Aspects of Mathematics and Its Applications (North-Holland mathematical library)

rr.,

36

J. Horvath I Life and Works of L. Nachbin

The continuous functions
I ,-I± f(x)-


E

for x EX.

If we write out the definition of
for xE X. Since each elements of (6.3)

W Yj

is a finite linear combination with real coefficients of

w: we can rewrite the inequality as

If(x)- ±'Yj(a\(x), ... , am (x»Wj(x)! <

E

for x E X,

J-I

where wj E W (1::s:;; j ::s:;; t) and the 'Yj are finite linear combinations with real coefficients of the f3;, and are consequently bounded continuous functions on R m. By hypothesis, for each j (1 ::s:;; j ::s:;; t) there is a fundamental weight bj on R m such that (6.4)

for xE X.

Since 'Yj is bounded on R m we have 'Yjbj E ceo(R m) and so, given 8> 0, for each j there exists a polynomial ~ (g) in m indeterminates such that

From here and from (6.4) we obtain (6.5) l'Yj(a\(x), ... , am (x»wj(x) - ~(a\(x), ... , am (x»wj(x)1 ::s:;; I'Y/a\(x), ... , am(x»-1j(a\(x), ... , am (x»lbj(a\(x), ... , am (x»

for x E X. Combining (6.3) and (6.5) we get

<8

J. Horvath I Life and Works of L. Nachbin

37

if 5 is sufficiently small. To conclude the proof it is enough to observe that p;(al , ••• , am)Wj E d'WC 'Wand so L;=,lj(a,(x), ... , am (x))wj E 'W. D Choosing particular classes of fundamental weights, as for instance the functions characterized by the result of Izumi and Kawata, Nachbin obtains from Theorem (6.2) simple sufficient conditions for 'W to be localizable under d, which he calls the 'analytic criterion' and the 'quasi-analytic criterion' ([49, Sections 28, 29]). A special case is that if W does not vanish anywhere on X and if for every a E A the series L:=,lIa n wr 11n diverges, then the module 'W = dw generated by W is localizable under d. This and similar results were proved by P. Malliavin {57}, who was the first to consider weighted approximation in a Stonetype setting, see also {58}. Malliavin also furnished Nachbin with an example of a fundamental weight w such that w P is not a fundamental weight for all p > 0; the only place where this example can be found seems to be [54]. Nachbin spent the academic years 1961/62 and 1962/63 in Paris. He lived under no. 61 of the tree-lined boulevard du Lycee in Vanves, opposite the park of the Lycee Michelet, in an apartment found for him by Marie-Helene Schwartz. Through him I got an apartment in the same building and stayed there from 1963 to 1965. We overlapped a month in the early summer of 1963 and rekindled our friendship started nine years earlier. Several other mathematicians lived at 61, bid. du Lycee during their stay in Paris: J. Gil de Lamadrid, J. Nohel, F. Treves, S. Mizohata. Unfortunately in 1970 the owner, Madame Lucienne Treheux died, the apartments were sold as condominiums and became unavailable to visiting mathematicians. Nachbin says in his Houssay-prize acceptance speech: 'I believe that the next four year period 1961-1965 was in many important respects the climax of my career. ... During the two years of 1961-63 and through the hands of Laurent Schwartz, I was a visiting professor of the University of Paris. . .. The many surprising invitations that I received to lecture at European centers ... led me to a sort of psychological and human maturity... .' As I already mentioned, Nachbin was an invited lecturer at the 1962 International Congress of Mathematicians.

38

J. Horvath I Life and Works of L. Nachbin

In Paris Nachbin lectured on approximation theory, in particular on his own work, and his point of view underwent a change. Indeed, Bernstein's original approximation problem can also be stated in the following equivalent way: Given a weight w > 0 consider a function g E ce(R) such that wg E ceo(R). When does there exist for every E > 0 a polynomial P(x) such that w(x )Ig(x) - P(x)1 <

E

for xER?

Corresponding to this formulation Nachbin introduces a class 'Y of positive upper semicontinuous functions v on the completely regular space X which has the property that given VI' V 2 E 'Y there exist A ;.:;. 0 and v E 'Y such that vl:os;; AV and v 2 :os;; Av. The elements of 'Y are now called weights and following a suggestion of W. Summers {79} such a class 'Y is referred to as a Nachbin family. With 'Y Nachbin associates two vector spaces: the space ce'Yb(X) consisting of all IE ce(X) such that [u is bounded for all V E 'Y, and the space ce'Yo(X) of all IE ce(X) such that Iv tends to 0 at infinity. Clearly ce'Yo(X) c ce'Yb(X), and one defines locally convex topologies on these spaces with the help of the family of seminorms (PV)VEV defined by Pv(f) = IIvlli = sup v(x)l/(x)/. xEX

The approximation problem now takes the following form: Let sti be a sub algebra of ce(X) and 'Wa subspace of cero(X) which is an sti-module. Describe the IE ce'Yo(X) with the property that given E > 0 and V E 'Y there exists awE 'W such that vex)/I(x) - w(x)1 <

E

for xEX.

The concept of the localizability of 'W under s1 can be introduced as before, and the results and their proofs are similar to the ones explained above. For details I refer to Nachbin's excellent accounts [48], [49], [50]. However, the slight shift of the point of view was heavy in other consequences. The spaces ce'Yb(X) and ce'Yo(X) became known as weighted spaces of continuous functions or simply as Nachbin spaces. They and their vector-valued analogues immediately caught the attention of several researchers, in particular of W. Summers {79}, {80} and of K.D.

J. Horvath / Life and Works of L. Nachbin

39

Bierstedt {5}, {6}, {7}, who studied them in detail, partly in collaboration with R. Meise. The reason for this sudden popularity was due to the fact that they unify different kinds of topological vector spaces of continuous functions, which until then had no connections between them. Let me quote four examples. (I). If "fI is the collection of characteristic functions of compact subsets of X, then C6'''fIb(X) = C6'''fIo(X) is the space C6'(X) with the compact-open topology. (2). If "fI is the set of all positive constants, then C6'''fIb (X) = C6'b (X), C6'''fIo(X) = C6'o(X) with their normed topologies. (3). If "fI is the set of all positive, bounded, upper semicontinuous functions which tend to 0 at infinity, then C6'''fIb(X) is C6'b(X) with the so-called strict topology. (4). If X is locally compact and the union of a countable family of its compact subsets, and if "fI consists of all positive continuous functions, then C6'''fIb(X) = C6'''fIo(X ) is the space C6'c (X) of all continuous functions with compact support, equipped with the 'locally convex inductive limit' topology for which the dual of C6'c (X) is the space of Radon measures on X. In particular the investigation of the dual of C6'''fIb(X) has led ProIla, Summers {80}, {81}, {82}, {83}, {84} and G. Kleinstiick {52} (a doctoral student of Bierstedt) to the solution of the weighted approximation problem in complex, non self-adjoint situations, when localization is defined not in terms of the sets where all the a Ed are constant but in terms of sets which are maximal antisymmetric with respect to d. Prolla's book {69}, which in a sense is a continuation of [49J, gives an account of these researches up to 1977 and has a useful bibliography. As I already hinted at, and as the titles of Prolla's book and Kleinstuck's paper indicate, the weighted approximation problem was considered also for vector-valued functions. This was done first in two papers Nachbin wrote jointly with Machado and Prolla [68], [70], and which are based in part on the doctoral dissertations of the last two. Before discussing these two papers I will return to [30J, written in Princeton in 1958, which I have passed over so far. In this paper Nachbin introduces a concept of convexity which reappears in [68J and [70J. An algebra A over R is said to be topological if it is also a topological space, and if both addition and multiplication are continuous maps from A A into A. Given a polynomial P(t) = ~ 7~o a.t' with real coefficients

x

40

J. Horvath / Life and Works of L. Nachbin

and an element x E A, we can associate with them the element P(x) = A. Nachbin says that A has an operational calculus with ee' (IR.) if the map (P, x) ~ P(x) can be extended to a continuous map (f, x)~ f(x) from ee'(IR.) x A into A. I will not state precisely Nachbin's main theorem ([30, p. 435]) since it requires a number of definitions from the theory of algebras, in particular the concept of an algebra of differential order n, introduced by Nachbin himself ([30, p. 429]), and also since there is no need to duplicate Nachbin's accessible, lucid and self-contained presentation. I only want to focus on one particular concept. Nachbin obtains a necessary and sufficient condition for ,a certain type of algebra A to have an operational calculus with ee' (R), where it is assumed that A is locally convex with respect to a certain category of algebras. It is this notion of local convexity that I want to describe. Let E be a topological vector space over Rand Y a collection of linear subspaces of E. A subset C C E is Y-convex if it is convex in the usual sense (i.e. x, y E C, a, {3 E R, a ~ 0, {3 ~ 0, a + {3 = 1 imply ax + {3y E C) and if C is the intersection of all the sets C + S = {c + s] c E c: s E S} as S varies in Y. The space E is locally Y-convex if the Y-convex neighbourhoods of 0 form a basis for the neighbourhoods of O. Let next A be a commutative algebra of continuous linear operators on the topological vector space E. If I is an ideal in A (i.e, a subspace I of A such that T 1 E I, T2 E A implies T 1 T2 E I), then let I(E) be the subspace of E spanned by all the vectors T(x) with TEl and x E E. Clearly I(E) is invariant under A. Letting ,1 be a collection of ideals in A and setting Y(,j) = {I(E)I I E ,I}, we say that C C E is convex with respect to ,1 (resp. that E is locally convex with respect to ,1) if ee is Y(,j)-convex (resp. if E is Y(,j)-locally convex). Finally, let ~ be a category of commutative algebras over R. Then C C E is convex under A with respect to ~ if C is convex with respect to the collection ,j(~) of ideals I in A for which the quotient algebra A/I belongs to ~. Similarly E is locally convex under A with respect to ~ if it is locally convex with respect to ,j(~). In [68] and [70] the authors consider functions which not only are vector-valued but the vector space is permitted to vary from point to point, i.e. they consider sections of vector bundles. Weights are then functions which with every point associate a seminorm. All definitions and results of the scalar case have appropriate generalizations, and the

L 7~o a.x' of

J. Horvath / Life and Works of L. Nachbin

41

authors even omit the proofs of the theorems concerning localizability. They apply their results to various questions (e.g. spectral synthesis) of which I want to explain that one where the above concept of local convexity plays a role. Like before, let E be a locally convex Hausdorff space over R and A a commutative algebra of continuous linear operators on E. The point co-spectrum X = X(A) of A is the set of all homomorphisms X: A ~ R for which there exists a non-zero fEE' such that f(u(x)) = x(u)f(x) for all u E A and x E E. One equips X with the coarsest topology for which all maps X ~ x(u) are continuous as u varies in A. For each X E X let Sx be the closed linear subspace of E spanned by the vectors u(x), where u E Ker(x) (i.e. x(u) = 0) and x E E. Then Ex = E/Sx ¥ {O} and assigning to X E X the vector space Ex we obtain a vector bundle F over X. Given x E E, let Xx E Ex be the class of x modulo Sx; then (XX)XEX is a section of F denoted by XC. For each u E A let u# denote the continuous map X ~ x(u) from X into R. Then u ~ u" is a homomorphism from A into «6(X) whose image will be denoted by d. The algebra .91 separates points of X, one has u# (X) = x(u) and (u(x))O = u# . XO for u E A, x E E. Given a continuous seminorm p on E, for any X E X let px be the quotient seminorm on Ex defined by px(xx) = inf yEx.l' p(y). The mapping X ~ px associates with each X E X a seminorm Px on Ex, i.e. it is a weight over X which we denote by p". At this point the authors introduce a hypothesis according to which there exists a set of continuous seminorms on E which determines the topology of E and is such that for every x E E and pEr the function x~ Px(xx) is upper semicontinuous and tends to 0 at infinity. Denote by r the set of weights p", where p varies in F, and analogously to the scalar-case introduce the space ..cero(X) of all cross sections XO equipped with the topology defined by the seminorms qp'(XO) = SUPxEx Px(xx). The authors' theorem then asserts that there exists a set r of seminorms on E which determines the topology of E such that x ~ XO is an isomorphism between the topological vector spaces E and 2"Ifo(X ), and u ~ u" is an isomorphism between A and .91 if and only if E is locally convex under A with respect to the category of all algebras isomorphic to R. During his stay in Paris Nachbin devised other approaches to the study of Bernstein's approximation problem. Thus in the note [43] he announces a Denjoy-Carleman type theorem for the quasi-analyticity of an infinitely differentiable map from an open subset of a locally convex topological vector space over R into another such space. In a lecture

r

42

J. Horodth / Life and Works of L. Nachbin

given in P. Lelong's seminar in Paris [44], and then again in a lecture given at a Conference in New Orleans [53] he points out that the 'quasi-analytic criterion' follows from the Denjoy-Carleman type theorem. In [53] he hints at other methods for proving the main result of [50]. In a later paper [79] he shows how the results concerning modules can be obtained from the results concerning algebras. To conclude the overview of Nachbin's work on approximation theory, let me observe that weighted spaces of not only continuous but also of differentiable functions have been considered by various writers, e.g. by B. Baumgarten {3}, another doctoral student of Bierstedt. The weighted approximation problem for differentiable functions was investigated by Zapata {92}.

7. Harmonic Analysis Since the beginning of his research career Nachbin was interested in harmonic analysis. In the early articles [2], [3], [5] he investigated questions concerning limits of integrals and convergence of series which arose from the theory of Fourier series. In [6] and [7] he introduces a sequence (cP n) of Lebesgue measurable functions defined on a subset Il of the real line, with positive Lebesgue measure. He considers the following five conditions: (CL). (Cantor-Lebesgue condition) If (An) is a numerical sequence which does not tend to zero as n ~ 00 then ~ AncPn (x) does not converge for almost every x E Il. (LD). (Lusin-Denjoy condition) If ~ IAnl = 00 then the series ~ jAnl·lcPn(x)1 diverges almost everywhere. (PI)' There exists a subsequence of (cPn) which converges to zero on a subset of positive measure of (P2) . There exists a subsequence of (cPn) which converges uniformly to zero on a subset of positive measure of n. (P3) . There exists a subset zl of n with positive measure such that liminfn-+",fJ l4>n(x)1 dx = O. Nachbin proves that (CL) and (LD) are equivalent to the negation of either one of the conditions (Pj ) , j = 1,2,3. The equivalence of (CL) and (LD) was proved earlier by M.H. Stone {78}. The logical pattern of Nachbin's proof is

n.

J. Horvath / Life and Works of L. Nachbin

43

Andre Weil published in 1940 his great classic {90}, in which he develops harmonic analysis on compact and on abelian locally compact groups. This work had of course a great influence on Nachbin. Article [26] calls a group G left separating if given two groups HI and Hz, and homomorphisms ex, {3 : Hz -+ HI such that ex ¥- {3, there always exists a homomorphism c/J: HI -+ G such that c/J 0 ex ¥- c/J 0 {3. Nachbin proves that G is left separating if and only if there exists an isomorphism of Q /I into G. He also proves that this property characterizes Q /Z up to isomorphism. The last result was given by S. MacLane without proof. In 1959/60 Nachbin lectured on the Haar integral first in Rio de Janeiro and then in Recife. The text of the course was published the same year in Portuguese [32] and English translation appeared in 1965 [51] and was reprinted in 1976. Nachbin gives an unhurried presentation, with many examples, of the theory of Haar measure. The first chapter is an exposition of the H. Cartan-Weil-Bourbaki treatment of integration with respect to a positive Radon measure on a locally compact topological space. The second chapter treats Haar measures, i.e. translation-invariant measures on a locally compact topological group. Relatively invariant measures are also considered. For the existence of a Haar measure Nachbin presents both the original proof of Alfred Haar, touched up by A. Weil and using the axiom of choice, and also a proof due to H. Cartan {I7}, which avoids the axiom of choice. The chapter starts with a very enlightening discussion in the case of IR n of the idea behind the HaarWei I construction. The last chapter deals with locally compact homogeneous spaces. It is motivated by applications of the invariant integral to questions concerning geometric probabilities and integral geometry. Not too surprisingly, the book became quickly popular among mathematicians and is used as a reference also by theoretical physicists and statisticians. In [36] Nachbin gives a new, simple proof of one of the classical theorems of harmonic analysis, according to which every irreducible unitary representation of a compact topological group is finite-dimen-

44

J. Horvath I Life and Works of L. Nachbin

sionaI. This proof has become standard, and figures e.g. in the treatise of Hewitt and Ross ({44. (22.13))). It also had an impact on the work of Siegfried Grosser and Martin Moskowitz.

8. Holomorphy

In 1963/64 and 1964/65 Nachbin was a professor at the University of Rochester, first as a visitor and then as a permanent member of the faculty. The University of Rochester wanted him to remain there on a full-time basis, and several other universities in the U.S.A. and in Europe had made him similar offers, including the University of Chicago which offered him an endowed chair. Though the son of immigrant parents (his father, Jacob Nachbin, was born in Poland, his mother, Lea, nee Drechsler, in Austria), he could not envision to be separated permanently from the country of his birth and from the city of Rio de Janeiro. He felt that he must remain a participant in the mathematical activities of Brazil and of Latin America. So an arrangement was found whereby Nachbin spent since 1966, with a few exceptions, the first semester of each academic year at Rochester and the remainder of the year in Rio. Since 1967 he has the title of a George Eastman professor at the University of Rochester. Of course Nachbin has also been a visitor at numerous other centers since 1965, e.g. at the Universities of Chicago, Miami, Texas, Madrid, Paris VI, at the Institute for Advanced Study in Princeton, at Rutgers University, at the Scuola Normale Superiore in Pisa and at the Institut des Hautes Etudes Scientifiques in Bures-sur-Yvette. In Rochester Nachbin lectured again on the theory of approximation and the notes of this course became the volume [49] to which I have referred repeatedly above. However, his main interest was already in another direction. Around 1963 Laurent Schwartz generalized Radon measures, which were originally defined by Bourbaki on locally compact spaces, to arbitrary topological spaces {74}. This suggested to Nachbin the idea to try to extend the theory of distributions, developed by Laurent Schwartz on finite-dimensional Euclidean spaces, to arbitrary Banach spaces. As a preparation for this program, Nachbin in his 1963 course in Rochester gave a coordinate-free treatment of the theory of distributions. He starts with a treatment of differential calculus on normed spaces of any dimension. The course resulted in the elegant book [47] which I hope will soon be reissued by Chelsea. A spinoff of the first part of the book is

J. Horvath / Life and Works of L. Nachbin

45

the detailed and very accessible exposition of differential calculus on Banach spaces, published first in Portuguese in the monograph series of the Organization of American States [57] and then in English [92]. As Nachbin says in his Houssay-prize acceptance speech [95a], he 'underestimated the difficulties' of developing a theory of distributions in infinitely many dimensions. However, the Paley-Wiener-Schwartz theorem, which is the ultimate goal of [47], establishes a bijective correspondence between distributions with compact support and certain entire analytic functions of exponential type. This led Nachbin to the theory of holomorphic functions on infinite-dimensional spaces which has been the central topic of his mathematical activity during the last twenty years. In the last third of [47] the elementary theory of holomorphic maps from a normed space into another normed space is presented without the restriction of finite-dimensionality. The study of holomorphic functions on infinite-dimensional spaces presupposes a familiarity with the theory of holomorphic functions of several complex variables. Nachbin lectured repeatedly on the subject and the course he taught at Rochester in 1967 resulted in the book [58] which appeared as the first volume of the collection 'North-Holland Mathematics Studies' but not yet in the sub series 'Notas de Maternatica'. Holomorphic functions figured already in one of Nachbin's earliest publications [4]. As is well known, an entire function f(z) is said to be of exponential type if there exist numbers M > 0 and T > 0 such that If(z)1 ,,;;;; Me 1"lzl for all z E C. The greatest lower bound of the numbers T for which this inequality holds is called the exponential type of f. Nachbin's generalization consists in replacing the exponential function in the above definition by a function '/t(t) = :L nEN '/tnt n, where 1/'n > 0 and '/tn+J/V'n tends decreasingly to 0 as n ~ 00. The ratio test for convergence shows that '/t is an entire function. An entire function f(z) = :L nEN c.z" is said to be of finite V'-type if there exist numbers M > 0 and T > 0 such that If(z)1 , ; ; MV'( Tlz Dfor z E C. The greatest lower bound of all numbers T for which this inequality holds is called the V'-type of f. Nachbin proves that the V'-type of f is given by limsup,... co Icn/V'nI J/n • R.P. Boas and R.c. Buck call this result 'Nachbin's theorem' in their book ({8, p. 6}) and give a complete proof for it. At the outset of my report on Nachbin's contributions to infinitedimensional holomorphy I must explain the notation introduced by him

46

J. Horvath I Life and Works of L. Nachbin

and used universally now. As is well known, if E 1, ••• ,Em and Fare vector spaces over the same field, a map A : (Xl' ••. ,xm)~ A(x l , ... ,xm) from E 1 x ... x Em into F is m-linear if for any index i (1::;; i,,;;;; m) and for arbitrary fixed vectors xj E E j (1::;; j::;; m, j ¥- i) the map Xi ~ A(x l , ••• , Xi' •••• x m ) from E, into F is linear. If all the spaces E, are equal to the same space E, then the vector space of all m -linear maps from Em = Ex", x E into F is denoted by !£a(mE, F). The superscript m is to the left of E to distinguish the space from the space !fa (Em, F) of all linear maps from Em to F; the subscript a stands for 'algebraic' and indicates that no continuity requirements are involved. An m -linear map A E !fa (mE, F) is symmetric if A(x l , ••• , Xm) = A(Xq(I)' ... , xq(m» for any permutation a of m objects. The vector space of all symmetric m-linear maps from Em into F is denoted by !fas (m E, F). The linear map s which associates with each A E !fa (mE, F) the map s(A)E !fasCE, F) defined by

s(A)(x 1,

••• ,

xm) =

1

-,

m.

2: A(xu(l)' ... , xu(m» o

, .

where a runs through the m! permutations of m objects, is a projector from !fa C E, F) onto !fas (mE, F), i.e. satisfies so s = s. A map P from E into F is an m-homogeneous polynomial if there exists an AE!faCE,F) such that P(X) = Ax", In this definition m~1 is an integer and Ax" stands for A(x, ... ,x) with X repeated m times. The preceding relation between P and A is expressed by writing P = A o. One has Ax" = s(A)x m , so there is always a symmetric m-linear map A for which P = A o. The map A ~ A ° is a bijection from !fase E, F) onto the vector space [j'Ja (mE, F) of all m -homogeneous polynomials, as the polarization formula

shows ({26, p. 4}). O-homogeneous polynomials are the constant maps from E into F, their space [j'Ja (0 E, F) can be identified with F The direct sum [j'Ja(E, F) = L m E N [j'Ja(mE, F) is the space of all polynomials from E into F, and its elements can be written uniquely in the form Po + PI + ... + Pm with Pm ¥- 0 and P; E f!/JaeE, F), 0,,;;;; i::;; m. Now let E and F be locally convex spaces over
J. Horvath / Life and Works of L. Nachbin

47

sc(E) and sc(F) the sets of continuous seminorms on E and F, respectively. These sets determine the topologies of the spaces in the sense explained above. Nachbin denotes by .2b CE, F) the space of those maps A E .20 (mE, F) which map bounded subsets of Em into bounded subsets of F, and by .2CE, F) the space of those A E .20 (m E, F) which are continuous. The definition of the spaces (m E, F), s; CE, F) and F), fJ(mE, F) is analogous. If B is a bounded, closed, absolutely convex subset of E, one denotes by E B the subspace of E spanned by B, and by gB the gauge of B defined by gB(X) = inf{A > 01 x E AB}. On E B the gauge gB is a seminorm and if E is Hausdorff, then gB is even a norm. One equips E B with the topology defined by gB' From now on we shall denote by U a non-empty open subset of E. A map f from U into F is said to be Silva-continuous (after the regretted Portuguese mathematician J. Sebastiao e Silva) if for every bounded, closed, absolutely convex subset B of E the restriction of f to un E B is continuous for the topology induced by E B • A map A E .20 C E, F) belongs to .2b CE, F) if and only if it is Silva-continuous ([93, 1.3, p. 439]); a similar characterization holds for the elements of fJb (mE, F) ([93, 1.8, p. 440]). Let me now formulate three basic definitions. (H). A map f : U ~ F is said to be holomorphic if for every (E U there exists a power series L mEN r; with r; E fJ ("E, F), having the following property: for every f3 E sc(F) there exists an open neighbourhood V of ( contained in U such that

s;

e,ee.

lim f3 (f(x) n-+ X

±

Pm (x - ()) = 0 ,

m=O

uniformly for x E V, i.e. given E > 0 there exists an no ~ 0 such that 1f3(f(x)- L Pm (x - ())/ ~ E whenever n ~ no and x E V. A topological vector space is Hausdorff if and only if {O} is a closed set. In general the closure N in F of the set {O} is a linear subspace of F. The quotient space FIN equipped with the quotient topology is called the Hausdorff space associated with F. Let K be the canonical surjection from F onto FIN which associates with each element of F its class modulo N. If u is a continuous linear map from F into a Hausdorff topological vector space G, then there is a unique continuous linear map u' : FIN ~ G such that u = u' K. A map f : U ~ F is holomorphic if and only if K of: U ~

:=0

0

48

J. Horvath / Life and Works of L. Nachbin

FIN is holomorphic. We shall therefore assume that F is Hausdorff. If this assumption is made, then the polynomials Pm' and thus also the maps Am E Its eE, F) such that Pm = A~, are uniquely determined by f and f Nachbin sets

and therefore the Taylor series of f at

~

can be written as

in harmony with the one-variable notation. (S). A map f: U ~ F is said to be Silva-holomorphic if for every ~ E U there exists a power series L mEN Pm' with Pm E {Jf>b (mE, E), having the following property: for every f3 E sc(F) and every bounded, closed, absolutely convex set BeE there exists a number. p > 0 such that ~+pBC U and

~i~ f3 (f(x) - ~o Pm (x - ~») = 0 , uniformly for x E ~ + pB. (A). A map f: U ~ F is said to be algebraically holomorphic if for every finite-dimensional linear subspace S of E the restriction of f to U n S is holomorphic, where S carries its unique Hausdorff topology for which it is a topological vector space. This concept is independent of the topology of E. lf X is a completely regular Hausdorff topological space and F a locally convex topological vector space, denote by 'e(X, F) the vector space of all continuous maps from X into F. For any compact subset K of X and any neighbourhood W of 0 in F let V(K, W) be the set of those 4> E 'e(X, F) which satisfy 4> (x ) E W for all x E K. Similarly as in the case of scalarvalued functions, the subsets of 'e(X, F) containing some set V(K, W) are the neighbourhoods of 0 for the compact-open topology on 'e(X, F). The vector space of all holomorphic maps f : U ~ F will be denoted by i!e(U, F). The compact-open topology of 'e(U, F) induces on i!e(U, F) its compact-open topology ;Yo' If U is an open subset of a finite-dimensional

J. Horvath / Life and Works of L. Nachbin

49

space, then go is the topology which has been used classically. In the infinite-dimensional case, however, go does not have all the properties one would desire, and therefore several other topologies have been considered on ~(U, F). To explain and motivate the first one of these, the ported topology ([55], [59, Section 8, p. 31]), let me return to the space ~(X; F). Consider a linear subspace Y of ~(X, F) and a seminorm p on Y. The compact set K C X is said to support p if, whenever lEY vanishes in some neighbourhood of K, one has p(f) = O. In a recent paper J.A. Barroso and Nachbin [98] examine under what conditions is p supported by some compact set, and when does there exist a smallest compact set which supports p. If this smallest compact set exists, it is called the support of p. The authors say that Y has uniqueness 01 continuation if lEY is identically zero whenever it vanishes on some non-empty open subset of X. For instance ~(U, F) has uniqueness of continuation if U is connected. It is obvious that if Y' has uniqueness of continuation, then any seminorm on Y' is supported by any compact subset of X. Barroso and Nachbin observe that conversely, if p is supported by anyone-point set in X; then Y has uniqueness of continuation. Thus in particular if U is connected, then the intersection of all compact subsets of U which support a seminorm p on ~(U, F) is empty, and therefore p has no support. In view of this, Nachbin has introduced a weaker concept to which I will turn after stating the two main results of [98]. A function <jJ E ~ (X) is a multiplier of Y if <jJ1 E Y for every lEY. Assume that given two points X o 'I'- XI of X there exists a multiplier <jJ of Y such that <jJ(x) = 0 in a neighbourhood of X o and <jJ(x) = 1 in a neighbourhood of XI' If there exists at all a compact subset of X which supports a given seminorm p on Y, then the intersection K of all the compact sets which support p also supports p, i.e. K is the support of p. On the other hand, let G be a locally convex Hausdorff space and let u : Y...." ~(X, G) be a linear map such that if lEY and g = u(f), then the interior of {x E Xl I(x) = O} is contained in {x E XI g(x) = O}. If Y is equipped with the inverse image topology by u of the compact-open topology on ~(X, G), then every continuous seminorm on Y is supported by some compact subset of X. Let us now proceed to the definition of the ported topology. Assume first that F is a normed space. Inspired by work of Martineau on analytic functionals, Nachbin says that a seminorm p on ~(U, F) is ported by the compact set K C U if for every open neighbourhood V of K

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contained in U there exists a number c(V) > 0 such that

p(f):so; c(V) sup IIf(x)/1

for all f E 'Je( U, F) .

xEV

The ported topology gO} on 'Je(U, F) is defined by all the seminorms which are ported by some compact subset of U. If F is a locally convex space and fi E sc(F), we denote by F/fi the quotient space of F modulo the subspace N(fi) of those y E F for which fi(y) = O. Let Kp be the canonical surjection from F onto F/fi. If Yl - Yz E N(fi) then fi(Yl) = fi(yz), so a norm fi' can be defined on F/fi by fi'(Kp(Y» = fi(y), and this norm yields the quotient topology of F/fi. (We shall use later the analogous notation for E and a E sc(E).) If the function f: U ~ F is holomorphic, then for every fi E sc(F) the function Kp 0 f : U ~ F/fi is holomorphic, and so we get a linear map lPp: f ~ K p 0 f from 'Je( U, F) into 'Je( U, F/ fi). Equip each space 'Je( U, F/ fi) with its ported topology gO}. Then the ported topology on 'Je( U, F), also denoted by gO}, is the inverse limit with respect to the maps C/Jp of the ported topologies of the spaces 'Je( U, F/ fi), i.e. the coarsest topology on 'Je( U, F) for which the maps lPp are continuous; this topology is locally convex. The next topology on 'Je(U, F), denoted by gl)' was introduced independently from Nachbin also by G. Coeure {I8} in the case that E is a separable Banach space. First let us, however, consider more generally, following Nachbin [66], a completely regular (Hausdorff) topological space X, a normed space F, and a linear subspace Y of ~(X, F). If m= (V;)'EI is an open cover of X; denote by Y'1l the subspace of g consisting of all fEY which are bounded on each \I., and equip Y'1l with the topology defined by all seminorms

f ~ sup IIf(x)1I

(L E 1).

xEVL

The space Y is the union of the subspaces Y'1l (indeed, given fEY, set Vn = {x E XII'f(x)1I < n}), and the topology gA on Y is defined as the finest locally convex topology for which all the canonical injections Y2J ~ Yare continuous. If we only consider countable coverings mof X, we obtain in a similar way the announced topology gl) on Y. R.M. Aron observed that gl) is defined by all seminorms p on Y which satisfy the following condition: for every sequence V o C VI C ... C V n C ... of open subsets of X such that

J. Honuith / Life and Works of L. Nachbin

x =U

nEN

51

V n there exists an mEN and a number c > 0 such that

p(f) ~ C sup IIf(x)'1

for all

f

E

9'

xEV",

(cf. {26, 2.35, p. n}). If F = C and fI is the whole space ~(X, C), then for any X, the compact-open topology coincides with :YA , but :YA is equal to fl a if and only if X is replete. If now F is a locally convex space, then the topology fla on :lC( U, F) is the inverse limit with respect to the maps 0 such that f3(f(x» ~ M for f E il' and x E V. Denote by ~a(U, F) the collection of amply bounded subsets of :lC(U, F), by ~o(U, F) the collection of those subsets of :lC(U, F) which are bounded for the topology flo' by ~w (U, F) the collection of subsets bounded for fl." and by ~a (U, F) the collection of subsets bounded for e; One has

Nachbin and his fellow writers prove, among others, the following results: (1). If E is equipped with the weak topology (T(E, E'), then ~.,(U, F) = ~a(U, F).

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(2). If (EJ'El is an arbitrary family of metrizable locally convex spaces and E = TI'El E" then [J30( U, F) = [J3a (U, F). This is remarkable because Barroso has proved that if I is uncountable, then already on :te(
J. Horvath / Life and Works of L. Nachbin

53

motivates the consideration of holomorphy on compact sets. Let K be a compact subset of E. In the union of all the spaces 7t'(V, F), where V is an open subset of E containing K, one introduces an equivalence relation by saying that liE 7t'(VI , F) and 12 E 7t'(V2 , F) are equivalent if there exists an open set W such that VI n V2:J W:J K and that the restrictions of II and 12 to Ware equal (see Fig. 2). The equivalence classes with respect to this relation are called F-valued germs of holomorphic maps on K, they form a vector space denoted by 7t'(K, F). For each open set V:J K there is a linear map 4J v from 7t'( V, F) into 7t'(K, F) which associates with each IE J'e(V, F) its equivalence class: the germ of I on K. The topology gw on J'e(K, F) is defined as the finest locally convex topology for which all the maps 4J v are continuous, if each 7t'(V, F) is equipped with its topology gw' If F is normed, then J'e(K, F) equipped with gw is bornological; if F is a Banach space, then 7t'(K, F) is barrelled. Let U be, as earlier, a fixed open subset of E. For each compact subset K of U denote by 4JK the linear map from 7t'(U, F) into 7t'(K, F) which associates with each IE 7t'(U, F) its germ on K. Equip each space 7t'(K, F) with its topology gw' and denote by gTT the inverse (= projective, hence the notation) limit topology with respect to the maps 4JK (i.e. the coarsest topology on 7t'(U, F) for which the maps 4JK are continuous), as K varies in the collection of compact subsets of U. Clearly e;» gTT'

Fig. 2.

