Table Of ContentBanach SpaceComplexes
Mathematics and Its Applications
ManagingEditor:
M.HAZEWINKEL
CentreforMathematicsandComputerScience,Amsterdam,TheNetherlands
Volume334
Banach Space
Complexes
by
Călin-Grigore Ambrozie
Institute ofM athematics,
Romanian Academy of Sciences,
Bucharest, Romania
and
Florian-Horia Vasilescu
U.F.R. de Mathematiques,
Universite de Lille J,
Villeneuve d'Ascq, France
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-94-010-4168-3 ISBN 978-94-011-0375-6 (eBook)
DOI 10.1007/978-94-011-0375-6
Printed on acid-free paper
Ali Rights Reserved
© 1995 Springer Science+Business Media Dordrecht
Origina11y published by Kluwer Academic Publishers in 1995·
Softcover reprint ofthe hardcover Ist edition 1995
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permis sion from the copyright owner.
Contents
Introduction.
I Preliminaries 3
1 Algebraic prerequisites 3
2 Algebraic Fredholm pairs .... 8
3 Paraclosed linear transformations 18
4 Homogeneous operators 25
5 Linear and homogeneous projections and liftings 33
6 The gap between two closed subspaces ..... 39
7 Linear operators with closed range, and finite extensions 48
8 Metric relations and duality ... 53
9 Operators in quotient Banach spaces 64
10 References and comments 67
II Semi-Fredholm complexes 69
1 Semi-Fredholm operators 69
2 Semi-Fredholm complexes 82
3 Essential complexes . . . . 94
4 Fredholm pairs . 119
5 Other continuous invariants 137
6 Reference;;and comments 151
IIIRelated topics 153
1 Joint spectraand perturbations . . . . . . . 153
2 Spectral interpolation and perturbations 173
3 Versions of Poincare's and Grothendieck's lemmas 176
4 Differentiable families of partial differential operators 187
5 References and comments 196
Bibliography 197
Subject index 203
Notations 204
Introduction
The aim of this work is to initiatea systematicstudy of those propertiesof Banach
space complexes that are stable under certain perturbations. A Banach space complex is
essentially an object of the form
... --+ XP-1o-p-+-l XP -o-P+ XP+1 --+ ...,
whereprunsafiniteorinfiniteintervalofintegers,XP areBanachspaces,and oP :Xp.....Xp+1
are continuous linear operators such that OPOp-1 = 0 for all indices p. In particular, every
continuous linearoperator S :X.....Y, where X, Yare Banach spaces, may be regarded as a
complex:
O.....X ~ Y.....O.
The already existing Fredholm theory for linear operators suggested the possibility
to extend its concepts and methods to the study of Banach space complexes. The basic
stability properties valid for (semi-) Fredholmoperators have their counterparts in the more
general context of Banach space complexes. We have in mind especially the stability of the
index (i.e., the extended Euler characteristic) under small or compact perturbations, but
other related stability results can also be successfully extended.
Banach (or Hilbert) space complexes have penetrated the functional analysis from
at least two apparently disjoint directions.
A first direction is related to the multivariable spectral theory in thesense of J. L.
Taylor. The use of the associated Koszul complex, whose exactness leads to the definition
of thejointresolvent (and therefore to that ofthejointspectrum),also leads to thestudy of
Fredholm-type phenomena, when the exactness is replaced by the finitness of the dimension
of the homology spaces.
Aseconddirectionisconnectedwith thea-problemfor theDolbeaultcomplex,whose
approach, whenestimatesfor thesolutions aresought,leads tosomeBanachor Hilbertspace
complexes.
That these two directions are, in fact, not disjoint, will follow from the discussion
presented in the third section of the last chapterof this work.
Unlike in the case ofone operator, the major difficulty when studying the stability
of various propertiesofcomplexes is that this class itselfis, in general, unstable under linear
perturbations. For this reason one should take care to remain within the same class, and
sometimes it is even necessary to enlarge the framework of Banach space complexes. The
reader will encounter a certain scale of complexes and extensions, whose introduction has
been dictated by the technical difficulties and the specificity of a given problem.
