Operator Theory Advances and Applications Vol.66 Editor I. Gohberg
Editorial Office: School of Mathematical Sciences Tel Aviv University Ramat Aviv, Israel
"
Editorial Board: A. Atzmon (Tel Aviv) J.A. Ball (Blackburg) A. Ben-Artzi (Tel Aviv) A. Boettcher (Chemnitz) L. de Branges (West Lafayette) K. Clancey (Athens, USA) L.A. Coburn (Buffalo) K.R Davidson (Waterloo, Ontario) RG. Douglas (Stony Brook) H. Dym (Rehovot) A. Dynin (Columbus) P.A. Fillmore (Halifax) C. Foias (Bloomington) P.A. Fuhrmann (Beer Sheva) S. Goldberg (College Park) B. Gramsch (Mainz) G. Heinig (Chemnitz) J.A. Helton (La Jolla) M.A. Kaashoek (Amsterdam)
T. Kailath (Stanford) H.G. Kaper (Argonne) S.T. Kuroda (Tokyo) P. Lancaster (Calgary) L.E. Lerer (Haifa) E. Meister (Darmstadt) B. Mityagin (Columbus) V.V. Peller (Manhattan, Kansas) J.D. Pincus (Stony Brook) M. Rosenblum (Charlottesville) J. Rovnyak (Charlottesville) D.E. Sarason (Berkeley) H. Upmeier (Lawrence) S.M. Verduyn-Lunel (Amsterdam) D. Voiculescu (Berkeley) H. Widom (Santa Cruz) D. Xia (Nashville) Honorary and Advisory Editorial Board: P.R Halmos (Santa Clara) T. Kato (Berkeley) P.D. Lax (New York) M.S. Livsic (Beer Sheva) R. Phillips (Stanford) B. Sz.-Nagy (Szeged)
Differentiable Operators and Nonlinear Equations
Victor Khatskevich David Shoiykhet
Translated from the Russian by Mircea Martin
Authors Victor Khatskevich Department of Mathematics University of Haifa Afula Research Institute Mount Carmel, Haifa 31905 Israel
David Shoiykhet Department of Mathematics International College of Technology Ort Braude, College Campus P.O.B.78 Karmiel20101 Israel
A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA
Deutsche Bibliothek Cataloging-in-Publication Data
Khatskevich, Victor: Differentiable operators and nonlinear equations / Victor Khatskevich ; David Shoiykhet. Trans!. from the Russ. by Mircea Martin. - Basel; Boston; Berlin: Birkhiiuser, 1994 (Operator theory; Vo!' 66) ISBN 3-7643-2929-7 (Basel ... ) ISBN 0-8176-2929-7 (Boston) NE: Soi~e!, Dllwid:; GT
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© 1994 Birkhiiuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Printed on acid-free paper produced from chlorine-free pulp Cover design: Heinz Hiltbrunner, Basel Printed in Germany ISBN 3-7643-2929-7 ISBN 0-8176-2929-7
Table of Contents
Introduction Chapter 0: Preliminaries 1. Sets and relations ................................................. . 2. Topological spaces ................................................ . 3. Convergence. Directedness ........................................ . 4. Metric spaces ..................................................... . 5. Spaces of mappings ............................................... . 6. Linear topological spaces ........................................ '" 7. Normed spaces .................................................... . 8. Linear operators and functionals .................................. . 9. Conjugate space. Conjugate operator .............................. . 10. Weak topology and reflexivity ..................................... . 11. Hilbert spaces ..................................................... . Chapter I: Differential calculus in normed spaces 1. The derivate and the differential of a nonlinear operator ........... . 2. Lagrange formula and Lipschitz condition ......................... . 3. Examples of Frechet differentiable operators ....................... . 4. Lemmas about differentiable operators ............................ . 5. Partial derivatives ................................................. . 6. Multilinear operators. Duality. Homogeneous forms ............... . 7. Higher order derivatives ........................................... . 8. Complete continuity of operators and of their derivatives .......... .
VI Chapter II: Integration in normed spaces 1. Riemann - Stieltjes integrals of vector-functions ....................... 61 2. Pettis integral and the connection with Riemann - Stiltjes integral ..... 65 3. Antiderivatives of vector-functions. Integral representations ............ 66 4. Integrals of operators in Banach spaces ................................ 71 Chapter III: Holomorphic (analytic) operators and vector-functions on complex Banach spaces 1. Differentiability in complex and real sense. Cauchy - Riemann conditions ......................................... 76 2. The p-topology and holomorphy ....................................... 80 3. Cauchy integral theorems and their consequences ...................... 85 4. Uniqueness theorems and maximum principles 5. Schwartz Lemma and its generalizations ............................... 92 6. Uniformly bounded families of p-holomorphic (holomorphic) operators. Montel property ...................................................... 98 Capter IV: Linear operators 1. The spectrum and the resolvent of a linear operator .................. 103 2. Spectral radius ...................................................... 108 3. Resolvent and spectrum of the adjoint operator ...................... 111 4. The spectrum of a completely continuous operator .................... 114 5. Normally solvable operators .......................................... 117 6. Noether and Fredholm operators ..................................... 119 7. Projections. Split able operators ...................................... 122 8. Invariant subs paces .................................................. 127 Chapter V: Nonlinear equations with differentiable operators 1. Fixed points. Banach principle ....................................... 133 2. Non-expansive operators ............................................. 137 3. Fixed points for differentiable operators .............................. 143 4. Some applications of fixed point principle ............................ 147 5. Implicite and inverse operators. Connection with fixed points ......... 160 Chapter VI: Nonlinear equations with holomorphic operators 1. s-fixed points for holomorphic operators. A converse of Banach principle ....................................... 171 2. Criterions for the existence of an s-fixed point and its extension with respect to a parameter ............................. 177 3. Regular fixed points. Geometric criterions ............................ 182
VII 4. Apriori estimates and the extension of an s-solution to the boundary of the domain ....................................... 189 5. Local inversion of holomorphic operators and a posteriori error estimates ........................................... 195 6. Single-valued small solutions in some degenerate cases ................ 200 Chapter VII: Banach manifolds 1. Basic definitions ..................................................... 211 2. Smooth mappings .................................................... 213 3. Submanifolds ........................................................ 214 4. Complex manifolds and Stein manifolds .............................. 218 Chapter VIII: Non-regular solutions of nonlinear equations 1. Ramification of solutions. Statement of the problem .................. 223 2. Equations of ramification ............................................ 225 3. Equations of ramification for an analytic operator. The problem of the coefficients ....................................... 231 4. The description of the set of fixed points for an analytic operator .................................................... 232 Chapter IX: Operators on spaces with indefinite metric 1. Spaces with indefinite metric ......................................... 239 2. Angle operators ...................................................... 242 3. Plus-operators ....................................................... 244 4. Symmetric properties of a plus-operator and its adjoint ............... 249 5. The problem of invariant semi-definite subspaces ..................... 258 6. An application of fixed point principles for holomorphic operators to the invariant semi-definite subspace problem .................................................... 262 References ..................................................................... 267 List of Symbols ................................................................ 277 Subject Index .................................................................. 279
Introd uction We have considered writing the present book for a long time, since the lack of a sufficiently complete textbook about complex analysis in infinite dimensional spaces was apparent. There are, however, some separate topics on this subject covered in the mathematical literature. For instance, the elementary theory of holomorphic vectorfunctions. and mappings on Banach spaces is presented in the monographs of E. Hille and R. Phillips [1] and L. Schwartz [1], whereas some results on Banach algebras of holomorphic functions and holomorphic operator-functions are discussed in the books of W. Rudin [1] and T. Kato [1]. Apparently, the need to study holomorphic mappings in infinite dimensional spaces arose for the first time in connection with the development of nonlinear analysis. A systematic study of integral equations with an analytic nonlinear part was started at the end of the 19th and the beginning of the 20th centuries by A. Liapunov, E. Schmidt, A. Nekrasov and others. Their research work was directed towards the theory of nonlinear waves and used mainly the undetermined coefficients and the majorant power series methods. The most complete presentation of these methods comes from N. Nazarov.
In the forties and fifties the interest in Liapunov's and Schmidt's analytic methods diminished temporarily due to the appearence of variational calculus methods (M. Golomb, A. Hammerstein and others) and also to the rapid development of the mapping degree theory (J. Leray, J. Schauder, G. Birkhoff, O. Kellog and others). These new methods were particularly attractive since they enabled the study of many classes of nonlinear equations, and therefore they were highly developed. (Important results were obtained by M. Krasnoselski, P. Zabreiko, V. Odinetz, Yu. Borisovich and B. Sadovski.) However, these new techniques retarded the development of spe-
x
INTRODUCTION
cific methods for solving equations with an analytic nonlinear part. That is why in the sixties some mathematicians (P. Rybin, V. Pokornyi, M. Vainberg, V. Trenogyn and others) interested in the theory of integral equations and their applications returned to the Liapunov-Schmidt and Nekrasov-Nazarov analytic methods. At the same time the theory of functions of one or several complex variables was enriched with more significant and subtle results. Parallel with these achievements, the first results on holomorphic mappings on infinite dimensional spaces appeared in the works of A. Cartan, R. Phillips, L. Nachbin, L.Harris, T. Suffridge, W.Rudin, M. Herve, E. Vesentini, J.-P. Vigue, P. Mazet, K. Goebel, and of many others. We consider that it is now about the right time "to set a bridge" between nonlinear analysis and the theory of holomorphic mappings on infinite dimensional spaces. Of course, to this end it is necessary to put together results and techniques from the homology theory, sheaf theory, vector fields theory and from a lot of other modern theories in analysis - a task difficult to achieve within the limits of but one book. That is why we decided to start this vast project, by presenting only the theory of differentiable and holomorphic mappings on Banach spaces, as well as some prerequisites from functional analysis and topology. In all chapters with the exception of Chapter 0 which has the character of a dictionary, we tried to give a complete account of definitions and proofs, and to make this book accesible not only to specialists, but also to students and to those engineers who are currently using the solutions of some specific integral and differentiable equations. We conclude the work by mentioning the interesting relationship between the theory of holomorphic mappings and the theory of linear operators on spaces with indefinite metrics. More precisely, our last chapter is a brief exposition of the theory of spaces with indefinite metrics and of some relevant applications of the holomorphic mappings theory in this setting. In closing, we draw our readers to a few technical points. Throughout the book we strove to use a uniform notation for objects of the same type. The most used notations are presented in Chapter O. At the end of the book we give a list of some standard symbols, and also a subject index. We used the symbols " 0
there exists £ E i" such that for d E i" with d;;: £ we have p(fd(a),j(a))
.x) = >.n H(x),
x
E
X,
(6.7)
for each >. E lK, and
IIH(x)11 ~ IIHllllxlln,
x
E
X.
(6.8)
It is easy to see that IIHII is the smallest constant for which (6.8) holds. Thus, the number IIHII is called the norm of the operator H. The set of homogeneous forms of order n will be denoted by Ln(x; !D). Clearly, Ln(X;!D) is a linear normed space relatively to the norm IIHII.
§7. Higher order derivatives A way of defining the derivatives and differentials of higher orders is the following. DEFINITION 7.1. Let F: X ----t !D be an operator and suppose that the first variation of(x, hi) exists in a neighborhood of a point x, for any element hi EX. If for each
Higher order derivatives
51
fixed hI and h2 the limit (7.1) exists in the space Z), then it is called the second variation of the operator F and is denoted by 82F(x, hI, h2)' The n-th variations are defined by induction. Namely, assume that 8n -dx, hI,
... , hn-d exists in a neighborhood of a point x for any hI, h 2, ... , h n - I E X. If for each fixed hI, h2' ... ,hn the limit (7.2)
exists in the space Z), then it is called the n-th variation of the operator F and is denoted by 8n (x, hI"'" hn ). Suppose now that the operator F is Gateaux differentiable in a neighborhood 1)
of a point x. Then, the variation 8F can be considered (on this neighborhood) as
an operator acting from X into L(X, Z)). Indeed, the equality
8F(x, h) = A(x)h maps x E
1)
into the operator A(x) = F'(x) E L(X, Z)). If this operator is Gateaux
differentiable in Xa, that is, if the limit lim A(x + th 2) - A(x)
t-a
t
=
B(x)h 2
(7.3)
exists, then, obviously, the value of B(x)h 2 at an element hI is exactly the second variation 82F(x, hI, h 2) defined by (7.1). Thus, in this case, 82F can be considered as an element of the space L (X, L(X, Z))), or, according to Theorem 6.2, as an element of the space L 2 (X; Z)), i.e., as a bilinear operator (relatively to hI and h 2 ) acting from
x-i-x into Z). This operator is called the second Gateaux derivative and its value at an element (hI, h 2 ) E X+X is called the second Gateaux differential and is denoted by D2 F(xa, hI, h2) (= 82F(xa, hI, h2))' The n-th Gateaux derivative and the n-th Gateaux differential of the operator F are defined by induction, using the reccurent relations
DIFFERENTIAL CALCULUS IN NORMED SPACES
52
The n-th Frechet derivative and the n-th Frechet differential of the operator
F are defined using Definition 1.3. Sometimes, it is more convenient to use Theorem
1.2. Thus, if for a Gateaux differentiable operator F the equality (7.3) is uniformly fulfilled for h2 with IIh211
=
1, where B(x) E L(X,L(X,~)), then the operator F is
said to be twice Frechet differentiable.
(n + 1)-th variations of an operator F exist at a point Xo E E X and at each point of an open interval]xo, Xo + h[, such that 0 there exists
8 = 8(c) such that IK(tl' s) - K(t2' s)1 < c for any s E [a, b] and any tl, t2 with It I - t21 < 8. But Y = Ax with x E lB, so taking into account that Ix(s)1 ~ Ilxll = max Ix(s)1 ~ 1, [a,b]
E
[a, b]
DIFFERENTIAL CALCULUS IN NORMED SPACES
56
we obtain
ifltt-t21O
since h is an arbitrary element, the existence of the first variation of the operator F at the point Xo will follow. For
fir/D)' we have fi (0) = .lim F(xo + i?h) 1)
-- ~. 1·1m F(xo 1 1)--->0
11)--->0
+ ry(ih)) "I
- F(xo) =
1"1
- F(xo) -- _." IvR. F( XO,l·h) - "vIFt F( Xo, h) .