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J. Horvath I Life and Works of L. Nachbin

The topology fY" has some desirable properties, e.g. It'( V, F) is complete when equipped with fY". It is therefore of interest to ask: when do and fY" coincide on It'(V, F)? In the case that E and F are Banach spaces, Nachbin gives two answers to the question. The set V is said to be balanced with respect to g E V if whenever x E V and AE
s,

s;

v:

0

v:

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55

K a : E -4 Eta is both continuous and open (i.e. K a maps open sets into open sets) is directed (i.e. for any ai' a z there exists a 3 such that a 1 ~ a3' az ~ ( 3 ) , and defines the topology of E. In [67] Nachbin lists a number of spaces which satisfy (C), and asserts that if E satisfies (C), then every holomorphic map f : U -4 F is uniformly holomorphic. He points out in [74J that this theorem generalizes the fact according to which a function f belonging to iIt(C I, C), which a priori depends on every component of (= «(,) ,EI E C I, in fact depends only on finitely many of the (,. Nachbin [99] says that E leads to pure uniform holomorphy if for every holomorphic map f: U -4 F there is an a E sc(E) such that U is a-open and f is holomorphic if U is considered as an open subset of Ea' He then lists eight conditions which are all equivalent to the fact that E leads to pure uniform holomorphy, under the assumption that E belongs to the class of (DFC)-spaces considered by R. Hollstein {45}, {46} and J. Mujica {63}, {64}. A locally convex Hausdorff space E is a (DFC)-space if there exists a Frechet space (i.e. a metrizable, complete, locally convex space) G such that E is isomorphic to the dual G' of G equipped with the topology of uniform convergence on the compact subsets of G. If E is a (DFC)-space, then on its dual E' the compact-open and the boundedopen ('strong') topologies coincide, and E' equipped with these coinciding topologies can be taken for G. Some of the above-mentioned equivalent conditions are the following: (1). E leads to pure uniform holomorphy. (2). The dual space E' equipped with the coinciding topologies is separable (i.e. has a countable everywhere-dense subset). (3). Every open subset of E is uniformly open. (4). There exists a continuous norm on E. (5). There exists a countable set of continuous seminorms on E which separate the points of E. (6). Every open subset of E is the union of a countable collection of compact sets. Obviously (1) implies (3) and (4) implies (5). Next Nachbin considers the space iIt(K, C) of germs of holomorphic functions on the compact subset K of E. On each space iIt(lI: C), where V is an open set containing K, he puts the compact-open topology flo, and on iIt(K, C) he considers the finest locally convex topology for which the linear maps cP v : iIt(V, C)-4 iIt(K, C) are continuous. This topology, introduced by O. Nicodemi {67}, is also denoted by flo. It is a fundamen-

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tal result of Mujica {64} that ~(K, C) equipped with ;Yo is a (DFC)-space. Nachbin proves that if E is metrizable and if ~(K, C) is equipped with ;Yo then ~(K, C) satisfies the above equivalent conditions (1)-(6) (with E replaced by ~(K, C» if and only if E is separable [99, Th. 2]. A map f: U -+ F is said to be equitably holomorphic [69, OM. 2] if in definition (H) the neighbourhood V of (E U. can be chosen independently of f3 E sc(F). If f is equitably holornorphic, then the power series ~ mE" Pm (X - () converges in the topology of F to f(x) uniformly for x E V. Clearly, if F is normed then every holomorphic map f : U -+ F is equitably holornorphic, but in [69] Barroso and Nachbin give examples of holomorphic maps which are not equitably holomorphic in the case when E is a (necessarily infinite-dimensional) Banach space and F is a Frechet-Montel space. The map f : U -+ F is locally bounded if every (E U has a neighbourhood V such that f( V) is a bounded subset of F. Every locally bounded f E ~(U, F) is equitably holomorphic [69, Prop. 3] but there exist equitably holomorphic maps which are not locally bounded: consider a nonnorm able space E, take F = E and let f be the identity map from E onto itself. From Hilbert to Grothendieck, functional analysts have introduced a great variety of linear maps, e.g. continuous, compact, Hilbert-Schmidt, Fredholm, nuclear and integral ones. The definitions of most of these carryover to multilinear maps and to polynomials. Thus a polynomial P E [lPCE, F) is said to be nuclear if the following ingredients exist: (1) An absolutely convex neighbourhood V of O in E, (2) A bounded subset B of F, (3) A sequence (Ak )kEN of complex numbers such that ~ IA k i < 00, (4) A sequence (X~hEN of elements of the dual E' of E such that Ix~(x)1 'S 1 for all x E V and kEN, (5) A sequence (Yk)kFN of elements of B, and if with these ingredients P has the expression P(x) = ~ Ak (xt(X)rYk

for x E E.

kE'J

Of course, just like the case of linear maps {74, Part II, Chap. IV, Section I}, this definition can be couched in terms of complete topological tensor products (cf. [93, 2.19 and 2.24]). Denote by [lPNCE, F) the subspace of (ijJ(mE, F) consisting of all nuclear m -homogeneous polynomials. One could then define nuclearly holomorphic functions f by

J. Horvath / Life and Works of L. Nachbin

57

requmng that in the Taylor series 2: r; of f the polynomial r; shall belong to PJN C E, F) for each m E r»j, and by adding some condition on the way the series 2: Pm converges. If, for instance, E is a Banach space and F = C, Nachbin [62] defines a nuclear norm P~ IIPIIN on each space PJNCE; C) and requires that the Taylor series at ~ = 0 of a nucleariy entire function satisfy the condition 2: Ilpm liN < 00. In [93, 4.2, p. 469] a definition of Silva-holomorphic functions of nuclear type can be found. Such examples have led Nachbin to another of his major contributions to infinite-dimensional holomorphy. A holomorphy type 8 consists of a choice for each 111 EN of a subspace PJoCE, F) of PJ(mE, F) such that PJo(lE, F) = PJ(lE, F). This choice has to satisfy two conditions. In the first place PJ(mE, F) and PJo(mE, F) are equipped with topologies and it is required that the canonical injection PJo(m E, F)4 PJ(mE, F) is continuous. In the second place the choices of the spaces PJo CE, F) for different values of In must be coherent in the sense that if P E PJoCE, F) then the derivative dip should belong to PJoeE, F) for O:s:; I:s:; 111. It is superfluous to reproduce here the exact definition of a holomorphy type since it can be found in very accessible and detailed publications. For Banach spaces E, F in [59] and for Silva-holomorphy types-where one considers subspaces PJbo(mE, F) of PJbCE, F)-in the joint article [93] with Matos. Let me also refer to the expository lecture [64], and to [56] which covers the same ground as [59, Sections 9-14]. Once a holomorphy type 8 has been selected, the space flto(E, F) of maps of 8-holomorphy type can be defined. If PJoCE, F) = PJCE, F) for all Ill, then Nachbin calls 8 the current holomorphy type. The space flto(E, F) can be equipped with the compact-open topology go,o and with the ported topology g w, 0 [59, Section 11], [93, Section 6]. For the ported topology characterizations of the bounded [5, Section 12], [93, Section 7] and of the relatively compact [59, Section 13], [93. Section 8] subsets of flto(E, F) are presented. The original motivation for considering nuclearly entire functions comes from work of C. Gupta and Nachbin ({37}, [62]) in which Malgrange's theorem is generalized to infinite-dimensional spaces. For any tEE the translation map 7, on the space fltN(E,C) of nuclearly entire functions is defined by (7J)(x)=f(x-t) for xEE, fEfltN(E,C). A convolution operator u on fltN(E, C) is a continuous linear map from fltN(E, C) into itself which commutes with translations, i.e. u 0 7, = 7,0 U for all tEE. Convolution operators are actually linear partial differential operators with constant coefficients of finite or infinite order. A nuclear

58

J. Horvath / Life and Works of L. Nachbin

exponential-polynomial on E is a linear combination of functions of the form r «: where P E ~ mEN (J}N (m E, C) and ¢ E E'. Malgrange's theorem states that for any convolution operator u on 'leN (E, C) the exponentialpolynomials g such that u(g) == 0 form a dense subspace of the kernel Ker(u) == {f E 'leN(E, C)! u(f) == O} of u. Malgrange proved his theorem originally for E == R", where 'leN(E, C) = 'le(E, C). Gupta {37} proved it in the case of a Banach space E for nuclearly entire functions of bounded type (i.e. which are bounded on bounded subsets of E), the extension to 'leN(E, C) is due to Gupta and Nachbin [62]. In a joint paper with Matos [94] a detailed presentation is given of convolution operators and of Malgrange's theorem on spaces of nuclearly Silva-entire functions on a locally convex space E.

The concepts of barrelledness and of being bornological, discussed above in the section on topological vector spaces, depend on the behaviour of linear maps. With the collaboration of Barroso and Matos, Nachbin developed an analogous theory in which holomorphic maps are substituted for linear ones. There came first a short expository lecture [76], then a longer announcement with definitions and theorems but without proofs [80], and finally a detailed and extremely readable account of the theory with full proofs [81]. As we know, E is bornological if every linear map I: E -'> F which maps bounded sets into bounded sets is necessarily continuous. It is not appropriate to model the definition of the analogous holomorphic concept on this property because a holomorphic map, though continuous, does not map bounded sets into bounded sets as continuous linear maps do. Fortunately, Nachbin observed that E is bornological also if every linear map 1 : E -'> F which maps compact sets into bounded sets is necessarily continuous. To prove the equivalence of the two properties, all one has to show is that if f is bounded on every compact subset of E, then it is also bounded on any bounded set BeE. Let /3 E sc(F) and let (x k ) «e». be an arbitrary sequence in B. Since B is bounded, for any sequence (AkhEN of scalars which converges to 0, the sequence (Akxk) converges to 0 in E. Now the set of the vectors AkXk together with 0 is compact in E, hence by assumption the set of positive numbers /3 (f(AkX k» = IA kl/3(f(xk » is bounded. But (Ak ) can tend to zero arbitrarily slowly, therefore (f3(f(xk») is bounded. Since every /3 E sc(F) is bounded on {f(xk)1 kEN}, this set is bounded in F. Finally, since (xk ) is arbitrary in B, it is clear that I(B) must be bounded in F. Now the road is open for the definition of the desired analogue. E is

J. Horvath / Life and Works of L. Nachbin

59

said to be holomorphically bornological if for every U C E, every algebraically holomorphic map j": U ~ F which maps compact sets into bounded sets, is necessarily holomorphic. Since every linear map is algebraically holornorphic, every hoi om orphically bornological space is bornological, but the converse is false [81, Example 18]. Any metrizable locally convex space and any Silva space is holomorphically bornological [81, Prop. 6, 10]. If E is holomorphically homological then for every U C E the space :le(U, F) is complete if equipped with flo [81, Prop. 16]. Next, E is barrelled if and only if every subset fie of the space .2(E, F) = .2CE, F) of continuous linear maps from E into F, such that for every gEE the set {J(g)l! E fie} is bounded in F, is necessarily equicontinuous, or-what amounts to the same-is amply bounded. In the definition given earlier we took F = C but the passage to a general F is immediate. Again this property is not appropriate to serve as a model for the definition of the holomorphic analogue, but Nachbin has observed that E is barrelled if and only if every f!l' C .2(E, F), which is bounded on every finite-dimensional compact subset of E, is necessarily- equicontinuous. Accordingly Barroso, Matos and Nachbin define E to be holomorphically barrelled if for every U every fie C :le(U, F), which is bounded on every finite-dimensional compact subset of U, is necessarily amply bounded. For the same reason as above, every holomorphically barrelled space is barrelled, and again the converse is false [81, Remark 45]. Any Baire locally convex space and any Silva space is holomorphically barrelled [81, Prop. 37, 41]. The Holomorphic Banach-Steinhaus theorem [81, Prop. 40] asserts that if E is a Frechet space and if 2f C :le(U, F) is bounded on every affine one-dimensional compact subset of U, then 2f is equicontinuous. In a similar vein Barroso, Matos and Nachbin introduce the concept of a holomorphically infrabarrelled space in analogy with the linear concept of an infrabarrelled space. Also, holomorphically Mackey locally convex spaces are introduced in analogy with the property of a space E to have the Mackey topology, i.e. the finest locally convex topology for which E' is the topological dual of E. Let me observe that C-barrelled spaces were introduced by P. Lelong {55}, in their definition plurisubharmonic functions playa role. Lelong also proves extensions of the Banach-Steinhaus theorem. The space C(l) equipped with the locally convex direct sum topology is

60

J. Honuith I Life and Works of L. Nachbin

holomorphically bornological or holomorphically barrelled if and only if I is finite. On the other hand, it is an open question of long standing, related to the Ulam-Mackey problem {54, Section 28.8}, whether C' is bornological for every index set I. In a recent paper [97] Barroso and Nachbin prove that e(I) with the topology induced by the product topology of I is always holomorphically bornological.

e

In the field of infinite-dimensional holomorphy Nachbin had many more students and followers than in any other field. While in approximation theory he had had only the three doctoral students I named earlier, in infinite-dimensional holomorphy he directed the theses of at least sixteen candidates, some of whom I mentioned above, and many of whom became distinguished and active workers in the discipline. In Brazil, France, Germany, Ireland, Spain, Sweden and the United States whole schools have been formed which study the concepts introduced by Nachbin. Infinite-dimensional holomorphy is also applied to build a solid mathematical foundation to quantum field theory. I would have to write a volume to report on all the research that has been done in the footsteps of Nachbin in infinite-dimensional holomorphy. Fortunately this is not necessary since several accounts have appeared. In the first place there are Nachbiri's own works: his early Ergenbnisse-volume [59] which reproduces lectures given at the Sixth Brazilian Mathematical Colloquium in Pecos de Caldas in 1967, and has served as an introduction to almost everybody interested in infinite-dimensional holomorphy, including the writer of these lines. Following this book many lectures and expository articles have appeared: [60], [62], [64], [65], [67], [69], [71], [74], [75], [76], [80], [83], [88], [89], [91]. Reference [74] is Nachbin's invited address at the annual meeting of the American Mathematical Society in Atlantic City on January 22, 1971, which I had the luck to hear in person. The lecture [76], delivered at the Kentucky meeting in 1973, includes topics I did not touch upon above: the behaviour of holomorphy with respect to the perturbation of the topology on E [75], [78], and the comparison of vector-valued and of scalar-valued analytic continuation [73]. Then there are the books on infinite-dimensional holomorphy written by Coeure {19}, Colombeau {20}, Dineen {26}, Franzoni and Vesentini {29}, Herve {41}, Mazet {59}, Mujica {62}, {65}, Noverraz {68} and by several others, which are due to appear in the near future. Not too surprisingly, most of these are published in the 'Not as de Matematica' series. The most ambitious of them, Dineen's monograph, has a biblio-

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61

graphy with 725 references, a substantial part of which is devoted to works by Nachbin, his students and his followers. But even on 431 pages Dineen covers only a part of the theory. Thus holomorphy types are barely mentioned on pp. 51 and 382 in spite of the fact that Dineen himself made essential contributions to it in his thesis {25}, which, incidentally, he presented to the University of Maryland. Let me mention that Dineen's a-f3-y holomorphy types were extended to Silva-holornorphic functions by M. Bianchini {4}.

9. Conclusion Since 1972 Nachbin, his students and collaborators have organized in Brazil on the average one conference per year on either approximation theory or on infinite-dimensional holomorphy, or, in general, on functional analysis with emphasis on these two subjects. I listed at the end the proceedings of these conferences, adding the proceedings of two more conferences which, though not held in Brazil, were inspired by Nachbin. A perusal of these proceedings shows the enormous influence Nachbin has had on a whole generation of mathematicians in many countries. Nachbin's outstanding achievements were rewarded by many honours. Already in 1942 as a student he received a Licinio Cardoso prize. In 1948 he became elected to the Brazilian Academy of Sciences as an associate member, and became a titular member in 1950. In 1962, during his two years in Paris, he was awarded the Moinho Santista prize. It was the first time that a mathematician obtained it, and Nachbin travelled from Paris to Sao Paulo for the presentation. In 1969 he became a corresponding member of the Academy of Sciences of Lisbon, and in 1970 he obtained a medal from the University of Liege. In June 1973 he received an honorary doctorate from the Federal University of Pernambuco. On October 14, 1982 Nachbin was awarded by the Organization of American States with the prestigious Bernardo Houssay Science Prize. This was again the first time that a mathematician got this prize. The awarding took place at the annual meeting of the OAS Interamerican Council for Education, Science and Culture in Washington, D.C. Again it was my good fortune to be present at the moving ceremony and to hear Nachbin's autobiographical remarks [95a] delivered in Portuguese. In 1983 Nachbin was elected member of the Academy of Sciences of Latin America founded in that year.

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I know that I speak in the name of all the contributors to this volume and of the many friends and admirers of Leopoldo Nachbin the world over when I wish him good health and many more years of fruitful, happy activity.

Added in proof (May 20, 1985). The following appeared or were called to my attention after the manuscript was sent to the editor. 1. Nachbin's characterization of Boolean algebras [10] figures on p. 65 of the book of G. Gratzer: General Lattice Theory, Pure and Applied Mathematics vol. 75 (Academic Press, New York, 1978) in the following different but obviously equivalent form: Let E be a distributive lattice with o-:;i 1. Then E is a Boolean algebra if and only if given two distinct prime ideals of E, none is contained in the other. 4. Simultaneously with G. Mehta, the concepts concerning ordered topological spaces have been applied to economic theory by G. Chichilnisky (Spaces of economic agents, J. Economic Theory 15 (1977) 160-173). The space of commodities is a Hausdorff topological space X which is assumed to be connected and normal. A preference () on X is simply a relation, i.e. a subset of X x X. Denote by 0 the set of all continuous preferences () on X, i.e. such that () is equal to the interior of its closure in X x X. The economic agents of the title are the elements of X x O. On (J an order is defined by setting ()l < ()2 if clos ()l C ()2' Chichilnisky defines the order topology T on 0 by choosing the sets {() COl () < ()'} and {() cOl () > ()'} as a subbasis of open sets «()' EO). Nachbin ([23, Introduction, no. 4]) calls a subset A of an ordered set E convex if x :s; z :s; y, x E A and yEA imply z E A. An ordered topological space is locally convex ([23, Chap. I, no. 1]) if each point has a fundamental system of convex neighbourhoods. Chichilnisky proves, among many other results, that if X is a compact metric space, then 0 equipped with :!I is a separable and metrizable locally convex ordered space. Any totally ordered, connected, bounded and closed subset of 0 is compact. Chichilnisky also considers the so-called closed convergence topology :!Ie on the space 0' of all closed preferences (), i.e. such that ()is closed in X x X. On 0' she defines order by inclusion, and proves, e.g. that if X is a locally compact separable metric space, then 0' equipped with :!Ie is a normally ordered space in the sense of Nachbin. Properties of the topology :!Ion (J are used in a later paper of Chichilnisky (Continuous representation of preferences, Review of Economic Studies 47 (1980) 959-963).

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6. A recent article of C.H. Cook (The finest Nachbin topology is a mixed topology, Manuscripta Math. 46 (1984) 83-95) is related to both [27] and Nachbin spaces. A subset C of a completely regular space X is (topologically) bounded if every f E '€(X) is bounded on C. One lemma in Cook's paper states that if :Je is a set of finite Radon measures on X, each having bounded support, and if clos U J.tE1f supp(zz) is not bounded, then for every sequence (an) of numbers there exist f E '€(X) and f.L n E ';/{ such that JL n(f) = an' n EN. The proof of this lemma is modelled after a portion of the proof of the first theorem in [27]. Denote by !:I the Nachbin family of all positive upper semicontinuous functions on X which tend to 0 at infinity, and have bounded support. Cook proves that '€'Vo(X) = '€(X) for a Nachbin family 'V if and only if 'V c 5/. For any positive f E '€(X) set Bf = {g E '€(X)llgl ~ f}. Cook's main result states that the topology defined by the Nachbin family !:I on '€!:Io(X) = '€(X) is the finest locally convex topology which coincides with the compact-open topology on each set B f · 7. The papers of S. Grosser and M. Moskowitz mentioned in the text are the following: On central topological groups, Trans. Amer. Math. Soc. 127 (1967) 317-340. Representation theory of central topological groups, Ibid. 129 (1967) 361-390. Harmonic analysis on central topological groups, Ibid. 156 (1971) 419-454. The authors extend in them results concerning compact groups to central topological groups, i.e. such that the quotient of the group module its center is compact. In particular they use Nachbin's method of [36] to prove in the second paper quoted above that every continuous irreducible unitary representation of a central topological group on a complex Hilbert space is finite-dimensional. 8. A second book by J.F. Colombeau follows {20}: New generalized functions and multiplication of distributions. NorthHolland Math. Stud. 84 = Notas Mat. 90 (North-Holland, Amsterdam, 1984). It bears the dedication: 'Dedicated to Professor Laurent Schwartz and Professor Leopoldo Nachbin; this work could not have been accomplished without their contributions to Distribution Theory and Infinite Dimensional Analysis'. Another new book on infinite-dimensional analysis is: Soo Bong Chae, Holomorphy and calculus in normed spaces, Pure and

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Applied Mathematics: A Series of Monographs and Textbooks 92 (Marcel Dekker, New York, 1984). It should be added to the second paragraph in section 8 that a theory of distributions and Sobolov spaces on infinite-dimensional spaces has been developed by several authors, e.g. Paul and Mirella Kree and B. Lascar; see the notes in the June 6,1983 issue of the C.R. Acad. Sci. Paris, Ser. 1296, pp. 833-840 by M. and P. Kree and by J. Diebolt, and the references quoted there. Nachbin's work is explicitly referred to in: P. Kree, Courants et courants cylindriques sur des varietes de dimension infinie, Linear Operators and Approximation, Proc. Conf. held at the Mathematical Research Institute at Oberwolfach, Black Forest, August 14-22,1971. International Series of Numerical Mathematics 20 (Birkhauser, Basel, 1972) 159-174. Sobolov spaces are treated in detail in: M. Kree, Propriete de trace en dimension infinie d'espaces du type Sobolev, Bull. Soc. Math. France 105 (1977) 141-163. The preprint {64} was published in a final form: J. Mujica: A Banach-Dieudonne theorem for germs of holomorphic functions, J. Funct. Anal. 57 (1984) 31-48. The quoted result is Proposition 5.2, p. 44.

Publications of Leopoldo Nachbin [I) [2]

[3] [4] [5] [6] [7] [8] [9] [IO] [I I]

Sobre a permutabilidade entre as operacoes de passagem ao limite e de integracao de equacoes diferenciais, An. Acad. Brasil. Cienc. 13 (1941) 327-335. (MR 3, 239). Un'estensione di un lemma di Dirichlet, Atti Accad. Italia Rend. CI. Sci. Fis. Mat. Nat. (7) 3 (1942), 204-208. (MR 8, 263). Sobre as series de funcoes quasi-sempre absolutamente divergentes, Univ. Nac. Tucuman. Revista A. 3 (1942) 311-315 (MR 5, 117). An extension of the notion of integral function of the finite exponential type, An. Acad. Brasil. Cienc. 16 (1944) 143-147. (MR 6,60). Algunos teoremas sobre las series de terrninos positivos, Math. Notae 4 (1944) 90-104. (MR 6, 47). On linear expansions I, Trans. Amer. Math. Soc. 59 (1946) 437-440. (MR 7, 518). On linear expansions II, Summa Brasil. Math. I (1946) 17-20. (MR 7, 518). Sur la combinaison des topologies pseudo-rnetrisables et metrisables, C.R. Acad. Sci. Paris 223 (1946) 938-940. (MR 8, 285). Combinacao de topologias metrisaveis e pseudo-metrisaveis, Notas Mat. 1 (1947). Une propriete caracteristique des algebres booleiennes, Portugal. Math. 6 (1947) 115-118. (MR 9, 324). Sobre el axioma de las sucesiones no convergentes en algunos espacios topol6gicos lineales, Rev. Un. Mat. Argentina 12 (1947) 129-150. (MR 9, 367).

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[12] Espacos vetoriais topologicos, Notas Mat. 4 (1948). (MR 10, 610). [13] Sur les espaces topologiques ordonnes, c.R. Acad. Sci. Paris 226 (1948) 381-382. (MR 9, 367). [14] Sur les espaces uniformisables ordonnes, c.R. Acad. Sci. Paris 226 (1948) 547. (MR 9, 367). (15] Sur les espaces uniformes ordonnes, c.R. Acad. Sci. 226 (1948) 774-775. (MR 9, 455). (16] On a characterization of the lattice of all ideals of a Boolean ring, Fund. Math. 36 (1949) 137-142. (MR II, 712). [17] On strictly minimal topological division rings, Bull. Amer. Math. Soc. 55 (1949) 1128-1136. (MR 11,368). [18] A characterization of the normed vector ordered spaces of continuous functions over a compact space, Amer. J. Math. 71 (1949) 701-705. (MR 11,39). [19] Sur les algebres denses de fonctions differentiables sur une variete, c.R. Acad. Sci. Paris 228 (1949) 1549-1551. (MR 11,20). [20] On the Hahn-Banach Theorem, An. Acad. Brasil. Cienc, 21 (1949) 151-154. (MR II, 114). [21] A theorem of the Hahn-Banach type for linear transformations, Trans. Amer. Math. Soc. 68 (1950) 28-46. (MR II, 369). [22) On the continuity of positive linear transformations, Proc. Internal. Congress of Math., Cambridge, Mass., 1950, vol. 1464-465 (Amer. Math. Soc., Providence R.I., 1952). (23] Topologia e ordem (Univ. Chicago Press, Chicago, 1950). (24) Algunos problemas de analisis funcional, Simp6sium sobre algunos problemas matematicos que se estan estudiando en Latino America Diciembre 1951 15-21 (Centro de Coop. Cient. de la Unesco para America Latina, Montevideo, 1952). (MR 14, 563). [25] Linear continuous functionals positive on the increasing continuous functions, Summa Brasil. Math. 2 (1951) 135-150. (MR 14, 288). [26) On a duality theorem for commutative groups, An. Acad. Brasil. Cienc. 24 (1952) 137-142. (MR 14,350). [27] Topological vector spaces of continuous functions, Proc. Nat. Acad. Sci. U.S.A. 40 (1954) 471-474. (MR 16, 156). [27a) Aspectos do desenvolvimento recente da maternatica no Brasil, Cienc. Cultura 8 (1956) 213-217, Soc. Parana. Mat. Anuario 3 (1956) 28-41. (MR 19, 1150). (28) A generalization of Whitney's theorem on ideals of differentiable functions, Proc. Nat. Acad. Sci. U.S.A. 43 (1957) 935-937. (MR 19. 753). [29) On the operational calculus with differentiable functions, Proc. Nat. Acad. Sci. U.S.A. 44 (1958) 698-700. (MR 20, 4610). [30) Algebras of finite differential order and the operational calculus, Ann. of Math. (2) 70 (1959) 413-437. (MR 21, 7444). [31] Alguns resultados recentes sabre equacoes diferenciais parciais lineares de coeficientes constantes, Rev. Un. Mat. Argentina 19 (1960) 187-191. (MR 24, A2135). [32] Integral de Haar, Textos de Mat. 7 (Inst. Ffsica Mat. Univ. Recife, Recife, 1960). (MR 22, 9571). [33) Distribuicoes e equacoes diferenciais parciais. Conf. 1959. Textos de Mat. 6 Conf. I (Inst, Fisica Mat. Univ. Recife, Recife, 1960). (MR 23, AI911). [34] Extensao de funcoes reais continuas. Topicos de Topologia, Exposicoes de Mat. 3 (Univ. Ceara, Fortaleza, Ceara, 1%1). [35] Forrnulacao geral do teorema de aproximacao de Weierstrass para funcoes diferenciaveis, T6picos de Topologia, Exposicoes de Mat. 3 (Univ. Ceara, Fortaleza, Ceara, 1%1).

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[36J On the finite dimensionality of every irreducible unitary representation of a compact group, Proc. Amer. Math. Soc. 12 (1%1) 11-12. (MR 23, A526). [37J On the weighted polynomial approximation in a locally compact space, Proc. Nat. Acad. Sci. U.S.A. 47 (1%1) 1055-1057. (MR 27, 513). [38J Some problems in extending and lifting continuous linear transformations, Proc. Internat. Sympos. Linear Spaces, Jerusalem, 1960 (Jerusalem Academic Press, Jerusalem; Pergamon Press, Oxford, 1%1) 340-350. (MR 24, A2826). [39J Algebras de operadores e de fun~6es continuas, Rev. Un. Mat. Argentina 20 (1962) 312-314. (MR 30, 5179). [4OJ Sur I'abondance des points extrernaux d'un ensemble convexe borne et ferme, An. Acad. Brasil. Cienc, 34 (1%2) 445-448. (MR 26, 6736). [41J Sur I'approximation polynomiale ponderee des fonctions reelles continues, Atti 2a. Riunione del Groupement de Mathematiciens d'Expression Latine, Firenze and Bologna, 1%1 (Edizioni Cremonese, Firenze and Bologna, 1963) 42-58. [42J Resultats recents et problemes de nature algebrique en theorie de I'approximation, Proc. Internat. Congress Math., Stockholm, 1%2 (Almqvist and Wiksells, Stockholm, 1%3) 379-384. (MR 31, 548). [43J Sur Ie theorerne de Denjoy-Carleman pour les applications vectorielles indefiniment differentiables quasi-analytiques, C.R. Acad. Sci. Paris 256 (1963) 862-863. (MR 26, 2864). [44J Fonctions analytiques et quasi-analytiques vectorielles et Ie probleme d'approximation de Bernstein, sem. Lelong (Analyse) 1%2/63 expo No.6, Faculte des Sc. Univ. Paris. (MR 35, 7061). [45J Regularite des solutions des equations differentielles elliptiques, S6m. Bourbaki expo 258 (1%3). [46J Topics on topological vector spaces, Univ. Rochester (1%3). [47J Lectures on the theory of distributions, Univ. Rochester (1%3); Textos de Mat. 15 (Inst. Fisica Mat. Univ. Recife, Recife, 1964). (MR 35, 4722). [48J Weighted approximation over topological spaces and the Bernstein problem over finite dimensional vector spaces, Topology 3 Suppl. I (1964) 125"'::130. (MR 28, 4286). [49J Elements of approximation theory, Univ. Rochester (1964); Notas Mat. 33 (1%5); Van Nostrand Math. Studies 14 (Van Nostrand, New York, 1967); (Krieger, Huntington WV, 1976). (MR 34, 8038; 36, 572; 54, 3261). [50J Weighted approximation for algebras and modules of continuous functions: real and self-adjoint complex cases, Ann. of Math. (2) 81 (1965) 289-302. (MR31, 628). [51J The Haar integral, Univ. Ser. Higher Math. (Van Nostrand, New York, 1965); (Krieger, Huntington WV, 1976). (MR 31, 271; 54, 2925). [52J Topology and order, Van Nostrand Math. Studies 4 (Van Nostrand, New York, 1965); (Krieger, Huntington WV, 1976). (MR 36, 2125; 54, 3667). [53J Weighted approximation for function algebras and quasi-analytic mappings. Function Algebras, Proc. Internat, Symp. Function Algebras, Tulane University New Orleans, 1%5 (Scott, Foresman, Glenview, IL, 1966) 330-333. (MR 33, 7835). [54J Aproximacao ponderada por polinornios, Atas do Terceiro Col6q. Brasil. Mat., Fortaleza, 1%1 (Inst. Mat. Pura Aplicada, Rio de Janeiro, 1966) 146-189. (MR 35, 4645). [55J On the topology of the space of all holomorphic functions on a given open subset, Nederl. Akad. Wetensch. Proc. Ser. A 70 = Indag. Math. 29 (1967) 366-368. (MR 35, 5910). [56J On spaces of holomorphic functions of a given type, Proc. Conf. Functional Analysis, Univ. California, Irvine, 1966 (Thompson Book, 1967) 50-70. (MR 39,3298).