The presentworkemergesfrom a text which has been used for someyears by thesec
ond named author as an advanced one-semester course inoperator theory. Theoriginal text
has been rewritten and completed with somerecent contributions ofthe first named author,
as well as with a few related topics. Several results have been obtained when elaborating
the actual version.
2
The first chapter contains algebraic and topological P.ferequisites. Part of the ma
terial included in this chapter is standard, but there are also less standard concepts and
facts, as for instance Fredholm pairs, paraclosed transformations, homogeneous operators,
and operators in quotient Banachspaces.
The main part of the present work is the second chapter. The major results con
cerning the stability of various invariants and other properties of Banach space complexes
under small or compact perturbations are herein presented.
The third chapter deals with somerelated topics. The conceptofjointspectrum, in
the spirit ofJ. L. Taylor but in an extended framework, as wellas somestability properties
of this concept are the contents of the first section. After a few facts about the spectral
interpolation, wepresentin thethird sectionsomeversionsofPoincare'sand Grothendieck's
lemmas, in an attempt to give a unifying image of the two main sources of the theory of
Banachspacecomplexes. Thelastsectionofthischapterisrelatedto thedeformationtheory
ofcomplex manifolds, whereour methods seem to lead to a new approach to certain results
ofthis field.
There are not too many works especially dedicated to the study of Banach space
complexes, altough we think that the subject has reached a certain maturity. Our bibliog
raphy reflects those contributions ofwhich weare aware. Afew related works and works of
general interest have been also quoted.
This book is written primarily for specialists in functional analysis. Some people
workinginpartialdifferentialoperatorsorincomplexmanifoldsmightfind certainappealing
topics. Tryingtowriteamaterialasself-containedaspossible,therearedetailswhich maybe
dullforaspecialist butwelcomefor abeginner. Wealsomentiontheexistenceofsomerather
technical parts, less enjoyablebut unavoid~ble,at least at this moment of the development
ofthe theory ofBanach space complexes.
Ciilin-Grigore Ambrozie
and
Florian-Horia Vasilescu,
Bucharest, June, 1994.
Chapter I
Preliminaries
1 Algebraic prerequisites
Throughout this work wedenote by F eitherthe real field R or thecomplex field C.
Thecategory whose objectsare F-linearspaces and whose morphismsare F-linearmappings
between F-linear spaces is denoted by Linr. In particular, LinR is the category of all real
linearspaces with real linear mappings, and Line is the categoryofallcomplexlinearspaces
with complex linear mappings.
A complex in the category LinF is a sequence (L,8) =(LP,8P)PEZ (Z is the ring of
integers), where L = (LP)PEZ are F-linear spaces, and 8 = (OP)PEZ are F-linear mappings
such that 8P: LP -t LP+1 and Op+10p = 0 for all p E Z (in other words, the image 1moP
of 8P is contained in the kernel Kerop+1of Op+l). Since the domain of definition LP of oP
is uniquely determined by bP, we often identify the complex (L,o) = (LP,OP)PEZ with the
sequence 0 = (OP)PEZ, which is also called a complex. The sequence L = (LP)PEZ will be
designated as the domain ofdefinition of the complex 0= (OP)PEZ,
A more suggestive representation of the complex (L,0),'or simply 0, is given by the
sequence
(1.1)
Fromnowon, ifnot otherwisementioned, weshallwork withobjectsand morphisms
in the category LinF.
A complex (L,8) = (LP,OP)PEZ with the property LP = {OJ for all except a finite
number ofindices is said to be offinite length. If(L,0) is not offinite length, then it is said
to be of infinite length.
If(L,o) = (LP,OP)PEZ is a complex of finite length, and if LP i= {OJ for at least one
index p, one can define the numbers
1+(0) := min{n E Z; LP = {OJ for all p> n}
and
qo):= max{n EZjIP = {OJ fOf all p<n}.
3
4 CHAPTER 1. PRELIMINARIES
Then the number
(1.2)
is called the lengthof the complex 8.
IfLP = {OJ for all p E Z, then, by definition, 1(8) :=1(0) = O. If(L,8) is ofinfinite
length, weset 1(8) := 00
For instance, the length ofthe complex
is :::; n +1, and it is exactly n+1if Lm =f. {OJ, Lm+n =f. {OJ, where n, m are integers with
n 2: O.