77
Cauchy-Riemann conditions
To complete the proof of the theorem, notice that if the operator F is Gateaux (resp. Frechet) differentiable, then the variation OIRF(xo, h) is real-linear in h. By (1.1) it is also complex-linear in h, i.e., it is the Gateaux (resp. Frechet) derivative in the complex sense.
~
Assume now that the space QJ (see §4 of Chapter 0), i.e.,
~ =
~
is the complexification of a real Banach space ~
QJ + iQJ and the topology of
topology of the product QJ x QJ. Then, any operator F: X
---* ~
coincides with the
can be represented as
F = P + iQ, where P, Q: X ---* QJ. Suppose further that the operators P and Q are real-differentiable on a domain :D containing the point xo. Then the operator F is also real-differentiable, and 'F(
vIR
Xo,
h)
l'
=
1m
P(xo
t--->O,tEIR
+ th) -
P(xo) . + I . l'1m Q(xo t t--->O,tEIR = OIRP(xo, h) + iOIRQ(xo, h).
+ th) -
Q(xo)
t
If condition (1.1) is fulfilled, then
OIRP(xo, ih) + iOIRQ(xo, ih) = = iOIRP(xo, h) - OIRQ(xo, h). Comparing the real and the imaginary parts, this infers the equalities
OIRP(xo, ih) = -OIRQ(xo, h) OIRQ(xo, ih)
=
(1.2)
OIRP(xo, h)
which are generalizations of the well-known Cauchy-Riemann conditions. Indeed, if
X = C = {AI A = T+i1]}, then P and Q can be considered as vector-functions oftwo real variables T,1] E IR. Writing Xo
TO + i1]o, h
= 5:
VIR
P( Xo, 'h) I =
lim t--->O,tEIR
For h
=
(D.T, 0)
E IR2,
=
D.T + iD.1], we obtain
we have
1m
P(xo
+ tih)
- P(xo) t P(TO - tD.1] + i(1]o + tD.T)) - P(TO + i1]o) t l'
t--->O,tEIR
78
HOLOMORHIC OPERATORS ON COMPLEX BANACH SPACES
On the other hand, the first equality in (1.2) implies, for the same h, that b~P(xo,
Q(TO
lim
ih)
=
+ tf:1T + i7]o) -
t-.O,tE~
-b~Q(xo,
Q(TO
h)
+ i7]o)
t
= 8Q 8T
I (
Xo= 70,1]0
).
Thus we obtain the first Cauchy-Riemann condition
8P
(1.3)
87] Analogously we obtain the second Cauchy-Riemann condition 8Q 87]
8P 8T'
(1.4)
(1.3) and (1.4) the condition that X is an one-dimensional space is not essential. It is sufficient to assume that X, like q), is the complexification of a real space i1. In this case, Theorem 1.1 has the following consequence. REMARK 1.1. Notice that in order to prove
Let il, QJ be two real Banach spaces and let P, Q: il x il --> QJ be two operators defined on a neighborhood of the point Xo = (Po, qo) E il x il. Assume that P and Q have continuous partial Frechet derivatives on that neighborhood. Then the operator F = P + iQ: X --> q), where X = il + ill and q) = QJ + iQJ, is Frechet complex-differentiable if and only if the equalities COROLLARY 1.1.
P;(xo) P~(xo)
= Q~(xo) = -Q~(xo)
(1.5)
are fulfilled. <J
The necessity can be established as above, in the same way in which we
deduced (1.3)-(1.4). For sufficiency, choose an arbitrary vector h
=
consider b~P(xo, ih). Notice that in il x il the vector ih has the form ih Using (1.3), we have
= b~P (Po, qo, (-f:1q, f:1p)) = = P~(xo)( -f:1q) + P~(xo)f:1p = -Q~(xo)f:1q - Q~(xo)f:1p = -b~Q(xo, h),
b~P(xo,
=
ih)
f:1p
+ if:1q and
=
(-f:1q, f:1p).
Cauchy-Riemann conditions
79
which is the first equality in (1.2). The second one can be proved analogously.
~
COROLLARY 1.2. If, in addition to the conditions of the previous corollary, one assumes that P and Q are twice continuously differentiable and satisfy the conditions (1.5), then
D.F = 0
and
D.P = D.Q = 0
where the symbol D. denotes here the Laplace operator, i.e., for any G: UxU --+
D.G = G~p
~x~,
+ G~q.
(In fact the requirement that P and Q are twice continuosly differentiable is superflous in the previous corollary; as we have already mentioned, the complexdifferentiability of F implies its infinite real-differentiability, and this means that P and Q are also real-differentiable of any order.)
At the end of this section we consider the notion of the complex-conjugate differential, just by mimicing the classical constructions. To this aim, for any element
x = P + iq in the space X = U + ill we introduce the conjugate element x this case, p
=
1
"2 (x + x),
1
q = 2i (x - x). Therefore, an operator F: U x U --+
gives rise to an operator F(x, x)
=
=
P - iq. In
!D formally
F(p, q) by replacing p and q with their expressions
in terms of x and x. (The problem is that
F
can not be considered as a function
of two independent variables, since the change of x implies a simultaneous change of x.) Further, if the operator F is Frechet differentiable on a neighborhood of a point
Xo = (Po, qo) (in the real sense), then, by the formula of the complete differential, we have where h
=
(D.p, D.q), so F'(xo)h = =
F~(xo) [~(h + h)] + F~ [;i (h -
h)]
~ [F~(xo) - iF~(xo)] h + ~ [F~(xo) + iF~(xo)] h = =
F~(xo)h + F/r(xo)h,
where F~(xo)h and F/r(xo)h are the formal derivatives of the operator F with respect to x and x, defined by
F~(xo) = ~ [F~(xo) - iF~(xo)] F/r(xo) =
~ [F~(xo) + iF~(xo)]
.
80
HOLOMORHIC OPERATORS ON COMPLEX BANACH SPACES Conditions (1.5) for the operator F
conclusion: if the operator F: il x il
~ ~
=
P
+ iQ lead directly to the following
is Frechet differentiable in the real sense,
then it is Frechet differentiable in the complex sense as an operator from into ~
= m + im if and
only if F~(x)
x = il + ill
= o.
§2. The p-topology and holomorphy As we have already noted, the class of operators having the first variation, the class of Gateaux differentiable operators, and the class of Frechet differentiable operators form a strictly decreasing chain with respect to the inclusion. Nevertheless, from the point of view of the complex analysis, these classes have many properties in common. It is therefore interesting to consider a characteristic of these classes which permits to
recover these common properties in a unified manner, and, on the other hand, which permits to distinguish between these classes of operators. For the Gateaux and the Frechet differentiable operators, such a characteristic seams to be a "connection" between the domain of definition of the operator and the operator itself. More precisely, we consider, on the domain of an operator, the so-called p-topology, which is weaker than the norm topology; in this topology both classes of differentiable operators have the same properties. Note that the interest in this approach arises not only from its generality, but also from the possibility of obtaining new results. As an example see §6 of Chapter VI, where some degenerate problems for implicite Frechet differentiable operators are transferred to nondegenerate problems for operators differentiable relatively to the p-topology. Our presentation and terminology mainly follow E. Hille and R. Phillips [1]. 1. For a set 1) ~ x and for elements x E 1), hEx, we introduce the following notion: p(x, h) = sup{p : x
+ Th E 1), TEe, ITI::;; p}.
Clearly, p(x, h) ~ 0, for all x E 1), hEx and for any set We will need the following property of p(x, h). LEMMA 2.1. Let 1) be a set in
x.
Then for any x E
have p(X,A-1h) =p(x,h)IAI·
1) ~
1),
(2.1)
x.
hEX, A E C (A i=- 0), we
81
Holomorphy
The equality is obvious if p(x, h) = o. Assume that p(x, h) > 0 for some and hEX. Denote h1 = A- 1h and take an arbitrary Po, O 0 for any hEX,
Ilhll = 1.
Then, by Lemma 2.1 we obtain
Since for any N > 0 there exists f30 E Q3 such that IT* - T,BI < p(x*, h)N- 1 for all f3 E Q3, f3 > f30 (the element h is fixed), the relation (2.3) implies
and the proof is complete.
~
2. Let us turn now to the notion of holomorphy. An operator F:::D --+ ~ is called p-holomorphic in ::D (in short FE 1ip(::D , ~)) if for any x E ::D and hEX the vectorfunction f(T) = F(x + Th) is complex-differentiable in the disk ITI < p(x, h) of the complex plane C. DEFINITION 2.4. Let::D be a p-open set in X.
Thus, any p-holomorphic operator F has at each point x E ::D the first variation 8F(x, h). DEFINITION 2.5. An operator F:::D --+ ~ is called Q-holomorphic if it is p-holo~or phic and for any point x E ::D there exists a linear operator A(x) E L(X,~) such that 8F(x, h) = A(x)h. In this case we will write F E 1i(::D, ~).
Holomorphy
83
If the set 1:J is open, i.e., condition (2.2) is fulfilled for all x E 1:J, then a
Q-holomorphic operator is called F-holomorphic or, simply, holomorphic. Obviously if F E H(1:J,!I)) then F is Gateaux differentiable at any point
x E 1:J. We will see in §3 below that if an operator F is F-holomorphic, i.e., the set 1:J is open, then F is Frechet differentiable at any point x E 1:J (this explains the term "F-holomorphic operator"). Since the notion of holomorphy depends on the notion of convergence, we may naturally consider the notion of weak holomorphy. However, it is not necessary to do that, because - as we have already mentioned - the strong holomorphy follows from the weak holomorphy. This result was obtained for vector-functions by N. Dunford
[1]. We will prove here a sligt generalization. An operator F: 1:J
---t
!I) is called locally bounded on 1:J if for each x E 1:J there
exists a neighborhood it of the point x such that F is bounded on it n 1:J. We make the supplementary assumption that there exists a Banach space !I) * such that (!I)*)*
=
!I).
THEOREM 2.1. Let 1:J be a p-open set in X. An operator F, locally bounded on 1:J,
is p-holomorphic (Q-holomorhic) if and only if (y*, F(·))
E
Hp(1:J, C) (H(1:J, C)) for
each y* E !I) *. (Maybe it is useful to explain that the value of the operator (y*, F(·)) at x E 1:J is the number (y*, F(x)), i.e., the value of the functional F(x) E !I)
= (!I)*)*
at the
vector y* E !I)*.) -i=p
holds for any x E:D, hEX and for any p < p(x, h). -i=p
F(x + Th)d.\, y* )
=
J i>-i=p
(F(x + Th), y*)d.\
HOLOMORHIC OPERATORS ON COMPLEX BANACH SPACES
86
holds for any y* E ~ * . Our theorem follows now from Theorem 2.1 and from the classical theorems of Cauchy and Morera (see, e.g., B. Shabat [1]) . • The analogue of the Cauchy integral representation formula can be obtained similarly. THEOREM 3.2. For an operator F E
21fiF(x
Hp(TJ, ~), the representation
J
+ Th) = (P)
F(x
+ '\h)('\
- T)-ld,\
(3.2)
I>-I=p
is true for any x
ETJ,
TEe
hE oX and
ITI < p < p(x, h).
with
In particular
J
21fiF(x) = (P)
,>.-1 F(x
+ '\h)d'\,
(3.3)
I>-I=p
that is, the integral in the right hand side of (3.3) does not depend on h if p is small enough, and it is equal to the value of the operator F at x multiplied by 21fi.
TJ, then it is p-analytic at each point of TJ, i.e., F admits a decomposition as a generalized power series COROLLARY 3.1. If the operator F is p-holomorphic and locally bounded on
00
F(x
+ Th) = ~ okF(x, h)Tk
(3.4)
k=O
which is norm convergent (in okF(x,h)
~)
ITI ~ p < p(x, h)
for
J
= ~(P) 2m
and
,\-k- 1 F(x+,\h)d,\.
(3.5)
I>-I=p 0
for all h with Ilhll = 1. Assume that the inequality IIF(z)11 ~ M is fulfilled for all points in the ball centered at x and with radius c(x). Then, for any u with Ilull ~ 0 = = min{o(x),c(x)}, and taking h = UT- I , Ilhll = 1, relation (3.4) implies that 00
F(x
+ u) = of(x, h)T + L
Ok F(x, h)Tk
+ F(x) =
k=2
= A(x)u + w(x, u) + F(x), where A(x) E L(X, ZJ). Using Cauchy inequality (3.8) and the well known formula for the geometric series, we obtain
89
Cauchy integral theorems for any u
E
X,
Ilull < ();
this estimate proves the Frechet differentiability of the
operator F. Hence, it follows that the vector-function F(X+TIh l + .. .+Tnhn): e n ---> ---> ~ ~ for any fixed hI, ... ,hn ~ has continuous partial derivatives with respect to TI, ... , Tn
in a polydisk it with a sufficiently small radius, i.e., it is holomorphic on
this polydisk with respect to each of its variables. By the fundamental theorem of Hartogs (see, e.g., Shabat [2]), the function (F (X
+ ~ Tkhk)
,y* ) is holomorphic
in the polydisk it with respect to all of the variables, and consequently it has partial derivatives of any order. By Theorem 2.2 it follows that the vector-function F(x
+
+ ... + Tnhn)
has the same property. So, it follows, in particular, the existence of the second order derivative +TIh l
The symmetry of d 2 F and its linearity in hI, implies the linearity in h 2 . Analogously, we have dkF(xI' hI"'" h k )
= aTI,
.~~, aTk F(x + TIhl + ... + Tk h k)I
T 1= ... =Tk=O'
(3.10)
for all k;;::: 2. Taking into account the uniqueness of the Taylor decomposition, we obtain equalities (3.9). ~ Let us remark that from (3.9) and (3.10) we can infer the following representation for the form {)k F (- , . ):
a
k k 1 () F(x, h) = k! aTI,"" aTk F(x
+ TIhl + '" + Tkhk) I Tl=' '=Tk=O
(3.11)
h 1=,,·=hk=h
(an analogue of Liouville Theorem). Let X and ~ be Banach spaces, F E 'H(X, ~), and assume that for a point x E X there exists a positive integer m > 0 such that
. COROLLARY 3.3
IIF(x +
h)11 ~ Mllhll m
(3.12)
90
HOLOMORHIC OPERATORS ON COMPLEX BANACH SPACES
for all elements hEX with sufficiently large norm. Then the operator F is a polynomial of degree ~ m, i.e., F is a finite sum of homogeneous forms in h. In particular, if the value IIF(x)11 is uniformly bounded for all x E X, then F(x) takes a constant value in !D. m, so our first assertion is proved. The second assertion follows from the first
one taking m = O.