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[56a) A brief report on mathematical education at universities in Brazil, Presented to the Second Inter-American Conf. on Math. Education, Lima, Peru, Dec. 5-10, 1966. [57) Introducao a analise functional: espacos de Banach e calculo diferencial, (Univ. Brasflia, 1%7); revised edition: Serie de Mat. Monograffa 17 (Organization of American States. Washington, D.C., 1976). [58) Holomorphic functions, domains of holomorphy and local properties, Univ. Rochester (1%7); North-Holland Math. Stud. I (North-Holland, Amsterdam, 1970). (MR 43, 558). [59) Topology on spaces of holomorphic mappings, Sexto Col. Brasil. Mat., Pocos de Caldas, 1%7; Ergeb. Math. Grenzgeb. 47 (Springer, Berlin, 1969). (MR 40, 7787). (60) Some aspects of holomorphic mappings (Center for Theor. Stud. Univ. Miami, Coral Gables, Florida. 1%8). (61) Metodos matematicos da fisica, (Inst. Mat. Pura Aplicada, Rio de Janeiro, 1968). (62) Convolution operators in spaces of nuclearly entire functions on a Banach space, Proc. Conf. on Functional Analysis and Related Fields, Univ. Chicago, 1%8, ed. F.E. Browder (Springer, Berlin, 1970) 167-171. (63) Convolucoes em funcoes inteiras nucleares, Atas de Analise Functional e Equacoes Diferenciais Parciais, Sao Jose dos Campos, 1969 (Soc. Brasil. Mat.). [64] Concerning holomorphy types for Banach spaces, Proc. Internat. Colloq. on Nuclear Spaces and Ideals in Operator Algebras, Warsaw. 18-25 June 1969, Studia Math. 38 (1970) 407-412. (MR 43, 3787). [65) Concerning spaces of holornorphic mappings (Rutgers Univ., New Brunswick, New Jersey, 1970). [66) Sur les espaces vectoriels topologiques d'applications continues, c.R. Acad. Sci. Paris 271 (1970) 596-598. (MR 42, 6593). [67) Uniformite d'holornorphie et type exponentiel, Sem, Lelong (Analyse) 1970. Lecture Notes in Math. 205 (Springer, Berlin, 1971) 216-224. (MR 52, 1304). [68) Weighted approximation, vector fibrations and algebras of operators (jointly with S. Machado and J.B. Prolla), J. Math. Pures Appl. (9) 50 (1971) 299-323. (MR 45, 2474). [69] Sur certaines proprietes bornologiques des espaces d'applications holomorphes (jointly with J.A. Barroso), Troisieme Colloq. sur L'Analyse Fonctionnelle, Liege, 1970. Centre BeIge de Recherches Math. (Vander, Louvain, 1971) 47-55. (MR 52, 15006). [70) Concerning weighted approximation, vector fibration and algebras of operators (jointly with S. Machado and J.B. Prolla), Proc, Internat. Conf. Approximation Theory, Related Topics and their Applications, Univ. Maryland, College Park, Md., 1970, J. Approx. Theory 6 (1972) 80-89. (MR 50, 2917). (71) Sur quelques aspects recents de l'holomorphie en dimension infinie, Sem. GoulaouicSchwartz (1971/72): Equations aux derivees partielles et analyse fonctionnelle, expo 18 (Ecole Poly technique, Centre de Math., Paris, 1972). (MR 52, 15549). (72) Entire functions of exponential type bounded on the real axis and Fourier transforms of distributions with bounded supports (jointly with S. Dineen), Proc, Internat. Symp. Partial Differential Equations and the Geometry of Normed Linear Spaces, Jerusalem, 1972, Israel J. Math. 13 (1972) 321-326. (MR 48, 4723). [73] On vector-valued versus scalar-valued holomorphic continuation, Nederl. Akad. Wetensch. Proc, Ser. A 76 = Indag. Math. 35 (1973) 352-354. (MR 48, 9395). (74) Recent developments in infinite dimensional holomorphy, Bull. Amer. Math. Soc. 79 (1973) 625-640. (MR 48, 871). (75) Limites et perturbations des applications holomorphes, Colloq. Internat. CNRS 208, Paris, 1972: Fonctions analytiques de plusieurs variables et analyse complexe, Agora Math. I (Gauthier-Villars, Paris, 1974) 141-158. (MR 58, 2279).

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[76) A glimpse at infinite dimensional holomorphy, Proc. Internat. Conf. Infinite Dimensional Holomorphy, Univ. Kentucky, Lexington Ky., 1973. Lectures Notes in Math. 364 (Springer, Berlin, 1974) 69-79. (MR 52, 15007). [77) On bounded sets of holomorphic mappings (jointly with l.A. Barroso and M.e. Matos), Proc. Internat. Conf. Infinite Dimensional Holomorphy, Univ. Kentucky, Lexington Ky., 1973. Lecture Notes in Math. 364 (Springer, Berlin, 1974) 132-134. (MR 52, 15003). [77a) Brazil's mathematical requirements, Gaz. Mat. (Lisboa) 34-35 (1973n4) no. 129-132, 1-5. (MR 52, 19). [78) Perturbation of holomorphic mappings revisited, Rend. Mat. (6) 8 (1975) 337-344. (MR 52, 6426). (79) On the priority of algebras of continuous functions in weighted approximation, Convegno sugli Anelli di Funzione Continue, INDAM, Roma, 1973. Symposia Mat. XVII (Academic Press, London, 1976) 169-183. (MR 53, 14102). [80) Some holomorphically significant properties of locally convex spaces, Proc. Brazil. Math. Soc. Sympos. Functional Analysis, Inst. Mat. Univ. Estad. Carnpinas, Sao Paulo, 1974. Lecture Notes in Pure Appl. Math. 18 (Marcel Dekker, New York, 1976) 251-277. (MR 58, 30240). (Zbl 344, 46100). [81) On holomorphy versus linearity in classifying locally convex spaces (jointly with l.A. Barroso and M.e. Matos), Infinite Dimensional Holomorphy and Applications, North-Holland Math. Stud. 12 = Notas Mat. 54 (North-Holland, Amsterdam, 1977) 31-74. (MR 57, 13478). [82) Algunas propiedades de topologia y partes acotantes en holomorfia, Bol. Departamento Cienc., Pontiffcia Univ. Cat6lica del Peru 9 (1977) 71-75. [S2a) The development of mathematical physics and related functional analysis, and its role in a developing country, Proc. Internat. Colloq. on the Role of Mathematical Physics in the Development of Science, College de France, Paris, June 13-15, 1977, Unesco, Bol. Soc. Paran. Mat. (2) 2 (1981) 17-26. (MR 83i. 01082). [83] Analogies entre I'holomorphie et la linearite, Sern. Paul Kree: Equations aux Derivees Partielies en Dimension Infinie 4 (1977nS) expo I, (Institut Henri Poincare, Paris). (MR 82e, 46064). [84) Sur la den site des sous-algebres polynomiales d'applications continument differentiables, Sem, Pierre Lelong-Henri Skoda (Analyse) 1976/77, Lecture Notes in Math. 694 (Springer, Berlin, 1978) 196-202. (MR BOd, 46(46). [85) On the closure of modules of continuously differentiable mappings, Rend. Sem. Mat. Univ. Padova 60 (1978) 33-42. (MR 81g, 580(4). [S5a) Aspects of the recent development of functional analysis in Brazil, Proc. Internat. Conf. Developing Mathematics in Third World Countries, Khartoum, March 6-9, 1978, ed. M.E.A. EI Tom, North-Holland Math. Stud. 33 (North-Holland, Amsterdam, 1979) 81-87; Bol. Soc. Paran. Mat. (2) I (1980) 49-57. (MR 83a, 01054). [86) A look at approximation theory, Approximation Theory and Functional Analysis, North-Holland Math. Stud. 35 = Notas Mat. 66 (North-Holland, Amsterdam, 1979) 309-331. (MR 81d, 46024). [87) Some topological properties of spaces of holomorphic mappings in infinitely many variables (jointly with l.A. Barroso), Advances in Holomorphy, North-Holland Math. Stud. 34 = Notas Mat. 65 (North-Holland, Amsterdam, 1979) 67-91. (MR 8Ii,46049). [S8) Some problems in the application of functional analysis to holomorphy, Advances in Holomorphy, North-Holland Math. Stud. 34 = Notas Mat. 65 (North-Holland, Amsterdam, 1979) 577-583. (MR 58, 30215).

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[89] Warum unendlichdimensionale Holomorphie?, Jahrbuch Uberblicke Math. 1979 (Bibliographisches Inst., Mannheim, 1979) 9-20. (MR 81i, 46053). [90] Introducci6n al algebra (Editorial Reverte, Spain, 1980). [91] Why holomorphy in infinite dimensions?, Enseign. Math. 26 (1980) 257-269. (MR 82i, 46(66). [92] Introduction to functional analysis: Banach spaces and differential calculus, Monographs and Textbooks in Pure and Appl. Math. 60 (Marcel Dekker, New York. 1981). (MR 82f, 58002). [93] Silva-Holomorphy Types (jointly with M.C. Matos), Functional Analysis, Holomorphy, and Approximation Theory, Lecture Notes in Math. 843 (Springer, Berlin, 1981) 437-487. (MR 83k, 46041a). [94] Entire functions on locally convex spaces and convolution operators (jointly with M.C. Matos), Compositio Math. 44 (1981) 145-181. (MR 83i, 46050). [94a] 0 desenvolvimento da maternatica na America Latina, Cienc. e Cultura 33 (1981) 649-651. [95] Some natural problems in approximating continuously differentiable real functions, Functional Analysis, Holomorphy and Approximation Theory, Lecture Notes in Pure and Appl. Math. 83 (Marcel Dekker. New York, 1983) 369-377. [95a) A glimpse at my career as a mathematician, Cienc. Cullura 35 (1983) 3-5. [95b] Massera, one of the greatest mathematicians of Latin America of all times, Cienc, Cultura 35 (1983) 917-919. [95c] The influence of Antonio Aniceto Ribeiro Monteiro in the development of mathematics in Brazil, Cienc, Cultura 35 (1983) 1976-1977. [%] A brief outline of stochastic order, Quinto Sem. Intern. Probabilidade e Estatistica (Univ. Sao Paulo, Sao Paulo, 1982). [97] A direct sum is holomorphically bornological with the topology induced by a Cartesian product (jointly with J.A. Barroso), Portugal. Math. [98] On the existence of the smallest compact support of a seminorm and a linear mapping (jointly with J.A. Barroso). Nederl. Akad. Wetensch. Proc. Ser, A = Indag Math. [99] On pure uniform holomorphy in spaces of holomorphic germs, Proc. Amer. Math. Soc.

Proceedings of Conferences [1972a] Functional Analysis and Applications, Proc. Symp. Analysis, Univ. Federal de Pernambuco, Recife, Pernambuco, Brazil, July 9-29 1972, ed. L. Nachbin, Lecture Notes in Math. 384 (Springer, Berlin, 1974). [1972b] Analyse Fonctionnelle et Applications, C.R. Coloq, Analise, Univ. Federal do Rio de Janeiro, Rio de Janeiro, RJ. Brasil, Aug. 15-24 1972, ed. L. Nachbin, Actualites Sci. Indust. 1367 (Hermann, Paris, 1975). [1973a] Proceedings on Infinite Dimensional Holomorphy. Internat. Conf. Univ, of Kentucky, Lexington, Ky.. U.S.A.. May 28-June I 1973, ed. T.L. Hayden and T.J. Suffridge, Lecture Notes in Math. 364 (Springer, Berlin, 1974). [1973b] Symposium on Infinite-Dimensional Function Theory. Proc. Northern Illinois Univ., DeKalb, Ill., U.S.A., Nov. 30-Dec. 1 1973, ed. T.A.W. Dwyer and c.r. Gupta, Northern Illinois Univ. [1975] Infinite Dimensional Holomorphy and Applications, Proc. Internat. Symp. on Infinite Dimensional Holomorphy, Univ. Estadual de Campinas, Sao Paulo,

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Brazil, Aug. 4-8 1975, ed. M.C. Matos, North-Holland Math. Stud. 12 = Notas Mat. 54 (North-Holland. Amsterdam. 1977). (l977a) Approximation Theory and Functional Analysis, Proc. Internat. Symp. on Approximation Theory. Univ. Estadual de Campinas, Sao Paulo, Brazil, Aug. 1-5 1977, ed. J.B. Prolla, North-Holland Math. Stud. 35 = Notas Mat. 66 (NorthHolland, Amsterdam, 1979). (l977b) Advances in Holornorphy, Proc. Sem. Holomorfia, Univ. Federal do Rio de Janeiro, Rio de Janeiro, RJ, Brazil, Sept. 26-28 1977, ed. J.A. Barroso, NorthHolland Math. Stud. 34 = Notas Mat. 65 (North-Holland, Amsterdam, 1979). [1978) Functional Analysis, Holornorphy, and Approximation Theory, Proc. Sem. Analise Funcional, Holomorfia e Teoria de Aproxirnacao, Univ. Federal do Rio de Janeiro. Rio de Janeiro, RJ, Brazil, Aug. 7-11 1978, ed. S. Machado, Lecture Notes in Math. 843 (Springer, Berlin, 1981). [1979] Functional Analysis, Holomorphy, and Approximation Theory, Proc. Internat. Sem. on Functional Analysis. Holomorphy, and Approximation Theory, Univ. Federal do Rio de Janeiro, Rio de Janeiro, RJ, August 6-10 1979, ed. G.l. Zapata, Lecture Notes in Pure and Appl. Math. 83 (Marcel Dekker, New York, 1983). [1980] Functional Analysis, Holomorphy and Approximation Theory, Proc. Sem. Analise Functional, Holomorfia e Teoria de Aproxirnacao, Univ. Federal do Rio de Janeiro, Rio de Janeiro, RJ, Aug. 4-8 1980, ed. J.A. Barroso, North-Holland Math. Stud. 71 = Notas Mat. 88 (North-Holland, Amsterdam, 1982). [1981J Functional Analysis, Holomorphy and Approximation Theory II, Proc. Sem. Analise Functional, Holomorfia e Teoria de Aproxirnacao, Univ. Federal do Rio de Janeiro, Rio de Janeiro, RJ, Aug. 3-7 1981, ed. G.!. Zapata. North-Holland Math. Stud. 86 = Notas Mat. 92 (North-Holland, Amsterdam, 1984). [1984) Functional Analysis, Holomorphy and Approximation Theory, Proc. Internat. Symp., Campinas, Sao Paulo, Brazil, 1984, ed. J. Mujica, North-Holland Math. Stud. = Notas Mat. (North-Holland, Amsterdam, to appear in 1985).

References {I} R.M. Aron and J.B. Prolla, Polynomial approximation of differentiable functions on Banach spaces, J. Reine Angew. Math. 313 (1980) 195-216. {2} J. Auslander, Generalized recurrence in dynamical systems, Contrib. to Differential Equations 3 (1964) 65-74. {3} B. Baumgarten, Gewichtete Raume differenzierbarer Funktionen (Thesis, Darmstadt, 1976). {4} M. Bianchini, Silva-holomorphy types, Borel transforms and partial differential operators, In: Functional Analysis, Holomorphy, and Approximation Theory, Proc. Sem. Analise Funcional, Holomorfia e Teoria de Aproximacao, Univ. Federal do Rio de Janeiro, Brazil, Aug. 7-11, 1978, ed. S. Machado, Lecture Notes in Math. 843 (Springer, Berlin, 1981) 55-92. {5} K.-D. Bierstedt, Gewichtete Raume stetiger vektorwertiger Funktionen und das injektive Tensorprodukt I-II. J. Reine Angew. Math. 259 (1973) 186-210; 260 (1973) 133-146. {6} K.-D. Bierstedt, Injektive Tensorprodukte und Slice-Produkte gewichteter Raume stetiger Funktionen, J. Reine Angew. Math. 266 (1974) 121-131.

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{7} {8} {9} {IO}

{ll} {12}

{13} {14}

{15} {16}

{17} {18} {19} {20}

{21} {22} {23} {24} {25} {26} {27}

71

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{48} S. Izumi and T. Kawata, Quasi-analytic class and closure of it"} in the interval (-00,00), Tohoku Math. J. 43 (1937) 267-273. {49} T. Kamae, U. Krengel and G.L. O'Brien, Stochastic inequalities on partially ordered spaces, Ann. Probability 5 (1977) 899-912. {50} T. Kamae and U. Krengel, Stochastic partial ordering, Ann. Probability 6 (1978) 1044-1049. {51} J. Kelley, Banach spaces with the extension property, Trans. Amer. Math. Soc. 72 (1952) 323-326. {52} G. Kleinstiick, Der beschrankte Fall des gewichteten Approximationsproblems fiir vektorwertige Funktionen, Manuscripta Math. 17 (1975) 123-149. {53} G. Kothe, Fortsetzung linearer Abbildungen lokalkonvexer Raume, Jahresber. Deutsch. Math. Verein. 68 (1966) 193-204. {54} G. Kothe, Topological vector spaces I, Grundlehren Math. Wiss. 159 (Springer, Berlin, 1969). {55} P. Lelong, Recent results on analytic mappings and plurisubharmonic functions in topological linear spaces, In: Several Complex Variables II, Proc. Internat. Math. Conf., Univ. Maryland, April 6-17 1970, ed. J. Horvath, Lecture Notes in Math. 185 (Springer, Berlin, 1971) 97-124. {56} J.G. Llavona, Approximations of differentiable functions, In: Studies in Analysis, Advances in Math. Supplementary Stud. 4 (Academic Press, New York, 1981) 197-221. {57} P. Malliavin, L'approximation polynomiale ponderee sur un espace localement compact, Amer. J. Math. 81 (1959) 605-612. {58} P. Malliavin, Approximation polynomiale ponderee et produits canoniques, In: Approximation Theory and Functional Analysis, Proc. Internat. Symp. Approximation Theory, Univ. Estadual de Campinas, Brazil, August 1-5, 1977, ed. J.B. Prolla. North-Holland Math. Stud. 35 =- Notas Mat. 66 (North-Holland, Amsterdam, 1984) 237-262. {59} P. Mazet, Analytic sets in locally convex spaces, North-Holland Math. Stud. 89 = Notas Mat. 93 (North-Holland, Amsterdam, 1984). {60} G. Mehta, Topological ordered spaces and utility functions, Internat. Econom. Rev. 18 (1977) 779-782. {61} G. Mehta, A new extension procedure for the Arrow-Hahn theorem, Internal. Econom. Rev. 22 (1981) 113-118. {62} J. Mujica, Germenes holomorfos y funciones holomorfas en espacios de Frechet, Pub!. Departamento do Teoria de Funciones 1 (Univ. Santiago de Compostela, 1978). {63} J. Mujica, Domains of holomorphy in (DFC)-spaces, In: Functonal Analysis, Holomorphy, and Approximation Theory, Proc. Sem. Analise Funcional, Holomorfia e Teoria de Aproximacao, Univ. Federal do Rio de Janeiro, Brazil, Aug. 7-11,1978, ed. S. Machado, Lecture Notes in Math. 843 (Springer, Berlin, 1979) 500-533. {64} J. Mujica, A new topology on the space of germs of holomorphic functions, IMECC, Relatorio intemo No. 193 (Univ. Estadual de Campinas, 1981). {65} J. Mujica, Complex analysis in Banach spaces; holomorphic functions and domains of holomorphy in finite and infinite dimensions (North-Holland, Amsterdam, 1984). {66} M. Neumann, Automatic continuity of linear operators, In: Functional Analysis: Surveys and Recent Results II, Proc. Conf. Functional Analysis, Paderborn, Germany, Jan. 31-Feb. 4, 1979, eds. K.-D. Bierstedt and B. Fuchssteiner, North-Holland Math. Stud. 38 =- Notas Mat. 68 (North-Holland, Amsterdam, 1980) 269-296. {67} O. Nicodemi, Homomorphisms of algebras of germs of holomorphic functions, In:

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l.A. BARROSO editor, Aspects of Mathematics and its Applications © Elsevier Science Publishers B.V. (1986)

77

TOEPLITZ KERNELS, SCA TIERING STRUCTURES AND COVARIANT SYSTEMS

R. AROCENA, M. COTLAR,

J. LEON Universidad Central de Venezuela, Facultad de Ciencias, Caracas, Venezuela To Leopoldo Nachbin with admiration and friendship

The notion and basic properties of generalized Toeplitz kernels lead to the consideration of coupled pairs of covariant systems. In particular coupled pairs arising from the LevyKhinchine theorem, and the corresponding theorem for those kernels, are considered.

o.

Introduction

The aim of this paper is to interpret the basic properties of the operator-valued generalized Toeplitz kernels, GTK, in terms of covariant systems and scattering structures. The scalar-valued GTK were studied in [6a], [4], [3a], [3c], [3d], [6b], [6c], [2b], [5], where they were linked to topics concerning weighted inequalities, prediction and interpolation problems, and Hankel operators. The study of operator-valued GTK was started in [3b], and in [2a], [2c] this study was linked to the Nagy-Foias commutator theorem and to linear systems and scattering operator functions. In the present paper, which is a self-contained complement to [2c], we propose to detail some of the relations between GTK and scattering structures from the point of view of covariant systems. This connexion with covariant systems is explained in detail in Section 1. In Section 2 it is shown that basic properties in the Adamjan-Arov paper [1], on classical scattering structures, can be rewritten for a wide class of covariant systems. In Section 3 covariant systems arising from the Levy-Khinchine theorem are considered, and this theorem is extended to GTK. Section 2 can be given a white-noise interpretation which will be detailed elsewhere.

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R. Arocena et al. / Toeplitz Kernels

1. Preliminaries and Motivations

Let .NI,.N2 be two Hilbert spaces, ?' I = {n E ?' I n ~ O}, l' 2 = {n E Z In < O}, and set .Nm if m E ?' a (ll! = 1,2). We shall consider functions A defined in Z with values A(n) E .Nn , and functions K defined in ?' x l' with values Kim, n) E 5£b (.Nm, .Nn) (the bounded linear operators

-:«,

from .Nm to .Nn). Such a kernel K(m, n) is said to be positive definite, p.d., if for every function A(n) E .Nn , of finite support,

2:

(1.1)

m.nez

(K(rn, n)A m , An)N. ~ O.

einr, t E T, n E Z, and if r!P is the vector space of all trigonometric polynomials ~ hmem(r) with coefficients hm E .Nm, then each such p.d. kernel K defines a scalar metric in r!P by If en (t)

=

(1.2) giving rise to a Hilbert space 'JI such that r!P is dense in 'V in an obvious sense. We say that K is a generalized Toeplitz kernel, GTK, if

(1.3)

K(rn + 1, n + 1) = Kim, n),

whenever m

,t.

-1, n

» -1.

(Observe that if .NI ,t..N2 then equation (1.3) makes sense only if m, n ,t. -1.) This is equivalent to the following requirement: there exist four operator functions Gafl (n) E .5£b (.Na , .Nfl)' n E ?', such that (1.3a)

Kim, n) = Gap(rn - n)

if m E Z a' n E ZfJ (a, f3

=

1,2).

In particular K(n, n) = K(O,O) if n ~ 0, K(n, n) = K(-l, -1) if n < 0, and for simplicity we shall restrict ourselves to kernels satisfying (1.3b)

K(n, n) = Id, = the identity operator in .Nn

('Vn) .

Let 'JI be as above and set 'JIk , the subspace spanned by the elements hkek (r): 'JI~, the subspace spanned by the elements hnen(t) with n,t. k (h k E .Nk) and set A", = r o, .N_ = 'JI_ I • From (1.3b) and (1.2) it follows that h~heo(t) (.NI~.N+) and h~he_I(t) (.N2~}(J are isometries, so that

R. Arocena et at. / Toeplitz Kernels

79

(1.3c) If hen E 'V~l then both elements hen and hen+1 are in 'V and we then set (hE.Hn , n,i-1).

(1.4)

If K is a p.d. GTK then Ila(hen)11 = Ilhenll, so that in this case there is a well-defined shift operator a : 'V~l -+ 'Vb which satisfies (1.4), and such that

(1.4a)

a : 'V~I-+ 'V~ is an isometry .

Moreover, setting (l.4b) and letting V" (l.4c)

=

the restriction of a"l to 'V", then

V" E L('V,,) are isometries,

so that

(l.4d)

'V- k - 1 = V~(.HJ

(k ;;. 0) .

From (1.4a) it follows that a can be extended to a unitary operator: there exists a Hilbert space 1e:J 'V and a unitary operator U E .Ph (1e), such that (1.5)

(1.5a)

U=a

in 'V".

Any unitary operator U E .Ph (1e) satisfying (1.5a), 'V" c 1e, is called a unitary coupling of the two isometries V+, V_. The above U is a special coupling satisfying 'V C 1e and extending a, and 'V+ ('VJ is an outgoing (incoming) space of U, that is (1.5b)

In the formulae that follow we use (1.3c).

so

R. Arocena et al. / Toeplitz Kernels

Let E(.::1) be the spectral measure of U and let fJ-afJ (0', f3 = 1,2) be the measure, In T, defined by (fJ-afJ (.::1 )ha, hfJ) = (E(.::1 )ha, hfJ) if ha E «; hfJ E Xf3' Since by (1.5a) U mha = amha if m E Z a and ha E X a, we shall have that if mEl a' n E Z f3' using (lAd), (;lafJ (m n)h a, h(3) = f ei(m-n)t d t(E(.::1)h a, hfJ) = (Umh a, UnhfJ) = (anh a, anhfJ) (haem(t), hfJen(t))K = K(m, n) (;laf3 = Fourier transform of fJ-afJ)' Thus

s; (Xa, XfJ )-valued

(1.6)

K(m, n)

=

JiafJ(m - n)

Moreover, for each .::1 C T, the matrix (fJ-afJ(.::1)) can be considered as an operator in 2 b (XI ttlXz), and from the construction of fJ-afJ it follows that this operator is non-negative. We express this fact by writing (fJ-afJ);?: O. From these considerations it is easy to deduce the following basic property. Theorem (1.1). ([2a], [3b], [3d]) Let K(m, n) E 2 b (Xm' X n) be a p.d GTK. Then (a). (Integral representation) There exists a matrix measure fJ- = (fJ-afJ);?: 0 (0', f3 = 1,2) in T, such that (1.6) holds. (b). (Dilation property) These exists a unitary operator U E 2 b ('Ie) and two isometries 4>a :Xa ~ 'Ie (0' = 1,2), 4>a(XJ = X±, such that (1.7)

where 7T"m is the projection of 'Ie onto 4>m(Xm), (c). (Lifting property) There exists a 2 b(XI ttlXz)-valued function G(n) such that the ordinary Toeplitz kernel G(m - n) E 2 b(XI ttlXz) is p.d., and such that if G(m) = (G af3 (m)) is the matrix representation of G(m) then K - (G afJ), in the sense of (1.3a). There are as many representations (1.6), or minimal dilations (1.7), as minimal unitary couplings U of (V+' VJ satisfying "f!' C 'Ie, and in all these representations fJ-11 and fJ-22 remain fixed, but not fJ-IZ = fJ- ;1· Consider now the special kernels K satisfying

(1.8)

K(n, m) = 0 if n

oj.

m and n, m E same Z a (0'

=

which is equivalent to (1.8a)

V~(X±)

is orthogonal to V;(X±) if n

oj.

m,

1,2),

R. Arocena et al. / Toeplitz Kernels

so that (1.8b)

nn>O

V~(.H±) =

81

{O}. In this case the fJ-ap in Theorem (Ll) satisfy (a = 1,2) ,

where dt is the Lebesgue measure of T. Unitary couplings U of isometries V::t satisfying (1.8a) were studied by Adamjan and Arov [1], and in this case (see (1.5b), we say that [U E 2 b (Jt'), r::t] is an Adamjan-Arov scattering structure. As we shall see in (2.14) of the next section, if K satisfies (1.8) then the operator functionfJ-12(t) of (1.8b) is just the scattering function of Adamjan and Arov. In these cases all the unitary couplings U E .Pb (Ye) which satisfy r eYe (and therefore give extensions of 0-), constitute an equivalence class of couplings of V::t, in the sense of Adamjan and Arov. All these couplings and the corresponding functions fJ-dt) were described and parametrized, in this case, by Adamjan, Arov and Krein [la]. Thus, we may say that the theory of classical Adamjan-Arov structures is essentially the study of the special p.d. GTK which satisfy (1.8). The above-mentioned parametrization was extended for· arbitrary scalar-valued GTK in [2b], and the generalization to operator-valued kernels is also in process. It is then natural to ask whether the above scattering interpretation of Theorem (1.1) can be extended for more general classes of kernels, not satisfying (1.8). That is, whether the basic properties in the Adamjan-Arov paper [1], concerning scattering functions and models, extend to a wider class of GTK. For this purpose observe first that (by (1.7) or (1.2» the kernel K, and the measures fJ-ap associated with U, are determined by the operator U and the subspaces U" (.H::t). If K satisfies (1.8) then .H::t = 'V± ~ VAr::t), so that in this case K and fJ-a{J are determined by the scattering structure [U, r+, rJ. In the general case K will be determined if we give U and the two sequences M±(n) = projection of Ye onto Un(.H±). And to give {M+(n)} is the same as to give a measure M+ in Z such that M+(.1) = M+(n) if .1 consists of a single point n. These M::t(n) satisfy. (1.9)

(for all n E l) ,

and similarly for M_. If K satisfies (1.8) then by (1.8a) .1 1 n .12 = 0 implies M+(.1I)M+(.1~ = 0, and M+(.1) is a projection for all .1 c Z. That is, in the Adamjan-Arov case M+ (or M_) is an ordinary orthogonal (or spectral) measure, and (1.9) expresses that [M+, U] is an imprimitivity system, with

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R. Aroana et at. / Toeplitz Kernels

base In I in the sense of Mackey. In case of an arbitrary GTK the measure M+ (MJ satisfies (1.9) but is not orthogonal any more (i.e. M+(Ll) is not a projection if Ll contains more than one point). Then we say that [M+, U) (and [M, U)) is a non-orthogonal imprimitivity system or a covariant system in the sense of Davies [7], and Holevo [Sa], [8b). Such a pair [M+, U], [M_, U] of covariant systems, with the same unitary U, is called a coupled pair of covariant systems. Two such coupled pairs will be said to be equivalent if they have the same subspaces M+(n)(:lt) (n;;;: 0), M_(n)(:lt) (n <0), as well as the same subspaces (Vn;;ooM+(n)(:lt», (Vn
2. Representations of Covariant Systems Let [M, U) be a covariant system, that is, U E 5£b (:It) is a unitary operator and M an 5£b(:lt)-valued measure in Z satisfying (see (1.9» (2.1)

UM(n) = M(n + l)U,

(VnEZ).

We will assume that (2.1 a)

M(O)l/2 has closed range, M(Oi/2;;;: 0 .

Of special interest will be the case when all the M(n) are projections. If the M(n) are projections and in addition M(n )M(k) = 0 for n ¥- k, then [M, U) is called a classical covariant system or an imprimitivity. A basic example of a classical covariant system is obtained by taking :It = L 2(T, X), U = S, M(n) = P(n), where L 2(T, X) is the space if all Xvalued functions l(t) in T satisfying I 11/(t)ll~ dt < 00, S is the shift defined by (Sf)(t) = ei'f(t), and P(n)1 = hnen(t) if 1 = L m b,»; (t). [P(n), S] IS called the X-canonical covariant system.

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R. Arocena et al. / Toeplitz Kernels

We set JltM = the subspace spanned by all the elements of the form M(n)h (n E 7L, hE JIt). From (2.1) it follows that JltM reduces V.

(2.1b)

Let IE L(JltM ) be a positive operator such that I: JeM necessarily isometric) topological isomorphism, so that

~

JeM is a (not

(2.2)

We say that [M, VJ is a I-couariant system if is satisfies

2:

(2.2a)

M(n)h

=

Ih

n=-oc

that is, if M is a finite measure of total mass I (in JltM ) . Every classical system is a I-system with I = Id = identity of JeM • From (2.1) and (2.2a) it follows that (2.2b)

VI=IV.

Recall that an operator if> from Je to another Hilbert space JIt' is a partial isometry with domain DC Je, if /Iif>(h)11 = /Ih/l for hE D and if>(h) = 0 if h .l.. D. In this case if> -I : if>(D)~ Je' is defined by (2.2c)

if>-lif>h

=

It,

for h E D,

With this notation we have Proposition (2.1). If [M, UJ is a I-covariant system and .JV = the range of M(O)II2, then there exists a partial isometry if> : Je ~ L 2 (T, .JV) with domain JeM such that

(2.3) (2.4) (2.5)

Vh = if> -I Sif>h

R. Arocena et al. / Toeplitz Kernels

84

where [P(n), S] is the K-canonical covariant system, 7T the projection of L 2(T, K) onto 4> (H). Moreover there is a measurable operator-valued t~ 7T(t) E L(K) such that 7T(t) is a projection for almost all t E T and such that

(2.6) Proof. Assign to each h E Je the element 4>(h) = f E L 2(T, .N') defined by

(2.7) f= 4>h

= 0,

where j is the Fourier-transform of f. Since M(Or(Je) = K, j(n) E K, and using (2.1) and (2.2a),

n

n

=

n

(L: M(n)r

l/2h,

r1/2h)

= (Jr 1/2h, rl/2h)

n

=

IIhIF,

if h E JeM , so that 4> is a partial isometry with domain JeM • Since (cfJUh)A(n) = M(O) II2U- n r Il2Uh = M(o) lI2 u - (n . I J r l/2h = (S4Jhf(n), for h E JeM , we have that 4>Uh = S4>h so that (2.3) holds. Since (2.7a)

(J1/24J -1 7TP (n )cfJJ I/2h, hi) = (P(n )4>J I /2h, 4>J I /2h l ) =

(y' M(O)

u- nr

l

/2 f/2h, y' M(O)U- n r ll2 J 1/2h)

for h, hi E JeM , it follows that (2.4) holds. Finally by (2.3), S4Jh = 4JUh E cfJ(Je), and the same is true for S-\ so that 4>(Je) reduces S, and (2.5) and (2.6) follows from well-known properties of S-invariant spaces. 0

R. Arocena et al. / Toeplitz Kernels

E L(:JeM ) , and U * = the restriction of U to :JeM • Then [M *, U *] is a covariant system in 'ltM ,

Corollary (2.2). Let M *(n)

=r

85

I/2M(n)rI/2

(2.7b)

so that [M *, U *] is unitarily equivalent to the contraction of the .N'canonical system to c/J(:JeM ) . Proof. Immediate consequence of (2.2b) and (2.4). 0 Corollary (2.3). If c/J* : L 2(T, .N')~:Je is the adjoint of c/J, then

(2.8)

c/J *f = c/J *1Tf = c/J -1 7Tf

CVfE L 2(T, .N'»,

(2.8a)

c/Jc/J *f = 7Tf

CV f

(2.8b)

c/J *c/Jh = Pnh = projection of h onto 'ItM,

(2.&)

c/J *(Sf) = Uc/J*f,

(2.8d)

c/J*fh

=h

2

E L (T, .N'» ,

CVh E.N'),

where I, is defined by (2.8e)

CVtE T).

