3
3
A1 U A2.
endowed with the topology of the
E’
uniform convergence on the elements of LEMMA.
A E G
$.
b.
E
there is an
E
E
G.
satisfying the following con-
474
1. 2.
Then,
MANUEL VALDIVIA
n A EA n F
F
F
is of finite codimension in
E‘[T]
is closed if Let
PROOF.
x
{xi : i E I]
A E G
is closed,
E A , A E Ci.
is complete.
be a vector in
E
which is not in
a family of vectors in
E
such that
is a Hamel basis of an algebraic complement of
I.
be the family of all finite subsets of A E Ci
we find a continuous linear form on
x
and
We give an order relation
s
in
p1,p2 E N
then
it takes the value
only if
and
1
A1,A2 E G
fl C f2
,
p1 L p2
and
If
F
if
(x,xi : i E I) in
E.
Let
f E 3, p E N
u(f,p,A)
(3,N,G):
Let
F.
on
E
C
and
such that
fl,f2 E 3 ;
(f19~1,A1) s (f2,p2,A2) A1
3
if and
A2.
We consider the net
We take that
2
m
0 and C .
Since
is a set
D E Ci,
contains
B.
If
B E G. EB
n
a part
z
E B
F
We find a positive integer
m
is of finite codimension in
g E 3
and an integer
we have that
q E N
such
EB
there
such that
OF F&CHET
A PROPERTY
SPACES
475
from where we get
-
I(U(fl’P1,A1)
implying that (4) is Cauchy in in
E’[T].
z
C ,
e,
be the limit of (4)
u
Let
= 1.
u(x)
On the other hand, if
> 0 there is a positive integer r
r
2