~
§4. Uniqueness theorems and maximum principles The next theorem easily follows from representation (3.4).
Let 1) be a p-open set which is a C-star relatively to a certain point Xo E 1) (i.e., x E 1) implies Xo + >.(x - xo) E 1) for all >. E C with 1>'1 ~ 1). If a p-holomorphic operator F and all its Taylor coefficients equal zero in Xo, then F(x) == 0 on 1). THEOREM 4.1.
If the domain 1) is arbitrary, but F is holomorphic, then the next result is true.
Let 1) be a domain in X and let F E H(1),!D) be a locally bounded operator. IfF and all its derivatives are zero at a certain point Xo E 1), then F(x) == 0
THEOREM 4.2.
on 1). ..) - f(llxll)
If(>..)f(llxll) -
1
I
~
I >.. -
=
o.
=
'l/J(f(..llxll - 1 .
98
HOLOMORHIC OPERATORS ON COMPLEX BANACH SPACES
Whence, we obtain the inequality
I : : :; If(A)7(lfXlf) If(A)A-- f(llxll) Ilxll Allxll - 1 Taking the limit for A ----+
Ilxll, we
have 1 -If(llxIIW 1 _ IIxl1 2
,
If (1IxlJ)1 : : :; But
1'(A)
=
11·
: : :;
1 1-
Ilx112'
(F'(AXllxll- 1 )xllxll- 1 , x*), so 1'(llxll) = (F'(x)x,x*)llxll- 1 = IJF'(x)xllllxll- 1 ,
hence
,
IIF (x)xll : : :;
Ilxll Ilx112'
1_
§6. Uniformly bounded families of p- holomorphic (holomorphic) operators. Montel property Let
1)
be a p-open set in X. Until now, we did not use the fact that
1)
itself can be
considered as a topological space with the p-topology defined in §2. Recall that in the initial topology of the space X, a p-holomorphic operator can have discontinuities (see Example 1.1.2). However, for the p-topology the next assertion is true.
Any p-holomorphic bounded operator is p-continuous. We prove below a stronger result.
Let 1) be a p-open set and let F = {F"'}"'EQl, F", E 1i p (1) , ~), be a uniformly bounded family (with respect to the norm topology of~), i.e., IJF",(x) II : : :; :::::; M < 00, for all a E 21. Then F is equicontinuous as a family of operators acting from the set 1) with the p- topology into the space ~ with the norm topology.
THEOREM 6.1.
Given an arbitrary c: > 0, take N ~ max{2c:- 1 M, I} and let x and x' be such that p(x, x' - x) > N. Then, setting x' - x = (,h, where Ilhll = 1, (, E C, and using Lemma 2.1, we obtain <J
p(x, (,h) = p(x, h)I('I- 1 >
N
99
Montel property or
1(1 < N-1p(x, h) ::;:; p(x, h). It follows that Fa(x') = Fa (x
+ (h)
has the Taylor representation (3.4). Then, by
Theorem 5.1, we have
IlFa(x') - Fa(x)11 ::;:; IlFa(x + (h) - Fa(x)11 ::;:;
2MI(1 [p(x, h)r 1 .
(6.1)
As h = (-l(X' - x), we actually have
COROLLARY 6.1.
If the set 1) is open, then a uniformly bounded family F =
= {Fa }aE'2l of operators holomorphic on 1), with values in a Banach space~, satisfies a Lipschitz condition, uniformly with respect to a 0 (see
2.2),
Ilx - x'il : ;:; 8(x).
21.
where hEX,
IlFa(x') - Fa(x)11 ::;:; 2MI(1 [8(X)]-1 ::;:; for all x' such that
E
Ilhll
=
1, (6.1) implies
2M [8(x)r 1 Ilx - x'il
~
Notice that the last relation implies the local uniformly boundedness of the Frechet derivatives of the operators Fa, a E
2(,
too.
We return now to the general case. Let us consider again 1) as a topological space with the p-topology. On the set of all continuous mappings from 1) into
~
we
introduce the topology Tp(1) , ~) of compact convergence (i.e., the uniform convergence on compact subsets in the p-topology of 1), see §5 of Chapter 0). Using Theorem 6.1 and Theorem 0.5.4, we obtain the following analogue of the Montel property.
Let FM be a bounded subset of Hp(1), ~), i.e., FM c::;; {F E : IIF(x)ll::;:; M < 00 for all x E 1)} and assume that for any x E 1),
THEOREM 6.2. E Hp(1),~)
the orbit FM(X) = {y E ~ : y = F(x), FE F M } is a sequentially compact subset of ~. Then the family FM is sequentially compact in the topology Tp (1) , ~). In contradistinction with the classical version of Montel theorem (X = en, ~ = em) we assume in Theorem 6.2 the compactness of the orbits; this compactness follows, for em, from the boundedness of the family F M. The next example shows the necessity of our assumption. Let X = ~ = £2. Consider the REMARK 6.1.
HOLOMORHIC OPERATORS ON COMPLEX BANACH SPACES
100
family {Bd ~ I of linear bounded operators (hence holomorphic operators) defined on the set {x = (0, ... ,0,
£2 : Ilxll < R < oo} by the equalities Bkx = Xn ,·· .). Obviously IIBkXl1 < R for all X with Ilxll < R, k= 1, 2, ....
= (XI,""X n , ... )
Xl,""
E
'--v-'" k
However the sequence {Bd~l is not compact; it is enough, for instance, to consider the values at the point x = (1,0, ... ,0, ... ). Suppose now that
~
is the dual space of some Banach space and that
~
is
metrizable in the corresponding ultraweak topology (this is the case, for instance, when
~
is separable).
0), any bounded set in
Then, by Alaoglu-Bourbaki Theorem (see §1O of Chapter ~
is sequentially compact in the ultraweak topology. Con-
sequently, we can use Theorem 0.5.4 and Theorem 6.1 in order to obtain the next result. THEOREM 6.3.
Suppose that
bounded set FM
~ HpCD,~)
If the space
~
~
is metrizable in the ultraweak topology. Then any
is sequentially compact in the topology Tp (:.D , ~).
is reflexive, the ultraweak topology in Theorem 6.3 can be
replaced with the weak topology of the
space~.
Thus, Theorems 6.1-6.3 provide a
Banach-Steinhaus type theorem (see §8 of Chapter 0). (Note, once more, the similarity between the geometric properties of holomorphic bounded operators and those of linear continuous operators.)
Assume that ~ is equipped with the norm (resp. ultraweak) topology. Let {Fn}~=l' Fn E Hp(:.D, ~), be a sequence of bounded operators. Then the sequence {Fn}~=l is convergent, in the topology Tp(:.D, ~), to a bounded operator F E Hp (:.D , ~) if only if the following two conditions are simultaneously fulfilled: THEOREM 6.4.
1) the family
{Fn}~=l
is uniformly bounded on :.D;
2) the sequence {Fn(X)}~=1 is fundamental (resp. ultraweak fundamental) for all x E :.D' , where:.D' is a subset of:.D, dense in the p-topology. We conclude this section with two generalizations of Vitali Theorem.
Let:.D be a p-open set which is a C-star relatively to a point Xo E :.D, and let {Fn}~=l' Fn E Hp(:.D,~), be a sequence of operators, uniformly bounded on :.D. Assume that for any hEX, Ilhll = 1, the sequence {Fn}~=l is fundamental on the disk Dp(xo, h) ~ 1) in the norm topology of the space~. Then the sequence is fundamental on :.D with respect to the norm topology of~, and, consequently, it is THEOREM 6.5.
Mantel property
101
strongly convergent to some operator FE Ji p ('1) , ~). 0 and 0 < q < 1, taking p> In
[cC1- r)(l- q)(2M)-lJ
(lnr)-l
(6.2)
and
{) > cq(r - d)dP- 1(r P - dP)-l, we obtain IlFn(u) - Fn+m(u) II
N({)). ~
For the case of ultraweak convergence, the proof of the analogous assertion can be sligthly simplified; it is enough to use Montel property. However, we will be mainly interested in the case of holomorphic operators.
!D is metrizabile in the ultraweak topology and let ~ be a domain in X. Let {Fn}~=l be a uniformly bounded sequence of operators holomorphic in ~. Suppose that {Fn}~=l converges in the ultraweak topology to an operator F E 1{(~I,!D), for any element of a certain open set ~' ~~. Then F admits a holomorphic continuation on the whole ~ and the sequence {Fn}~=l' converges to this continuation in the ultraweak topology, for each element of ~. THEOREM 6.6. Assume that the space
By Theorem 6.3, the sequence {Fn}~=l contains a subsequence {Fnk}k=l which is convergent in the ultraweak topology, for any element of ~, to a certain <J
F E 1{(~, !D). F I ~' = F.
F is a holomorphic continuation of the operator
operator
The operator
F, i.e.,
The uniqueness theorem implies that any other convergent
subsequence {Fn"J~=l has the same ultraweak limit
F.
The proof is complete. ~
In spite of the fact that the previous proof is very simple, it is not at all constructive; it provides no estimates analogous to those in (6.2) and (6.3), which are very useful in many applications (see §2 of Chapter VI below).
Chapter IV Linear operators
This chapter, as well as Chapter 0, is mainly auxiliary. However, compared with Chapter 0, the topics considered here are more special. Some of the subsequent results were presented only in research papers, but not in monographs. Therefore, our exposition in this chapter will be quite detailed, and will include proofs.
§1. The spectrum and the resolvent of a linear operator 1. CLOSED OPERATORS. Let X and ~ be complex Banach spaces and A: X ----+ ~ a linear operator (see Chapter 0, §8 for definition). Recall that the operator A is equally well described as the set of pairs (x, Ax), with x E D(A) ~ X. The operator A is invertible (i.e., A has an inverse operator A-I) if and only if its kernel N (A) = = {x E D(A) : Ax = o} equals {O}. In this case, the operator A-I is defined from R(A) into D(A); moreover, A-I Ax = x for all x E D(A) and AA-Iy = y for all y E R(A).
An operator A is called closed if it is closed as a subset of the topological product X x ~ (i.e., if Xn E D(A), Xn ----+ x and AXn ----+ Y for n ----+ 00, then x E D(A) and Ax = y). If A is closed and invertible, then obviously A-I is also closed. If an operator A is bounded and defined on the whole X, and if B is a closed operator, then the operator A + B ~ defined on D(B) ~ is also closed. THEOREM 1.1. A bounded linear operator A is closed if and only if its domain D(A)
LINEAR OPERATORS
104
is closed. A is bounded then Xn E D(A) and Xn ~ x for n Ax. Therefore A is closed if and only if D(A) is closed. ~
0 is an arbitrary positive number, we have n--->oo
lim IIA n II;'; ~ r. Thus we proved the existence of the limit
n-+oo
lim \lIIAnll = inf \lIIAnll = rCA).
n-+oo
n
~
1
THEOREM 2.1. For any A E L(X), we have
rCA) = sup IAI.
(2.3)
AEa(A)
r, hence, for any n-->oo
c > 0, there exists m (= m(c)) EN such that
IIAnl1 ~ (c+r)n
for all n ~ m. It follows
that r(A) = lim IIAnll~ ~ r. n-->oo
COROLLARY 2.1. If
For
IAI < r(A),
then the Neumann series diverges.
IAI > r(A)
the Neumann series is convergent in the operatorial norm and
= r(A).
By Theorem 2.1, Corollary 1.1 and Corollary 1.2, a point from
00
- LA1-iAi i=O
the boundary of the disk of radius r(A) belongs to the spectrum of A. Therefore, the number r(A) is called the spectral radius of the operator A, since r(A) is the radius of the smallest closed disk centered at the origin of the complex plane C, which contains the spectrum of the operator A. From the inequality IIAn I ~ IIAlln, n E N, it follows that r(A) ~ IIAII. It is not difficult to find an example of an operator A for which r(A) < EXAMPLE 2.1.
{el,e2},
IIAII.
Let X be a two-dimensional real or complex space with a basis
Ilelll = IIe211 = 1, and let
A be the operator given by the matrix
A=
(~ ~)
with respect to this basis. Since An = 0 for n
~
2, we obtain that r(A) = 0, while
IIAII ~ 1. The next related result is interesting. THEOREM 2.2 (Ya. B. Rutitzky). Let A E L(X). Then for any c
norm
II . II"
r(A) ~ <J
> 0 there exists a
on X, equivalent with the initial norm, such that
Choose n such that
From the inequalities
IIAII" ~ r(A) + c.