Proof. (2.8) is an immediate consequence of (2.2c), and (2.8a) follows from (2.8). Since (c/J*c/Jh, hI) = (c/Jh, c/Jh l) = (c/JP~, c/JP~I) = (P~, P~I) = (P~, hI)' (2.8b) follows. Using (2.8), (2.5), (2.3) we get that c/J* S = c/J -1 7TS = c/J -I S7T = c/J -I Sc/Jc/J -1 7T = Uc/J -1 7T = Uc/J * so that (2.8c) holds. Finally, by (2.7), (c/J*fh' hI) = (fh' c/Jh l) = (!h(O), (c/Jhlf(O» (M(Or l12J I12 h, M(O)112r l12h j ) = (h, hI)' and (2.8d) follows. 0 Corollary (2.4). If A E L(:JeM

) satisfies UA = A U then there exists a bounded L(.N')-valued function A(t), t E T, such that

(2.9)

(c/JAh )(t) = A(t)(c/Jh)(t)

R. Arocena et at. / Toeplitz Kernels

86

In particular if E. is the restriction to JeM of the spectral measure of U, then (2.9a)

(¢E.h)(t) = 1.(t)(¢h)(t) ,

and for every h E JeM we have (2.9b) Proof. Since ¢A¢-I7TS = ¢A¢-IS7T = ¢AU¢-I7T = ¢UA¢-I7T = S¢A¢-l7T, ¢A¢-I7T commutes with S and there exists A(t) such that (¢A¢-I7TJ)(t) = A(t)(7TJ)(t), and (2.9) follows. Similarly (2.9a). From (2.9a) and (2.7) it follows that Jo(¢h)(t)dt=J;"(¢E.h)(t)dt= 27TM(OY/2F I/2E.h, which gives (2.9b). 0 Consider now a pair of covariant structures (2.10) with the same unitary operator U E 2?h(Je), where M+(n) E 2?h(Je), M_(a) E 2?h(Je) (that is a coupled pair). Let Je+ = JeM + , Je_ = JeM - and assume that J±E 2?h(Je±) are two topological isomorphisms such that [M+, U] (respectively [M_, UJ) is a J+(J_)-system. We may also assume that (2.1Oa) Let ¢±: Je -+ L 2(T, X ±) be the corresponding partial isometries with domains Je± as in Proposition (2.1) and 7T±(t) the corresponding projections. In the special case where [M±, U] are classical systems, Adamjan and Arov [1] defined the scattering function of the pair (2.10) and through it gave a functional model of the pair. Since their arguments only use the properties of Corollaries (2.3) and (2.4) (which become simplified in their case), only slight modifications are required to rewrite their procedure in the present situation. We only recall the main steps, and to simplify suppose that dim X+ = dim X_, so that we may identify both spaces L 2(T,X+)=.L2(T,X_), and consider that ¢_: Je-+L 2(T,X+) and that ¢_¢: acts in e(T, X+). By (2.Bc), (2.3), ¢_¢: commutes with S, hence there is a 2?h (X+)-valued function So(t), t E T, such that

R. Arocena et al. / Toeplitz Kernels

87

(2.1Ob) and by (2.9b) and (2.8d), SoU) is given (with fh as in (2.Be» by (2.1Oe) This SoU) is called the scattering function of the pair (2.10). Let At = '(f{ 8 '(f{_, so that by (2.1Oa), At = Pm'(f{+, and '(f{+, '(f{_. At reduces U. If t. = Pmh i, hi E '(f{+ (i = 1,2) then as in [1] one obtains, using (2.8b) and (2.1Ob), that

where (2.11)

It follows that there exists an isometry

lj :

At ~ L 2(T, K+) such that

(2. 11a)

Similarly (2.11b) Let

(2.11e)

and let Q : '(f{ ~ L~ be the operator defined by (2.11d) Then from Proposition (2.1), (2.7b), (2.11)-(2.11 d), one gets Theorem (2.5). (Model Theorem.) Let (2.10) be a J",-coupled pair of

88

R. Arocena et al. / Toeplitz Kernels

covariant structures, with .N>=;f X, and So(t) its scattering function. Then there is a linear isometry 0 of 'Je into L~, given by (2.11d), such that

(2.12b)

(Oh)(t) = (4)_h)(t)(f) 0

(h E 'JeJ,

and such that, under 0, U passes into multiplication by e", M *(n)P:x_ into 7T_P(n)(f) 0 and M *+(n)P:x+ into 7T_P(n)(f) 7T+P(n).

Corollary (2.6). The scattering function So(t) of the pair (2.10), and the two projection functions 7T+(t), 7T-
Remark (2.7). From (2.lOa) and (2.10c) it is not difficult to deduce (see [1], [2cD that the Fourier-transform of So(t) is given (within a constant isometry) by (2.13)

Suppose now that K is a p.d. GTK satisfying condition (1.8), and that (2.10) is the classical pair of covariant systems defined by K (see Section 1), and let f.LrxfJ be the representing measures of K (see Theorem (1.1». Then from (a) and (b) of Theorem (1.1) (or from (1.6», it follows that Iilin)=Px_Unlx+' Hence, by (2.13), (2.14) Formula (2.14) is still true for more general non-classical coupled

R. Arocena et al. / Toeplitz Kernels

89

covariant systems (see [2c]). Thus the considerations of this section provide an interpretation of Theorem (1.1) in terms of covariant structures, for a wide class of GTK. Using the special coupling of [1] it is easy to extend Theorems (2.5) and (2.lOb), to the case where J{l 'I- J{2'

3. Covariant Systems Arising in the Levy-Kinchine Theorem In this section conditionally positive definite GTK will be considered (for the theory of ordinary Toeplitz and conditionally positive kernels, see Parthasarathy-Schmidt [9]). We shall only sketch the basic steps and for the sake of simplicity take J{l = J{2 = J{. Thus, let J{ be a Hilbert space, K a GTK with values K(m, n) E !£b(J{), so that (3.1)

K(m + 1, n + 1) = K(m, n)

(m 'I- -1, n 'I- -1),

and suppose that K is conditionally positive definite, c.p.d., that is (3.1a)

2: (K(m, n)A(m), A(n»;;'O, m,n

for every J{-valued function A(n) E (3.lb)

J{

of finite support such that

LA(n)=O. n

K is said to be normalized if K(n, n) = O. Setting (3.2)

L(m, n) = K(m, n) - K(m, 0) - K(O, n) + K(O, 0) ,

it is easy to see that (3. la)-(3. lb) imply that L is p.d. Note that L(m, 0) = L(O, n) = 0

(ym, n).

L(m, n) is not a GTK, that is, L doesn't satisfy (3.1); however L is an affine GTK:

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R. Arocena et aJ. / Toeplitz Kernels

(3.3) L(m+l,n+I)=L(m,n)+f3(m)+f3(n), form¥ -I,n¥ -1,

where (3.3a)

f3(m) = K(m, 0) - K(m + 1,0) .

Let [Jj> be the vector space of all trigonometric polynomials ~ hnen(t) with coefficients h; E .JV. Since L is p.d. it defines in (l} a scalar metric given by (3.4) Let 0 be the set of all polynomials ~ hnen(t) with ~ h; = O. 0 with the metric (3.4) gives rise to a Hilbert space v such that 0 is dense in u, in an obvious sense. Really, since Ilheoll L = 0, 0 and [Jj> give rise to the same Hilbert space v and 0 = (l} in v. Let a be the shift, that is the linear operator a : 0 -+ 0 such that (3.5)

It is easy to deduce from (3.3) and (3.4) that, if m ¥ -I, n ¥ -1,

Let Ok be the set of all ~ hnen E 0 satisfying hk = 0, and vk the closure of Ok' Since hoeo = 0 in v, every element in can be written in the form ~ hn(en - eo) with n., = 0, and from (3.6) it follows that

o.,

(3.7)

a : V~I-+ v~ is an isometry of V~I onto v~.

Hence there is a Hilbert space 'Ie C 'V and a unitary operator V E .2b ('Ie) such that (3.8)

V

- I

= a

-I,

,

In Vo'

Let us set (3.9)

Then v is spanned by the elements hn1Jn (t), V~I by the elements hn1Jn with

R. Arocena et al. / Toeplitz Kernels

n;t.

-1,0, and

v~

91

by those with n ;t. 0, 1. Set

(3.9a) (3.9b) Then U±I(V±) C v±' so that if V± = U±llv+ then V+ E L(v+) and V_ E L(v_) are isometries, v± = Vk;>o V~.N'±. Moreover (since .N'+ C v+ C V~I) .N'_ C v~, ir' = o-±I in .N'±, and U" hrux = hTJa+m if m E 7L a (a = 1,2). As in Section 1, we see that if the operators JL a,8 (.::1) E .Ph(.N') are defined by (3.10)

(a,{3=1,2),

where £(.::1) is the spectral measure of U, then

Setting M±(O) = projection onto .N'±' M;l:(n) = projection onto U" (.N'±), M±(n) = UnMAo)u- n, then (3.11) are covariant systems, associated with our kernel K, and from (3.lOa) it is dear that this pair of covariant systems determines the measure <JLa,8);;': 0, and vice versa. Let us see that (JLa,8)' and therefore the pair (3.11) determines 'practically' the kernel K. Let us see first that (JLa,8) determines the kernel L(m, n) or what is the same the scalar product (hmem, hnen)L' Consider first the case m ;;.: 0, n ;;.: O. By (3.10), (3. 11a) n

(hnen, h; (em - em-I) = =

2: (h n(ek -

k=1

~

L.J

J

ei(k-mlt

k=1

= Jeimt (eint -

ek _ I), hm(em - em-I) dt (JLII ha' h) ,8

1)(1 -

e

it)

d (h h ) 2(1- cos t) JLII k' m

•

92

R. Arocena et al. / Toeplitz Kernels

Writing similarly hm(em - eo) = L ;~1 hm(ek - ek - t ) we get, after replacing this in (3.11a) and after some algebraic transformations (3.11b) ifmEZ1,nEl!. Similarly, using Lim, n) get that (3.11c)

=

Lin, m )*, and assuming K normalized, we

(L(m, n)h m, hn> = (hmem, hnen>L =J(eimt-1)(eint-1)1

if mEl a' n E Z /3 (a,,8 be rewritten as

d (J..ta/3 hm,h u>' - cos t

= 1,2). Setting T(n, t) = e inl - 1- int, (3.11c) can

(3. 11d)

(L(m, n)h m, hn> = J [T(m - n, t)- T(m, t)- T(n, t) + T(O, t)] d(J..t a/3hm, hn>. Taking into account (3.2), as in the classical Toeplitz case (d. [9], [10]), one deduces from (3.IId) Theorem (3.1). (Levy-Kinchine formula for GTK.) If Kim, n) is a normalized conditionally positive definite GTK, then there exists a positive matrix measure (J..taP)~O, J..ta/3(J)E2'b(.Ha,.H/3) (a,,B=1,2) (.H1= .H+, .H2 = .HJ, such that (3.12)

2,,-

+ J (ei(m-n)t - 1- i(m - n)t) o

d

1- cos t

(J..ta/3h 1, h2>,

if mEl a' n E Z /3' where A a/3 E 2'b(.H) is a constant operator.

93

R. Arocena et al. I Toeplitz Kernels

With the notation of (c) in Theorem (1.1), Theorem (3.1) can be restated as Corollary (3.2). (Lifting property.) If K is a normalized c.p.d. GTK, then there exists a c.p.d. 5£b (Jr (£)Jr)-valued Toeplitz kernel G(m - n)(Ga/3(m - n)) (a, f3 = 1,2), such that

(3.12a) We express (3.12a) by writing (3.12b) In the particular case where dim Jr = 1, i.e. when K(m, n) is scalar-valued, Theorem (3.1) can be complemented as follows: As is well known [9] the scalar kernel K is c.p.d. iff for every a > 0 eaK(m, n) is a p.d. kernel, and in our case also a GTK. Therefore by Theorem (1.1) there is, for every a >0, a matrix-measure. (j.t:/3);;:'0 (a, f3 = 1,2), such that (3.12c)

eaK(m, n) -

(l1a/3 (m - k)) .

Observing that 71. a -1 a = 1 (a = 1,2), but 11 -1 2 = {n E 11 n > O}, it follows from (3. 12c) that if b = 2a, 0 < b ~ 1, and if K:/3(m, n) = 11:p(m - n), K:a = (K:ai, but K~2(m) = (Kfz<m)i only if m > 0 (a = 1,2). Therefore II b r-aa

= r-aa a* II

0

IIa

rr oa

= (II a )(2) rr aa ,

II b _ r-12 -

(II

Q

,-12

)(2)

+ h" ,

where (haf (m) = 0 for m > O. From this fact it is easy to deduce the following: Corollary (3.3). If K is a normalized scalar-valued conditionally positive definite GTK, then for each a > 0 there exists a 2 x 2-matrix-measure (v:/3);;:' 0 in T (a, f3 = 1,2), such that

(for all a > 0) ,

(3.13) where u"

=

(v:/3) is a convolution semigroup in the sense of Schur,

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R. Arocena et al. / Toeplitz Kernels

(a, f3 = 1,2) .

(3.14)

Proof. In fact, letting a = 1/2, 1/4, ... , 1/2p, ... we have

_ ( .. (J..t 2-p)(2P) ap \.fA ap , I

(zP(J..t 2-P) ap

)

1) _ (

1/2)

J..t aft ' ...

P) The measures v; = (J..ti;P)(2 have uniformly-bounded total mass and there is a subsequence v; converging weakly to some Vi. Define (v~/3) by v;2= Vi, v'; = J..t~a' so that (v~p)- (J..t~p). 1 Let v~2= (J..ti;P)(zP- ) and let Pjk be a subsequence such that v~~~ v l /2 weakly. Define 1/2 _ V aa -

1/2

/-L a a

,

V

1/2 _

l2 -

V

1/2

, .••

and so on. Using the diagonal process, we can obtain a subsequence Pq 1 such that V~-I ~ v 2- I (for all I), where V~-I = (J..ti;Pq)(zPq- ) . Setting v~~ = J..t~~, 1 ) ( 2 ) ' /) 1 1 1) 222-1+lq (2() v 2weq have (v 212 = v aP - (J..taP , v a/3 = v ap , WhICh proves 3.14 for diadic a. Passing to the limit, it is extended to all real a > O. 0 Application (3.4). If the ordinary Toeplitz matrix-valued kernel (Kap(m - n» (a, f3 = 1,2) is conditionally positive definite, we cannot assure that (eakop (m-n»a.P=I,2 is positive definite (as a vector-valued kernel) for each a > O. Therefore we cannot associate with the first kernel a convolution semigroup by the usual procedure. However this matrix-valued Toeplitz kernel (Ka /3 (m - n» gives rise to a (scalar-valued) GTK K(m, n) ~ Kap(m - n) (see (3.12b) for the definition of -) where K will be a c.p.d. GTK, and by Corollary (3.3) K has an associated convolution semigroup of matrix-measures (in the sense of Schur), which can be called the convolution semigroup generated by (Kap(m - n This can be applied to ground-state properties for 2 x 2-matrix-valued c.p.d. functions.

».

Remark (3.5). In [3e] is indicated a general procedure which allows to transport Theorem (1.1) to p.d. GTK K(x, y) defined in (x, y) E IR x IR. The same procedure also allows to extend Theorem (3.1) and Corollary (3.3) to a c.p.d. GTK K(x, y) on the real line, that is, to extend the Levy-Kinchine theorem to GTK's on IR.

R. Arocena et al. / Toeplitz Kernels

95

References [1] V. Adamjan and D. Arov, Amer. Math. Soc. Transl. (2)95 (1970) 75-129. [La] V. Adamjan, D. Arov and Krein, J. Funct. Anal. Appl. 2(4) (1968) 1-17.

[Za] R. Arocena, Toeplitz kernels and dilations of intertwining operators, Integral Equations Operator Theory (to appear). [2b] R. Arocena, On the parametrization of Adamjan, Arov, Krein, Publ. Math. Orsay (1983). [2c] R. Arocena, GTK, scattering functions and linear systems (preprint). [3a] R. Arocena and M. Cotlar, Proc. Conf. in honour of A. Zygmund, Chicago (1981). [3b] R. Arocena and M. Cotlar, Lecture Notes in Math. 908 (Springer, Berlin, 1982) 169-188. [3c] R. Arocena and M. Collar, Integral Equations and Operator Theory 4 (1982) 37-55. [3d] R. Arocena and M. Collar, Acta Cient. Venezolana 33, 89 (1982). [3e] R. Arocena and M. Collar, Portugalia Math. 39 (1-4) (1980) 419-434. [4] R. Arocena, M. Collar and C. Sadosky, Adv. Math. Suppl. Studies 7A (Academic Press, New York, London, 1981) 95-128. [5] R. Arocena and J. Leon, Generalized stationary processes (preprint). [6a] M. Cotlar and C. Sadosky, Proc. Symp. Pure Math. (35)1 (Amer. Math. Soc., Providence, RI, 1979) 383-407. [6b] M. Collar and C. Sadosky, Lecture Notes in Math. 908 (Springer, Berlin, 1982) 150-169. [6c] M. Collar and C. Sadosky, Proc. Conf. in honour of A. Zygmund, Chicago (1981). [7] E.B. Davies, Funct. Anal. 6 (1970) 318-346. [8a] A. Holevo, Rep. Math. Phys. 13 (3) (1978) 287-307. [Sb] A. Holevo, Rep. Math. Phys. 16 (3) (1980) 289-304. [9] Parthasarathy and K. Schmidt, Lecture Notes in Math. 272 (Springer, Berlin, 1970). [10] M.D. Moran, Univ. Central Venezuela Fac. Cienc. (1982).

l.A. BARROSO editor, Aspects of Mathematics and its Applications © Elsevier Science Publishers B.V. (1986)

97

OPERATORS AND THEIR SYMBOLS

W. AMBROSE Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, U.S.A. Dedicated to Leopoldo Nachbin in appreciation of forty years of friendship

O. Introduction

Horrnander and Duistermaat have introduced an important class of operators, the Fourier integral operators, with which they have obtained many new results in the theory of linear partial differential equations. Their results depend upon the notion of the symbol of such an operator and a functional calculus relating operators to their symbols. The class of Fourier integral operators is not linear and is not closed under composition in general. Nevertheless these properties hold in sufficient generality to yield deep results in the theory of partial differential equations. In this paper we attempt to improve this calculus to a class of operators containing the Fourier integral operators, and which is closed under linear combinations and composition. We are not concerned here with applications of this calculus but expect to turn to that in a subsequent paper. Our symbol is not the same as that of Hormander and Duistermaat in the case of Fourier integral operators but it is suggested by theirs; it is easy to obtain their symbol from ours in that case. In some ways ours is simpler. For example, our symbols take values in (equivalence classes) of complex-valued functions whereas that of Horrnander and Duistermaat takes values in (equivalence classes) of functions with values in a Maslow bundle. And the class of operators for which we define a symbol is intrinsically defined (rather than by phase functions and amplitudes). Also, our symbol is a complete symbol rather than just a principal symbol. The new complications that we encounter are: (1) Our symbols are defined on the bundle .!(X, Y) (defined below). (2) We need to assume X and Yare Riemannian. (3) Our symbols take values in the spaces Zm/Z P+ (defined below) instead of the more usual Sm/S-'".

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W Ambrose / Operators and Their Symbols

These complications are forced on us by the fact that our operators are attached to no particular Lagrange submanifold of the tangent (or cotangent) bundle. This is necessary in order to have a class of operators that is closed under addition and composition. That the values of the symbol lie in such spaces is not new because if p(D) is a pseudo-differential operator and p(x, 0 is what is colloquially called the symbol of p(D) then p(x,~) gives rise at each (x,~) to the function u ~ p(x, u~) on ~ + and, strictly speaking, the symbol of p(D) is the map of the tangent or cotangent bundle (minus its a-section) into sm/s-oo defined by (x, ~)~ p(x, . 0 mod Our symbol, even for pseudodifferential and Fourier integral operators, will take values in Zm/Z P+ with p ¥ -00; it is defined on a larger set than the usual symbol and it is at some of these additional points that it will have such values. We express here our thanks to D. Marcus who showed us that a previous version of this paper was not sufficiently general, who pointed out that the spaces S" were not sufficient for our purposes, and who stated and proved Theorem (2.2) below. We also express thanks and appreciation to G. Uhlmann for many conversations about the matters discussed here, and for much necessary encouragement. We mention some notation that will be used here. We use the usual letters qjJ, '(;, [f', qjJ', '(;', [f" for the usual Schwartz spaces on a Riemannian manifold. If T is a distribution arising from a locally integrable function / we say that / is a density for T (rather than saying that / is T). We use j and t for the Fourier transforms of the function / and the distribution T (in case these are defined on a Euclidean space) and we use f and t for their inverse Fourier transforms. If X is any manifold we write Xx for the (real) tangent space to X at x; and ax will always denote the zero element of Xx' TO(X) will always denote the tangent bundle to X minus its a-section. We denote the exponential map of Xx into X (if X is a Riemannian manifold) by exp., If X is any metric space, x E X, and r E ~ + we write (x, r) for the open ball about x of radius r. Hence, if X is a Riemannian manifold, we write qjJ(x, r) for the Schwartz space qjJ of functions on this ball, etc. If X is a Euclidean space, i.e. a finitedimensional linear space over R with a non-degenerate positive definite scalar product, we let M(X) = the symmetric linear transformations 0/ X into X. This paper consists of four sections. Section 1 consists of the basic definitions of the symbol and related notions. Section 2 proves completeness of our symbol. Sections 3 and 4 establish the essentials of our functional calculus.

sr.

W Ambrose / Operators and Their Symbols

1. Definition of L,,(X, Y) and

99

UF

In the following X, Y, Z will always be real cgoo-Riemannian manifolds, hence X x Y; X x Z, Y x Z are also such. If (x, y) E X x Y we write expxy for exp(x,y)' If f is any complex-valued function on X we write I, for the function on X, defined by fx = foexpx' We define

R, = the largest R E IR+ U {oo} such that expx is a diffeomorphism of (Ox' R) onto (x, R). We define

It is trivial that if f E cgOO(X), g E cgOO(Ox' R x)' T E 3"(x, RJ then (1.1) We define R~= min{l, hup{R E lR+j for all y E (x, R),

expy is a diffeomorphism of (Oy' R) onto (y, R)}} . Thus y E (x, R~) implies expy is a diffeomorphism of (Oy, R~) onto (y, R~). The existence of convex neighbourhoods of x implies that R ~ > O. If T E !?lJ'(X) and c/J E (x, R~) we then have that c/JT E 3"(y, R y) for all y with d(y, x) < R~, hence (c/JT)y is defined if d(y, x) < R~ and c/J E !?lJ(x, R~). We define $(X) = {(x,

s, A)j x E X, s E Xx, /ls/l = I, A E M(XJ} ,

so $(X) is a cgoo-bundle over X. If Q is any open set in X we let $o(X) = 1T- 1(Q ) n $ (X ) where 1T is the projection of $(X) into X. We define for each (x, s, A) E $(X) a real-valued function gxsA on Xx by

100

W Ambrose / Operators and Their Symbols

gxsA(V) = (v, s)- (Au, v).

If T E ~'(x, R x ) we define a map T+ of .!ex,R.iX) into the complex-valued functions on IR + by

(T+(x, s, A»(u) = T,.(exp(-iugxsA» = T(exp(-iu[(Lx

' ,

s) - (ALx

' ,

L, .)])).

So if T E ~'(X) then (cf>Tt is defined on .!e.,RJX) if cf> E ~(x, R~). We write for (~r if / is a locally integrable function. We shall use below the spaces Z'", Zm+ defined, for m E IR U {±oo}, by: if m E IR then

r

Z" = {f E 'eOO(1R +)1 for every k E Z + U {O} there exists a

ck E IR +

such that I(Ok f)(u)1 ,,;;;; ck (I + ll)m for all u E IR +} ,

z-:

and if m = ±oo then Z-oo = n mZ", Zoo = Urn Z", Then we define if mE IR by zr: = p>m Z" and define Ze-oo)+ = Z-oo, Zoo+ = Zoo. We use on

n

Z" (if m E IR) the topology given by the seminorms

II/Ilk' =

sup I(Okf)(U)/I(1

+ ut

(IE Z", k E I

+

U {O}).

We let ::l = the set 0/ all

also use below the spaces S" = S" (IR +) of Hormander, i.e. is a point and IR N is IR +. that if T E ~'(x, R.) and T has order N then for cf> E (cf>Tt(x, t, A)E ZN and, letting

We shall S" = S" (X replaced by We note ~(x, R x ) '

zrtz»: for p, q E IR U (±oo), p ~ q.

X IR N) for the special case where X

sup over lal";;;; k, y E (x, R x ) we have, for each k E Z+ U {O}, u E IR+,

If X and Yare Riemannian manifolds we define

'

IV Ambrose / Operators and Their Symbols

2'(X, Y)

=

101

{(x, y, s, t, A, B)I x E X, Y E Y, s E Xx, t E Yy,

II(s, t)11 = 1, A E M(Xx ) ' BE M(Yy)} , so 2'(X, Y)c2'(Xx Y). If TE!?lJ'(XX Y) and XE!?lJ«x,y),R~y) we now define (XTt to be the restriction of the previously defined (XTt to 2'«x,Y).R~,><X, Y). So for such x, T, (x, y, s, t, A, B) we have «xTt(x, y, s, t, A, B»(u) = T(X exp(-iu[(Lx" s) + (L y.. , t) - (ALx' , Lx') - (BLy'" L;. .)])).

Let .5t'(X, Y) be the linear space (over C) of all continuous linear transformations F of !?lJ(X) into !?lJ(Y). If FE .5t'(X, Y) we write F T for its transpose, i.e., F T is the linear transformation of !?lJ'(Y)~ !?lJ'(X) defined by (1.3)

(FTT)(¢» = T(F¢»

If T = 1f with then have

I

(for all ¢> E !?lJ(X» .

locally integrable we shall write

FTI for FT 1f, so

we

(1.4) We also use the notation {FT(l/J)} for the distribution FT(l/J) so (1.4) becomes

If FE .5t'(X, Y) we denote its Schwartz kernel by K F so K F E !?lJ'(X x Y) and (if ¢> E !?lJ(X), l/J E !?lJ(Y»,

(1.5)

K F(¢>®

J

l/J) = (F¢»l/J = T",(F¢» = (FTT",)(¢» = (FTl/J)(¢».

Hence, if supp ¢> and supp l/J are sufficiently close to x and y, «(¢>® l/J)KF t(x, y, s, t, A, B»(u) = (¢>®

l/JKF )(exp(-iu[(Lx' , s) - (ALx' , Lx') + (Ly. . , t) - (BLy. . , L; . .)]))

102

=

W Ambrose / Operators and Their Symbols

J

F(4)(.)(exp(-iu[(Lx·' s)- (ALx·, Lx·)])))(y')

y

. t/J(y')(exp(- iu[(Lyy', t) - (BLyy', Lyy'»») dy' = (FT(t/J exp(-iu[(Ly. . , t) - (BLy. . , L y. .)])))

. (4) exp(-iu[(Lx' , .) - (ALx' , Lx')])). We now define the subspace 2,,(X, Y) with which we will be mainly concerned. Definition (1.1). 2,,(X, Y) is the set of all FE 2(X, Y) which have the

following two properties: (1). For each (x, y) E X x Y and p E IR there exists a dense open set Q/x,y) in M(Xx)XM(~) such that if (A,B)EQp(x,y) then the following is true. For each A E IR + there exists an RpA(x, y, A, B) E IR +, with R pA(x, y, A, B) < R ~y, such that for every EO with a < EO < R pA(x, y, A, B) there is a neighbourhood Q£ of {(x, y, s, t, A, B)II/(s, t)1/ = I} such that if E is any bounded subset of 0J«x, R pA(x, y, A, B)) x (y, R pA(x, y, A, B))) then {(XKF t(x', y', s', t', AA T, ABT)I (x', y', s', t', AT, B X E E, X =

T)

E Q£ '

a in (x, EO) X (y, EO)}

is a bounded subset of Z". (2). For each m E IR the following holds. For each (x, y) E X x Y there exists an'R~(x,y)EIR+ and an TJ = TJ~(x,y)EIR+ such that if we define Q~(x, y) = {(x', y', s', t', A', B') E X(X, Y)I dix, x') < TJ, d(y, y')

< TJ,

IIA'II < 1, IIB'I/ < 1, Ils'l/ < TJ}, then for each bounded subset E of 0J«x, y), R~(x, y)) {(XKFt(x', y', s', t', A', B')I (x', y', s', t', A', B') E Q~(x, y), X E E}

is a bounded subset of Z": Remark (1.2). It is easily seen that condition (1) implies the following: for each (x, y) E X x Y and p E IR there is a dense open set Qp(x, y) in M(Xx) x M(Yy ) such that if (A, B) E Qp(x, y) then the following is true.

w:

Ambrose / Operators and Their Symbols

103

For each compact K C IR + there exists an RpK(x, y, A, B) E IR +, with RpK(x, y, A, B) < R ~y, such that for every E with 0 < E < RpK(x, 'y, A, B) there exists a neighbourhood Q£ of {(x, y, s, t, A, B)III(s, t)11 = I} such that if E is any bounded subset of f0«x, RpK(x, y, A, B) x (y, RpK(x, y, A, B» then {(XKFr(x', y', s', t', AA', AB')I (x', y', s', t', A', B') E O£, A E K, X E E, X = 0

in (x, c) x (y, E)}

is a bounded subset of Z". Remark (1.3). Using Theorem (2.3) below we also see that condition (2)

implies the following: For each In E IR the following holds: for each (x, y, A, B) (x E X, Y E Y, A E M(Xx ) ' BE M(Yy» and compact J C IR there exist an R~J(x, y, A, B) E IR + and an 17 = 17~Ax, y, A, B) E IR + such that if we define Q~Ax, y, A, B)

== {(x', y', s', t', AA', AB') E 1:(X, Y)I A E J, d(x, x')

< 17, dey, y') < 17,IIA - A'II < 1,

liB - B'II < 1, Ils'li < 17} , then for each bounded subset E of f0«x, y), R~Ax, y, A, B», {(XKFrcX', y', s'; t', A',

B')I (x', y', s', t', A', B') E

O~Ax, y, A, B), X E E}

is a bounded subset of Z'". We now define, for each FE ::£(X, Y), a symbol aF' a F will be a map of 1:(X, Y) into ~. Although it is defined for all FE ::£(X, Y) it will only be useful for FE ::£u(X, Y). For the definition of a F we first define, for FE ::£(X, Y), and (x, y, s, t. A, B) E 1:(X, Y), a set that we denote by QF(x, y, S, t, A, B), OF(X, y, s, t, A, B) = {p E IR I there exists an R p = Rp(x, y, s, t, A, B) E IR +

such that for each bounded subset E of f0«x, R p) x (y, R p» the following holds: if 0< E < R p then there exists a neighbourhood Q£ of (x, y, s, t, A, B) such that

104

W. Ambrose / Operators and Their Symbols

{(XKFt(x', y', s', t', A', B')I (x', y', s', t', A', B') E Q., X E E, X = 0 in (x, E) x (y, E)}

is a bounded subset of ZP}. We also define qF(X, y, S, t, A, B), an element of IR U {-oo}, by qF(X, y, s, t, A, B) = inf QF(X, y, S, t, A, B) .

Definition

(lTF(x, y, S, t, A, B).) Choose a sequence (qk) in QF(X, y, S, t, A, B) such that qk > qF(X, y, S, t, A, B) and the qk descend monotonely to qF(X, y, S, t, A, B) as k ~ 00. Choose R; = R qk (x, y, s, t; A, B) as in the definition of QF(X, y, S, t, A, B) and such that R k+1 ":;; R k. Choose Xk E 9fi«x, y), R k) with Xk = 1 on some neighbour(1.4).

hood of (x, y). Then

for all (x', y', s', t', A', B') in some neighbourhood of (x, y, s, t, A, B). Consider the set n«XkKFt(X, y, S, t, A, B)+

Zqk)

k

(which is clearly non-empty). If I and g are any two elements of this set then 1- g E Zqk for all k, hence 1- g E Zqr<x.y.s,I,A.B)+. Hence the preceding intersection is contained in some element z in Z"'/zqpX,y,s,I.AB)+ and we then define lTF(x, y, S, t, A, B) = z.