1.1.Example. Let LO, L1 be two linear spaces and let 00 : LO -+ L1 be a linear
mapping. Ifwe set LP = {OJ ifp f/. {O,1} and 8P= 0 ifp=f. 0, then (L,8) = (LP,8P)PEZ (or
8= (8P)PEZ) is obviously a complex, which can be represented as
Thiscomplexwill bedesignated as the complex associatedwith the linear mapping8° :LO -+
Ll.
Let 8 = (oP)PEZ be a complex. The homology H(8) of 8 is the sequence of linear
spaces (HP(8))PEZ, where
(1.3) W(8):= KeroP/lmoP-1,pE Z.
If HP(8) = 0for all pE Z, i.e. 1m8P-1 = Ker 8Pfor all pE Z, one says that the complex 8
is exact.
For a linearspace E we denote by dimFE the algebraic dimensionofE, which may
be finite or infinite.
1.2.Definition. Let8= (8P)PEZbeacomplex. Wesaythat8is(algebraica/ly)Fredholm
ifdimFHP(8) <00for allpE Zand dimFHP(8) = 0for allexceptafinitenumberofindices.
For a Fredholmcomplex 8we may define the index (or the Euler characteristic) of8by the
formula
(1.4) indF8:= L(-I)PdimFW(8).
pEZ
If8 = (oP)PEZ is a Fredholm complex, and HP(8) =f. {OJ for at least one p, we can
define the numbers
n+(8) := min{n EZjW(8) = {OJ for all p >n}
and
n-(8):= max{n EZjW(8) ={OJ for all p <n}.
Then the number
(1.5)
1. ALGEBRAICPREREQUISITES 5
measures the length ofnonexactnessof6.
If HP(6) = {OJ for all p E Z, weset n(6) := O.
As a matter of fact, the natural number n(6) is the length of the complex
(HP(6),OP)PEZ, where OP :HP(6) ~ HP+I(6) is the zero mapping.
Note that if the complex 6is exact, then 6 is Fredholm and indF6= O.
1.3.Example. Let {fl : LO ~ LI be a linear mapping and let 6 be the complex
associated with {fl (seeExample1.1). Notethat6is Fredholmifand onlyifdimFKer6° <00
and dimFU 11m{fl < 00, since HO(6 ) = Ker6°, HI(6) = LllIm{fl and W(6) = 0 if
p~ {O,l}. Then,
If6is Fredholm, weshall say that 6° is Fredholm, and the number
is called the indexof6°.
= =
Notealso that 6 is exact ifand only if Ker6° {O} and 1m6° LI, i.e. ifand only
jf{fl is injective and surjective.
1.4.Remark. It is somewhat customary to use, for complexes, lower indices instead
of upper indices. More precisely, a complex can also be a family 6 = (6P)PEZ, where 6p :
Lp ~ Lp_1 satisfy 6p6p+l = 0 for p E Z. If we set 6~ := 6_p and L~ := L_p for all p E Z,
then 6. = (~)PEZ is a complex with upper indices, ~ : L~ ~ L~+I, and whatever has
been defined for complexes with upper indices can be transposed to complexes with lower
indices, via the one-to-one correspondence 6 6•. For instance, the homology H(6) of6 is
t-+
the family (H (6))PEZ, where H (6) = H-P(6.) for all p E Z. Note that we obviously have
p p
Hp(6) = Ker6p/lm6p+l·
If6. is Fredholm, wesay that 6is Fredholm and the indexindF6of6is bydefinition
the number indF6•. It is easily seen that
IJ
indF6= -l)PdimFH (6).
p
pEZ
Nevertheless, weprefer to use complexes written with upper indices throughout this
text.
Let 6 = (fJP)PEZ and E = (EP)PEZ be two complexes and let L = (LP)PEZ, M =
(MP)PEZ be the domains of definition of 8, Erespectively. A morphism r.p of the complex
6 into the complex E, denoted by r.p : 6 ~ E(or r.p : L ~ M) is a family (r.pP)PEZ of linear
mappings r.pP :LP ~ MP such that
(1.6)