IIAnl1 ~ < r(A) + c,
set r
(2.4)
= r(A),
and let
LINEAR OPERATORS
110
it follows that the norms II . II and II . lie are equivalent on X. Further, IIAlle
=
sup IIAxll e
Ilxll e=l
=
sup ((r + c)n-11IAxll + (r + c)n-21IA 2xll + ... + IIAnxll)· Ilxll e=l
According to the choice of the integer n, it follows that IIAnxll:::;; (r+c)nllxll. Hence, after a rearrangement of the terms, we obtain: IIAlle = (r + c) sup ((r + ct- 11lxll + (r + c)n-21IAxll + ... + IIAn-1xll) = Ilxll e =l
= (r + c) sup Ilxll e = (r + c) II x ll e =l
which proves the right hand side part of inequality (2.4). By (2.3), rCA) does not change if we replace the norm II . lion X with the equivalent norm II . lie. Setting IIAnll e instead of IIAnl1 in (2.3), we obtain that
rCA) :::;; IIAlle. ~ We conclude this paragraph by considering a family (called an analytic operatorial sheaf- see, for instance, A. S. Markus [1]) of linear bounded operators A(A) on X, which depends analytically on a parameter A. Our goal is to study r(A(A)) = rCA) as a function of the parameter A. The next result is true.
Let A(A) be an analytic operatorial sheaf defined C. Then the spectral radius rCA) is a subharmonic function.
THEOREM 2.3 (E. Vesentini [1]).
on a domain
=
eEN ~ ME.
we have 8 /2 contradiction.
Since
E
00
115
and a fixed s, and using (4.1),
is an arbitrary positive number, it follows that 8
=
0, a
~
Let A and B be two completely continuous linear operators on X. Then any linear combination of A and B is a completely continuous operator, too.
THEOREM 4.2. a)
b) If A is a completely continuous linear operator on X and C E L(X) then AC and C A are completely continuous operators.
c) If A = lim An in the norm topology of L(X) and An are completely n--->oo continuous linear operators on X, then A is completely continuous. A. If {Xj}jEN is a bounded sequence in X then each sequence {AnXj}jEN is precompact. Using the diagonal method we get the precompactness of the sequence {Axj} jEN' ~ The main theorem of this section deals with the structure of the spectrum of a completely continuous operator. In order to prove it, we need a preliminary result.
A bounded projection on a Banach space is a finite-dimensional operator if and only if it is completely continuous. LEMMA 4.1.
oo lim Ilu-xnll.
xE9J1
Since the space
wt is finite-dimensional, the sequence {Xn}nEN is bounded and, therefore, precompact. We may assume that Xn ----> Z E wt as n ----> 00; then d = Ilu-zll. Put Vk+1 = d-I(u-z). Clearly Ilvk+11l = 1 and }~Jn IIVk+1 - xii = }~Jn
II u -
z - dx d
II = dllu 1 - zll =
1.
LINEAR OPERATORS
116 Further
IIVk+1 - vkll ~ inf
xE9JI
Ilvk+1 - xii
~ 1,
for m = 1, ... , k. Thus there exists a sequence {vn}nEN satisfying all the above mentioned properties. Clearly none of its subsequences is convergent. On the other hand PV n = Vn , n E N, which rontradicts the complete continuity of P. Consequently,
dimPX
x and AX n ----> Y as n ----> 00. For any f E D(A*) we have lim (Axn' f) = lim (xn' A* f) = (x, A* I) and n---+oo
lim (Axn' f)
n--+oo
= (y, I).
So y
=
n---+oo
Ax and therefore A is closed. By Theorem 1.2 the
operator A is bounded. Consequently, its kernel N(A) is closed. The adjoint operator A* is bounded, too, and everywhere defined (see §4 of Chapter 0). We prove now equality 3). The inclusion N(A) S;; -LR(A*) can be obtained by arguments similar to those in the proof of equality 1). Let x E -LR(A*). Then (x, A* I) = 0 for all f E ~*. Therefore Ax = 0, i.e., x E N(A), hence -L R(A*) S;; N(A). Equality 4) follows from 3) using Theorem 0.4.8. ~ A densely defined linear operator A is called normally solvable if R(A)
=
-L N(A*).
(5.3)
By equality 2) in (5.1) we have: THEOREM 5.2 (Hausdorff). A densely defined operator A is normally solvable if and
only if its range is closed, i.e., R(A)
= R(A).
COROLLARY 5.1. If an operator BE L(X) is completely continuous then A = 1- B is normally solvable. <J If 1 E pCB) then R(A) = ~ (= R(I - B)) is closed. Assume that 1 E a(B). Then, by Theorem 4.3, we have that 1 is an eigenvalue of the operator B of a finite
Noether and Fredholm operators
119
multiplicity, i.e., the corresponding eigenspace Xl is finite-dimensional. Let PI be the projection onto Xl, defined by (1.2), corresponding to the obvious spliting of the spectrum: UI = {l} and U2 = u(B) \ UI· Then Xl = PIX and X = PIX + (I - PI)X. Setting X 2 = (1 - PdX and B2 = B I X 2 we obtain that 1 E p(B2). Therefore (I - B 2)X2 = X 2 and, consequently, R(A) = (I - B 2)X = (1 - B 2)X2 = X 2 is a closed subspace of X. ~ The next result follows from (5.3). THEOREM 5.3. If an
operator A : X
----+ ~
is normally solvable then the equation
Ax=y
has a solution, if and only if (y, 1)
=
0 for all solutions j of the equation A*j = O.
COROLLARY 5.2. If equation (5.1)
is solvable for any y E
(5.4)
(5.5)
has the only solution j = 0, then equation (5.4)
~.
§6. N oether and Fredholm operators The following classes of operators play an important role in the theory of linear equations with normally solvable operators. A normally solvable operator A : X ----+ ~ is called a Noether operator, or an N-operator, if both the spaces N (A) and N (A *) are finite-dimensional. For a Noether operator A, both n(A) = dimN(A) and m(A) = dimN(A*) are non-negative integers, called the number of zeros and the defect of the operator A, respectively. The integer X(A) = n(A) - m(A) is called the index of the operator A. A Noether operator A satisfying X(A) = 0, i.e., n(A) = m(A), is called a Fredholm operator, or an F -operator. Let A : X ----+ ~ be a Noether operator. If {gdk=l is a basis of IJ!(A*) then the condition of solvability for equation (5.4) can be written as a system of equalities:
(y,gk)
=
0,
k = 1,2, ... ,m.
If, in addition, A is a Fredholm operator then, using Corollary 5.2, we obtain the next
result, usually referred to as the Second Fredholm Alternative.
LINEAR OPERATORS
120
A : X ----t ~ is an F-operator then, either the equation (5.4) has a unique solution for any vector in the right hand side of that equation, or equation (5.5) has only the zero solution.
THEOREM 6.1. If
(5.3) the solvability of the equation (5.4) for any vector in its right hand side is equivalent with the equality ~ = J.. N(A*), which leads to N(A*) = {a}. From dimN(A) = dimN(A*) we obtain N(A) = {o}. On the other hand, let us assume that equation (5.5) has only the zero solution, i.e., N(A) = {a}. Then N(A*) = {a}. By (5.3) we obtain R(A) = ~, i.e., equation (5.4) has a solution for any vector y in its right hand side. Since N(A) = a such a solution is unique. ~ <J By
Suppose now that A E L(X, ~). If A is a Fredholm operator and, moreover, N(A) = {a}, then by Theorem 6.1 its range R(A) coincides with the whole space ~. By the Banach Inverse Mapping Theorem, the operator A is a linear homeomorphism between the spaces X and~. Assume, on the contrary, that N(A) =f. {a}. In this case the operator A can be perturbed by a finite rank operator K such that the corresponding perturbation B = A + K becomes a homeomorphism between the spaces X and ~. The construction of such a perturbation, which is given in the sequel, was obtained for the first time by E. Schmidt [1] in the case of integral operators. Let A E L(X,~) be a Fredholm operator, with n = n(A) ~ 1. Let {adk=l be a basis of N(A), and let {gdk=l be a basis of 5Jt(A*). By the Hahn-BanachSuchomlynoff Theorem we can choose two collections of elements Udk=l and {bdk=l in the spaces X* and ~, respectively, such that
(6.1) Define a linear operator K by the formula n
Kx
= 2)x, fi)bi ,
x
E
X.
(6.2)
i=l
The operator K acts from the space X onto the finite-dimensional subspace £ = clin{b 1 , ... , bn } ~ ~ and satisfies the conditions
Kai=bi,
i=l, ... ,n.
(E. Schmidt). Let A E L(X,~) be a Fredholm operator with n(A) ~ 1. Then the operator B = A + K, where K is defined by (6.2), has a bounded inverse
LEMMA 6.1
Projections. Splitable operators
121
operator B- 1 defined on the whole ~, i.e., B is a homeomorphism of the spaces X and~.
Assume that Bx = 0 for some x E X. Then, by (6.2), we have
<J
n
Ax
= -
(6.4)
2.Jx, fi)bi. i=1
From (6.4) and using the equality N(A*) = R(A).l (see Theorem 5.1) we obtain that (Ax, gk) = 0 for any gk E N(A*), k = 1, ... , n. Consequently n
O=L(x,fi)(bi,gk)
=
(X,fk),
(6.5)
k=l, ... ,n.
i=1
n
From (6.4) and (6.3) it follows that Ax = 0, i.e., x E N(A), and therefore x = L D:iai. i=1
By (6.5) we find D:k = (x, fk) = 0, hence x = O. So we conclude that N(B) = {O}. Let us prove now that R(B) =~. Let y E ~ be fixed. Consider the element n
f} = y - L(y,gi)bi . By (6.1) we have (f},gi) = 0, hence f} E .IN(A*) = R(A). This i=1
means that there exists an element x E X which satisfies f} = Ax. Set n
x=x+ L((y,gi)-(x,fi))ai. i=1
Then Ax
=
Ax, and using (6.3) we obtain
n
= f}
+L i=1
n
(x, fi)bi + L( (y, gi) - (x, fi) )bi = f} i=1
n
+L
(y, gi)bi = y.
i=1
Thus the equality R(B) = ~ is proved. Now the conclusion of our lemma follows from the Banach Inverse Mapping Theorem. ~
§7. Projections. Splitable operators 1. PROJECTIONS. We have already dealt with projections when we defined the direct sum of spaces (see §7 of Chapter 0). Recall that by a projection on a Banach space X
122
LINEAR OPERATORS
we mean a linear operator P defined on the whole space X, with the property p 2 = P. We will consider below only continuous projections which, for brevity, will be simply called projections.
If P is a projection then Q = I - P is a projection, too, since obviously Q is linear, is defined on the whole X, is continuous, and Q2 = (I - p)2 = 1- P = Q. The restriction of the projection P on its range R(P) = PX is the identity operator P on this subspace, so that, if P =1= 0 then IIPII ~ 1. Indeed, for any U E R(P) we find an element x E X such that u = Px and, therefore, Pu = P Px = p 2 x = Px = u. Thus R(P) is closed, i.e., it is a subspace in X. Any element x E X has a unique representation x = u + v where u = Px, v = Qx. Therefore the space X decomposes into the direct sum of subspaces R(P) and R(Q), i.e., X = R(P)+R(Q). By Theorem 0.4.2 this direct sum is topological. Thus any projection P gives rise to a decomposition of the space X into the topological direct sum X = X l +X 2 , where Xl = R(P) is the subspace onto which P acts, and X 2 = R(I - P) is the subspace along which the operator P projects the elements of X. It is clear that R(I - P) = N(P) and R(P) = N(I - P). The projections P and Q = I - P are called mutually complementary. According to the previous considerations, a subspace £ of a Banach space X is called topologically complementable if there exists a projection P on X such that R(P) = £. Generally speaking, not all the subspaces of a Banach space are topologically complementable. There are a lot of papers and monographs devoted to this topic - the existence of topological complements (see, for instance, M. Kadec, B. Mityagin [1]). However, in the case of a Hilbert space X = 5), any subspace £ has not only a topological complement, but also an orthogonal complement (see Theorem 0.11.2). More precisely, any proper subspace £ can be obtained as the range of a projection P, with minimal norm IIPII = 1, such that the subs paces R(P) and R(I - P) are orthogonal. Let us return now to the general case of a Banach space X. A projection P on X will be called proper if IIPII = 1. Since the finite-dimensional subs paces of a Banach space have always topological complements, they play an important role in some problems concerning projections. To be more specific, let X be a Banach space and (!; a finite-dimensional subspace of X, with dim (!; = n. Choose a basis {ai}f=l in (!; and define the linear functionals Ii, 1 ~ i ~ n, as follows. For u = alaI + ... +anan E (!; we put Ii(u) = ai, so that (ai, Ij) = Dij, i,j = 1, ... , n. Obviously the functionals Ii are well-defined on the subspace £ and are bounded. By Theorem 0.8.7 they admit extensions on the
123
Projections. Splitable operators whole X, denoted also by
Ii, i = 1, ... , n, with
the same norms. For x E X we set
n
Px
=
L(x, Mai.
(7.1)
i=l
Then the operator P is a projection on X, with R(P)
=
IE. Indeed,
n
=
L (x, li)ai = Px, i=l
for any x E X, hence p 2 = P. Further, it is clear that R(P) 9)1i is the restriction of the operator EiAPj on the subspace £j, = 1,2. The matrix representation (7.3) of the operator A is called non-degenerate if the operator A22 is a homeomorphism from the space £2 onto 9)12. Provided that this condition is fulfilled and, in addition,
i,j
(7.4) then the representation (7.3) is called regular.
124
LINEAR OPERATORS
An operator A : X -+ ZJ defined on the whole space X is said to be splitable if it admits a regular representation (7.3). THEOREM 7.1. A normally solvable operator A E L(X, ZJ) is splitable if and only if its kernel N(A) and its range R(A) have topological complements.