2. Completeness of the Symbol We now prove, for FE !t(X, Y), that F has a '€'"-kernel if and only if (TF = O. We write 0_ for the zero element of Z"'/Z-'" and write (TF(X, y, S, t, A, B) = 0_ to mean both: lTF(x, y, S, t, A, B) E Z"'/Z-OO and (TF(X, y, S, t, A, B) = 0_",. The object of this section is to prove 00

00

Tbeorem (2.1). liFE !t(X, Y) then the following are equivalent: (1). K F E

«:

w: Ambrose I

Operators and Their Symbols

105

(2). O'F(X, y, S, t, A, B) = 0_00 for all (x, y, s, t, A, B) in .!(x' Y). (3). For each (x, y, s, t) E TO(X X Y) with I/(s, t)1I = 1 there exists an A E M(Xx ) and aBE M(Yy ) such that O'F(X, y, S, t, A, B) = 0_00' Theorem (2.1) is a consequence of Theorems (2.2), (2.4) and (2.5) below. In these three theorems, X is a Riemannian manifold and T E '@'(X). One obtains Theorem (2.1) by applying them with X replaced by X x Y and T replaced by Kp-

Theorem (2.2). Let XO E X, A °E M (Xxo), ~ E IR + , RO = min{R~o, QO = {(x, t, A) E .!(X)llIx -

(811AOII + MOr x011 < ~Ro, IIAII < IIAol1 +~}. l

} ,

If E is any bounded subset of ,@(xo, RO) then

is a bounded subset of S" for each m E IR. This is proved by integration by parts as in [5] so we omit the proof. Applied to K F, with FE .2(X, Y), it shows that O'F(X, y, S, t, A, B) = -00 for all (x, y, s, t, A, B) E .!(X, Y) if K F E cgoo, hence (1) => (2) in Theorem (2.1). Obviously (2) => (3). To prove that (3) => (1), we prove Theorem (2.4) below. The following lemma will be used in the proof of Theorem (2.4). The n occurring here and below is dim X,

Lemma (2.3). If T E ~'(x, RJ, l/J E ,@(x, R x ) and l/J = 1 on supp T then for all (x, t, A) E .!(X) and u E IR + we have

(2.1)

(r+(x, t, A»(u) = un

J J (T+(x, v, A/r)(ur) R+ I/vl/~l

·l/J+(X, (t- rv)/iit - roll, {A'- A)/llt- rolD . (ullt - roll) dv

Proof. A trivial calculation. 0

r:' dr.

106

W Ambrose / Operators and Their Symbols

If X is a Riemannian manifold, T E 0J'(X), and (x, f, A) E S(X) we define WT(x, t, A) = {m E

RI there

exists an R E R+ with R < R: and a neighbourhood 0 in S(X) of (x, t, A) such that for every bounded set E in 0J(x, R) the set {(¢Tt(x', t', A')/ ¢ E E, (x',

i; A') EO}

is a bounded subset of Z"} . If E and F are subsets of R we let EEBF = {u

+ vi u E E, v E F}.

Theorem (2.4). Let X be an n-dimensional Riemannian manifold and T E 0J'(X). If (x, t, A) and (x, t, B) E S(X) then (2.2)

Wr(x, t, B);;) wr(x, t, A)EB {u E R I u > n}.

Proof. Let m E wr(x, t, A), R E R + and let 0 be a neighbourhood of (x, t, A) such that for every bounded set E in 0J(x, R) {(¢Tnx', t', A')I ¢ E E, (x', 1', A') E O}

is a bounded subset of Z'". We shall prove the theorem by showing the existence of R 1, 0 1 which satisfy the corresponding conditions for (x, t, B) and m + n. Let X' be a local cross section of S(X) over a neighbourhood 0' of (x, f) in TO(X) such that X'(x, f) = A. Define A(x ', t '), for (x', t') E 0' by X'(x', f') = (x', t', A(x ', f')). Let X be a local cross section of TO(X) over some (x,8 0) contained in 0 ' and such that X(x) = (x, t). Define t(x') for x' E (x, 80) by X(x ') = (x', f(X')). Choose 8 E R + with 8 < 80 and a neighbourhood 0 0 of (x, t, A) such that (x', t', A'/r) E 0 if d(x, x') < ~8, IIt(x')t'll < ~8, Ir - 11 < 8 and (x', t', A') E 0 0 • If E is any bounded subset of 0J(x, R) it follows that (2.3)

{(¢Tt(x', t', A'/r)1 d(x, x') < ¢, IIf'- f(x')11

< 8,

Ir-ll < 8, (x', t', A') E 0 0 , ¢ is a bounded subset of Z'".

E E}

W. Ambrose / Operators and Their Symbols

Let M

= 818, R I = min{8, (8MIIA - BII + Mrl,

107

R~, R}, and

0 1 = {(x', t', B') E .!(X)I d(x, x') < ~ 8, lit' - t(x')11 < ~8, /lA(x', t') - B'/I < IIA - BII + I}.

We shall prove the Theorem by showing that if £1 is any bounded subset of ~(x, R I ) , then {(c/>Tr(x ', t', B')I (x', t', B') E 01' c/> EEl}

(2.4)

is a bounded subset of zr». To show this we first choose if; E ~(x, R I ) such that if; = 1 on a neighbourhood of U (supp c/» (this union over all c/> E £1)' By (2.1),

II

«c/>Tr(x', t', B'))(u) = un

R+

«c/>Tr(x', v, A(x', t')lr))(ur)

IIvlH

. (if;+(X ', (t' - rv )//lt' - rvll, (B ' - A(x ', t'))//lt ' - rvlD)

. (u/lt ' - rvlDrn-1 dr dv .

Using the 8 chosen above we write the preceding integral as a sum of 5 integrals, II' ... ,15 , each with the same integrand as above but defined over the regions indicated below, so each I, = f;(c/>, x', t', B '),

12 = integral over the region: Ilv -

Ir- 1/ < ~8, t(x')II;;;;. ~8, Ir - 11 < ~8,

13 = integral over the region:

t(x')/I < ~8, r ~ 1 - ~8,

14 =

t(x')11 < ~8, r » 1 - ~8,

II = integral over the region:

15 =

Ilv -

/Iv integral over the region: Ilv integral over the region: /Iv -

t(x')11 < ~8,

t(x')II;;;;. ~8,

Ir-

11;;;;. ~8.

We now prove Theorem (2.4), by proving for each i, (2.4.i)

{!;(c/>, x', t', B')I (x', t', B ') E 01' c/> EEl}

is a bounded subset of Z'".

Proof of (2.4.1). By (2.3) we have for all I E Z + U {OJ a c, in IR + such that

108

W Ambrose / Operators and Their Symbols

(2.5) We also have, for all k E Z + U to}, (2.6)

(Okt//(X', (t' - rv)/llt' - rvll, (B' - A(x', t'»/llt' - rvlD)(.llt' =

Ok

f

rolD

l/Jx(w) exp(-i.llt' - rol/[(w, (t' - rv)/llt' - rvll) - (l/llt' - rvll«B' - A(x', t'»w, w)])) dw

=

=

Jl/JAw)Ok(exp(-i.[(w, t' J t' -

rv) - «B' - A(x', t'»w, w)])) dw

rv) - «B' - A(x', t'»w, W)])k

I/JAw)(-i[(v,

. (exp(i.[(w, t' - rv) - «B' - A(x', t'»w, w)])) dw. Obviously this function of '.' is, for each k, uniformly bounded in t', r, v, ¢ (in the region of integration of II)' This with (2.5) shows, since this region of integration has finite measure, that (2.4.1) is true. 0

Proof of (2.4.2). We have in this case, (2.7)

lit' -

roll =

lit' - t(x) + t(x) - rt(x) + rt(x) -

~ rllt(x)-

vII - lit' -

~ (1- io)~o

t(x)II

- ~o - ~o

roll

-11- rlllt(x)11

~~o.

Hence liB' - A(x', t')i/I(I/t' - rolD os; (liB - All + l)/IIt' - rvll os;M(IIB-AII+ 1)= MIIB -AII+M.

Now we apply Theorem (2.2) with (B' - A(x', t'»/llt' and M(B - A) instead of AO, to obtain, using (2.7), (2.8)

{I/J "(x', (t' - rv )/lIt' - roll, (B' - A(x', t'»/IIt' -

IIv -

t(x/)II ~ ~o,

roll

rvll)(.IIt' - rolDI

Ir - 11 < ~o, (x', t', B') E Ol}

is a bounded subset of SP for all p E IR.

instead of A,

W. Ambrose / Operators and Their Symbols

109

We also have (Tt(x', v, A(x', t'»(ur) = (T)x.(exp(-iur[(., v)- (A(x', t')., .)/r])) =

(T)Aexp(-iu[(. , rv) - (A(x', t'). , .)])),

and these, by the preceding, form a bounded subset of Z'". This, with (2.8), gives (2.4.2), since 12 is an integral over a set of finite measure. 0 Proof of (2.4.3). Same as proof of (2.4.1). 0 Proof of (2.4.4). As in (2.4.1) we have (2.5). We also have

(2.9)

lit' -

rvll ;;;.llt' -

rt'li =

11- r] ,

and, because r;;': 1 + ~8, we have (lIr):,;;; 1/(1 + ~8) which implies r - 1 ;;.: r(8/(4+ 8» and with (2.9) this gives lIt' - rvll;;.: r8/(4 + 8) .

(2.10)

Hence, for all p E IR + and u E IR + (2.11)

/(Dkl/J+(x', (t'- rv)/lIt' - rvll, (B' - A(x', t'»/llt' :,;;; ck(1

+ uPllt' -

:,;;;ck«(4 +

rvll-

rvll)(ullt' - rvll)1

P

8)18)pr-P (1 + up.

Using (2.5) and (2.11) we get (2.4.4). 0 Proof of (2.4.5). We get, as in the proof of (2.4.2), that if N is the order of

(c/JT) then

We also get (2.8) as in the proof of (2.4.2) if r:';;; 1- ~8, and get (2.11) as in the proof of (2.4.2) if r;;': 1 + ~8. This completes the proof of Theorem (2.4).0 Theorem (2.5). If T E 0J'(X) and wT(x, t, 0) = IR for all (x, t,O) in l'(X) then TE

e:

110

W Ambrose / Operators and Their Symbols

Proof. Note that «V(x, t, O»(ll) = «¢T)xf(llt). If wr(x, t, 0) = IR then, using compactness of the unit sphere, for each x E X and m E IR with m < - n - 1 there exist an R mx E IR + and a neighbourhood Omx of x such that «¢T)J(ut) decreases more rapidly, as u ~ 00, than u", uniformly for Iltll = 1 and ¢ in any bounded subset of 0J(x, R mx ) . This implies in standard fashion that (¢T)x E cg[lmlJ-n-l, hence ¢T E cg[lmlJ- n- 1 • Using a partition of unity this implies (¢T) E cgllmlJ-n-1 if ¢ E 0J(X). Being true for all m, this proves the theorem. 0 It is trivial from Theorems (2.4) and (2.5) that (3):? (1) in Theorem (2.1).

3. Multiplicative Properties of Lu(X, Y) and the Symbol In this and the following section we shall prove Theorem (3.1). If FE !fu(X, Y) !fu(X, Z).

and

G E !fu( Y, Z)

The proof also yields a formula for (TGoF in terms of bound for WGoF in terms of WG and WF.

(TG

then

and

Go FE

(Tp

and a

We now begin the proof of Theorem (3.1). It will be proved in this and the following section. Throughout these sections it will be understood that X, Y, Z are Riemannian manifolds of dimensions nx, ny, nz. The reasons for dividing the proof between this and the following section is that we must first prove the desired facts about (XKGoF when X has the special form

r

(3.1)

X=4>08

(4) E 0)(X), 8 E

0) (Z»

,

and the core of the proof in this case is formula (3.3) below. In this section, (3.1) will always be assumed. For such X the conditions (1) and (2) of the definition of !fu(X, Z) (for Go F) are obviously expressible in a modified form in which we replace the bounded set E in 0J(X x Z) by a bounded set E 1 in 0J(X) and a bounded set E 2 in 0J(Z), and prove the conclusion for the «¢0 8)K GoFr for all ¢ EEl and 8 E E 2 . We call the resulting statements the modified form of the definition.

111

W. Ambrose / Operators and Their Symbols

For each compact K in X there exists, by the continuity of F, a smallest compact K' in Y such that 4> E gg(K) implies F4> E gg(K'). We denote this K' by F(K). We now make a calculation, valid under the following assumptions:

(3.2) FE oPu(X, Y),

4> E

gg(x, R x ) ,

G E oPu(Y' Z) , fJ E ~(z, R z )

,

= F(cIos(x, R~»,

K

Yl""'Yk E K,

cover K,

n; ... ,RkEIR+,

R;
l/J1' ... ,l/Jk is a '€~ partition of unity over K, subordinate to the (y;, R;),

l/J? E gg(y;, R;), l/J? = 1 on supp l/J;, AEM(Xx>,

BEM(Yy)

,

CEM(Z.),

J, is the Jacobian determinant of L y

Fj E oPu(X,

Y) is defined by

Fj4>

, }

= ~ (F4»

.

The purpose of this calculation is to show

(3.3)

«(4>® fJ)KaoFt (x,z, s, q, A, C»(u) _

- u

v) AE/1B II '7' J«(4>® l/J)K +( x, Yj' II(s,(s, v)II' 11(5, v)11 (u!l(s, v» o +( (-v, q) -BjEB C) . «(l/Jj ® fJ)K o» Yj' z, I/(v, q)II' II(v, q)1I (ull(v, q)/I) dv,

n '"

j)

F)

where n, here and below, is n y. It is understood here that s E Xx' q E Zz' I/(s, q)11 = 1, and v E Y y. In • this calculation we write L.J for L Yj . We also use the following notation: 1= 1(.) = 4> exp(-iu[(Lx., 5) - (ALx" Lx.)]) E gg(X), I~ = I~( . . , v) = l/Jj exp(-iu[(Lj . . , v) - (BjLj . . , L j . .)]) E gg(y) ,

IIi

= IIi(··,

v)

= l/JJ exp(-iu[(Lj .. , -v)- (-BjLj .. , L j . . ») E

III = III( .. .) = fJ exp(-iu[(Lz • • • , q)- (CLz ••

•,

gg(y),

L, ... )]) E gg(Z).

Hence F(I) E gg(y), G(II) E ~(Z), GT(III) E @'(Y).

w:

112

Ambrose / Operators and Their Symbols

If T is a distribution and e/> E g; we write, in this calculation, {THe/» instead of Te/> or T(e/», to help make clear which elements here are distributions. Thus in particular we write {G T (III)He/» instead of (GT(III)(e/> ). We also let P be the determinant of the Jacobian matrix of Lx' The calculation:

«e/>® O)KG oF r(x, z, s, q, A, C) =

J(G (2; r/JJ r/JjF(I) )(z')III(Z'») dz' J

=

{GT(III)}(L r/JJr/JjF(I») J

=

L {r/JJGT(III)Hr/JjF(I) j

=

L {(exp iu(Bj . . , . .»(r/JJ GT(III»y)«exp -iu(Bj . . , .. »( r/JjF(I»y) J

=

L {(exp iu(BjLj. . , Lj .. »r/JJGT(III)}y/(exp -iu(BjLj . . , L j.. »r/JjF(I)Yj j

=

L J [{«ex p iu(Bj .. , .. »r/JJGT(III)y)(exp -i( .. , u) I

.J«exp -iu(BjLj. . , Lj' .))r/JjF(I»Yj (exp i( w, u) d w] d v =

2; J [{«exp -iu(BjLj.. , i; .»r/JJGT(III»y)(exp -i( .. , v» I

.J«exp iu(BjL

j . "

=

u"

L J[{«ex p I

i; .»!/JjF(I»yj(w)(exp -i(w, e) dw ] dv

iu(BjLj, . , L j, .»r/JJ GT(III)y)(exp iu( . . , v»

W Ambrose / Operators and Their Symbols

= un

113

~ J [{(exp -iu(BjLj . . , L j . . »)I/JJGT(III)}(exp iu(Lj" . , v») )

.J«exp -iu[(Lj . . , V) - (BjLj . . ,Lj .. )])I/JjF(I)) eXpy/w) dw ] dv = un L J [{GT(III)}(I/JJ exp(-iu[(Lj .. , -V)- (-BjLj .. , L j . .)])) 0

}

.J(I/J/~F(I))oexpYj w dW] dv = un

~ J[f G(IIj( .. , V ))(Z')(I/JJIII(z', q)) dZ'] )

.[f ~(y')I~(y', v)F(I(., S))(y')1j(y') dY'] dv and this proves (3.3). We now prove (1). If F and G satisfy the second condition in the definition of L" then Go F satisfies the modified form of that condition. Fix any mER; we may suppose m < O. Let (x, z) E X x Z. With an m l to be determined later we consider, for each Y E 1'; R~l(X, y), 17~I(X, y) and R~I(Y' z), 17~I(Y' z ). Let K = F(clos(x, R~)). Choose for each Y E K a positive number R(y) such that «x, y), R~I(x, y)) d (x, R(y)) x (y, R(y)), «y, z), R~l(Y' z)) d (y, R(y)) x (z, R(y)) .

Choose Yl>'''' Yk such that (YI' R(YI))'" ., (Yk' R(Yk)) cover K. Let I/JI' ... , I/Jk be a ~~ partition of unity over K subordinate to the (y;, R(y;}) and choose, for each i, I/J~ in f»(y;, R(Yj)) with I/J~ = 1 on supp I/Jj' Define R~(x, z) = min R(y), Q~ = {(x', z', s', A', C')I d(x, x') < 17~(X, z), d(z,Z')<17~(X,Z), IIA'II
infj(inf{17~I(x, Yj)' 17~l(Yj' z)})z.

We now prove that the modified form of condition (2) holds for GoF and if we choose m l = min{m - N 1 - n - 1, m - N z - n - I}, where N 1 is any integer .> the order of all (c/J® ~)KF' and N z is any integer > the ) 01 order of all (I/J/lY 8)Ko, where t/> E f»(x, Rml(x, z)) and 8 E 0J(x, R~l(X, z)).

w:

114

Ambrose I Operators and Their Symbols

So let E 1, E z be bounded sets in 0J(x, R~I(x, y), 0J(z, R~I(Yj' z ) and let E lj = {C/>0l/Jjl c/> E E t }, E Zj = {l/Jj0 0/ E E z}. By (3.3),

°

«(C/>0 8)Ko oF

_

r (x', z', s', q', A', C'»(u)

(s', v) A'EBO)) II ' '7 J(«c/>0l/Jj )Kry )+(,x, Yj' lI(s', v)1/' lI(s', v)1I (u (s, v)lI)

n ""

- U

lj

+( Yj' z, II(v, (-v, q) 8EB C')) I I q')II' II(v, q')11 (u (v, q') ) dv.

(

. «l/Jj0 8 )K o )

This integrand is the product of two factors that we now denote by fl(u, v), fz(u, v) (our bounds will not depend on the other variables). Note that fl(U, v) = «c/>0l/Jj )Kry );,.yJexp(-iu[(. , s') +(.. , v) - (A'. , .)])) , so by (1.2), (3.4) and similarly for fz(u, v). We have then, for (x', z', s', q', A', C')E Q~l' the following inequalities (in which the v's are various positive constants), independent of x', z', s', q', A', C', C/>, 8 when these vary as described above. By (3.4) we have the desired bounds for 0,;;;: u ,;;;: 1 so we now consider only u > 1. We also treat only the case 1=0, the other cases being essentially the same. So we now suppose Ils'li < 77~l(X, z) = (77')z if we let 77' = infj(inf{77~I(x,Y),77~I(Yj'z)}), and hence Ilq'II;;:(1-(77't)l/z. Let 77"= 77'(1- (77't)lIz. We consider separately the regions 0,,;.;; Ilvll,,;.;; 77", 77",;;;: IIvll,,;.;; 1/77', IIvll> 1177'· If Ilvll> 1/77' then Ils'II/I/(s', v)II< 77', hence Ifl(u,

v)1 ,;;;: y(1 + ull(s', v)ll)m\,,;.;; y(1 + ullvllt

l

,

Ifiu, v)/ ,,;.;; y(1 + ull(v, Q')llt2 ";';; y(l + u(llvll + 1)t2 , hence if

nil

+ N z + n + 1 ';;;:0,

W. Ambrose / Operators and Their Symbols

IJ

!J(U, v)liu, v) dv

I::; J 'Y

II"II>I/~'

115

(1 + u/iv/l)m 1+N2 dv

1I"1I>1/~'

J

::;; 'Y

(1 + u)ml+~+n+I(I + /lvllr n - I dv

II"II>I/~'

::;; y(1

+ u)m l+N2+ n+ 1 •

If 0::;; IIvll::;; n" then IIv/i//I(v, q')11 < ",', hence

hence

I J

!J(u, v)/z(u, v) dv

I::; 'Y(I + ut

1 Nt

+

•

O""lIvll"'~'

If ","::;; I/vll::;; 1/1]' then

Ils'II/I/(s', v)11 < (1]'f//lvll::;; 1]'. Hence

1/1(u, v)/ ::;; 'Y(1 + u/l(s', v)lI)m l

::;;

'Y(I + uti,

hence

I J

II(u, v)lz(u, v) dv

I::; y(I + ur+N2.

~'''''II"II''''I/~'

These show that condition (2) is satisfied. We next prove (2). If F and G are in .20' then Go F satisfies the modified form of the first condition in the definition of .20" Let (x, z) E X x Z, A E R +, and P E IR - . These will remain fixed throughout this proof. We seek the Op(x, z) and then, for each (A, C) E Op(x, z), we seek the RpA(x, Z, A, C) required in the definition of

.2O'(X, Z).

Let PI E IR ", to be determined (in terms of p) below. Let K F(clos(x, R~». For y E K let

=

116

W Ambrose I Operators and Their Symbols

0Pj(X' y, z) = {(A, B,

C)/ A

E M(Xx), B E M(Yy ), C E M(ZJ,

(A, B)E 0Pt(x, y), (-B, C) E 0Pt(y, z)}.

For each

yEK

choose

(A(y), B(y), C(y» E 0P1(x, y, z).

Let

J=

[V2A, V2A]. For y E K define R(y) = min{R~:(x, y), R~~AY, z, -B(y), C(y»}.

K is covered by the balls (y, R (y» (y E K) so there exists a finite subset Yl' ... ,Yk of K such that K C U (y;, R (Yi We define O/x, z) (with PI still to be determined) by

».

Op(x, z) = {(A, C)/ for all i, (A, B(yJ) E 0Pj(x, z)

and

(-B(yJ, C)E 0Pt(y, z)}.

This Op(x, z) is open and dense in M(Xx) x M(Zz)' Now let (A, C) E Op(x, z ): we seek the needed RPA(x, z, A, C). For each y;,.let

O, = {(x', Yi' S, t, A', B(y;)/I/A - A'l/ < 1, lI(s, t)1/ = 1, d(x', x) < 1J~:(x, Yi)}' Choose M; E IR + such that if Iisl/ ~ 1 and

I/vll;;;:, M; then

{(x', Yi' AA'/I/(s, v)ll, AB(y;)/I/(s, v)II)1 (x', Yi' s/I/(s, v)l/, v/I/(s, v)l/, A', B(y;) E

OJ

Let

0'; = {(Yi' z', t. q, - B(y;), C')IIIC - C'II < 1 , l/(t,q)l/= 1,d(z,Z')<1J~:(Yi'Z)}.

Choose m. E IR+, with m, ~ 1, such that if

v1 ~ I/qll ~ 1 and Ilvll ~ m. then

{(y;, z', v/ll(v, q)ll, q/ll(v, q)ll, -AB(y;)/II(v, q)ll, AC'/II(v, q)/I)I (Yi' z', v/ll(v, q)l/, q/I/(v, q)lI, -B(y;), C')E O'a

~ O~:AYi' z, - B(Yi)' C) .

117

W Ambrose / Operators and Their Symbols

Choose M = max M i , m = min mi , L = [Am, AM]. Define R pA (x,

z, A, C) = m,in min{R(y;), I

RpIL(Y.

RpIL(X,

Y;, A, B(y;),

z, -B(y;), C)}.

Let {I/tJ be a C(;'~ partition of unity over K, subordinate to the covering I/I? E gg(Yi' R(y;) with I/I? = 1 on supp 1/1;. Now consider any 8 with 0 < 8 < RpA (x, Z, A, C), we shall show the existence of the O. demanded in the definition of £'O'(X, Z). Let 'O~ and l,A,B(Y;)II/(s',t')II=l} "Q~ be the neighbourhoods of {(x,y;,sl,t and {(Yi' z, t', q', - B(y;), C)III(t', q')11 = I} associated with R pA (x, Yi' A, B(y;), R pA (Yi' Z, - B(Yi)' C) and 8. We define {(y;, R(y;»} and let

O. = {(x', z', s', q', A', C')III(s', q')11 = 1 and (x', Yi' s', t', A', B(y;) E 'O~ n O~ for allll(s', t')11 = 1 and all i and (Yi' z, t', q','-B(y;), C') E "0: n for aIlIIU', q')11 = 1 and all i} .

o;

To prove that this O. does what is required, let E I be any bounded subset of gg(x, R pA (x, Z, A, C» and let E 2 be any bounded subset of gg(z, R pA (x, Z, A, C». Consider ¢> EEl' () E E 2 such that X = tg; () is 0 in (x, E) X (z, 8). Then either = 0 in (x, 8) or () = 0 in (x, 8) (or both). Suppose the former (the other case being similar). By (3.3), «(
= un

~ J((¢>tg; ~)Kfjt(x', Yj'II~::: ~~II' ~I~,~~j»)) (ull(s', v)lI) Yj

0

)

. ( «I/Ij tg; () KG)

+(Yj' z , lI(v, (-v,q') -B(Yj)EBC))( II( ')II)d q')/I' II(v, q')11 u v, q v. I

We write, as before, II(u, v) and 12(u, v) for the two factors in the integrand and have the following inequalities, uniformly for
118

W. Ambrose / Operators and Their Symbols

() E £2' such that if> = both) of

Case I:

a in

(x, to). Because

IIs'II;?; ~v'2,

II(s', q')11 =

1 we have either (or

Case II: IIq'II;?; ~v'2

.

The r's below represent various positive numbers. We only consider u ;?; 1 because other u are taken care of by (1.2). In Case I, if IIvll :s;M,

1/1(u, v)l:s; r(l + ull(s', v)IIYI:S; r(I + u~v'2Yl:S; r(I + Uyl, 1!2(U, v») :s; r(I + ull(s', v)llt2:S; r(I + ut

2

•

Hence in this case,

IJ

I

ft(ll, v)Nu, v) dv :s;r(I + uYI+N2 •

IIvll"'M

In Case I or II, if IIvll;?; M,

'/1(u, v)l:s; r(I + ull(s, v)II)PI:S; r(I + ullvllY':s; r(I + IIvlly/2(I + uy,/2, 1!2(1l, v)1 :s; r(I + llllvll)N:!:s; r(I + IIvllt2 • Hence

, J ft(u, v)Nu, v) dv I:s;r(I + u y l J(1 + IIvllYP l2

l/2)+N2

dv .

Ilvll"M

In Case II, if Ilvll:s; m,

I!I(U, v)1 :s; r(I + ull(s, v)Ii)Pl :s; r, 1/2(u, v)/:s; r(I + ull(v, q)IIYl:S; r(l + U~v'2Yl:S; r(I + uy'. Hence

I J 11(u,v)!2(U,v)dv) :S;r(I+uyl. IIvll"'m

W Ambrose / Operators and Their Symbols

119

In Case II, if m :;;; IIvll :;;; M,

v)IIYI :;;; y(1 + Ul1 yl :;;; C(1 + u yl , Ifz(u, v)l:;;; y(1 + ull(s, v)llt

INu, v)1 :;;; y(1 + ull(s,

2

Hence

I J

•

Nll,V)Iz(U,V)dV! :;;;y(l+uyl.

Derivatives with respect to II are bounded in the same way, hence Condition (1) holds, if we choose PI:;;;P - 2(Nz + n + 1) and PI:;;; p-2(NI + n + 1).

4. Multiplicative Properties of 2" and

U F'

Part 2

In the previous section we proved that if FE 2,,(X, Y) and G E 2,,( Y, Z), then Go F satisfies a modified form for being in 2,,(X, Z). We now show that if this modified condition holds then the full condition is satisfied. This section completes the proof of Theorem (3.1). In doing this we also establish bounds for w(KOoF ) and a formula for (J'OoF in terms of (J'o and (J'p We only consider the first condition in the definition of 2,,(X, Z) because the second condition is proved by a simplification of the same argument. So we now turn to the first condition. Let P E IR, we first need the existence of an open Op(x, z). This we define to be the Op(x, z) of the previous section. We have an R pA (x, Z, A, C) from the previous section that henceforth will be denoted by R~A (x, Z, A, C). We now choose the desired R pA (x, Z, A, C) to be any positive number less than R~A (x, Z, A, C). Next consider any E, 0 < E < R pA (x, Z, A, C). We must show the existence of a certain neighbourhood 0, of {(x, z, s, q, A, C)III(s, q)11 = I}. Henceforth all the preceding will be fixed (except the 0, that we seek). Let E be any bounded subset of ~«x, R pA (x, Z, A, C)) x (z, R pA (x, Z, A, C))) and we must find the 0, such that

(4.1) {(XKooFt(x', z', s', q', AA', AC')! (x', z', S', q', A', C')E 0" X = 0 in B(x, E) x B(z, E), X E B}

is a bounded subset of ZP.

120

W Ambrose / Operators and Their Symbols

Let Q~ be the neighbourhood chosen in the previous section such that the restricted condition of the definition holds for Q~. We now define Q = Q~I3' and will prove (4.1). We choose a function a in ~(Xx x Zz) with the following properties:

(4.2)

(i) a E ~«x, R~A (x, z, A, C)) x (z, R~A (x, z, A, C))) , (ii) a=Oin (x,~e)x(z,~e), (iii) a = 1 in (x, R pA (x, Z, A, C)) x (z, R pA(x, Z, A, C)) n «x, ~e) X (z, je)t , (iv) a = L bi x ci where, for each i. either bi = 0 in (x, ~e) or c' = 0 in (z, ~e), (v) each bi E 0J«x, R~A(X, z, A, C))) and each c' E 0J«z, R~A (x, Z, A, C))) .

;=,

Notation (4.1). In the remainder of this section we adopt the following conventions. X will be a function in 0J«x, R x) x (z, R z)) and f will be Xxz (the lift of X to Xx x Z, under the map expx.z)' Whatever ornaments (i.e. superscripts, subscripts, etc.) that X may have, the same! with the same ornaments is the lift of the corresponding X. We also let a = aoexp~.

We next define, for X E ~«x, R pA (x, Z, A, C)) x (z, R pA (x, Z, A, C))), a sequence of functions {X k } in QO«x, Rp(x, z, A, C)) x (z, Rp(x, z, A, C))). Each X k will be a sum of functions of the type considered in the previous section and we will prove the needed facts about X by approximation by the x k • The procedure here is a variant of that used in [4] to show that every bounded set E in QO(R m x R n) is in the closure of E, 0 E 2 for some bounded sets E, in ~(Rm) and E 2 in QO(lR n ) . Note that

(4.3)

if X = 0 in (x, e)

X

(z, e) then af = f.

For each k E I + consider the usual coverings of Xx' Z, by nets of closed cubes {C;}, {D:} of side lengths of So, denoting the volume of km a cube C by vol C, vol C~ = vol D~ = T (if X has dimension nand Z has dimension m). Choose (~E C;, (: ED:. For X E 0J«x, R pA (x, Z, A, C)) x (z, R pA (x, Z, A, C))) we define on Xx X z, by

z>.

z".

r

P.q

. e«()(exp(-i( (, (;») vol D:,

w. Ambrose / Operators and Their Symbols

121

where L~~ here and below, is the sum over all p, q such that I~:I ~ k and Ic;1 ~ k. The d and e here, and the a, b', c' below, are those chosen in (4.2). In the following, when we write zllzl, with z a complex number, we understand this to be 0 if z = O. Let

gk

'g:4(n =

=

at,

itu: C;)/lf(t;, C;)/)I!(~;, C;W

/2

j

d(g)bj(g) vol

"g~(C) = I!(g;, C;W/2e(C)c « ) vol so

gk = L: j

C; exp(-i(g, g;»,

D; exp(-i(C, C;»,

L:(k l('g:4 x "g:4). p,q

It is easily verified that if X runs through any bounded set in g; then the

l

(g obtained from f corresponding to X and k varying in I +) also run through a bounded set. And it is easily seen (and proved in [4]) that, as k ~OO,

(4.4)

t

~ fin g;, uniformly for X in a bounded set.

It follows from (4.3) and (4.5) that (4.5)

l

~ g in g; uniformly for X in a bounded set and

such that X

=

0 in (x, E) X (z, E).

It then follows from (4.3) and (4.4) that (4.6)

gk ~ g in g; if X = 0 in (x, E) X (z, E) .

Hence, for each fixed u E R +, and all (x', z', s', q', A', C') E l'(X, Y), (4.7)

«ax kK a oFt(x', z', s', q', A', C»(u) ~ «axKaoF t(x', z', s', q', A', C'»(u) ,

if X = 0 in (x, E) X (z, E). By (4.2.iv) and (4.2.v) we also know, for all k, that (4.8)

(axkKaoF )(x', z', s', q', AA', AC') E ZP,

W Ambrose l Operators and Their Symbols

122

for all (x', z', S', q', A', C') E Qe' We must prove that

(4.9)

{(XKO'F )(x', z', s', q', AA', AC')/ (x', z', s', q', A', C') E Q., X = 0 in (x, e) X (z, e), X E E}

is a bounded subset of ZP. To prove (4.9) it is clearly now sufficient to prove (4.10)

the (axkKooFt(x', z', s', q', AA', AC') form, as k ~ 00, a Cauchy sequence in ZP and this holds uniformly for X E E, X = 0 in (x, E) X (z, e), (x', z', s', q', A', C') E Qe'

To prove (4.10) we must prove (4.11)

for every e' > 0 and r E Z + U (0) there exists a k o = ko(e', r) such that, uniformly in X E E, (x', z', s', q', A', C') E Q., X = 0 in (x, e) x (z, e), IO'«axkKooFt(X', z', s', q', AA', AC') - (axIKooFt(X', z', s', q', AA', AC'»(u)1 ~ e'(1

+ uY',

for all u E R +, (x', z', s', q', A', C') E O, and k, l?:- ko' These follow in routine fashion from the following easy facts, proved in [4]: (4.12)

For each NEZ + and multi-indices a, f3 there exists a cNa /3 E R + such that for all k E Z+, X E B,

L IJ(~;, C;)(~;t(C~)/3I(vol C;)(vol D~) ~ cNa/3 (1 + I~;I + Ic~lrN. p,q

(4.13)

For all N, k E Z + and multi-indices a, f3 there exists a cNk a /3 E R + such that for all X E B

for all (~, C) and (f,

t) such that I(!f, C)- (f, t)1 ~ u:'.