<J Suppose that the subs paces N(A) and R(A) have topological complements, i.e., there exist the projections Pi on X and El on ZJ such that R(Pd = N(A) and R(E1) = R(A). Set P 2 = I -Pi, E2 = I -E 1. Then formula (7.2) is true for all x E X. Since PiX E N(A), then E 1AP1x = 0, E 2AP1x = and Ax = E 1AP2x + E 2AP2x. Moreover, y = AP2x E R(A), so E 2y = 0, hence Ax = E 2AP2x. Thus, in our case, representation (7.3) has the particular form
°
A=
(0° 0) A22
(7.5) '
where A22 is the restriction of the operator E 2AP2 on R(I - Pd - the topological complement of the subspace N(A). It follows that R(A 22 ) = R(A), N(A 22 ) = {O} whence by the Banach Inverse Mapping Theorem we obtain that A22 is a homeomorphism. Obviously equality (7.4) is satisfied. So we proved that the corrditions of the theorem are sufficient. Conversely, assume the existence of the projections Pi on X and El on ZJ such that representation (7.3) is regular. Then the equation Ax = is equivalent to the system of equations { Al1P1X + A 12 P2X = (7.6) A 21 P 1X + A 22 P 2X = 0.
°
°
The second equation in (7.6), on its turn, is equivalent to (7.7) From (7.7) and (7.4) it follows that the first equation in (7.6) is an identity. Thus the kernel N(A) consists exactly of those elements which satisfy (7.7). Let us define a linear operator P on X by (7.8)
We have
125
Projections. Splitable operators
i.e., P is a projection (recall that A2"l : £2 -+ 9J1 2 and P l 9J1 2 then by (7.7) we obtain Px = (Pl + P 2)x = x, whence N(A) show that R(A) --> X with D(A) = X. A subspace ,c ~ X is called invariant relatively to A if A,C ~ 'c.
Invariant subspaces
127
1. Let A E L(X). Assume that the kernel N(A) has a topological complement, i.e.,
X = N(A)+9J1, where 9J1 is a subspace of X. Let us consider the following question: when is the topological complement 9J1 of N(A) an invariant subspace relatively to A? If, in particular, the operator A is splitable, then, by Theorem 7.1, the existence of two decompositions of the space X follows: X
= N(A)+9J1,
X
= 'c+R(A).
When these two decompositions coincide, i.e., N(A) = ,c, R(A) = 9J1, then the subspace 9J1 is invariant relatively to the operator A and X = N(A)+R(A). Let now A be an arbitrary operator in L(X). Consider the following two sequences of lineals: (8.1) {O}, N(A), N(A 2 ), ... ,N(Ak ), ... , X, R(A), R(A 2 ), ... , R(A k ), ... .
(8.2)
Relatively to inclusion, the sequence (8.1) does not decrease and the sequence (8.2) does not increase, i.e., N(Ak) 0 there exists n E N such that IIBnYl1 < c. We have y - BnY E R(A); indeed
y-BnY= (I-n- 1 tBm)y=n-1 t(I-Bm)y= m=l
m=l
n
= n-1(I - B)
L (I + B + B2 + ... + Bm-1)y = (I -
B)Yn,
m=l
n
where Yn = n- 1
L (I + B + B2 + ... + Bm-1)y. m=l
Since y - BnY E R(A) and R(A) is closed, taking into account that c was an arbitrary positive number, we obtain y E R(A), i.e., N(P) ~ R(A). The converse inclusion R(A) ~ N(P) follows directly from (8.9). ~ The arguments in our proof of Theorem 8.2 above are, actually, suitable modifications of the arguments used by K. Yoshida in the proofs of the Statistical Ergodic Theorem and its consequences (see, for instance, K. Yoshida [1]). This theorem asserts the following: REMARK 8.2.
Let B E L(X) be an operator satisfying condition (UB). Consider the operators B n , n E N, defined by (8.8) and assume that the sequence {BnX}nEN is weakly compact for x E X. Then this sequence is convergent in the norm topology of the space X.
132
LINEAR OPERATORS This theorem and Theorem 8.2 lead to the next simple but important result.
COROLLARY 8.3. Let X be a reflexive space. Assume that an operator B E L(X)
satisfies condition (UB) and the operator 1- B is normally solvable. Then the equality (8.6) is true. Some authors (see, for example, F. Riesz [1]) refer to Corollary 8.3 as the Statistical Ergodic Theorem.
Chapter V Nonlinear equations with differentiable operators §1. Fixed points. Banach principle 1. Let 1) be a topological space. Assume that F is an operator defined from 1) into 1),
i.e., (1.1)
DEFINITION 1.1. A point
z E
1)
is called a fixed point for the operator F if F(z)
= z.
Condition (1.1), which is refered to as the invariance of the set 1) for the operator F, enables us to define the iterations Fn of F, by the recurrent relations Fl = F, F n+1 = Fn 0 F, where "0" denotes the composition of operators. DEFINITION 1.2. 1) A fixed point z E 1) ofthe operator F is called locally attractive if there exists a neighborhood U ~ 1) of the point z, such that the iterations Fn(x) converge to z for all x E U. If U = 1) we shall say that z is an attractive (or globally attractive) fixed point. 2) The point z is called an s-fixed point (successful approximative fixed point) for the operator F in 1) if it is attractive uniformly on each closed and bounded subset of 1), i.e., the iterations Fn(x) converge uniformly on each closed and bounded subset of 1). 3) A fixed point z is called repulsive if the sequence Fn(x) converges to z if and only if x = z. 4) A fixed point z will be called neutral if it is neither locally attractive nor repulsive.
NONLINEAR EQUATIONS
134
DEFINITION 1.3. A fixed point z of F is called isolate if there exists a neighborhood
of this point which does not contain other fixed points for the operator F. REMARK 1.1. A locally attractive point z is an isolate fixed point. Moreover, a globally attractive point is unique. Indeed, assume that on a neighborhood ti of the point z the iterations Fn(x) converge to z for all x E ti. If we suppose that the operator F has another fixed point y in ti, then Fn(y) ----) Z. But y is a fixed point for the operator Fn for any n, since Fn(y) = FoFn-l(y) = F(y) = y. Hence z = y. We notice also that for a continuous operator F the convergence of the iterations implies
the existence of fixed points. A lot of work has been done to obtain criterions for the existence of fixed points and to study the properties of these points (uniqueness, isolation, attractiveness, and so on) for various operators. These criterions are frequently keystones in the theory of nonlinear operator equations. The theorems which establish conditions for the existence of fixed points are usually called fixed point principles. As a rule, the formulation of a fixed point principle starts with the condition of invariance (1.1). However, in many cases this condition is insufficient for the existence of a fixed point, and, even more, it is insufficient for the existence of an s-fixed point. Notice that the presence of an s-fixed point guarantees not only its uniqueness, but also the possibility of approximating it (by approximative computations). Let us remark that the development of the computer techniques enables one to approximate a fixed point by iterations even for rather complicated operators. At the same time the computer methods do not insure against accidental computational errors, which practically do not matter as soon as we know in advance that the iterative proccess converges uniformly.
One of the most effective fixed point principle is the Banach principle for contractive operators. Let (X, p) be a complete metric space with the metric p( " . ) (see §4 of Chapter 0). An operator F defined from a subset ~ 1, is an s-fixed point for F, too. However, as we shall prove below, this is true for holomorphic operators. In order to approximate a fixed point by iterations under the assumptions of Corollary 1.1 it is advisable, if it is possible, to start with an explicite computation of the operator FP and then to use its iterations. REMARK 1.3. The condition q < 1 in (1.2) is not necessary for the existence of s-fixed points. For example, consider the operator Fx = ax2, where 1/2 < a < 1, which maps the interval [0, 1] into itself and has the s-fixed point 0, though q = 2a > 1. Nevertheless, in the general case condition q < 1 can not be weakened by setting q ~ 1. For example, the operator Fx = x+a, a -I- 0, which maps IR into IR and satisfies condition (1.2) with q = 1, does not have fixed points. In this example the diameter of the set on which the operator F acts is infinite. However, there are examples of operators acting on domains with bounded diameter, in infinite dimensional spaces, which satisfy condition (1.1), or condition (1.2) with q = 1, and do not have fixed points (for more details see §2 below).
Let us return now to the concept of s-fixed points and state, without proof, a remarkable theorem due to P. R. Meyers [1], which explains the substance of this notion and represents the most complete converse of the Banach principle. (In connection with this result see also A. Levin and E. Lifsic [1].)
(P. R. Meyers). Let X be a complete metric space and 1) a closed subset of X. Assume that an operator F satisfies the invariance condition (1.1) and has an attractive fixed point z E 1), which is an s-fixed point in a neighborhood of z. Then z is an s-fixed point in 1) and, moreover, for any q with < q < 1 there exists a metric on the space X equivalent with the original metric, such that F satisfies the contraction condition (1.2) on :D with respect to this new metric. THEOREM 1.2
°
Although Meyers' result is merely an existence theorem, it shows that in some
Non-expansive operators
137
concrete problems, if we know a piori about the existence of an s-fixed point, then, for estimating the rate of convergence of the iterative process, it is convenient to try to find an equivalent metric, with respect to which F is a contraction. 2. One of the most important fixed point principles is the well known principle due to J. P. Schauder, which is a generalization of the finite-dimensional fixed point principle of Brouwer. THEOREM 1.3 (J. P. Schauder). Let F be an operator which maps a closed convex subset:D of a Banach space X into itself and is completely continuous on:D. Then F has at least one fixed point in :D.
All the proofs of this theorem we know about rely on subtle topological and geometric considerations. As a rule, these proofs use Brouwer Theorem, finite-dimensional approximations, and also some results from the degree theory for mappings and the vector fields theory (see, for instance, V. A. Trenogin [1], M. A. Krasnoselskii [1]). This is why we stated Schauder Theorem without proof. Obviously, neither Theorem 1.1, nor Theorem 1.3, are related to each other. Let us remark also that Schauder principle has no constructive features; it fails to indicate the number of fixed points, as well as methods for their approximation. In the following section we will consider a larger class of operators, which includes the class of contractive operators. At the same time, under the assumptions of Schauder principle, we will indicate some approximation methods for the fixed points.
§2. Non-expansive operators Let (X, p) be a metric space and :D a closed subset of X. DEFINITION 2.1. An operator F : :D strictly non-expansive, if the inequality
---+
p(Fx, Fy) is fulfilled for all x, y
for any x, y
E
:D, x
E
=1= y.
~
X is called non-expansive, respectively
p(x, y)
(2.1)
p(Fx, Fy) < p(x, y),
(2.2)
:D, respectively, if
NONLINEAR EQUATIONS
138
A large amount of work is devoted to the theory of non-expansive operators, either on metric, or on Banach spaces (see for example Z. Opial [1]). Here we treat only those aspects which are related to the fixed points of differentiable operators. As we have already noticed in Remark 1.2, the operators satisfying condition (2.1) may have no fixed points. In addition, let us present a few other examples.
2.1 (T. Hayden, T. Suffridge [1]). Let Co be the space of all sequences x = (Xl> ... , X n , ... ) of real or complex numbers which converge to zero, with the norm Ilxll = max IX n I· Define an operator F on the unit ball of this space by the equality
EXAMPLE
n
Fx =
(!, Xl, X2,·· .), for
any X = (Xl, ... , Xn , .. . ). Clearly IIFxII ,,;; 1 if IIxil ,,;; 1, i.e., F satisfies condition (1.1), and IIFx - Fyll = 11(0, Xl - YI,···, Xn - Yn," .)11 = IIx - YII, Le., F satisfies condition (2.1), too. At the same time, if we assume that there exists an element Z = (ZI, ... ,Zn, ... ) such that Z = Fz then, necessarily, Zl = Z2 = ... = = Zn = ... = i.e., Z ~ Co, a contradiction.
!,
We can show that condition (2.2), as well, is not enough for the existence of fixed points. EXAMPLE
2.2. Let X =
~
and :D
= [1, (0).
The operator F defined by Fx = X +
dearly does not have fixed points, while F(:D) jFx - Fyi
for all x, Y ~ 1,
X
= Ix -
Y+
~-
tl
=
~
Ix -
1
-
X
:D and
YII x Yx ; 11 < Ix - YI,
i=- y.
Related to the example above we easily notice that for any q, with 0 < q < 1, there exist x, Y E [1,(0) such that jFx - Fyi> qlx - YI. An analogous example can be produced on a ball with a finite radius in an infinite dimensional space, too. EXAMPLE 2.3 Let
F defined by Fx
1
+-n sin n.
=
X = Co and let :D be the unit ball in X. Consider the operator n-1
(YI, ... , Yn," .), where X
=
(Xl, ... , x n , ... ) and Yn
=
- - xn+
n We can easily prove that the operator F satisfies both conditions (1.1) and
(2.2) on:D. At the same time, F has no fixed poins in:D. Assume, on the contrary, n -1 1 that Z = Fz, where Z = (Zl"'" zn) E:D. This means that Zn = --Zn + - sin n. n n Therefore, Zn = sin n, which contradicts the definition of the space Co. All these examples show that the existence of a fixed point for a non-expansive operator requires some additional conditions.
Let (X,p) be a complete metric space, :D a closed subset of X, and F a strictly non-expansive operator on :D satisfying the invariance condition
THEOREM 2.1.
139
Non-expansive operators
(1.1), i.e., F(1») 1. Since in many concrete problems the function y(t) has a well-determined meaning, it is important to find an estimate for the absolute value of y(t), which guarantees that equation (4.20) has solutions. In order to accomplish this we will consider the auxiliary, so-called, majorant numerical equation: (4.21) where N is a positive number. Since the function in the right hand side of (4.21) is concave upward for r > 0, equation (4.21) has only two positive solutions r* and r*, r* < r*, for m-1 (4.22) Ilyll < l = m rn-VmH; for Ilyll = l it has the unique positive solution ro 1 below).
=
r*
=
r*
= (mN)-rn"-l (see Figure
NONLINEAR EQUATIONS
158
r*
r*
r
Figure 1
Thus, if we set N = K,(b - a), where
Ilyll ~
1'0,
=
max
a ~ lI,t ~ b
11(0", t)l,
then for
m-l m
=-y'mK, (b -
(4.23)
a)
the operator F defined by the right hand side of (4.20) satisfies the following condition
IIF(x)11 ~ Nr*m + Ilyll
= r*,
for Ilxll ~ r*. According to Schauder principle, there exists at least a solution of equation (4.20) satisfying the estimate max Ix(t)l~r*.
(4.24)
a~t~b
Whenever y satisfies the stronger inequality (4.22), with N the operator F is a contraction on the ball Ilxll ~ r*. <J
=
K,(b - a), then
Indeed, under these conditions we have r* < ro, hence
IIF(xd -
F(X2) II ~ K,(b - a)
::;;; mK,(b -
Ilxi"(t) - x;"(t) II ~ a)r:,,-lllxl - x211 = qllxl - x211,
where q = mK,(b-a)r:,,-l < 1. Moreover, IIF(x)11 < r* for IIxll < r*. Therefore, under condition (4.22), equation (4.20) has a unique solution x(t), satisfying the estimate (4.25)
Some applications of fixed point principles
159
In addition, the sequence {Xn (t)} nEN given by b
xn(t) =
J
K(O", t)X~_I (t)dO" + y(t),
n = 1,2, ... ,
(4.26)
a
converges uniformly on the interval [a, b] to x(t), for any continuous function xo(t) which satisfies the condition
Let us point out the next important remark. REMARK 4.5. If condition (4.22) is fulfilled, then between the numbers r * and r*
there exists a "clearance" (see Figure 1), which guarantees the existence of at least one solution for equation (4.20) continuous on [a, b]. However, in this case, following the same arguments, we can not infer from condition (4.27) the existence of a unique solution for equation (4.20) satisfying the above mentioned estimate. To achieve this goal we have to use essentially the properties of analytic operators. REMARK 4.6. An upper-bound of the norm of y that guarantees the existence of a unique continuous solution can be obtained using the classical Cauchy-Goursat majorant series method (see V. A. Trenogin [1]). Using Cauchy inequalities (see §3 of Chapter III) for the coefficients ak(O"), we can estimate the Lipschitz constant L for the function f(O", u) on a certain disk of radius PI < P and centered at zero, as follows:
M P
(
~)
m-I
(m _ ~ (m _
1))
(1- ~f
Obviously PI can be chosen such that LK,(b-a) < 1. Now, assuming that PI is chosen and fixed, we can find a number it :( l such that, for
liyll :( it,
NONLINEAR EQf{ATIONS
160
equation (4.22) has two solutions r * and r* satisfying the relations
o < r * ~ PI ~ r*. Then, the operator F, defined by the right hand side of (4.22), maps the ballllxli ~ r* into itself and is a contraction on it. For example, assume that m = 2. Then the number PI must satisfy the quadratic inequality
~ f), which is invariant relatively to the operator P A for (x, A) E i> X itl, where itl is a ball with a radius larger than that of it. We must point out that we have already followed this approach in Subsection 4 of §4. Indeed, let us return to equation (4.27): x = Fx + y, where F is a Hammerstein integral operator. The solution x = tJt(y) of this equation can be considered as an inverse of the operator P = I-F. Therefore, the estimates for Ilyll obtained there are nothing else but estimates for the domain of existence of the inverse operator tJt. Finally, let us remark that the problem concerning the fixed points of a certain operator S can be solved as well, by reducing it to an implicit operator problem. This can be done, as we will see below, by introducing a parameter in the right hand side of the equation x = S(x). To be more specific, let us assume that we have already
NONLINEAR EQUATIONS
164
x
constructed an operator P : ::£) x st ---7 which satisfies the equality P(x, >'9) = S(x) for a certain >'0 E st. In the case when, for some reasons, there exists an implicit operator G(x) defined on a certain set it 3 >'0, which satisfies equation (5.1) for F(x, >.) = x - P(x, >'), then the point x* = G(>.o) is obviously a fixed point for the operator S. If the set ::£) is a ball centered at the zero element of the space x, we can almost always choose the operator P as P(x, >.) = >'S(x), where>. E [0, 1] if is a real Banach space, or >. E C, 1>'1 :::; 1, if is a complex space. Taking into account the convexity of::£) we can also set P(x, >.) = >'S(x) + (1 - >')y, where y is an arbitrary element of::£) and>' E [0, 1]. In this case, if the operator G(>') exists on a set containing the interval [0, 1], then the point x* = G(1) is a fixed point for the operator S.
x
x
2. In the rest of this section we present some local theorems on implicit operators. A standard theorem which generalizes the classical case, can be formulated in the following way.
Assume that the operators F : x x A ---7 ZJ and F~ : x x A ---7 ZJ are defined and continuous on the set {(x, >.) E x x A : IIx - xoll :::; R, II>' - >'011 :::; T}, for some (xo, >'0) E x x A such that F(xo, >'0) = 0, and the linear operator A = F~(xo, >'0) is continuously invertible. Then there exist the numbers p, 8> 0, p:::; R, 8:::; T, and a unique continuous implicit operator x = G(>') satisfying equation (5.1) and condition (5.4), which is defined on the ball {>. E A : II>' - >'011 < 8} and takes values in the ball {x E x : IIx - xoll < pl. Moreover, if the operator FHxo, >.) exists, then the operator G is differentiable at the point >'0 and THEOREM 5.1.
(5.8)
P = I - A-I F. Obviously, the operator P acts from equation (5.1) is equivalent to equation (5.7): .) = x - A-I F(x, >.). Then we have Xo
= P(xo, >'0)
P~(x, >.)
=
and
I - A-I F~(x, >.)
= A-I
(F~(xo, >.) - F~(x, >.)).
In particular, P~(xo, >'0) = 0. Since the operator F~(·, .) is continuous at the point (xo, >'0), then for any 0 < q < 1 we can find two numbers /1, TJ > 0, with /1:::; T, TJ:::; R, such that the inequality
(5.9)
Implicit and invertible operators
165
is fulfilled for Ilx - Xo I :s; 'f} and I A - AO II :s; /J. In addition, taking into account the continuity of the operator F, we can choose a number 0 < v < T such that
IIA-III for
IIA -
AO I < v. Let us denote min{/J, v} =
IIF(xo, A)II
T.
:s; (1- q)'f)
Then
IIP(x, A) - xoll :s; IIP(x, A) - P(xo, A)II + IIP(xo, A) - xoll :s; sup IIP~(x, A)II Ilx - xoll + IIA-III IIF(xo, A)II :s; :s; Ilx-xoll::;; TJ 11>'->'011::;; T
(5.10)
The last inequality shows that for A with I A - Ao I < T the operator PC, A) maps the ball {x EX: Ilx - xoll :s; 'f}} into itself, and inequality (5.9) guarantees that the operator P(·, A) is a contraction on that ball. According to Banach principle, for any A with IIA - Aoll:s;; T there exists a unique solution x = G(A) of equation (5.7), hence, of equation (5.1), which satisfies the condition IIG(A) - xoll :s; 'f}. To get the first assertion of the theorem it remains to put p = 'f} and 8 = T. The continuity of operator G follows by some standard arguments. The proof of this fact, as well as the proof of the second assertion of the theorem, are left to the reader. Notice however that (5.8) can be obtained directly differentiating the identity F(G(A), A) == 0 with respect to A, for IIA - Aoll :s; 8. ~ The numbers 8 and p play an essential role in different applications. Therefore, it is desirable to find some estimates for 8 and p. To this end we need some additional assumptions. Suppose that the operators F~(·, . ) : X x A -+ !D and F(xo, . ) : A -+ !D satisfy the Lipschitz conditions with constants Land N, respectively, i.e., 11F~(x, A) - F~(xo,
(we considered here that the norm and
IIF(xo, A) for (x, A) E X x A with are fulfilled for
where m =
IIA-III.
Ao)11 :s; L(llx - xoll + IIA - Aoll) II(x, A)II
is equivalent with the norm
F(xo, Ao)11
:s; NIIA - Aoll,
Ilx -xoll :s; R, IIA - Aoll :s; T.
(5.11)
Ilxll + IIAII) (5.12)
Then inequalities (5.9) and (5.10)
NONLINEAR EQUATIONS
166
Solving this system of linear inequalities we obtain T) ,,::. -
1
'" mL
.
qmN ---=-----
mN + 1 - q ,
T ,,::. -
1
'" mL
.
q(1 - q) mN + 1 - q
~--'-----'--'--
T), T are functions of the parameter q, where q E [0, 1]. For q = and q = 1 we have T = 0. Therefore, the largest value of T is reached for some q* E (0, 1). It is easy to verify that
It is clear that the numbers
°
y'1 +mN q* = -v771=+=m=7N'+-vrm=N~ and T*
= T(q*) = q~~) T(q)
(5.13)
1 =
mN( y'1
(5.14)
+ mN + vmN)2·
Set also
y'mN T)*
= T)(q*)
=
(y'1
(5.15)
+ mN + vmN)mL·
Finally, we obtain the next result.
Assume that the conditions in Theorem 5.1 and inequalities (5.11), (5.12) are fulfilled. Then, on the ball
THEOREM 5.2.
11,\ - '\011::;; b = min{r, T*}, there exists a unique continuous implicit operator G('\) defined by equation (5.1) and relation (5.4), which satisfies the condition
IIG('\) - xoll ::;; p = min{R, T)*}, where the numbers T*, p* are defined by formulas (5.14), (5.15), respectively. Taking into account the arguments developed in Subsection 1, from Theorem 5.2 we straightforwardly obtain the next result.
Let 1) = {x EX: Ilx - xoll ::;; R} and let
'0) : X on the ball {>. E A : II>' - >'011 ~ p}, where
~ ~
~
~
R}.
is continuously invertible. Then
(5.20)
( A B = 0
F~ (xo, fA
Uo
00
>'0)) '
=
(xo, >'0),
h
=
(x - Xo, >. -
>'0),
and fA is the identity operator on A, there exists a unique implicit operator G(>.) satisfying the identity F(G(>'), >.) == 0 and the condition G(>'o) = Xo.
170
NONLINEAR EQUATIONS
REMARK 5.1. The method of constructing the inverse operator by formula (5.17),
described in the proof of Theorem 5.3, is called the method of undetermined coefficients. It was used more than once by different authors, in order to solve some concrete integral equations (see, for example, N. N. Nazarov [1] and A. I. Nekrasov [1]). It seems that this method in its general form was used for the first time by K. T. Ahmedov [1] for an implicit operator depending on a numerical parameter, and by J. Orava and A. Halme [1] for the inverse operator problem. Formulas (5.13)-(5.15) were established by V. A. Trenogin (see [1]). Using the same approach we can obtain some other, yet equivalent, estimates for T, 'f] if we set II(x,A)11
= max{llxll, IIAII}
or
In §5 of Chapter VI below we will establish some more precise - and more convenient for computations - estimates for the numbers p, 8, in the case of complex spaces.
Chapter VI Nonlinear equations with holomorphic operators Throughout this chapter we will consider complex Banach spaces only. In this case, many local features related to the solvability of equations with operators that are differentiable in the complex sense, turn out to be global.
§1. s-fixed points for holomorphic operators. A converse of Banach principle In this section we deal with differentiable operators and their fixed points that have the property of "succesful approximation", i.e., s-fixed points (see Chapter V). Let f> be an arbitrary bounded domain in X and F an operator holomorphic in f> (i.e., F is Frechet differentiable in the complex sense on a neighborhood of each point of f». Assume that F satisfies the invariance condition
F(f» is a fixed point for F, that is, Fz = z. As we have already noticed, if F E C(f>, X), i.e., F admits a continuous extension on of> - the boundary of the domain f> - , and if the point z is an attractive fixed point for the operator F (see §1 of Chapter V), then, as soon as z is a local s-fixed point, it is a global s-fixed point too. Indeed, by Meyers Theorem (Theorem V.1.2), the next assertion is a consequence of our assumptions:
(*) for any q with 0 < q < 1 there exists a metric p on the set f>, equivalent to the metric given by the norm of X, relatively to which the operator F satisfies the
NONLINEAR EQUATIONS
172
Lipschitz condition with constant q, that is, p(Fx, Fy)
~
qp(x, y),
x, y E 1:>.
(1.2)
Thus the assumptions in Theorem V.1.1 - the Banach principle - are fulfilled. If inequality (1.2) is satisfied on some closed subset II ~ 1:> and F(ll) ~ ll, then we will say that the operator F is a q-contraction on ll. The following result holds.
Let F be a holomorphic operator on 1:>, satisfying condition (1.1), let zE1:> be a fixed point for F, and let A=F'(z) be the Fhkhet derivative of Fat z. a) The following assertions are equivalent: 1) Fn x -+ Z for any x E 1:> and the operator e iO I - A is normally solvable (see §5 of Chapter IV) for all () E [0, 27rJ, where I is the identity operator on X; 2) rCA) < 1, where rCA) denotes the spectral radius of the operator A; 3) there exist a number r > 0 and a norm II . II * equivalent to the original norm of X, such that F is a ql-contraction on the ball II,. = {x E 1:> : Ilx - zll* ~ r} for a certain ql with 0 ~ ql < 1; 4) there exist a number q2 with 0 ~ q2 < 1 and a metric equivalent to the metric given by the norm of X relatively to which the operator F is a q2-contraction on some neighborhood of the point z. b) If, in addition, F E C(1:>, X) and Fn x -+ Z for any x E 81:> then conditions 1)-4) are equivalent to condition (*). THEOREM 1.1.