W Ambrose / Operators and Their Symbols

123

References [I] [2] [3] [4] [5] [6] [7] [8] [9]

W. Ambrose, On the symbol of a distribution, Mathematical Analysis and Applications, Adv. in Math., Suppl. Stud. 7A (Academic Press, New York, 1981). J. Chazarain and A. Piriou, Introduction a la theorie des equations aux derivees partielies Iineaires (Gauthier-Villars, Paris, 1981). J. Duistermaat, Fourier integral operators (Courant Inst., New York, 1973). L. Garding and J. Lions, Functional Analysis, Novo Cimento 14, Supplement (1959). L. Hormander, Fourier integral operators I, Acta Math. 127 (1971). R. Melrose and G. Uhlmann, Lagrangian intersection and the Cauchy problem, Comm. Pure Appl. Math. (1979). F. Treves, Pseudo-differential and Fourier integral operators (Academic Press, New York, 1980). A. Weinstein, The order and symbol of a distribution, Trans. Arner. Math. Soc. 241 (1978). L. Garding, T. Kotake and J. Leray, Uniformisation et developpment asymptotique de la solution du probleme de Cauchy Iineare, Bull. Soc. Math. France 92 (1961).

l.A. BARROSO editor, Aspects of Mathematics and its Applications © Elsevier Science Publishers B.V. (1986)

125

DUALITY FOR THE SUM OF CONVEX FUNCTIONS IN GENERAL BANACH SPACES

Hedy ATrOVeH Dept. de Mathematiques, Uniuersite de Perpignan, 66025 Perpignan Cedex, France

Haim BREZIS Dept. de Mathematiques, Unioersite Paris VI, 75230 Paris Cedex 05, France Dedicated to L. Nachbin

O. Introduction

Let E be a Banach space. Let tp, ljJ : E ~ (-00, +00] be convex' functions. A well-known result due to Fenchel (in the finite dimensional case) and to Rockafellar [9] (in the general case) asserts that inf{cp (x) + ljJ(x)} + min{cp *( - f) + ljJ*(f)} = 0

(0.1)

xEE

fEE'

provided (0.2)

cp or ljJ is continuous at some point x o where both functions are finite.

(Assumption (0.2) is sometimes called a constraint qualification or a Slater stability condition.) Our purpose is to show that the same conclusion holds under the weaker and more geometrical assumption (0.3)

U A(Dom(cp)-Dom(ljJ)) is a closed vector space.

A__O

Next, we discuss some corollaries and we point out a connection with the well-known 'closed range Theorem' for linear operators.

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H. Attouch, H. Brezis / Sum of Convex Functions

Our work was motivated by several earlier results dealing with the sum of convex functions (or non-linear operators), in particular H. Brezis and A. Haraux [5], H. Brezis and L. Nirenberg [6], J.P. Aubin and I. Ekeland [3], J.P. Aubin [2], and H. Attouch [1]. We recall first some classical notations and definitions. Given a function q; : E -+ (-00, +00], the domain is Domfe) = {x E

EI q;(x) < +oo},

the conjugate function is q;*(1) = sup {(f, x) - .q;(x)}

for fE E*,

xED(",,)

the subdifferential is aq;(x) = {fE E*/ q;(y);;,; q;(x) +
The indicator function of a set K is

xEK,

xfl. K. The inf-convolution of two functions q;, l/J is: (q;Vl/J)(x) = inf{q;(x - y) y

+ l/J(Y)};

we say that the inf-convolution is exact if for each x there is some y such that (q;VrjI)(x)

=

q;(x - y) + l/J(Y).

1. The Main Result

Theorem (1.1). Assume and satisfy (0.3). Then

tp,

l/J : E

-+ (-00, +00] are convex l.s.c., q;,

l/J '¢ +00,

H. Attouch, H. Brezis / Sum of Convex Functions

(cp + 1/1)*

(1.1)

= cp *V 1/1*

127

on E*

and, moreover, the inf-convolution is exact. Remark (1.2). It is obvious that

(cp + 1/1)* ..:;; cp *V 1/1*

on E*

even without assumption (0.3). One says that a duality gap occurs if

(cp + 1/1)* < cp*VI/I*

somewhere.

We recall a simple example with a duality gap. Let E = R

cp(X) =

{

-vi x,x 2 +00

2

, X

= (XI'

xz),

on the set {Xl ~ 0, X 2 ~ O} , otherwise.

So that (cp + 1/1)* = I/xz"'o} , while cp*VI/I* = I/ x2
Remark (1.3). It is clear that assumption (0.3) is weaker than assumption (0.2). However, the conclusion of Theorem (1.1) holds if E is a locally convex Hausdorff space and tp, 1/1 are any convex functions satisfying (0.2) (we emphasize that lower semicontinuity is not required). It is easy to see that, in general, the conclusion of Theorem (1.1) fails if E is a (noncomplete) normed space or if one of the functions tp, 1/1 is not l.s.C. Proof. We divide the proof of Theorem (1.1) into 3 steps. Step 1. We claim that U A"O A (Dom(cp ) - Dom(I/I» = E => cp*VI/I* is l.s.c. on E* for the w*-topology. Indeed let j..t E R and let C = {f E E*I (cp *VI/I*)(I) ~ j..t}. We have to show that Cis w*-closed. For each e > 0 we consider the convex set

CE It is clear that

=

{g+ hE E*I cp*(g) + I/I*(h)":;;j..t + s},

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H. Attouch, H. Brezis / Sum of Convex Functions

c= £>0 n c£ and therefore it suffices to prove that for each e > 0, C£ is w *-closed. If we establish that (1.2)

C£

n rBB' is w*-closed for each

r>

°,

it will follow from the Banach-Dieudonne-Krein-Smulian theorem (see e.g. [7, Th. V.5.7]) that C£ is w*-closed. We consider the sets K= C£ nrBE • =

{g+ hE E*I cp*(g) + l/J*(h)~f.L + e and IIg+ hll~ r}

and H

=

{(g, h) E E* x E*I cp*(g) + l/J*(h) ~ f.L

+ e and Ilg + hll ~ r}.

We claim that H is bounded in E* x E*. In view of the uniform boundedness principle it suffices to check that for each element (x, y) E E x E there is a constant C(x, y) such that (g, x)+ (h, y)~ C(x, y)

(1.3)

(for all (g, h) E H) .

Using the assumption U A"O A(Domte) - Domte) = E we write x - y = A(u - v) for some A;;;. 0,

u E Domfe ), v E Domte},

and then (g, x) + (h,

y) = A(g,

u)

+ A(h, v) + (g + h, y - AV)

~

A«(jO * (g ) + (jO(u) + l/J*(h) + l/J(v» + IIg + hlilly - Avll

~

ACJL + e + (jO(u) + l/J(v» + rlly - Avll = C(x, y).

Therefore H is bounded in E* x E*. On the other hand, H is clearly w*-closed and thus H is w*-compact. Finally we note that K = T(H) where the mapping T : E* x E* -+ E* is

H. Attouch, H. Brezis / Sum of Convex Functions

129

defined by Ttig; h» = g + h. Since T is continuous for the w*-topologies it follows that K is w*-compact and the proof of (1.2) is complete. Step 2. We claim that

U A (Dom(cp ) - Dom(l/J» = E:? (cp + l/J)* = cp*Vl/J* on E* ,\;.0

and, moreover, the inf-convolution is exact. Indeed we always have (cp*Vl/J*)* = cp** + l/J** = cp + l/J and thus (cp + l/J)* = (cp*Vl/J*)** = clos(cp*Vl/J*), where the closure is in the w*-topology. We know from Step 1 that cp*Vl/J* is I.s.c. for the w*-topology and consequently

Finally, we see that for each fE E* inf {cp *(g) + l/J*(f - g)}

gEE'

is achieved. Indeed the argument used in Step 1 shows that for each constant J-L, the set {g E E*I cp*(g) + I/I*(f- g) ~ J-L} is bounded. In the final step we shall use the following simple Lemma (1.4). Let A be a convex set in a vector space E. Assume U ,\;.0 AA is a vector space. Then 0 E A and thus U ,\>0 AA = U ,\;.0 AA. Proof. Let a E A; we may write -a = Xb for some A ~ 0 and some bE A. Then 0 = 1/(1 + A)a + A/(1 + A)b E A. D

Step 3. (Proof of Theorem (1.1) concluded.) It follows from assumption (0.3) and Lemma (1.4) that Dom(cp) n Domte) =I- 0. After a translation we may always assume that 0 E Dom(cp) n Dom(l/J). Set F = U ,\;.0 A (Dom(cp) - Dom(I/I» so that Dom(cp) C F and Domre) c.F. Let j: F ~ E be the canonical injection, so that j* : E* ~ F*. We consider the functions tp'; l/J' : F ~ (-00, +00] defined by cp' = cp 0 j and 1/1' = l/Jo j. Note that U ,\>0 A (Dom(cp')- Dom(I/I'» = F and thus we may apply Step 2. It follows that (1.4)

on F*,

130

H. Attouch, H. Brezis I Sum of Convex Functions

and moreover the inf-convolution is exact. On the other hand we clearly have for IE E*

cp*(I) = cp'*(j*(I»,

l/J*(I) = l/J' *(j* (I» ,

(1.5)

(cp + t/J)*(I) = (cp' + l/J')*(j*(I» ,

(1.6)

(cp*Vl/J*)(I) = (cp' *Vl/J' *)(j * (f) .

(In the proof of (1.6) we use the fact that j* is onto.) Combining (1.4), (1.5), (1.6) we obtain (1.1). 0 Remark (1.5). We emphasize that Theorem (1.1) holds in general Banach spaces. In the special case of a reflexive Banach space the proof is slightly simpler, in particular it is not necessary to invoke the BanachDieudonne-Krein-Smulian theorem.

2. Some Applications Corollary (2.1). Let cp and t/J be as in Theorem (1.1). Then (2.1)

Dom«cp + l/J)*) = Dom(cp*) + Dom(l/J*)

and (2.2)

Proof. It is obvious that Dom(cp*) + Dom(l/J*)CDom((cp + l/J)*), even without assumption (0.3). Let f E Domte + t/J)*. We deduce from (1.1) that (cp*Vl/J*)(f) < 00 and thus IE Dom(cp*) + Dom(t/J*). It is obvious that acp + al/J C a(cp + l/J), even without assumption (0.3). Conversely let IE a(cp + l/J)(x), so that (2.3)

(cp + l/J)(x) + (cp + l/J)*(I) =


Using Theorem (1.1) we find some g E E* such that (2.4)

(cp + l/J)*(I) = cp *(1 - g) + l/J*(g) .

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H. Attouch, H. Brezis / Sum of Convex Functions

Combining (2.3) and (0.3) we see that ",(x) + t/J(x) + '" *(f - g) + t/J*(g) = (f - g, x) + (g, x)

and therefore

f-

g E a",(x), g E at/J(x). Hence f E a",(x) + at/J(x).

0

Corollary (2.2). Let tp : E ~ (-00, +00] be convex l.s.c., '" ~ +00, Let Me E be a closed vector space. Assume that U A"'0 A (Dornfe ) - M) is a closed vector space. Then inf ",(x) + min", *(f) = 0 .

xEM

fEM~

Proof. Apply (Ll) at 0 with t/J

=

1M , 0

Corollary (2.3). Let X, Y be Banach spaces and let T E .f£(X, Y). Let '" : X ~ (-00, +00] and t/J: Y ~ (-00, +00] be convex l.s.c., tp, t/J ~ +00. Assume that U A [Dom(t/J)- T(Dom(",»]

(2.5)

= Y.

A ",0

Then (2.6)

inf {",(x) + t/J(Tx)} + min{", *(- T* f) + t/J * (f)} = 0 .

xEX

fEY'

Proof. Let E = X x Y and let (/) : E

~

(-00, +00] be defined by

",(x) + t/J(y). Let M = Graph(T). We claim that

(2.7)

U

A;OO

(/)«x,

y»

=

A (Dom«(/) - M) = E.

Indeed, given [x, y] E E we may write (using (2.5) and Lemma (1.4», y - Tx = A(v - Tu), for some A> 0, v E Dom(t/J), u E Dom(",). Then we have x = A(u - a),

y=A(v-Ta),

with a = u - (xl A), and thus (x, y) E U A"'O A (Dom( (/) - M). Therefore we may apply Corollary (2.2) (to (/) and M) and deduce that

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H. Attouch, H. Brezis / Sum of Convex Functions

inf cP + min cP* M

M.J.

= O.

This is just (2.6). D Corollary (2.4). Assume M, NeE are closed vector spaces such that M

+ N is closed.

Then MJ. + NJ. is closed.

Proof. Set cp tp

= 1M , I/J = IN and note that

+ I/J = I MnN'

Using Theorem (1.1) we see that M + N closed ~ (M n Nl = MJ. + NJ.. 0

Remark (2.5). It is known (see [8, Th. 4.8] or [4, Th. 11.15]), that M

+ N closed ¢:> MJ. + NJ. closed.

Moreover, this statement is equivalent to the 'closed range Theorem', that is, Range(T) closed ¢:> Range(T*) closed, for any operator T E 5£(X, Y) (see [8], [4]). Such a result suggests the following question: Question. Assume tp, I/J : E can one conclude that

~

(-00, +00] are convex l.s.c.,

(cp*+ I/J*)*

=

tp,

I/J

¢

+00. When

cpVI/J on E?

In view of Theorem (1.1) the answer is clearly positive if E is reflexive and

H. Attouch, H. Brezis I Sum of Convex Functions

(2.8)

133

U ;\ (Domte *) - Dom(l,lJ*» is a closed vector space in E*.

A>O

In a general Banach space assumption (2.8) is not sufficient to conclude that on E. This may be seen on the following example. Let E be a (non reflexive) Banach space, let B = {x E Elllxli ~ I}. Let He E be a closed hyperplane in E such that 1 = inf zEH IIzll is not achieved; it is standard (and easy) to construct such an example. Let Ip = Id B and l,lJ = Id w Then Domte ") = E*-so that (2.8) holds-and nevertheless

since IpVl,lJ = Id B +H is not I.s.c. (note that 0 ~ B + Hand 0 E clos(B + H». It is however tempting to look for a statement of this kind which would include the 'closed range Theorem' in its full generality.

Added in proof After this work was completed we have been informed by R.T. Rockafellar that, in the special case where E is reflexive and U A>O A(Dom(Ip)- Dom(l,lJ)) = E, our Theorem (1.1) can be derived from Theorems 17 and 18 in [10]. References [1] H. Attouch, On the maximality of the sum of two maximal monotone operators, Nonlinear Anal. 5 (1981) 143-147. [2] J.P. Aubin, Mathematical Methods of Game and Economic Theory (North-Holland, Amsterdam, 1979). [3] J.P. Aubin and 1. Ekeland, Estimates of the duality gap in non convex optimization, Math. Oper. Res. 1 (1976) 225-245. [4] H. Brezis, Analyse fonctionnelle (Masson, 1983). [5] H. Brezis and A. Haraux, Image d'une somme d'operateurs monotones et applications, Israel J. Math. 23 (1976) 165-186. [6] H. Brezis and L. Nirenberg, Characterizations of the ranges of some nonlinear operators and applications to boundary value problems, Ann. Scuola Norm. Sup. Pisa 5 (1978) 225-326. [7] N. Dunford and J.T. Schwartz, Linear Operators Part I (Interscience, New York, 1964). [8] T. Kato, Perturbation theory for linear operators (Springer, Berlin, 1966). [9] R.T. Rockafellar, Extension of Fenchel's duality theorem for convex functions, Duke Math. J. 33 (1966) 81-90. [10] R.T. Rockafellar, Conjugate duality and optimization, SIAMlpubl., Conf, Board of Math. Sci. Ser. 16 (1974).

l.A. BARROSO editor, Aspects of Mathematics and its Applications © Elsevier Science Publishers B.V. (1986)

135

VARIATIONAL INEQUALITIES REVISITED

Jean-Pierre AUBIN International Institute for Applied Systems Analysis, Laxenburg, Austria

To Leopoldo Nachbin

Let K be a closed convex subset of a reflexive Banach space X and A be a set-valued map from K to X* satisfying

(Ll)

A is finitely upper sernicontinuous' with non-empty closed convex bounded images.

Our purpose is to solve variational inequalities (or generalized equations) (1.2)

x'EK,

oE A(x') + NK(x') ,

where NK(x) = {p E X*I SUPyEK (p, x - y);a. O} is the normal cone to K at x E K, by balancing (a). the lack of boundedness of K, measured by its 'barrier cone' b(K), defined by (1.3)

b(K) = {p E X*/ sup(p, x) < +oo} xEK

(because the larger b(K), the lesser is K unbounded), I The finite topology on a convex subset N of a vector space is the topology for which the maps 13K from the simplex S· = {A E R ~I L 7= 1 Aj = 1} to N defined by

f3K(A) =

•

2: AjXj,

i-I

are continuous for all finite subsets K = {x], ... , x.} of N. It is stronger than any vector space topology and any affine map is continuous for the finite topology [2, Section 7.1.3]. A finitely upper semicontinuous map from K to X* is an upper semicontinuous set-valued map from K supplied with the finite topology to X* supplied with the weakt-topology. When A is finitely upper semicontinuous, then the map x --+infuEA(x) (u, x - y) is lower semicontinuous on K for the finite topology [2, Section 13.2.4].

.L-P. Aubin / Variational Inequalities Revisited

136

(b). with the degree of monotonicity of A, measured by a non-negative proper lower semicontinuous function f3 from X to R U {+oo} satisfying (1.4)

V(x,p),(y,q)EGraph(A),


We shall say that such a set-valued map A is f3-monotone. We denote by f3* its conjugate function? For instance, we can take (and thus, f3* =

f3(z) = 0

Dom f3* = {O}) 3,

(and thus, f3* = l/!B.' Dom f3* = B*)4,

(1.5) f3(z) = [z]

1 f3(z)=-lIzll a

l/!{Ol'

a

( andthus,f3*=-/I-Ila',-+-=l,Domf3*=X* 1 1 1 ). a*

a

a*

In the following theorems, we shall measure the degree of monotonicity of A through the size of the domain of f3 *; the larger Dorn f3 *, the more 'monotone' is A. Theorem (1.1). We posit assumptions (1.1). Assume that A is f3-monotone and that (1.6)

oE Int(b(K) + A(K) + Dom f3*).

Then there exists a solution x' E K to the variational inequality 0 E A(x') + NK(x') (i.e. a solution of (1.2».

2 The

conjugate function 13* of a function 13 : x

.... R U {+oo} is defined on X*

f3*(p) = sup {(p, x) -

by

13 (x)} .

xEX

A function 13 is convex and lower semicontinuous if and only if Fenchel inequality (p, x),,;;; 13 (x)

13 = 13*.

It satisfies the

+ f3*(p).

3 The indicator of a subset K is the function I/JK defined by I/JK(X) = 0 when x E K and I/JK (x) = +00 if not. • B * denotes the unit ball of the dual.

l.-P. Aubin / Variational Inequalities Revisited

137

Assumption (1.6) shows how the lack of boundedness of K is compensated by the degree of monotonicity of A. We point out that (1.6) is satisfied when one of the following instances is satisfied:

K is bounded (b(K) = X*) , A is surjective (A(K) = X*), (1.7)

A satisfies (1.4) with f3(z) =

~ lizII", c > 0, a> 1 , a

A satisfies (1.4) with f3(z) = c1lzll, A(K) n -b(K) ~

c >0 and

0.

Naturally, these examples are known [3], [9], [7]. The novelty lies in the introduction of the function f3 as a parameter in assumption (1.6). We recall that NK(x), the normal cone to K at x, is the subdifferentiaf of the indicator !/JK' Therefore, variational inequalities are particular cases of inclusions of the form (1.8)

IE

A(x') + aV(x') ,

when V: X ~ R U {+oo} is a proper lower semicontinuous convex function and A is a set-valued map from the Banach space X to X*, which were studied by Brezis-Haraux [5], when A is maximal monotone, for solving Hammerstein equations (see [4]). We shall extend Theorem (1.1) to this case. To this end, we assume once and for all that (1.9)

Dom VCDomA.

We observe that a necessary condition for the existence of a solution x' to (1.8) is that (1.10) 5 The

IE Dom

V*+ A(Dom V).

subdifferential of a convex function V is the subset av, JV(x)={pEX*I(p,x)= V(x)+ V*(p)},

of gradients of the affine functions x -+ (p, x) - V*(p) below V and passing through (x, V(x». When V is Gateaux-differentiable at x, then aV(x) = {VV(x)}. The set of points x E X for which aV(x) ¢ 0 is dense in Dom V.

f.-P. Aubin / Variational Inequalities Revisited

138

We shall prove that this condition is 'almost sufficient'. Theorem (1.2). We posit assumptions (1.1) and assume moreover that A is {3-monotone. Then there exists a solution x' to the inclusion (1.8) when

(1.11)

f

E Int(Dom V*

+ A(Dom V) + Dom {3*) .

Remark (1.3). The size of Dom {3* balances the interiority condition in assumption (1.11), as the following corollary shows. Corollary (1.4). We posit assumptions (1.1). (a). If A is monotone (i.e. {3 = 0), then (1.12)

Int(Dom V* + A(Dom V)) C Range(A + aV) C Dom V*

+ A(Dom V).

(b). If there exists a c > 0 such that

(1.13)

V(x, p), (y, q) E Graph(A), (p - q, x - y);a. cllx -

yll,

then (1.14)

Range(A + aV) = Dom V* + A(Dom V).

(c). If there exists a c

(1.15)

> 0 and an

£I'

V(x, p), (y, q) E Graph(A),

> 1 such that (p - q, x - y);3 ~lIx £I'

yll" ,

then (1.16)

Range(A + aV) = Dom V* + A(Dom V)

= X* .

Before proving Theorem (1.2), we shall characterize Problem (1.8) by equivalent problems. For that purpose, we associate to the function ~ to the map A and to an element f E X the function C/J defined on Dam V by: (1.17)

C/J(y) = V(y)+ inf {V*(f-u)-(f-u,y)}. uEA(y)

139

l.-P. Aubin / Variational Inequalities Revisited

We observe that

v YE Dom V,

(1.18)

4>(y) ;;. 0 ,

since, for all uEA(y), V(y)+V*(f-u)-(f-u,y);;.O, thanks to the Fenchel inequality. We can also characterize the set-valued map A by the function y defined on Dom A x Dom A by

y(x, y)

(1.19)

=

inf (p, x - y).

pEA(x)

Proposition (1.5). Assume that the images A(x) are non-empty, closed, convex and bounded for all x E Dom v: The following problems are

equivalent: (i) 3x' E Dom V such that f E Ax' + aV(x'). (ii) 3p' E Dom V such that f E p' + A(aV*(p'». (iii) 3x' E Dom V such that Vy E Dom V, (1.20)

y(x', y) - (f, x' - y) + V(x') - V(y):s;; O. (iv) 3x' E Dom V such that 4>(x') = 0 (= min 4>(y». yEDom V

Proof. Let x' be a solution to (1.20i). Then there exists a p' E aV(x') such that f - p' E Ax' C A(aV*(p'». Conversely, let p' be a solution to (1.20ii). Then there exists an x' E aV* (p') such that f E p' + Ax'. Since p' E aV(x'), f belongs to aV(x') + Ax'. Let x' be a solution to (1.20i). There exists a u' E A(x') such that f E aV(x') + u', i.e. such that

(u', x'> y) - (f, x' - y) + V(x') - V(y):s;; 0

(Vy E Dom V).

By taking the infimum on A(x'), we deduce inequalities (1.20iii). Inequality (1.20iii) can be written sup

yEDom

inf {V(x)-V(y)-(f-u,x-y)}:S;;O. v uEA(x')

Since Dom V is convex, A(x') is convex and weakly compact. The

l.-P. Aubin / Variational Inequalities Revisited

140

lopsided-minimax-theorem now implies that inf

sup {V(x')-V(y)-(f-u,x'-y)}

uEA(x') yEDom V

= inf

{V(x')+ V*(f - u)- (f - u, x')} =
uEA(x')

Hence
This is equivalent to saying that

o

f-

u' E aV(x'), i.e. that x' solves (1.20i).

The equivalence between (1.20i) and (1.20iv) allows to interpret the solutions to problem (1.8) as solutions to a minimization problem (minimization of the functional cf» and provides a variational principle. The equivalence between (1.20i) and (1.20iii) allows to solve problem (1.8), (and, in particular, variational inequalities) by applying minimaxinequalities to the function defined by (1.21)

cf>(x, y) = y(x, y) - (f, x - y) + V(x) - V(y).

We observe that (1.22)

'fIx,

y ~ cf>(x, y) is concave,

'fly,

cf>(y, y) = 0,

that cf> is 'monotone' in the sense that (1.23)

'fix, y Dom(V),

cf>(x, y) + cf>(y, x)

~ 0,

and that (1.24)

'fly E X,

x ~ cf>(x, y) is lower semicontinuous for the finite topology. (Cf. the footnote on page 135.)

J-P. Aubin I Variational Inequalities Revisited

141

Therefore, if Dom V were compact, we could apply the generalization of the Ky Fan inequality [8] due to Brezis-Nirenberg-Stampacchia [6], which would imply the existence of a solution x' E Dom V to the inequalities (1.20iii), i.e. a solution x' to problem (1.8). When Dom V is not compact, we shall prove by approximation that assumption (1.11) is sufficient for the existence of a solution to inequalities (1.20iii). Proof of Theorem (1.2). We set K; = {x E Dom VI V(x):so; nand Ilxll:so; n}. The subsets K; are weakly compact and convex and Dom V= U:~I K;

because X is reflexive. Since K; is weakly compact and convex, Ky Fan's inequality for monotone functions implies that, for all n ;;;. 1, there exists an xn E K n , solution to (1.25) thanks to properties (1.22), (1.23) and (1.24). We shall now use assumption (1.11) for proving that x, remains in a weakly compact subset of X. For that purpose, thanks to the. uniformboundedness theorem, it is sufficient to prove that (1.26)

'tJpEX*

3n(p) such that sup (p,xn)<+oo. n:bn(p)

By assumption (1.11), there exist 'TJ > 0, r E Dom 13 *, q E Dom V*, y E Dom V, u E A(y) such that 'TJp

f + liPII = r + q + u .

(1.27)

We choose n(p) to be the smallest n such that y E Kno By taking the duality product with x n we get (1.28)

'TJ IlpH (p, xn) =

(r,

x; -

y)+ (q, xn)+ (u, xn - y)

- (f, xn

-

y) + (r + u -

f, y)

0

We use Fenchel's inequalities (r, xn - y):so; f3(xn - y) + f3*(r) and (q, xn):so; V(xn) + V*(q). We obtain

142

J.-P. Aubin / Variational Inequalities Revisited

(1.29)

1]

IIpil (p, Xn ) ~ (u, Xn -

y) + V(Xn ) - V(y)- (f, Xn - y) + f3(Xn

-

y)+ f3*(r) + V*(q)

+ V(y) + (r + u - f, y) . Since A is f3-monotone, we deduce that

Therefore, inequality (1.29) becomes 1]

IIpll (p, xn ) ~ (y(xn , y) -

(f, xn - y) + V(xn ) - V(y»

+ f3*(r) + V*(q)+ V(y)+(r+u-f,y). Consequently, for all n ;;;: n (p), we deduce from (1.25) that (1.30)

(p, xn )

Ilpli (f3*(r) + V*(q)+

~1]

V(y) + (r+ u - t. y».

The right-hand side is finite because r E Dom f3 ", q E Dom V* and y E Dom V. Hence the sequence is bounded and thus weakly relatively compact. So, a subsequence of elements x n converges weakly to some x' EX. Since V is lower semicontinuous, we deduce from the monotonicity of A and from the variational inequalities that V(X') ,;;;; Iiminf V(xn ) n

,;;;; Iiminf {(V(y) +
-

n

~ limsup n

{V(y) + (f,

Xn -

y) + y(y, x n

»- y(y, x

y) + y(y, x n ) }

,;;;; V(y) +
n) -

y(xn , y)}

f.-P. Aubin / Variational Inequalities Revisited

143

Therefore, x' belongs to Dom V and (1.31)

Vy E Dom V, O:s; cP(y, X').

We deduce from properties (1.22) and (1.23) that (1.32)

vz E Dom

V,

cP(x ', z):s; O.

Indeed, if this conclusion is false, there would exist a z E Dom V such that 0 < cP(x', z) and by (1.24) there would exist a t' E (0,1) such that 0< cP(x ' + (I(Z - x'), z). By taking y = x' + t'(z - x'), inequality (1.31) implies that O:s; cP(x' + (I(Z - x'), x'). Hence, the concavity of cP with respect to the second variable yields that 0 < cP(x ' + t'tz - x'), x' + (I(Z - x')), a contradiction to (1.22ii). Now Proposition (1.5) implies that the solution x' of (1.32) is a solution to the problem (1.8). 0 References [1]

[2] [3] [4]

[5] [6]

(7] [8] [9]

H. Attouch, On the maximality of the sum of two maximal monotone operators, Nonlinear Anal. 5 (1981) 143-147. J.-P. Aubin, Mathematical Methods of Game and Economic Theory (North-Holland, Amsterdam, 1979). H. Brezis, Equations et inequations non lineaires dans les espaces vectoriels en dualite, Ann. Inst. Fourier 18 (1968) 115-175. H. Brezis and F. Browder, Nonlinear integral equations and systems of Hammerstein type, Adv. in Math. 18 (1975) 115-147. H. Brezis and A. Haraux, Image d'une somme d'operateurs monotones et applications, Israel J. Math. 23 (1976) 165-186. H. Brezis, L. Nirenberg and G. Stampacchia, A remark on Ky Fan's minimax principle, Boll. Un. Mat. Ital. 6 (1972) 293-300. F. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Proc. Symp. Pure Math. 18 (2) (Amer. Math. Soc., Providence, RI. 1976). Ky Fan, A minimax inequality and applications, In: Inequalities 3 (Academic Press, New York, 1972) 103-113. J.-L. Lions, Quelques methodes de resolution de problernes non lineaires (Dunod, Paris, 1969).

1.A. BARROSO editor, Aspects of Mathematics and its Applications © Elsevier Science Publishers B.V. (1986)

KOROVKIN CLOSURES IN

145

~o(X)

Heinz BAUER Mathematisches Institut der Universitiit Erlangen-Niimberg, Erlangen, Fed. Rep. Germany

Klaus DONNER Fakultiit fur Mathematik und Informatik der Universitiit Passau, Passau, Fed. Rep. Germany

Dedicated to Leopoldo Nachbin

o.

Introduction

In a preceding paper [1] we have characterized the Korovkin closure Kor(:it) of a linear subspace :it in the space ~o(X) of those continuous real-valued functions on a locally compact space X which vanish at infinity. We proved that the upper envelope j of a function f E ~o(X), defined by j(x) = sup inf{h(x)1 h E:it, h ~ f .>0

d

(x E X),

and the corresponding lower envelope

J= -i-r. lead to several, highly satisfactory characterizations of Kor(:it) in spite of the fact that j attains the value +00 in general. These envelopes will be called here the free upper and lower envelope, respectively. In [1] Kor(:it) was identified with the space 'Ie of :it-affine functions on X. A function f E ~o(X) is called :it-affine if the two envelopes j and! coincide. The aim of the present paper is twofold. First we intend to define a family of envelopes M and!M which depend on a real parameter M ~ 1. In contrast to the fact that the free envelopes may attain the value +00 or -00, these envelopes are real-valued. We call them the M-contractive

1

146

H. Bauer, K. Donner / Korovkin Closures

upper and lower envelopes. While f~ !(x) is a lower semicontinuous hypolinear (= sublinear with values in (-00, +00]) form on !(6o(X),f~ M (x) is a continuous sublinear form for each x E X. So, analytically, the M -contractive envelopes behave much nicer. In particular, the classical Hahn-Banach theorem can be applied to them. There is an intimate connection with the free envelopes which allows new approaches to results of our preceding paper. In fact, it turns out that the free envelopes correspond in a natural way to the parameter M = +00 : I is the increasing limit of the 1M for M ~ +00 and for all f E !(6o(X). Secondly, we show that each M;;=: 1 leads to a Korovkin closure Kor M(JYe), namely to the space of all f E !(6o(X) such that (Tal) converges uniformly to f (on X) whenever (Ta ) is a net of positive linear operators on !(6o(X) of norm ~M such that (Tah) converges uniformly to h for all functions hE JYe. For M> 1 we show that KorM(JYe) equals the set ieM of JYe-affine functions of order M which is the set of all f E !(6o(X) satisfying

1

1 =1 M

M

•

However, for M> 1 this Korovkin closure is independent of M and is equal to the free Korovkin closure Kor(JYe) (M> 1).