EXPLANATIONS: 1. Under condition b) the implication 1) =} (*) is a global converse of the Banach principle with respect to a metric equivalent to the metric given by the norm of X. The implication 1) =} 3) is a local converse of that principle, but this time, with respect to a norm equivalent to the norm of X. In A. A. Ivanov's book [1] a construction of a metric such as in condition 4) above is given. In many concrete problems this construction turns out to be rather difficult. If we know already the spectral radius rCA), then it is very simple to find the norm II . 11* and the number r appearing in condition 3) (see Lemma V.3.1). This enables us to estimate the rate of convergence to z of the iterations {Fnx }nEN, starting with an arbitrary x E II,.. 2. The implication 2) =} 1) means that condition rCA) < 1 is, in fact, a global feature of the s-fixed point z (see §3 of Chapter V) and, consequently, it guarantees the uniqueness of the fixed point z on the whole 1:> and also the convergence of the iterations, for any x E 1:>. . E C. : I>' I < 1}). Then z = 0 is an s-fixed point for the operator F. Suppose, on the contrary, that z = 0 is not an s-fixed point for the operator F. Then, by Lemma 1.1, we get r(A) = 1, and, consequently, there exists a number () E [0, 27r] such that e iO is an eigenvalue of the operator A. It follows that we can find an element x E 81) satifying Ax = e iO x. Choose a linear functional 1 E X* such that (x, I) = 1, 11111 = 1, and consider the analytic function .x), I) = (A (h)
= >.eiO(x, I) +
(Q(>.x), I)
+ Q(>.x), I) =
= >.e iO +
(Q(>.x),I),
where Q = F - A and>' E C. with 1>'1 < 1. From (1.5) it follows that 11')II < 1, for all >. E c. with 1>'1 < 1, and : 1) x fl ---> 1) defined for any (x,..\.) E 1) x fl, such that 1>(.,..\.) E 1i(1), X) for all fixed..\. E fl, and 1>(x, .) E 1ip(fl, X) for all fixed x E 1) (Le., the operator 1> is holomorphic on x and p-holomorphic on ..\.). A solution of the equation
x(..\.) = 1>(x(..\.) , ..\.) is called an s-solution, if for any operator Xo : fl the sequence {xn(..\.)}nEN defined by Xn+l (..\.) =
1>(xn(..\.), ..\.),
---> 1)
(2.1) such Xo (= Xo (..\.)) E 1ip(fl, 1»),
n = 0, 1,2, ... ,
(2.2)
converges in the strong topology of X to x(..\.) uniformly on each compact subset of fl, and, for a fixed..\. E fl, uniformly relatively to those xo(..\.) with values inside 1); in other words, x(..\.) is an s-fixed point of the operator 1>(., ..\.), for each..\. E fl. Now let us state the main result of this chapter.
Let 1> : 1) x fl ---> 1) be an operator which satisfies the above mentioned conditions. If, for at least one value ..\.0 E fl there exists an s-fixed point x* for the operator 1>(., ..\.0), then there exists an s-solution for equation (2.1). Moreover, for any class of operators IU s:;; 1ip(fl,1») which is closed in the topology of uniform convergence on the compact subsets of fl, and contains the orbit ~ = {v n : V n +1 = = 1>(vn' ..\.), n> N ~ 1, Vo E 1ip(fl, 1»)} of a certain element Vo, we have x(..\.) E IU. In particular, it is always true that x(..\.) E 1ip(fl, 1»). THEOREM 2.1.
u (= u(· )) E 1ip (fl, 1») the composed operator 1>(u(-),·) belongs to the class 1ip(fl, 1»), too. For any x E 1), ..\. E fl, TEA and ( E admits the representation .) maps the ball 1.4 centered at x* into itself, and (.,>.) is a ql-contraction on 1.4. Consequently, the sequence {Xn(>')}nEN converges to a certain element x(>.) E ~, for all >. E D p2 (>'0, 7). Since Ilxn(>')11 ~ M, for all n = 0,1,2, ... , it follows that the sequence converges uniformly, so our claim is proved. By Theorem III.6.4 we have x n (-) E Hp(D, ~). It is clear that x(·) E sn whenever sn is a subset of Hp(D,~) closed in the topology of uniform convergence on the compact subsets of D, and containing the orbit ~ = {vn+l = ( V n , >.) : n > N} of a certain element Vo E Hp(D, ~). ~ REMARK 2.1. The last assertion of our theorem turns out to be interesting in the
following situation. Consider an open set Dl ::J D. The class sn = H(Dl'~) is contained in Hp(D, ~). Although the operator (x, .) E Hp(D,~) is not necessarily holomorphic in D l , it is possible for the orbit ~ of an element Vo E H( D l , ~) to be contained in H(Dl' ~). Then, taking into account that the class sn = H(Dl'~) is closed in the topology of uniform convergence on compact subsets, we obtain that the solution x( >.) of equation (1.1) is holomorphic in D l . Some related examples will be presented below (see §6). As a consequence of Theorem 2.1 we obtain the next criterion for the existence of an s-fixed point for a holomorphic operator. THEOREM 2.2. Let ~ be a bounded domain which is (>star-shaped relatively to
oE
X. An operator F E H(~, X) satisfying the condition F(~) S;;; ~ has an s-fixed point Z E ~ if and only if there exist a subset i> s;;; ~, a number pEN, and a number c > 0 such that dist(FP(i», ai» =
inf _ Ilx - yll > c.
(2.6)
yEFP(1J) xE8i>
<J
Necessity. Let z E
~
be an s-fixed point for F in ~ and set r
=
inf Ilx-yli.
xE81J
Then, by the uniform convergence of the iterations F (x) to z it follows that for any c with 0 < c < r and any set i> completely contained in ~ together with its boundary, n
Criterions for the existence of an s-fixed point
181
there exists a number p such that IIFP(x) - zll < r - E, for all x E i). Clearly (2.6) is fulfilled for such E and p. Sufficiency. Consider the operator .. E C : 1>"1 < 1 +E}. By (2.6) we have ..) E :D for all (x, >..) E :D x fl and .. E fl. In particular, setting z = x(l) we obtain z = FP(z). It follows that z is an s-fixed point for the operator FP, whence, as above, we conclude that z is an s-fixed point for the operator F, too. ~ If, in particular, :D is the open ball of radius R centered at the zero element of X, then the following corollary provides a sufficient condition for the existence of an s-fixed point. COROLLARY 2.1. Let F be an operator holomorphic in :D which satisfies the strict invariance condition, i.e., there exists r < R such that
IlFxll ~ r, for all x with
Ilxll < R.
Then the operator F has an s-fixed point z E :D.
It seems that this result was obtained for the first time by M. Helve [1] in the finite-dimensional case and by C. Early and R. Hamilton [1] in the general case of a Banach space, using the generalized Poincare metric (see also L. Harris [4], T. Hayden and T. Suffridge [1], and K. Goebel and S. Reich [1]). The results in this section have various applications which will be listed in the sequel. Now we return to an example of an integral equation studied in the previous chapter. Let us consider equation (V.4.20):
J b
x(t)
=
K(u, t)xm(u)du + y(t).
a
As it was proved in §4 of Chapter III, for y (= y(t)) with relation
IIF(x)11
~
K,(b - a)rm
+ lIyll < r,
Ilyll
~(X(A), A),
and, from conditions 1), 2), we obtain
IIX'(A) II ~ ML(l-IAI)-"'.
A priori estimates Therefore
1
(P)
191
1
J
x' (rei'P)dr
:::; M L
o
J
(1
~rr )
ML I-a'
0
and, consequently, the limit 1
lim x (re i6 ) = (P)
r-+1
J
x' (rei'P) dr
o
exists and is finite for all 'P E [0, 27r]. Thus the vector-function X(A) is defined for all A E r = aLl. Let us show first that (4.2) is fulfilled for all A, A' E r, with a suitable constant K. This will prove also the continuity of X(A) on r. Without losing the generality, we may assume that larg A-arg A'I :::; 7r. Consider the vector-function X(A) = (2R)-l x (A). Using the relation
'P 2 - 'P' 7r Ie1'P . - e1'P. I , I'P - 'P, I :::; 7r Isin - I ="2 I
for all I'P - 'P'I :::; 7r, we obtain that it is enough to prove the inequality Ilx(A) - x(A')II:::; K1 largA - argA'1 1 -
(4.3)
with a suitable constant K 1. To this end we may assume that Iarg A - arg A'I < 1, since, on the contrary, (4.3) with K 1 = 1 is obvious. Let us represent the left hand side of (4.3) as
X(A) - X(A') = (P)
J
x'(()d(,
I
where l is the piece-wise smooth curve consisting of the line segments [A, tAl and [tA', A'], and of the arc which connects tA and tA' along the circle 1(1 = t = 1-Iarg A - arg A'I < 1. Then
Ilx(A) - x(A')11 :::;
J
(1Ix'(rA)11
+ Ilx'(rA')II)dr + a/Alt Ilx' (tei'P) II d'P argA argA '
J
argA
MLt(l_t)-d'P 2R
:::;
NONLINEAR EQUATIONS
192
Thus (4.3) is fulfilled for all A, A' E
r, with
KI =maX{1,
ML 2R
(_2 +1)}. 1- IX
This means that (4.2) is fulfilled, for the same A, A', with
We fix now the points A, X E r and consider the vector-function Xl (() analytic in the disk 1(1 < 1 and continuous on the closure of this disk. Since Xl (0) = 0, by Schwartz Lemma we obtain
= X((A) - X((A'),
It follows that for any r E [0, 1] we have
Ilx(rA) - x(rX)11 ~ r sup Ilx(~A) ~
rK 2 sup I~A
-
-
x(~X)11 ~
1~1=1
~A'II- =
K2rlr(A - A'W- ~
(4.4)
1~1=1
Finally, let us prove inequality (4.2) - with an appropriate constant K -, for all A, A' E ,1 lying on the same ray, that is, arg A = arg A'. In this case
Ilx(A) - x(A')11 ~
1>.'1
JIlx'
(te iarg >.) II dt
~
1>'1 1>.'1
J
(4.5)
MLdt
(1 - t)
1>'1
Now let A and X be arbitrary elements of ,1. Choose a point A" such that and arg A" = arg X. It is easy to show that in this case we have
IA"I = IAI
IA - Alii + IA" - AI ~ 31)' - )"1·
A priori estimates
193
From inequalities (4.4) and (4.5) it follows that Ilx(A) - x(A')11 ~ Ilx(A) - X(A")II
~ K21A -
+ Ilx(A") -
+ _2a1 MLIA"
A"1 1 -a
x(A')11 ~
_ A'1 1-a
-0:
~ 31- a 2a max {K2' 1 ~a 0: ML} IA ~ 3 (~) a K21A _
A'1 1-a
~
~
A'11-a.
~
Setting K = 3(2/3)a K2 we get (4.2).
COROLLARY 4.1. Let F E H(::D, X) n C(::D, X), where::D is a C-star-shaped domain in X, and assume that for all possible solutions of the equation
B(A)Z where B(A)
= y,
AE
[0,
1], Y E X,
(4.6)
= 1- AF'(x) and x is a fixed element of::D, the a priori estimate Ilzll ~ 1'IIYII
(4.7)
is true. Then the operator F has a fixed point x* which can be approximated by the sequence Xt of the s-fixed points for the operators tF with t E [0, 1), and the rate of convergence is determined by the estimate
Ilx* - xtll ~ ~(1 - t), where ~ = (3/2)71" R . max{2, 31'} and R is the radius of a ball centered at the origin and containing ::D. Moreover, if the point x* belongs to ::D, then it is regular.
Indeed, the existence of the point x* E ::D and the estimate of the convergence rate follow straightforwardly from Theorem 4.1 if we set tP(x, A) = AFx, L = R, M = l' and 0: = o. If x* is an interior point of ::D then we can define the operator B = B(l) = 1- F'(x*). By estimate (4.7) and by the extension with respect to a parameter principle for the solutions of a linear equation (see V. A. Trenogin [1]), equation (4.6) has a unique solution for x = x*, A E [0, 1] and y E X. Consequently, the operator B is continuously invertible and A = 1 is not an eigenvalue of the operator F'(x*). ~ - R - M >- R - - = -. is differentiable on the ball ofradius R with center at the origin, and let m = IIA -111. From the already obtained results it follows that if the unbinding lJ>(0) satisfies the estimate E: =
11lJ>(0) II ~ 2~ min{1, R(2M)-1},
where
M
=
m· sup 11lJ>(x) - lJ>' (O)x - lJ>(0) II, IIxll..rG(x) +v = FI(>..,x,v).
(6.14)
We perform now a few formal transformations, the substantiation of which will become apparent below. Let us replace h in (6.7) by >..rG(x) +v. Based on (6.14) we get G(x)
= G(z) + >"ArG(x) + Av + Q(z, >..rG(x) + v).
Applying the operator P to the both sides of the last equality, and taking into account the equalities (6.13), PG(z) = 0 and PAv = Av (recall that Av E Sj) we obtain 0= Av
+ >"PArG(x) + PQ(z, >..rG(x) + v),
or Av = -[>"PArG(x) +PQ(z,>..rG(x)
+ v)],
whence v
= _[>"A- I PArG(x) + A-I PQ(z, >..rG(x) + v)] = F2(>'" x, v).
Consider now the Banach space ~
= max{llxll, Ilvll}, for any w = (x,v)
E~.
(6.15)
=
X-t-N(B) with the norm given by Ilwll = Denoting F = (FI ,F2) we may rewrite the
system (6.14), (6.15) as w = F(>",w).
(6.16)
We show that the mapping F is defined and analytic in some domain in ~, and for all >.. E fl = {>.. E IC : 1>"1 < b} (see (6.8)) it satisfies the conditions in Theorem 2.l. The mappings Fl and F2 are defined and analytic if Ilhll (= II>..G(x)
+ vii) < r:::;; R.
(6.17)
NONLINEAR EQUATIONS
208
Consider the domain ::Dr = {w : IIx - zll < r, Ilvll < rc} included in !C. Inequality (6.17), for all w = (x, v), is obviously fulfilled under the following conditions: 0< c < 1 and
r(1 - c)
(6.18)
IAI < IlrIIM(G)'
Let us establish the existence of such an r = f > 0 and of a function 8(r) : [0, R] ----+ ----+ [0, 8] so that the mapping F leaves the domain ::Df invariant for A with IAI < 8(f). From (6.14) and (6.18) it follows that IIF1 (A,w)11 < r for any r with 0 < r < Rand IAI < p(r). Consider the inequality (6.19) It is not difficult to convince ourselves that inequality (6.19) follows from
IAI
O. A direct computation shows that
{Rdc
(Rdc)2}
O~~R tp(r) = tp(r) = mm 21IArIIM(G)' 41IArIIM(G)M(Q) A
•
where f
= min { R,
,
2M~ Q) } .
Thus the mapping F is analytic and F(A, w) E ::Df for all w E ::Df and all IAI < 8(f) = min{p(f), tp(f)}. Clearly 8(f) depends also on c, and, as one can easily see, it attains its maximum value for c = 2- 1 , whence (6.8) follows. From (6.14) and (6.15) it is not difficult to remark that the operator F(O, . ) is a q-contraction on some neighborhood of the point Wo = (z,O) for a certain q with 0 < q < 1. By Theorem 2.1 we obtain an s-solution W(A) = (X(A), V(A)) of equation (6.16) and the iterative sequence Wn(A) = F(A,W n-1(A)) = (Xn(A),Vn(A)) which appears in formulas (6.9), (6.10). ~ To conclude this section let us show what the conditions in the theorem above mean in the case of a finitely degenerate operator B. Assume that N (B) has dimension m and ind B = O. Futher, let {V1, . .. , v m } be a basis of N(B) and let SJk = clin{Av1, ... ,Avk-1,Avk+1, ... ,Avm} be the linear hull of the elements AV1"'" AVk-l, AVk+1,"" Avm. Consider the spaces 'ck = = R(B)+-SJk which are proper subspaces of the space X.
Single-valued small solutions
209
The following conditions are equivalent: 1) the operator A = A I N(B) is an isomorphism from N(B) onto Sj; 2) d k = P(AVk' ~k) = inf IIAvk - xii> 0, k = 1, ... , m;
PROPOSITION 6.1.
XE£k
3) the equalities
are fulfilled for a certain basis
{'Ij;d~l
of the space N(B*) (B* is the adjoint of B).
<J From 1) it follows that the system of elements {Avd~l forms a basis of Sj. Assume that condition 2) fails for some k = 1, ... , m. Then there exists a non-zero element Y E R(B) such that
AVk = Y +
L if-k
1:::;; i
or
~
(XiAVi,
(Xi
E
0, x > 0. There are two homeomorphisms defined on ilj2, namely, 'PI : (x, y) ----* x and 'P2 : (x, y) ----* y. The overlap mapping 'PI 0 'P2I : Y f---7 X is given by the equality x = and is a homeomorphism from the intervaljO, 1 [ onto itself. Analogously we can show that the sphere is a smooth manifold of dimension 2. In the above mentioned examples it is worth to remark that the circle and the sphere are subsets of the spaces JR2 and JR3, respectively, which, on their turn, are also manifolds. We will be concerned with such aspects in §3.
+JT=Y2
§2. Smooth mappings The main reason for introducing the notion of manifolds consists in the possibility they offer to differentiate and integrate functions or mappings defined on "nonlinear" spaces. This is done by identifying locally these spaces with theirs models. Thus, for example, a function defined on a circle can be considered locally as a function given on an interval of the real line. Therefore it is convenient to think of a circle not as a set of points in the two-dimensional space JR2, but as a set of points which is identified locally with subsets of the one-dimensional space JR I . Let us give the precise definitions.
em,
m ~ 1, modelled on the space !D, and let f be a mapping from Sj into a Banach space 3. The mapping f is called m-times continuously Frechet differentiable at a point a E Sj if there exists a chart (lla, 'Pa) with a E lla 3 defined on some neighborhood of the point a E Sj is m-times differentiable at a in the sense of Definition 2.1.
DEFINITION 2.2. Let Sj and 9)1 be two manifolds of class C m
In this case it is also easy to show that the differentiability of g, in the indicated sense, does not depend on the choice of the local system of coordinates '¢b. Moreover, we have the next obvious result. PROPOSITION 2.1. A mapping 9 : Sj ---> 9)1 is m-times differentiable at a point a E Sj
- in the sense of Definition 2.2 - if there exist two local systems of coordinates, 'Pa at the point a E Sj and '¢b at the point b = g(a) E 9)1, such that the mapping '¢b 0 9 0 'P;; 1 : lD ---> 3 is m-times Frechet differentiable at the point 'Pa (a) ElDin the usual sense. Analogously we introduce the notion of continuously differentiable mapping and the notion of analytic mapping (if the manifold is analytic) at a point, or, on some open subset, of a manifold Sj. DEFINITION 2.3. Let Sj and 9)1 be two manifolds modelled on the spaces
lD and 3,
respectively, and let 9 be an one-to-one mapping from Sj onto 9)1 such that both 9 and g-l are smooth mappings at any point. Then 9 is called a diffeomorphism between
the manifolds Sj and
9)1,
and the manifolds Sj and
9)1
are said to be diffeomorphic.
Let us note that the model spaces of some diffeomorphic manifolds coincide, up to a linear isomorphism. In particular, two finite-dimensional diffeomorphic manifolds have the same dimension.
§3. Submanifolds Let
9)1
be a manifold modelled on the space X and let 5j be a subset of 9)1.
DEFINITION 3.1. The set 5j is called a submanifold of 9)1 if for any a E 5j there
Submanifolds
215
exists a chart (f)a, a), f)a ~ 9J1, such that a(a) = 0 and a(f)a n Sj) = lU a n Ita, where lU a = a(f)a) and Ita is a linear subspace of X. A sub manifold Sj is called a direct submanifold if each of the spaces Ita has a direct complement in X, i.e., X = = 1- T is locally invertible, i.e., we can find the numbers ri,r2 > 0 such that for all y with Ilyll ~r2 the equation 1>(x) = y has a unique solution x = 1>-l(y) in the ball Ilxll ~ ri, which depends analyticaly on y and satisfies the condition x(O) = O. Then for all x with Ilxll = ri inequality (4.5) is fulfilled for a suitable p. By Lemma 4.2, we obtain that ~ = {O}, which contradicts the assumption in 2). ~
Proof of Theorem 4.2. 1) Let 9J1 be a connected component of the set ~. Choose a neighborhood ti c ~ which contains 9J1n~ and such that tin (~\9J1) = 0. Then 1>(x) =I- 0, whence, by the fact that 1> is proper, inequality (4.5) follows with a suitable p. Our assertion is a consequence of Lemma 4.2. 2) If we assume in addition that 1> is a Fredholm operator, then, according to Theorem 4.1, it follows that 9J1 is a finite-dimensional complex-analytic submanifold of ~. Again by the fact that 1> is proper we conclude that ~ is countable at infinity and thus, by Theorem VII.4.2, it is a Stein manifold. Finally, let us suppose J n a~ = 0. Then there exists a neighborhood ~i (~ ~) of the set ~ such that all the values of the operator 1> (= 1- T) on a~i are
NON-REGULAR SOLUTIONS
238
different from zero. Consequently, for
o.
(Compare with Definition 3.4.)
Symmetric properties
253
A focusing operator A is said to be double-focusing, if A * is also a focusing plus-operator in the Jq-space 1l3*.
In the case of a Hilbert J-space Sj any strict plus-operator A with the property A .c ~ E 9J1+ is double-strict and JL(A) LEMMA 4.2.
= JL(A*)
for some
.c ~
E
9J1+
(4.2)
(see M. G. Krein and Y. A. Shmulian [1]).
In a Jp-space Il3 every strict plus-operator A with property (4.2) satisfies
the condition
(4.3) Assume that the conclusion is false. Then we find a sequence {x~ n E N} such that x~ E 1l3_, Ilx~11 = 1 and IIAl/A12X~11 -+ 1 for n -+ 00. Set an = = IIA1/A12x~11 and Xn = a~lAl/A12X~ -x~. Then Xn E.Ito (C Jt+) and 0, for n
~ no.
Setting
(4.6) Let M~ = (p~ (Ayn,
+ K~*) 1J3~.
M~) =
Then A* M~ = (p~
(yn, A* M~)
=
((p_
+ K~*) 1J3~.
Further
+ kr:.) yr:., (p~ + kr:.*) 1J3~) = {O},
which, by (4.6), contradicts condition M~ E OO1~; so (4.5) is proved. It remains to show that the plus-operator A* is strict. For any x* E .It~ we have: Jq(A*x*)
= Ilp~A*x*llq
-llp':A*x*ll q ~ (1- ,B*q)
II(Ai1 + A;lK:') x~llq =
= (1- ,B*)q IIAi1 (I + (Ai1)-1 A;lK:') x~llq ~ ~ Ilx~llq ~ ~Jq(x*),
Symmetric properties
255
Theorem 4.4 assures that any focusing strict plus-operator on a reflexive space, which satisfies condition (4.2), is collinear to a double uniformly -expansive operator, i.e., to an operator which is uniformly -expansive simultaneously with its adjoint. To conclude this section we take into consideration some conditions under which a Jp-expansive operator A is double Jp-expansive, i.e., both A and A * are J p-expansive.
J:
J:
Let A be a strict plus-operator on the Jp-space 23. Then the operators P_ ± AP+ and P_ ± P+A are continuously invertible.
LEMMA 4.3.
We consider the operators P_ ± P+A. Assume that Ilxnll = 1 and (P_± n ±P+A)x ----; 0 as n ----; 00. Then x~ ----; 0 and P+Ax n ----; O. Since Ilxnll = 1 we have Ilx+11 ;;:: a > 0 for n;;:: no. Further, P+Ax n = Anx+ + A12x~ and, since A12X~ ----; 0, then Anx+ ----; 0, too. On the other hand, by Lemma 3.3, we have IIA l1 x+ll;;:: 811x+11 ;;:: 8a > 0, for n;;:: no, a contradiction. Hence it follows that II(P-± ±P+A)xll ;;::~±llxll, for all x E 23, where ~± > O. By Theorem 0.4.5 we conclude that the operators (P- ± p+A)-l exist and II(P- ± p+A)-lll ~ ~±l. Assume now that (P_ ± AP+)yn ----; 0 as n ----; 00, where Ilynll = 1. Then y~ + P_Ay+ ----; 0 and P+Ay+ ----; O. As above, from Lemma 3.3 it follows that y+ ----; o. This means that P_Ay+ ----; 0, whence y~ ----; 0, too. Thus we obtain again that II(P+ ± AP+)yll ;;:: '/]±llyll for all y E 23, where '/]± > O. Hence the operators (P- ±AP+)-l exist and II(P- ±AP+)-lll ~'/]±l. ~ <J
LEMMA 4.4. If A
is a double-strict plus-operator on a Jp-space 23, then P+AP+23+
= 23+.
(4.7)
We notice that for a strict plus-operator on a Jp-space 23 condition (4.7) is equivalent to condition (4.2). <J By Lemma 4.3 the operator P_ + AP+ is a homeomorphism and therefore the lineal (P_ + AP+)23 is closed. If (P_ + AP+)23 =I- 23, then we find an element x* =I- 0 in 23* such that ((P- + AP+)23,x*) = {O}, that is (P::" + PtA*) x* = O. But A* is a strict plus-operator on 23*, hence, by Lemma 4.3, the operator P::" + P-i'-A* is invertible. It follows that x* = 0 ~ a contradiction. Thus (P_ + AP+)23 = 23, a relation which clearly leads to (4.7). ~
LEMMA 4.5.
Let A be a Jp-expansive operator on lB. Then the operator (4.8)
SPACES WITH INDEFINITE METRIC
256
is well-defined and
IIUxll p
~
Ilxll p
for all x E :D(U).
1. We consider a bounded strict plusoperator A defined on the whole space 23 and satisfying condition (4.7): P+AP+23+ = = 23+. By Corollary 3.2 the operator A maps any subspace I: + E 9)1+ onto the subspace A I: + which belongs to 9)1+, too. Let now K+ E K+. Then the subspace 1:+ = (P+ +K+)23+ belongs to 9)1+, therefore AI:+ E 9)1+. By Theorem 2.1 we know that AI:+ = (P+ +K+)23+, where ~ -1 K+ = P_(p+ I AI:+) E K+. We have:
= (All + A 12 K+)23+, P_A23+ = (A21 + A22K+)23+.
P+AI:+
Therefore, setting K+
= FA(K+)
we obtain: (6.1)
Thus to each strict plus-operator A with property (4.7) it corresponds the linear fractional transformation FA: K+ ....... K+ given by (6.1). From Lemma 4.2 it follows that IIAll A1211 < 1. Therefore
FA(K+)
= (A2l +A22 K+) (I + A IlA 12 K+)-1 All =
= (A21 +A22K+)
C~o(-l)n (AIlA12K+t) All,
(6.2)
i.e., the mapping FA is holomorphic on K +. From (6.1) it follows that any fixed point K+ E K+ for the transformation FA is the angle operator for a subspace 1:+ = (P+ + K+)23+ E 9)1+, invariant relatively to A, and, conversely, if A I: + c I: + for some I: + E 9)1+ then the angle operator of the subspace 1:+ is a fixed point for FA. Hence, based on Corollary VI.2.1, we obtain: THEOREM 6.1. Let A be a focusing plus-operator with property (4.7). Then A has
in 9)1+ a unique invariant subspace I: ~ E 9)1~ satisfying A.c ~ = .c ~, and there exists a number r < 1 such that IAI :::; rlJ.t1 for all J.t E a(A I .c~) and A E a(A) \ a(A I .c ~).
An application of pixed point principles
263
Consider the transformation FA given by (6.1). From Definition 4.3 of a focusing plus-operator, it follows that