So only the case M = 1, called the l-contractive case, needs further treatment. We determine Kor 1(JYe) and find a new approach to results of Flosser [4]. In contrast to [4] we stick to the framework of spaces !(6o(X). However, our methods are flexible enough to treat also more abstract situations (see [3]).

l. M -contractive Envelopes

In what follows we shall denote by X an arbitrary locally compact space (always assumed to be Hausdorff), by !(6o(X) the linear space of continuous real-valued functions on X vanishing at infinity, and by JYe an arbitrary linear subspace of !(6o(X). We shall consider !(6o(X) as a normed space with respect to the usual sup-norm 11·11. For functions g E !(6o(X), g + stands for positive part of g. For an arbitrary real number M;;;.o 1, we will associate to any f E !(6o(X) a new function

H. Bauer, K. Donner / Korovkin Closures

147

(upper semicontinuous, but discontinuous, in general) as follows: (1.1)

jM (x) = inf{h (x) + MI/(f - h till h E ~} ,

for x E X. This function will be called the M -contractioe upper envelope of f (with respect to ~). Correspondingly, the M-contractive lower envelope of I is defined by (1.2)

1

M

=

-i-It'.

It is immediate that these envelopes are real-valued functions on X since

(1.3)

h + MI/(f - h til ;a. h + (f - h

t ;a. I·

The map I ~ jM defined on <:go(X) possesses the foIlowing elementary properties (f, g always in <:go(X»: (1.4) (1.5)

(1.6) (1.7)

1

M

/'.....

A

(f+g)M~/M+g

M

,

(ij)M = AjM is bounded with sup-norm ~MIlt1I. More generally,

(l.8) Consequently, for x E X and M;a. 1,

is a continuous, increasing, sublinear form on <:go(X). These properties are almost evident from (1.1) to (1.3). The proof can be left to the reader. In the case IE ~ one can choose h = I in (1.1). In view of (1.4), we

148

H. Bauer, K. Donner I Korovkin Closures

then obtain (1.9)

f

for all h E 'Jt .

It is also evident that the map M ~ jM (defined on [1, +00) for all E ~o(X)) is increasing. The pointwise supremum

(xEX)

l.

turns out to be the free upper envelope X ~ (-00, +00] which stands in the centre of our paper (1] and which is defined as follows: (1.10)

j(x) = sup inf{h(x)/ h E 'Jt, h ;;;. f - s}. .>0

In other words, we claim: Proposition (1.1). For every function f E ~o(X), the M -contractiue upper envelopes of f converge pointwise on X to the free upper envelope of f as M~+oo:

(1.11)

sup M"J

l" = j.

Proof. Let us show first that jM ,;;;; j for all M;;;. 1. The condition h ;;;. f for h E 'Jt is equivalent to (f - h e and hence to

r ,; ;

-

e

11(f-hn';;;;e. We may assume) that for every e > 0 there exist functions h E 'Jt satisfying h ;;;. f - e, because otherwise j(x) = +00 for all x E X. Consequently, for every e > 0 and h E 'Jt satisfying h ;;;. f - e, we have h

+ MII(f- hn,;;;; h -r Me ,

which implies, for x E X, 1 In the terminology of [I)-to be repeated after Proposition (2.2)-this means that almost .?t-majorized.

f is

H. Bauer, K. Donner I Korovkin Closures

1M(X) ~ inf{h(x)1 hE 'X, h ~ f -

149

e}+ Me

~/(x)+ Me.

1

This proves M ~ j So, for the converse, we may assume that for the given point x E X sup 1M (x) < +00 M;>l

holds. We choose ex E IR such that sup 1M (x) < ex < +00, M;»

and, for a given e > 0, a real number M> 1 such that a - f(x) < e(M - 1). For functions h E 'X satisfying h(x) + Mjl(f -

1I(f- h til> e, we then

htll > h(x)+ I/U ~ f(x) + ex

have

htll + a - f(x)

- f(x) = ex ,

according to (1.4) for the l-contractive case. Consequently,

1

M

(x)

= inf{h(x) + MII(f ~ inf{h(x)1 h E

htlll h E 'X, IIU- h rII ~ e]

'X, IIU - htll ~ s},

which implies sup 1M (x) ~ inf{h(x)1 hE 'X, IIU - htll ~ s},

M;>I

for every

E

> O. According to the definition of I this yields sup 1 M (x) ~ I(x).

M;>l

The equality (1.11) now follows. 0

150

H. Bauer, K. Donner / Korovkin Closures

Proposition (1.1) tells us that the free upper envelope I could be called the «-conrractive upper envelope of f. For another interesting interpretation of this envelope the reader is referred to Lembcke [5]. Since f --+1M (x) is a continuous, increasing, sublinear form on cgo(X) for every M ~ 1, we obtain as a corollary a fundamental observation from [1]: f--+ I(x) is a lower semicontinuous, increasing, hypolinear form on cgo(X) for every x E X. The importance of the M -contractive upper envelope comes from a simple application of the Hahn-Banach theorem. As in [1] we denote by Atx = AtAJJe) the set of all JJe-representing measures for a given point x E X which, by definition, is the set of all bounded positive Radon measures )L on X satisfying

Jh dJL

= h(x)

for all h E JJe.

We shall denote now by At ~ = At ~ (JJe) the set (1.12)

of all JJe-representing measures for x of total mass :o;;;,M (M ~ I a real number). The announced consequence of the Hahn-Banach theorem is then the following: Lemma (1.2). The set

coincides with the compact interval and x E X. Proof. For arbitrary h E JJe and

)L

[/M (x), 1M(x)] for all f

E cgo(X), M ~ 1

E At ~, integration of (1.3) yields

Jf dJL Jh dJL + J(f - h r dJL = h (x) + J(f - h r dJL :0;;;,

:0;;;,

h (x) + MII(f - hrll.

H. Bauer, K. Donner / Korovkin Closures

Consequently, by the definition of

151

r.

and, by applying this result to - f,

Conversely, let a be an arbitrary number in [fM(X),jM(X)]. Then the Hahn-Banach theorem states the existence of a linear form ip on eeo(X) satisfying ip(f) = a and for all g E eeo(X) . It follows from this and (1.5) that ip(g) ~ 0 for all g ~ 0 in eeo(X). Hence ip is a positive linear form and, consequently, of the type ip(g)

=

I

g

d~

where ~ is a (uniquely determined) bounded positive Radon measure on X. Because of (1.9) ~ E Att x (1t). Since

for all g E eeo(X), the total mass II~ II of ~ is ~M. So we have ~ E Att ~ and f fd~ = ip(f)= a. 0 Corollary (1.3). For every function f E eeo(X) and every x E X the following inclusions hold:

(/(x),j(x»C

{I fd~1 ~ e .«, }c[f(x),j(x)].

Indeed, this isan immediate consequence from Proposition (1.1) since, obviously,

152

H. Bauer, K. Donner / Korovkin Closures

(1.13) It should be pointed out that this Corollary appears In [1] as the counterpart to Lemma (1.2). It follows here from (1.1) and (1.2). Conversely, (1.2) and (1.3) lead via (1.13) back to (1.1). The approximation of the free upper envelope by the M -contractive upper envelopes (Proposition (1.1» is useful also for other purposes. Lemma (1.4). Let f-L be a bounded positive Radon measure on X. Define J1 : C6'o(X)~ (-00, +00] as follows:

J1(f) =

(1.14)

J1dzz .

Then J1 is the increasing pointwise limit of the family (J1 M )M;;>I of continuous, increasing, sublinear forms on C6'o(X) where (1.15)

Consequently, C6'o(X).

J1

is a lower semicontinuous, increasing, hypolinear form on

Proof. It suffices to write (l.8) in the form

and to integrate. We then obtain

IJ1 M (f) - J1M (g)/ ~ M 1If-Lllllf - gil, for all f, g E C6'o(X). Consequently, jiM is continuous on C6'o(X). The fact that J1M is an increasing sub linear form follows from (1.5)-(1.7) by integration. Furthermore, M ~ J1 M (f) is increasing since M ~ M is increasing (f E C6'o(X». Hence Proposition (1.1) finally leads to

1

sup M;;>I

jiM(f) = J1 (f)

for all

f

E C6'o(X). 0

This Lemma simplifies considerations of Becker [2]. It leads via HahnBanach arguments directly to a result generalizing (1.3):

H. Bauer, K. Donner / Korovkin Closures

153

Corollary (1.5). Let f..L be as in Lemma (1.4). Denote by All'- the set of all bounded positive Radon measures v on X satisfying

Jh dv Jh df..L =

for all h E Je .

Then,

,1(f) = sup{v(f)1 v E All'-} . This result appears in Becker [2] as theorerne 23. The proof can be left to the reader.

2. Je-affine Functions of Order M and Almost Je-bounded Functions

In [1] we have studied the set (2.1)

ie = {f E

cgo(X)1 j = j}

of all Je-affine functions. In an analogous manner we define for every real number M;;. 1 the set (2.2)

of all Je-affine functions of order M. It follows from Proposition (1.1) that every Je-affine function is also one of order M ;;. 1. Furthermore, ieM is a linear subspace of cgo(X) according to the properties (1.6) and (1.7). Also, ie is a linear subspace of cgo(X) containing Je as we know from [I]-or, as can be concluded from (1.9) and (1.11). So we have (2.3)

(I~K~M),

where the last inclusion follows from the fact that M ~ jM is increasing for every f E cgo(X). Much more is true however: Proposition (2.1). For every M > 1 the Je-affine functions of order M coincide with the Je-affine functions, i.e. (2.4)

154

H. Bauer, K. Donner / Korovkin Closures

Proof. It suffices to prove that for given condition

fE

~o(X)

and x E X

the

(i) implies

(ii) According to the definition of of a function h E ~ such that

J(x)

=

f(x).

JM,

(i) yields for given e >

a the existence

h(x)+ MII(f - htll
MIIU - h til < f(x) -

h (x) + E ~ /lU - h til + E.

Since M > I this proves

IIU-htll<_E_, M-l

and hence E

h ";3f---. M-I

So we have

for every

E

> O. For

E

~

a this gives us J(x) ~f(x)

and hence (ii), since So

A

~

1

and

A

~

r«! according to (1.4) and (1.11). 0

are the only spaces to be studied further. The space

A

~

of

H. Bauer, K. Donner I Korovkin Closures

155

JYt'-affine functions has been characterized in different ways in [1]. So we concentrate on analogous characterizations of fel. One characterization is an almost immediate consequence of Lemma (1.2). The corresponding result for fe can be found in [1]. Proposition (2.2). A function f E C€o(X) is JYt'-affine of order 1 if and only if for every point x E X and every JYt'-representing measure J1- e « ~ of total mass ,,;;;1

Jf dJ1- = f(x) . Proof. Since the set {f f dJ1-1 J1- E .itt ~} coincides with the interval [p(x), l(x)] and since f(x) is in this interval, the result follows. 0 In [1] we have introduced as follows: A function f E almost JYt'-majorized as well means that, for every 8 > 0,

the set JYt'* of all almost JYt'-bounded functions C€o(X) is called almost JYt'-bounded if it is as almost JYt'-minorized. Almost JYt'-majorized there exists a function h E JYt' satisfying

h?::f-e.

(2.5)

Almost JYt'-minorized means that -f is almost JYt'-majorized. We know from [1] that JYt'* is a closed linear subspace of eeo(X) which satisfies (2.6) In contrast to this, we shall see that, in general, fel is not contained in JYt'*, even for compact spaces X. Obviously, a function f E C€o(X) is almost JYt'-majorized if and only if, for every 8 > 0, there exist functions h E JYt' satisfying

11(/- h til , ; ; e .

r

This formulation and the appearance of the norm of (/ - h in the definition of the envelope suggests the following procedure: We embed X into its one-point compactification

l'

X",=XU{w},

156

H. Bauer, K. Donner / Korovkin Closures

where w is the point at infinity (isolated in X", for compact X). As usual we extend all functions from ~o(X) continuously to X", by assigning to w the value zero. If we do so, the l-contractive envelopes jt and are also defined at w in a canonical way:

P

jt(w)=inf{J1U-htlll hE~},

Consequently, a function

fE

~o(X)

~-majorized

if and only if

jt(w) = O.

(2.7)

A function

is almost

f

E ceo(X) is almost ~ -bounded if and only if

jt(w) = p(w) = a.

(2.8)

So, in a very precise sense, almost ~-boundedness is a condition at infinity. At the same time, the use of the two envelopes at infinity suggests to look for an analogue to the crucial Lemma (1.~). The result is the following: Lemma (2.3). Denote by .,t( = Al:(~) the set of all positive Radon measures J.L on X of total mass :::::;1 satisfying

Jh dJ.L Then, for all functions f E

Proof. We have

=

a

for all h

E~.

~o(X),

f - h :::::; IIU - h til and hence

Jf dJ.L :::::; Jh dJ.L + IIU - htllllJ.L11 ~ /lU - htll since f h dJ.L = a and IIJ.L/I ~ 1. This proves f and, consequently, f f dJ.L E rP(w),jt(w»).

f dJ.L

~ jt(w) for all f E ceo(X)

H. Bauer, K. Donner / Korovkin Closures

157

Conversely, it is evident that g -+ gt(w) is an increasing sublinear form on ceo(X), The Hahn-Banach argument in the proof of Lemma (1.2) then leads for every a E [p(w), l(w)] to a positive Radon measure j..L on X of total mass ".;; 1 satisfying

Jf dj..L

=

a

Jg dj..L ".;; gt(w) ,

and

for all g E ceo(X). This proves the converse since, obviously, for all h E 'J(.

0

So in view of (2.8) we obtain as a corollary a measure-theoretic characterization of the space 'J(", Proposition (2.4). A function f E ceo(X) is almost 'J(-bounded if and only if

Jfdj..L

=

0

for all measures j..L E

.it{ ~ .

Example (2.5). Let j..Lo be a positive Radon measure 7"on a locally compact space X. Consider

a of total mass

".;;1

I Jh dj..Lo = a} .

'J( = {h E ceo(X)

Then the functions in 'J( are the only ones which are almost 'J(-bounded, i.e. 'J(* = 'J(. Indeed, we have j..Lo E .it{ ~('J(), and the result follows from Proposition (2.4). Example (2.6). Continuing Example (2.5), we choose in particular X (the one-point compactification of the discrete space N) and

j..Lo =

h;w + ~ L

= Nw

(~)n e,

n=l

(where ex is the unit mass at x V The corresponding linear subspace 'J( of functions hE ce(Nw ) satisfying I h dj..Lo = 0 coincides with 'J(* and is thus 2 This

is a modification of an example of Flosser [4, p. 122].

158

H. Bauer, K. Donner / Korovkin Closures

different from 'e(N.J. However, :leI = 'e(N w ) ' In fact, it is easy to see that condition (PP) of [4] is fulfilled, i.e. for two given different points x, yin N w there exist a function h E ;It and a real number a ~ 0 satisfying h + a ~ 0, h (y) + ex > 0 and h (x) + ex = O. It then follows easily (d. [4]) that Jtt~ = {Ex} for all x EN ... So Proposition (2.2) yields :leI = 'e(N os) which proves that even for compact spaces X we do not have :leI C 'Ie* in general. A simple modification of this example (take X = N as in [4]) yields the same phenomenon for X locally compact, but not compact. In [1, p. 229], the 'Ie-affine functions were characterized as those functions f E cgo(X) which can be trapped by 'Ie. This means that given E > 0 there exist finitely many functions h;, ... , h~ and h~, ... , h~ in ;It such that the two functions h = sup{h;, ... , h~} and ii = inf{h~, ... , h~} satisfy h - E ~f~ ii + E and llii - hll ~ E. We shall see that there are similar characterizations of :leI for X compact and of :leI n 'Ie* in the noncompact case. We first remark that the l-contractive upper envelope can be written in a slightly different form. Lemma (2.7). For every function f E 'eo(X) and every x E X

rex) = inf{g(x)1 g E 'Ie+ R+, g ~ f}. (;It + IR+ denotes the set of functions h Proof. In the original definition of

+a

with hE ;It and a E IR+.)

p the functions

h + IIU - h til E 'Ie + IR + are used. According to (1.4) they majorize f. If g = h + a E ;It + IR+ majorizes f we have f - h ~ a and hence I/U - h til ~ ex. So r(x) can be written as claimed. 0 We are now in the position to characterize ;It-affine functions of order 1 by means of a trapping procedure. Theorem (2.8). For X compact, a function f E 'eo(X) lies in :leI if and only if, for every E > 0, there are finitely many functions g;, ... .s; E 'Ie- R+ and g';, ' .. , s; E 'Ie + IR+ such that the corresponding two envelopes

H. Bauer, K. Donner / Korovkin Closures

8 = suplg], ... , g~}

and

159

g = inf{g~, ... , g:}

satisfy

For X locally compact, but not compact, the same condition characterizes the functions in :it I n :Jt *.

= f. So by the preced> 0, there is a function gx E :Jt + R+

Proof. Suppose X compact and f E :it l • Then f

ing Lemma, for every x E X and satisfying

E

By continuity, gAy)
For g'; = gx. and I

g = inf{g';, ... , g:} we then f(x) :s;: g(x) < f(x) + ~E

have

for all

X

E X.

By applying this result to - f, we arrive at the above mentioned condition. The converse is an immediate consequence of Lemma (2.7) since

Let us assume now that X is locally compact but not compact, and consider fE:it 1 n :Jt*. Since f is almost :Jt-bounded, there exists a function ho E :Jt satisfying ho ~ f - E12. Because ho - f vanishes at in~flity there exists a compact set K C X such that ho(Y)- f(y) < E/2 for all y E X\K. On K we can argue as above in the compact case. So there are functions g';, ... , s; E :Jt + IR+ such that their lower envelope g satisfies f(x):s;: g(x) < f(x) + ~E By joining

g~ =

ho + EI2 to

g~,

for all

X

E K.

... , g: as a new function, we end up with

160

the functions

H. Bauer, K. Donner / Korovkin Closures

g~ • .

. . • g: E 'J't + R+ whose lower envelope g satisfies

f(x):s;; g(x)
for all x E X.

By applying this result to - f, we arrive at the condition in question (with instead of e). The same argument as in the compact case shows that, conversely, this condition implies f E 1e1• In order to see that f is also almost 'J't-bounded. we refer to Proposition (2.4). So let JL be a measure in Al~. According to our condition the functions g;. gj are of the form g; = h/,and g j = h j + a j with h;, h j E 'J't and a;. a j E R+. Therefore we obtain from S :s;; f:S;; g by integration 215

a;

(2.9) where S(w) = sup{-a;, ... , -a~} and g(w) = inf{a~, ... , a:} are the values at w of the unique continuous extension of g and g to X." respectively. From IIg - sll:s;; e which is equivalent to Ig --sl.";; E. we obtain for x~w, x E X. (2.10) Since S(w).,;; 0.,;; g(w) the two conditions (2.9) and (2.10) imply for all

15

> 0,

and hence Jf dJL = O. This proves f E 'J't*. (It should be observed that a unique continuous extension of functions from 'J't + IR+ to X., is not available in the compact case. 0

3. M-contractive Korovkin Closures Again let X be a locally compact space and 'J't a linear subspace of A net (Ta)aEA of positive linear maps Ta : ~o(X)~ 't'o(X) will be called 'J't-admissible if (as in [1]) ~o(X),

(3.1)

sup !I Tall < +00 aEA

H. Bauer, K. Donner / Korookin Closures

161

(equicontinuity of the net) and if (3.2)

for all h E 'Je .

lim IITah -hll= 0

aEA

Given a real number M ~ 1, we define the Mrcontractine Korovkin closure of 'Je, Kor M('Je), as the set of all functions f E ~o(X) such that lim IITa f

aEA

holds for all 'Je-admissible nets (3.3)

- III =

(Ta)aEA

0

for which

supIITaII~M.

aEA

Obviously, KorM('Je) is a linear subspace of ~o(X). For 1 ~ K ~ M one has (3.4)

'Je C KorM('Je) C KorK ('Je) .

In [1] the Korovkin closure (3.5)

Kor( 'Je) =

n

KorM('Je)

Mibl

was studied. Since there is no restriction of the type (3.3) for the nets (Ta ) , we call it here the free Korovkin closure of 'Je. According to [1] it is equal to 'ie, i.e. (3.6)

Kor('Je) =

'ie.

There is essentially only one other case of importance, namely the J-contractive case, i.e. the determination of Korj('Je). This is due to the following simple result-a counterpart to Proposition (2.1):

Proposition (3.1). For every M> 1, one has

(3.7)

KorM('Je) = Kor('Je) = 'ie.

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H. Bauer, K. Donner / Korovkin Closures

Proof. According to (3.4)-(3.6) we only have to prove that 1 < K:so; M implies Kor K(.?'t) c Kor M(.?'t). In order to see this, let (Ta )aEA be an .?'t-admissible net such that II Tall :so; M for all a E A. Then (a E A),

where 0< A = (K - l)/(M - 1):SO; 1, and where Id is identity map of C€o(X) into itself, defines a net of .?'t-admissible, positive, linear operators on C€o(X). Obviously, sup IIT~II:so; AM + (1- A) aEA

K.

=

Since T~ - Id = A(Ta - Id) for all a E A, (Taf) converges uniformly to f for fE KorK(.?'t). Consequently, f lies in KorM(.?'t). 0 In the remaining case we claim Theorem (3.2). One has for X compact and for X noncompact .

Proof. Let (Ta )aEA be an .?'t-admissible net of operators on 't8 o(X ) satisfying II Tall :so; 1 for all a E A. Assume

for given e > 0, h E .?'t and a E A. For fin we then have (3.8)

fel

or

it l n .?'t*,

respectively,

Taf:SO; h + IIU - h til + e

since TaU - h):so; TaU - ht :so; IIU - h til. According to Theorem (2.8), there exist finitely many functions g I ' . . • , gn E .?'t + R + such that f:SO; g :so; f + t:, where g = inf{g" ... , gn}' Each gi is of the form hi + a i (hi E .?'t, a i E R+) where IIU - hitll:so; a i because of f:SO; gi' i = 1, ... , n. Therefore, (3.8) implies

H. Bauer, K. Donner / Korovkin Closures

for all i

= 1, ... , n

163

and hence

provided that II Tah i - hill < e for i = 1, ... ,n. This, however, is true by assumption for eventually all a E A. In the same way it follows from Theorem (2.8) that 1- 2s :,;; Taf for eventually all a. So lim IITal - III = 0 follows, and we have proved that it l or it l n fIt* is contained in Korl(fIt), respectively. The converse will be proved first for the case where X is not compact. We denote by f!F the net of all finite, non-empty subsets of fit ordered by inclusion c. Let I be a function in Korl(fIt). For compact K C X and A E f!F, there exists a compact set K A ::> K such that

1 I/(x)1 < 2/AI

and

1

Ih(x)/ < 2/AI'

for all h E A and all x E U o = X\KA (where elements of A). We then have (3.9)

1 I/(x)- l(y)1 < IAI

and

IAI stands for

the number of

1

Ih(x)- h(y)1 < IAI'

for all h E A and all points x, y E U o' K A can be covered by finitely many open, relatively compact sets U I , ••• , U; such that (3.9) holds for all points x, y lying in the same set U;, i = 1, ... , n, and all h E A. Since is upper semicontinuous, there exist Xi E clos(U;) such that

P

p(x;) = supP(clos(U;»

(i=I, ... ,n).

In addition, we choose X o = w (the point at infinity). Next we choose a partition of the unity subordinate to the covering U I , ••• , U; of K A , hence functions ql"'" qn E ~o(X) such that each qi is ;;;.0 and has compact support C U; and such that n

L qi :,;; 1 on i=1

X, and

=1 on K A

.

164

H. Bauer, K. Donner / Korovkin Closures

We also choose a function qo E ~o(X) with compact support disjoint from clos(v;) such that 0 ~ qo ~ 1 and %(xA) = 1 for some xAE V o' This implies

U7~1

n

2: qi ~ 1 on X, and

=1 on K A

•

;=0

Finally, we apply Lemmas (1.2) and (2.3) in order to obtain measures ILi E Ai ~I satisfying

ff

dj..t;

=

(i = 0, ... , n).

p(x;)

After these preparations let us consider the net (TA ) A E .90 of positive linear operators on ~o(X) defined as follows:

(g E The net is admissible. Indeed, for all h E

I±(f

ITAh(x)- h(x)1 =

~

~o(X».

and x E X, we have

h dj..t;) q;(x) - h(x)1

,=0

I±

h(Xi)qi(X)- h(x)j .

=

•=0

With the notation q~= 1- L qj'

q; = q;

(i

= 1, ... , n),

;=1

this leads to ITAh(x)- h(x)1 = )

±

;=0

n

h(x;)q;(x)- h(x)

±

;=0

q;(x)1 1

~ ~ Ih(x;) - h(x)lq;(x) ~ IAI'

H. Bauer, K. Donner / Korovkin Closures

165

Here, one has to observe that {q; =I O} C U; and Ih(x;) - h (x)1 ~ IAr t for x E U; and i = 0, ... , n. This proves that (TA ) is ,re-admissible. For g E "6io(X ) and x E X we also have ITAg(x)1 ~ Ilgll L ;=0 q;(x) ~ Ilgll, and hence II TAli ~ 1. Because of f E KOft(,re), this leads to the conclusion (3.10)

But n

TAf(x) =

L l(x;)q;(x) ;~o

and l(x;) ~ l(x) for x E U; as well as {q; =I O} C U; lead to for i

=

1, ... , n ,

and hence to for all x E K A (since qo(x) = 0). Passing to the limit along g; we obtain from (3.10) f(x) ~ l(x) and thus !\x) = f(x) Ior all x E K. Since K was an arbitrary compact set, this proves (3.11)

l=f

on X.

Another simple calculation yields

Because of

ITAf(x)j ~ ITAf(x) - f(x)1 + If(x)1 <

1

E

+ 21AI

for sufficiently large sets A E g; and x E X\KA , this implies

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H. Bauer, K. Donner / Korookin Closures

for A sufficiently large, whenever e > 0 is given. So we end up with

p(w)= o.

(3.12)

(3.11) and (3.12) remain valid for Kor)(.rt'). So we finally obtain

P=P=f and

-f

which is also a function from

p(w)=p(w)=O.

According to the definition of it) and to (2.8) this proves that f lies in it1n.rt'*. In the compact case the proof (of the converse) follows the same lines, but with simplifications. One can start with K = X from the beginning. One chooses qi' Xi' f-Li for i = 1, ... , n as before. However, there is no need for %' XA and f-Lo. This simplified version of the above proof can be left to the reader. D As usual in Korovkin approximation, we now obtain: Corollary (3.3). The L-contractioe Korovkin closure Kor)(.rt') equals ~o(X) if and only if At ;(.rt') = {ex} for all X E X and if (jor noncompact X) in

addition all functions in

~o(X)

are .rt'-bounded.

This is an immediate consequence of Theorem (3.2) in connection with Proposition (2.2). Remark (3.4). A necessary and sufficient condition for .rt'* = ~o(X) can be found in [1, p. 227]. A sufficient condition is that .rt' contains a strictly positive function. Remark (3.5). A simple change of the proof of Theorem (3.2)-following the lines indicated in [1, 3.3 Corollary 1]-shows that in the definition of Kor)(.rt'), nets can be replaced by sequences provided that the subspace .rt' is separable. In other words: the sequential contractive Korovkin closure then coincides with the contractive Korovkin closure. Remark (3.6). By means of Lemma (2.3), Theorem (3.2) can be identified with a theorem of Flosser [4, p. 125] proved by different methods. The

H. Bauer, K. Donner / Korovkin Closures

167

fundamental notion of almost .rt'-bounded functions does not appear explicitly in [4]. Example (3.7). For X = (0, 1] and .rt' generated by the two functions x ~ x and x ~ x 2 we have .rt'* = ~o(X) by Remark (3.4). According to Corollary (3.3), .Jl ~ = {eX(J for all Xo E (0, 1], since x ~ (x - X O)2 peaks at Xo and since

for all J.L E.Jl~. SO J.L = AeX{J with 0 ~ A ~ 1, and integration of h(x) = x yields A = 1. So Korl(.re) = ~o(X). In contrast to this the free Korovkin closure is much smaller. In fact, it follows from a general result about the free Korovkin closure of twodimensional subspaces .rt' in [3, p. 130] that Kor(.rt') = .rt'. Example (3.8). For X = [1, +00) and .rt' generated by x ~ X-I and x ~ x- 2, one can prove in a similar way that Korl(.rt') = ~o(X).

In fact, the examples in [1], suitably modified, lead to corresponding examples of two-dimensional spaces .rt' with ~o(X) as l-contractive Korovkin closure. (Cf. [4].)

References [1) [2)

[3) [4) [5]

H. Bauer and K. Donner, Korovkin approximation in '6'o(X), Math. Ann. 236 (1978) 225-237. R. Becker, Sur les cones inf-stables de fonctions continues sur un espace compact, Seminaire Choquet, Initiation Ii I'Analyse, 17" annee, 1977/78, Fasc. 2, n" 25, 13 p. (1978). (Universite Pierre et Marie Curie, Institut Henri Poincare, Paris.) K. Donner, Extension of positive operators and Korovkin theorems, Lecture Notes in Math. 904 (Springer, Berlin, 1982). H.O. Flosser, Sequences of positive contractions on AM-spaces, J. Approx. Theory 31 (1981) 118-137. J. Lembcke, Note zu 'Funktionenkegel und Integralungleichungen' von H. Bauer, Sitzungsber. Bayer. Akad. Wiss. Math. Naturwiss. KI. 1977 (1978) 139-142.

J.A. BARROSO editor, Aspects of Mathematics and its Applications © Elsevier Science Publishers B.V. (1986)

169

DISTINGUISHED ECHELON SPACES AND THE PROJECTIVE DESCRIPTION OF WEIGHTED INDUCTIVE LIMITS OF TYPE rd r5 (X ) Klaus D. BIERSTEDT Uniuersitdt-Gll-Paderbom, Paderborn, Fed. Rep. Germany

Reinhold MEISE Unioersitat Diisseldotf, Math. lnstitut, Dusseldorf, Fed. Rep. Germany Dedicated to Leopoldo Nachbin Contents O. Introduction

1. Sequence Spaces 1. Notation and Preliminaries 2. Condition (D); Statement of the Main Theorem 3. General Results; Proof of the Main Theorem Appendix

II. Function Spaces 4. Notation and Preliminaries 5. Some Additional Results 6. The Main Theorem and its Corollaries Appendix References

O. Introduction The general problem of a projective description for weighted inductive limits of spaces of continuous and holomorphic functions was raised in [4], but there the investigations were concentrated on the case of the locally convex inductive limit 'Ydo~(X)=indlimn .... ~(vn>O<X) for a decreasing sequence 'Yd = (Vn)nEN of strictly positive weights on a locally compact space X. As it turned out, 'Yd OC(5'(X) always is a topological subspace of its associated weighted space C(5''Vo(X), where 'V = 'V'Yd

170

K.D. Bierstedt, R. Meise / Distinguished Echelon Spaces

denotes the maximal Nachbin family relative to "JI'd' and hence the inductive limit topology is given by the system (qiJve'Y of weighted supnorms. Whenever "JI'd = (vn)n consists of continuous functions, C(J'Vo(X) is exactly the completion of "JI'do ~(X). Moreover, it was proved in [4, Section 2] that the (algebraic and topological) identity "JI'dOC(J(X) == C(J'Vo(X) holds if and only if the sequence "JI'd is regularly decreasing (a condition which coincides with a property already introduced in the appendix of [3]). In the case "JI'dC(J(X) == il1dlimn-> C(Jvn(X), the situation is completely different: We always have the algebraic equality "JI'd~(X) = C(J'V(X), and the two spaces also yield the same bornology whence "JI'd~(X) is the bornological space associated with ~r(X), but, in general, the inductive limit topology of "JI'dC(J(X) is strictly stronger than the weighted topology of C(J'V(X). In the subsequent paper [5], similar questions were studied in the setting of sequence spaces on general index sets 1. While, in that article, echelon spaces and co-echelon spaces of arbitrary order p, p = 0 or I .;;; p .;;; 00, were considered, we now want to restrict our attention to the classical case of the echelon spaces A( of order 1 and the co-echelon spaces (NO and) Noo of order (0 and) 00, respectively, where the previous results can be applied directly. But there also exists a well developed (and easily established) topological duality theory for sequence spaces, and thus better insight and more information were obtained in the restricted setting. E.g. it was pointed out that the regularly decreasing condition is equivalent with the echelon space Al being quasi-normable (a result which was discovered independently by M. Valdivia). And the fundamental topological isomorphism "JI'dC(J(X) == C(J'V(X), or rather Noo == K; in our notation for sequence spaces, holds if and only if A( is distinguished. It is an interesting problem to characterize the class of the distinguished echelon spaces A( = A(1, stJ) on the index set 1 in terms of a necessary and sufficient condition on the Kothe matrices stJ of its elements. This problem remains open, but we present a new, very weak, condition (D) sufficient for distinguishedness of echelon spaces in Part I, Section 2 of the present paper. Property (D) generalizes both the quasi-normable and the reflexive (or, equivalently, Montel) cases of distinguished echelon spaces Ai - In fact, it is modelled on the condition (wG) of Grothendieck [7, II, p. 102]; it arises from that condition by replacing the 'Schwartz type' part by a 'Mentel type' assumption. And while Grothendieck's claim that (wG) implies that Al is quasi-normable was shown to be erroneous in [5, Example 3.11], we now prove (Theorem

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171

(2.3) and Section 3) that even the weaker hypothesis (D) is sufficient to imply that AI is distinguished. Moreover, in Part II of this article, we demonstrate that the arguments used in the sequence space setting can be modified to yield a similar, rather general, result (Theorem (6.9» on the topological equality 'Vd'€(X) == '€'Y(X). In fact, for a decreasing sequence 'Vd = (vn)n of continuous weights on a completely regular space X (resp., on a locally compact and zr-compact space X), this also proves that the regularly decreasing property (resp., property (M), the analogue of the Montel condition (M) for echelon spaces) implies the desired projective description 'Vd'€(X) == '€'Y(X) (Corollaries (6.10) and (6.11». Concerning the structure of the paper, let us mention at this point that, in Part I (and, especially, in the beginning of Section 3), we consider one half of condition (D) in greater generality than the actual proof of our Main Theorem (2.3) on co-echelon spaces of order 00 requires. But, in this way, direct applications to function spaces become possible in Part II (d. Section 5 and the beginning of Section 6). Next, comparing Part I and Part II of the article, parallels between the two parts become apparent; function space analogs sometimes arise from the corresponding results on sequence spaces simply by replacing the system lS of finite subsets of the index set I by the system ~ of all compact sets in a completely regular space X. It is quite useful to exploit this analogy as far as possible even though this may require new proofs or modifications of proofs. Hence, besides considering co-echelon spaces and weighted inductive limits as being parallel, we also demonstrate the utility of the general approach by noting two results for weighted Frechet spaces of continuous functions, characterizations of '€OU(X) = '€OUo(X) (by condition (M'), d. Proposition (5.5» and of quasi-normable spaces '€OU (X) and '€OUo(X) (by the regularly decreasing property, cf. Proposition (5.8». Finally, we note that the weighted (or Nachbin) spaces of continuous functions were introduced, and studied in approximation theory, by Leopoldo Nachbin (see e.g. [10]) and his students. Since then, the general framework of weighted spaces has proved important, and very stimulating, for other studies, including research by the authors of this article, e.g. in connection with topological vector spaces, duality, and topological tensor products. In fact, the present investigations are based on a consequent use of the associated weighted space '€'Y(X). Therefore, we have been especially grateful for the opportunity to dedicate this article to Leopoldo Nachbin on the occasion of his 60th birthday.

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I. Sequence Spaces 1. Notation and Preliminaries

From [5], we briefly recall the notation and some relevant results, referring to that paper for any unexplained terminology and more details. 1.1. Kothe Sets and Associated Sequence Spaces

In the sequel, we will let I denote an arbitrary index set. Definition (1.1). A set (i} of non-negative real-valued functions a = (a(i»iEl on I is called a Kothe set on I if the following two properties are satisfied: (1). For each pair (a, {3) E (i} X (i}, there exists ayE (i} such that max{a(i), {3(i)} ~ y(i) for all i E 1. (2). For each i E I, there exists an a E (i} with a(i) > O. Corresponding to each Kothe set

(i},

we associate the spaces

Aoo(I, (i}) = {x E C 1 (or IR 1)1 qa(x) = sup(a(i)/x(i)l) < 00 for each a E (i}} iEl 1 Ao(I, (i}) = {x E C (or IR 1)/ for each a E (i}, (a (i )x(i ))iEl converges to O} . (a(i)x(i»iEl is said to converge to 0 if for each e > 0, there is a finite subset J = J(e) C I with a(i)lx(i)/ < e for all i E 1\1. If (i} consists of a single strictly positive function a = (a(i))iEl on the index set I, we sometimes prefer to write t,(I, a) instead of A",(I, (i}) and co(1, a) instead of Ao(I, (i}). Under the system {qal a E (i}} of seminorms, Aoo(I, (i}) is a complete (Hausdorff) locally convex (I.e.) space, while Ao(1, (i}) is a closed subspace of Aoo (1, (i}) which will be endowed with the corresponding induced system of seminorms. In fact, Ao(I, (i}) is the closure in Aoo(1, 9J» of the space ({J(I) all finite sequences on 1. (The finite sequences on I are the functions which are zero except for finitely many points in 1.) 1.2. Echelon and Co-echelon Spaces

Now let d = (an )nEN be an increasing sequence of strictly positrve functions on the index set 1. Then d is a countable Kothe set which will

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173

be termed a Kothe matrix on 1. In particular, )..",(/, d) and ;"0(/' d) are Frechet spaces with the sequence (qn)nEN of norms, where we write qn instead of qa (n = 1,2, . . .). • By ).. 1(/' d), we denote the corresponding echelon space )..1(1, d)

= {x

EreI (or IR 1)1 Pn(x) = ~ an (i)/x(i)1 <

00

for each n

EN}.

;.. 1(/' d) is a Frechet space with the sequence (Pn )nEN of norms.

Taking 'V = (Dn)nEN to denote the associated decreasing sequence of functions Dn = I/a n , we put k ",(I, 'V) == indlim 1",(1, D.) and

k 0(/' 'V) == indlim co(1, Un) ;

i.e. k",(/, 'V) (resp. ko(/, 'V» is the increasing union of the Banach spaces 1",(1, D.) (resp. co(/' o; endowed with the strongest I.e. topology under which the injection from each of these Banach spaces is continuous; this topology is Hausdorff. k",(/, 'V) is called the co-echelon space associated with A1(/' d). Note that, instead of fixing a Kothe matrix d = (an)nEN on I and putting 'V = (D.). for u. = l/a., it is equivalent to fix a decreasing sequence 'V = (U.).EN of strictly positive functions on I and to put d = (a.). for an = I/Dn • Also, if E and F are two I.e. spaces, we will write E == F in case E and F are topologically isomorphic while E = F means that E and F are (canonically) algebraically isomorphic.

»,

1.3. The Kothe Set

r = )..",(1, d)+

At this point, we introduce the system r

for each n

r

=

r('V) by

EN} . r»

Then is a Kothe set on I, and we shall write K",(I, 'Y) (resp. Ko(I, instead of >...,(/, (resp. >"0(1, The topology of these spaces is given by the system (qii)iiE'Y of seminorms. K.,(/, r) is a complete I.e. space, and

r)

r».

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K.D. Bierstedt, R. Meise I Distinguished Echelon Spaces

Ko(I, 'V) is a closed subspace of K,,(I, 'V); viz. the closure of rp(I) in K,,(I, 'V). Even though all the functions Vn are assumed to be strictly positive, the system 'V need not contain any strictly positive element (if I is not countable, d. [5, 1.6-1.8]). If I and the Kothe matrix stl (or, equivalently, the decreasing sequence 'V) are fixed, we will sometimes write Ap (p = 0,1, (0), ko' k"" K o, K", instead of A/I, stl) (p = 0, I, (0), ko(I, 'V), k",(I, 'V), Ko(I, 'V), K",(I, 'V), respectively. 1.4. Two General Results The following natural injections are continuous: k",4Koo, k o4Ko' and k o4 k oo• However, much more can be proved (see [5, Section 2]).

i o of ko. In particular, the l.c. inductive limit topology of k o coincides with the topology induced by the system (qV)VEY of seminorms. However, k o can be a proper subspace of K o. (2). k oo = K; (algebraically), moreover, the weighted topology of K; and the l.c. inductive limit topology of k", have the same bounded sets. Hence k", is the bornological space associated with K"" but, in general, the inductive limit topology is strictly stronger than the topology of K",. (3). The canonical injection k O 4k", is a topological isomorphism into. Theorem (1.2). (1). K o is the completion

Among other things, the next theorem describes the full topological duality between echelon and co-echelon spaces (again, see [5, Section 2]).

«

Theorem (1.3). (1). (ko)~ == (Ko)~ == A) and (A))~ == K",. Whence ko)b)~ == «Ko)b)~ == K", and k '" is the bomological space associated with (A I);' (2). It follows that the I.e. inductive limit topology of k", is f3«A))', (A 1)")'

and k oo is always complete. (3). The following assertions are equivalent: (i) (Al)~== k"" (i') K oo == k",. (ii) K", is barrelled and/or bornological. (iii) A I is distinguished. The class of all distinguished echelon spaces A) is treated in Theorem

(1.3) (3). At this moment, a characterization of this class in terms of a necessary and sufficient condition on the Kothe matrices stl of its elements

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175

Al = A1(1, s1) remains open. By Theorem (1.3) (3), a characterization of the distinguished echelon spaces amounts to characterizing when K" equals k", topologically, and this question is interesting on its own. 1.5. Two Classical Conditions

Next, we recall some conditions on the Kothe matrices s1 which we prefer to formulate in terms of the decreasing sequencer = (vn)n' (S). For each n E N, there exists an m > n such that vm/vn converges to 0. (M). For each n E N and each infinite subset 10 of I, there exists an m = m(n, Io»n such that infjElo(vm(i)/Vn(i)) = O. We note that, clearly, (S) ~ (M) and that, in the presence of condition (M), the index set I is at most countable [5,4.2]. For the following characterizations, see e.g. [5, Section 4]. Proposition (1.4). (1). The following assertions are equivalent: (i). "II satisfies condition (S). (ii). k O = k",. (iii). Al (or, equivalently, ,10 or A",) is a Schwartz space. (iv). k O is a (DFS)-space (or, equivalently, semi-reflexive). (v). k", is a (DFS)-space. (2). The following assertions are equivalent: (i). "II satisfies condition (M). (ii). = K"" (ii'). Ao = A",o (iii). AI (or, equivalently, ,10 or A",,) is a (semi-) Montel space (or, equivalently, semi-reflexive). (iv). K o (or, equivalently, K",,) is a (semi-) Montel space (or, equivalently, semi-reflexive). (v). k", is a (semi-) Montel space (or, equivalently, semi-reflexive).

x,

1.6. Other Conditions

The next conditions are 'less classical'. (ON). "II is said to be regularly decreasing if, for each n EN, there exists an m » n such that, for any loCI with infiElo(vm(i)/Vn(i)) > 0, we also have infiElo(vk (i)/vn(i)) > 0 for all k ~ m. (wS). For every n EN, there is an m ~ n such that, for each set 10 C I with infiElo(vm(i)/vn(i))>O, it is possible to find aBE with B ~ Vm on 10 ,

r

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(0). For each n EN, there exists an n' > n and an increasing sequence (/:)mEN of subsets of I such that: (i) inf iE 1 " (Vk(i)/Vn(i» > 0, k = n + 1, n + 2, ... , m = 1,2, ... , while (ii) lim; ....: SUPiEI\I" (vn.(i)/vn(i» = O. (wG). There exists"an increasing sequence (/m )mEN of subsets of I such that: (i) for each mEN, there is an nm E N with inf iE Im (vk(i)/v nm (i»>0, k = nm + 1, nm + 2, .... (ii) for each n EN, there exists an n' > n with

lim, .... oo SUPiEIII)Vn.(i)/vn(i» = O. It is not hard to see that (ON)~ (wS)~ (0)

and

(S)~ [(M)

+ (ON)] .

However, we only have (O)=? (wO), and the converse does not hold. (See [5, 3.2, 3.9-3.11, 4.8].) Various characterizations of condition (ON) were obtained in [5, Section 3]. Theorem (1.5). The following assertions are equivalent:

(1). "V is regularly decreasing. (2). Al (or, equivalently, Ao or Aoo) is quasi-norm able. (3). K; satisfies the strict Mackey convergence condition. (4). k oo == indlirn, [00(/' vn ) is a boundedly retractiue inductive limit. (5). k o == indlim, co(l, vn ) is a regular (or, equivalently, boundedly retractive) inductive limit. (6). k o is complete. (7) k o = tc; (8) k o is closed (or equivalently, stepwise closed) in k oo ; in other words k o = koo-clos !p(/)= closure of !p(/) in k oo. We note that condition (ON) clearly implies any of the equivalent assertions of Theorem (1.3) (3) as does (M), by the way. Now, it is the purpose of Part I of this article to approximate the characterization of the distinguished echelon spaces Al = AI(I, sIi) in terms of sIi by constructing a condition (D) which is implied by (ON) as well as (M) and which, conversely, implies Al distinguished. Condition (wO) and the equivalence (iii)~ (i') of Theorem (1.3) (3) are the key to our approach. The new condition (D) will be formulated at the beginning of the next section, and, after some remarks, we will then state our Main Theorem (2.3) concern-

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ing condition (0). The rest of Section 2 is devoted to a consequence of Theorem (2.3), while the proof of Theorem (2.3) is postponed to Section 3 (i.e. after Corollary (3.14».

2. Condition (0); Statement of the Main Theorem The following new condition (0) is of the same type as (wG) above, but the assumption (ii) of (wG), which resembles condition (S), is replaced by a weaker assumption similar to condition (M). (0). There exists an increasing sequence ;:5 = (1m )mEN of subsets of I such that (N, ;:5). For each m EN, there is an n m EN with infiEl)vk(i)/vn)i»>O, k = nm + I, nm + 2, .... (M, ;:5). For each n E N and each subset 10 C I such that 10 n (I\Im ) ¥c 0 for all mEN, there exists an n' = n'(n, 10) > n with infiE~(vn.(i)/ vn(i» = o.

Remark (2.1). (1). For 1= Nand ;:5 = (Im)mEN with 1m = {l, ... , m}, m = 1,2, ... , condition (N, ;:5) above is satisfied trivially while (M, ;:5) holds if and only if 'Y satisfies condition (M). (2). In view of (1), and recalling that condition (M) forces the index set I to be at most countable, we obtain (M) ~ (0), while (ON)¢:> (G)=? (wG)~ (0) is obvious. Hence we have the following diagram:

~

(S)

(QN)~ (wG)~

~

~

,~ "

(0)

~(M)~

At this point, it remains to exhibit the counterexamples which show that none of the implications .~' in the diagram can be reversed. In fact, (ON)~ (S) is trivial, (M)~ (S) by the classical counterexample due to Kothe and Grothendieck, and (O)~ (M) by Grothendieck's example of an echelon space Al = A1(N x N, sd) which is quasi-normable, but not reflexive. (See e.g. [5, Examples 4.11, 1 and 2].) Moreover, we have already noted (in (1.6» that (wG)~ (G) [5, Example 3.11], and for (O)~ (wG), we may argue as follows:

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Clearly, in the presence of condition (M), each set 1m C I with (i) of (wG) must be finite, and then (ii) of (wG) implies condition (S); i.e. we have obtained (d. (1.6» (S)¢> [(M) + (wG)] .

But at this point, any "V = (vn)n which satisfies condition (M), but does not satisfy condition (S) also serves as an example where (D) holds, but (wG) is violated. Remark (2.2). Ifcondition (D) holds, then there exists a strictly positivefunction

vE

r.

This remark generalizes [5, 3.12], and, in fact, the proof of that result works in the generality needed here. (In particular, if ~ = (1m )mEN is an increasing sequence of subsets of I as in (D), we must have I = U mEN 1m , ) In [5, Example 1.6], we found a decreasing sequence "V = (Vn)n on an uncountable index set I such that each element v E = 'Y('V) must have a zero on 1. By Remark (2.2), this 'V cannot satisfy condition (D). Also, for the Kothe matrix d on N x N in Kothe's example of a non distinguished echelon space (d. e.g. [5, Example 4.11, 3.]), the associated sequence "V does not satisfy (D). (This can be verified directly, but it clearly follows from Theorem (2.3) below, too.) The following is our main result (of Part I). It justifies the introduction of condition (D).

r

Theorem (2.3). In the presence of condition (D), k", == K" holds topologically; i.e. (D) implies that Al is distinguished. It is not known whether the converse of the implication in Theorem (2.3) is true so that the problem of characterizing the class of all distinguished echelon spaces Al = A1(1, d) in terms of a necessary and sufficient condition on the Kothe matrices d of its elements is still open. However, (D) is the weakest sufficient condition known so far (in terms of the decreasing sequence "V). And moreover, for all practical purposes, if one wants to prove that some specific echelon space A 1 is distinguished, condition (D) is usually verified quite easily. The next section contains some general results which, after Corollary (3.14), lead to a (rather simple) proof of Theorem (2.3). For a fixed

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sequence ~, treating the two parts (N,~) and (M,~) of condition (0) separately, we start with (M,:J) and will first consider a more general form of this condition. As we go on, two different countability assumptions will be added where necessary, and thus we will end up with the present form of condition (M,:J). But before proceeding to the next section (and the proof of Theorem (2.3», we note a consequence of the main result, concerning topological tensor products of echelon spaces and their strong duals. The notation used at this point is as follows. For two decreasing sequences VI = (v ~)n and V 2 = (v~)n of strictly positive functions on the index sets II and 12, respectively, let V, ® V z denote the decreasing sequence (v~ ® v~)n on I, x l z' where (v~ ® V~)(il' iz) = V~(iI)V~(i2) for all (iI' iz) E II x l z. Similarly, the tensor product of the associated Kothe matrices sill = (a~)n on II and silz = (a~)n on l z, a~ = l/v~ and a~ = l/v~, is defined to be the Kothe matrix sill ® silz = (a ~ ® a~)n on I) x I z. Furthermore, we will consider the associated systems 'Y(V\) on II' 'Y(Vz) on l z' and 'Y(VI ® V 2 ) as well as the tensor product

We note that

whence, clearly,

(ct. [4, Lemma 3.5]).

Finally, our terminology concerning 7T- and s-tensor products as well as s-products is quite standard (see, e.g. Grothendieck's thesis [7], [3], and [4, Section 3]). Lemma (2.4). If both VI and V z satisfy condition (0), the same holds for V.®Vz·

The proof of Lemma (2.4) is straightforward and will therefore be omitted. Now, turning to the complete zr-tensor product of two echelon spaces,

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we clearly have

by a combination of some well-known results of Grothendieck [7]. Hence Theorem (2.3) and Lemma (2.4) imply

Corollary (2.5). If both VI and V z satisfy condition A](.91] ® .91z) == Alll1,) ®". A,(.91z) is distinguished; i.e.

(D),

then

(A.(.91 1) ®".A 1(.91z»~ == (AI(.91 1 ® .91z»~ == K",('V(V I ® V z» == Aoc(I1 x t; 'V(V,) ® 'V(Vz» is topologically isomorphic to k",("If I® "Ifz) (= k",(II x l z' "If,® "If2». In particular, this holds for A1(.91]) Mantel and A](.912) quasi-normable. For purposes of a comparison. let us mention the only general result which is known in this direction [7, II, p. 77]: If E and Fare Frechet spaces, then E nuclear and F distinguished implies E®".F distinguished. In fact, since it is not known if condition (D) is equivalent to A, distinguished (and since a 'good' characterization of the distinguished echelon spaces is not available). it is not clear whether the rr-tensor product A1(.91,) ®". A,(.91z) of two distinguished echelon spaces A1C«1,) and A1(.91 z) is always distinguished. At least in one specific case, it is possible to look at this question from the point of view of the topological duality of the 7T- and s-tensor products and to point out a connection with a hereditary property of the s-tensor product. To start with, here is the general (somewhat intricate) situation. By [I, II] and [2, 2], (A,(.91,»~E(A,(.91z»~ == K",(17'(V,»EK",('V(Vz» (= K",(II' 'V("If I»EKoc(I2' 'V("Ifz»)

== (A,(d',»~0E (A,(.91z»~ = Koc('V("If,» 0 . Koo('V("Ifz»

is topologically isomorphic to the following topological subspace of

K",(17'("If1 ® "Ifz»:

A~('V("If,) ® 'V("Ifz» = {x = (xU" iZ»iIElloi2Eh E Aoc(I1 x i; 17'("IfI ) ® 'V("Ifz»1 for each Vz E 17'("Ifz), {vzUz)x(., iz)1 iz E IJ is relatively compact in K",('V("If,»} .

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(If AI (S'i) and AI (S'i z) are distinguished, we also have equality with

r.

However, if satisfies condition (M), i.e. if Koo(,V(rJ» is a (semi-) Montel space (see Proposition (1.4) (2», the situation becomes more transparent. From the preceding considerations (or from a direct argument which we omit), we obtain

r

Proposition (2.6). If l satisfies condition (M), there are the following topological isomorphisms:

(A 1(S'iI))~

®. (A1(S'iZ»~ == (A1(S'i)))~e(A 1(S'iZ))~ == s; (AI(S'iI), (AI(S'iZ»~) == Aoo(I1 X t; r(r.)0 r(rz» == K",(r(r, 0 r z)) == (A 1(d I 0 S'iz»~ == (AI(S'i I) ®1TA1(S'i~)~·

Clearly, under the hypothesis of Proposition (2.6), the complete 7[tensor product A)(S'iI)®1TAI(S'iZ)==Al(S'i10S'iz) of the Frechet-Montel space AI(S'il) with a distinguished echelon space AI (S'iz) is distinguished if and only if the complete s-tensor product (or, equivalently, the £product) of the (DFM)-space (AI(S'iI»~ == K",(r(r l » == koo(rl ) with the ultrabornological space (A.(S'iz))~ == K",(r(rz)) == k",(rz) is bornological or barrelled. (And Corollary (2.5) shows that this is true whenever r z satisfies condition (D).) But, at this point, it is interesting to observe that the method of proof of our Theorem (2.3), together with an application of results on topological tensor products from [I] and an abstract theorem (due to Hollstein [8]) which permits commuting inductive limits and s-tensor products, actually yields an affirmative solution of the last question on a hereditary property of the s-tensor product of co-echelon spaces. Le., by tensor product methods we are able to obtain a solution of the specific case of our general question on the 7T-tensor product of distinguished echelon spaces, even though a characterization of distinguished echelon spaces is not available! Proposition (2.7). If r J satisfies condition (M) and koo(r z) == K",(r(rz)) topologically, then koo(rl ) k",(rz) is ultrabornological, and hence k oo (r l 0 r z) == Koo(r(r l 0 r z)) (topologically). In particular, the complete tr-tensor product A.(S'iJ)®1TA.(dz)==A.(S'i,0S'iz) of a Frechet-Montel

®.

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space '\1(.91 1) with a distinguished echelon space '\1(.912) is always distinguished. The proof of Proposition (2.7) requires vector valued generalizations of some results in this paper. The method is similar to the one used in [4] to treat the case of functions with values in an (LB)-space (e.g. see [4, Cor. 3.3]). Therefore, we leave it to the reader to verify the details in the following Sketch of proof. (1). For a decreasing sequence r = (vn)n of strictly positive functions on an index set I and an arbitrary Banach space E, let ,(,,(E) = ~00(1, v', E) denote the (LB)-space indlim., 1 (1, vn ; E) of Evalued functions on I and define Koo(E) = Koo(I, 'V(r); E) (as well as Ko(E) = K o(l, 'V(r); E» in the canonical way, replacing the modulus in C (or IR) by the norm of E. We claim that, in the presence of condition (D), ~oo(E) == Koo(E) is true. Clearly, this requires a vector-valued version of Theorem (2.3), but it can be proved in essentially the same way as that result (see Section 3 especially Proposition (3.3) (1)~ (2), Proposition (3.7), and Lemma (3.12) to Corollary (3.14». For the last part of the proof, we note that, in the case of sequences with values in a Banach space E, it is not clear (a priori) whether ~oo(E) is complete. But a refinement of the method of proof (similar to the one used in the case of function spaces, see Section 6 below) works here, too. (2). Now, let the sequence r = (vn)n satisfy condition (M) (so that the index set I is at most countable and r satisfies condition (D) as well). From Proposition (1.4) (2) and the results of [1], we conclude the following topological isomorphisms: 00

(3). At this point, returning to the hypotheses of Proposition (2.7), we remark that, by (2) above, n_

= 1,2, ... , is an (LB)-space and hence ultrabornological. But here ~oo(rl) == Koo('V(r j » == Ko('V(r l » is a (DFM)-space by Proproposition (1.4) (2) and an s-space by [8, Prop. 2.3]. Hence, applying

m

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[8, Th. 4.1], we obtain i,,(rl)E k,fV2) == i,,(r,) 0, i,,(r2) == i,,(r1)

0, (indlim 1,,(12' v~))

== indlim (i ,,(rt ) 0, t(l2' v:')) . I.e. i,,(r1)

m-

0, i,,(r2) is uItrabornological, as we wanted to prove. 0

3. General Results; Proof of the Main Theorem In the sequel, we fix a non void system @; of subsets of the index set I which is directed upward with respect to inclusion. Clearly, the case @; = ~ = ~(I) = all finite subsets of I is of special interest, as is the case of an increasing sequence @; = ~ = (1m )mEN. Definition (3.1). Corresponding to a Kothe set pjJ on I, we associate the space Ao(!' @:i; pjJ) = {x E A,,(!, pjJ)1 for each a E pjJ, (sup (a (i)!X(i)/))SE6 converges iEIlS to 0 (i.e. for each E > 0, there is a set S E @3 with a(i)lx(i)1 < E for all i E l\S)}. Ao(1, @;; pjJ) is a closed subspace of A",(l, pjJ) which will be endowed with the induced system of seminorms. In fact, Ao(l , @3; pjJ) is the closure in A",(l, PJ) of the space !p(l, @3; PJ) of all @:i-finite sequences in A",(l, pjJ), where !p(1, @;; pjJ) = {x E A",(l, pjJ)/ for some S E @3, xl Ils == O}. If pjJ consists of a single strictly positive function a = (a(i))iEI on the index set l, we sometimes prefer to write co(!' @3; a) instead of Ao(!' @3, pjJ) and !p (l, @;; a) instead of q;(1, @;; pjJ).

Under the usual pointwise order, a Kothe set pjJ is directed upward, and hence Ao(l, @3; pjJ) == projlim co(l, @3; a). -..E~

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(The terminology here is obvious, even though slightly more general than introduced above.) The spaces Ao(I, (S; [5}) interpolate between Ao(I, [5}) and A.AI, gil) since

while, for

~

= {I},

Definition (3.2). The Kothe set [5} on I is said to satisfy condition (M, G) whenever, for each a E gil and each subset 10 of {i E II a(i) y!. O} with 10 n (I\5) y!. 0 for all 5 E ~, there exists an a' E gil with a' ~ a such that infiElo(a(i)/a'(i» = O. Condition (M, ~) will be abbreviated by (M). Clearly, in the presence of condition (M, (S) we must have I = U SEG 5 by property (2) in the definition of a Kothe set. Also, if (SI and 8 2 denote two directed systems of I and if ~I:';; 8 2 in the sense that, for each 51 E @iI' there is an 52 E @i2with51 C 52' then, obviously, Ao(I, I~\; gil) is a topological subspace of Ao(I, @i2; [5}), and condition (M, (SI) implies condition (M, (S2)' In particular, (M) implies (M, is) for each directed system @i with I = USE€: 5. At this point, we can state our first general result, a characterization of condition (M, S). Proposition (3.3). The following assertions are equivalent: (1). [5} satisfies condition (M, S). (2). 11. 0 (1, @i; [5}) = gil). (3). On each bounded subset of Aoc(I, [5}) (or, equivalently, of Ao(I, G; gil», the induced topology is given by the system (qa,StEiJ'.SEG of seminorms, where

«u.

qa.S(x) = sup(a(i)lx(i)/) iES

for all x E A.AI, [5}).

Proof. (1) ~ (2). If there is an element y E Aoc(I, gIl)\Ao(I, (S; [5}), we can find a o E [5} such that (SUPiEI\S(ao(i)/y(i)l))sEG does not converge to O. Hence there is an E > 0 so that, for each 5 E (S, it is possible to find an

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index is E I\S with ao(is)! y(is) I;?: S (whereby, obviously, ao(is) ~ 0). Now, for every a' E rJP with a';?: a o and every is' we have

where Ca,;?: SUPiEI(a'(i)/ y(i)I) is a positrve constant. Hence infiElo(ao(i)/a'(i));?: > 0 holds for 10 = {isl S E @3}, and condition (M, 6) is violated. (2) ~ (1). If condition (M, @3) is not satisfied, we can find a o E rJP and 10 C {i E II ao(i) ~ O} with 10 n (I\S) ~ 0 for all S E @3 such that infiElo(ao(i)/a'(i)) = Sa' > 0 for each a' E [JP with a';?: ao. Define y : I ~ R+ by y(i) = (ao(i))-I for i E 10 , y(i) = 0 elsewhere. Since, for every a';?: a o and every i E 10 , 1,;:;; a'(i)y(i) = a'(i)/ao(i)';:;; 1/sa" we clearly have y E Aoo(1, rJP )\Ao(I, @3; rJP). (2)~ (3). We suppose that Ao(1, @3; rJP) = Aoo(I, rJP) and fix a bounded set B C A",,(I, rJP). To show that (qa,S )aEg>,SE(; induces a stronger topology on B than A",,(1, rJP) does, let Y. E B, a E rJP and let S > 0 be given. Now yo(i) = SUPyEB Iy(i)1 for all i E I clearly defines a function Yo E Ax (1, '?P) with Iy I,;:;; Yo for all y E B. By hypothesis, Yo E Ao(I, rJP; CS), and hence there exists an S E @3 such that sUPiEI\S(a(i)yo(i)) < sl2. Putting

ac;

we have the desired neighbourhood of Y. with respect to the topology induced by (qa,S)a,S for which

un B C {y E BI sup(a(i)1 y(i) -

Yl(i)/),;:;; S},

iEI

since, for each y E U

n B,

sup(a(i)1 y(i) - Yl(i)/),;:;; max{sup(a(i)1 y(i) - y.(i)i), sup (a(i)1 y(i)- y.(i)/)} iEI

iES

iEllS

,;:;;max{q s(y- YI)' sup(a(i)ly(i)/)+ sup(a(i)jYl(i)1)} a,

iEIIS

,;:;; maxjs, 2 sup (a(i)yo(i))}';:;; e . iEIIS

iEIIS

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(3)? (2). We suppose that (qa.S)a,S induces the topology (and hence the uniform structure) of Aoo(1, g» on each (absolutely convex) bounded subset of Ao(1,~; g» and fix x E A,,{l, g». Defining xs(i) = x(i) for all i E Sand xs(i) = 0 elsewhere, we note that, for Xs = (XS(i))iEI' (XS)SEI5 is a net of functions in q;(1,~; g» C Ao(l,~; g». By IXsl ~ Ixl for arbitrary S E~, (xs)s is bounded in Aoo(l , g», and hence also in the topological subspace Ao(1,~; g». Moreover, (xs)s is a Cauchy net in the topology given by the system (qa.S)a,S of seminorms. But (xs)s converges to x, whence x E Ao(1,~; g», i.e. Aoo(l , g» = Ao(1,~; g». 0 Note that, if each a E g> is bounded on every set S E g and if, for each there is an a E g> with infiEsa(i»O, then the system (qa,S)a,S of Proposition (3.3) (3) induces the topology of uniform convergence on all sets S E~. In particular, the case ~ = i5 of Proposition (3.3) yields a well-known result (e.g. see [5, Cor. 4.5]). SE~,

Corollary (3.4). The following assertions are equivalent: (1). g> satisfies condition (M). (2). Ao(1, g» = Aoo(l, g». (3). Aoo(l, g» (or, equivalently, Ao(1, g») is a semi-Mantel space. Remark (3.5). For a given Kothe set g> on 1, let us denote the Kothe set

Aoo(l , g»+ = {a

= (a(i))iEI E R ~I for each

a E g>, sup(a(i)a(i)) < oo} iEI

by pj. Then the following holds (1). A set B C Aoo(1, g» is bounded if and only if there exists an a E pj with SUPxEBlx(i)1 ~ a(i) for all i E 1. (2). Ao(1,~; pj) = Aoo(l, pj) implies Ao(1,~; g» = Aoo(1, g», and hence g> satisfies condition (M, ~) whenever pj does.

Proof. (1). If a E pj, then B = {x = (x(i))iElllx(i)1 ~ a(i) for all i E I} is a bounded subset of Aoo(l , g» (by definition of pj). Conversely, if Be Aoo(l , g» is bounded, then a(i) = SUpxEB Ix(i)1 for each i E 1 clearly defines a function a E pj with the desired property. (2). We suppose that Ail, ~; pj) = A,,{l, g». Fix y E Aoo(l, g» and a E g>. But a E Aoo(1, pj) by definition of pj, and thus our assumption implies a E Ail, ~; pj). Now, since Iy IE Aoo(l , g»+ = rJP, (suPiEI\s(a(i)/ y(i)I))SEI5

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converges to 0; i.e. y E A. o(1, @;; 9»). The second assertion follows from the first one in view of Proposition (3.3), (l)~ (2), applied to both g> and &>.

o

The converse of Remark (3.5) is true whenever g> is countable. Thus, we make our first countability assumption at this point; in the sequel, we will let .st1 = (an)nEN denote a (strictly positive) Kothe matrix on 1. .st1 satisfies condition (M, @;) if and only if, for each n E N and each subset 10 of 1 with 10 n (1\5) oi 0 for all 5 E @;, there exists an m ~ n such that infiEIo(an(i)/am(i» = O. In particular, condition (M) = (M, lS) for .st1 = (an)n is nothing else but condition (M) of Section 1 for the associated decreasing sequence r = (v n)n' and the special case of Corollary (3.4) yields (i)~ (ii'j e> (iii) of Proposition (1.4) (2). Next, for g> =.st1, rjj = A. (1, g»+ was denoted by 'V = 'V('V), and in analogy with the previous terminology (introduced in Subsection 1.3), we will now abbreviate A. o(1, @;;.st1) by A. o(@; ) and will write K o(@; ) = K o(1, @;; 'V) instead of A. o(1, @;; &» (and = K oo(1, 'V) instead of A. (1, &»). 00

x:

00

Proof. In view of Remark (3.5) (2), it remains to show (I):} (2). Fixing y E K; and is E 'V, we note that, because of K; = k oo algebraically (see Theorem (1.2) (2», there exists an noEN with SUPiEI(V'l(P) I y(i)I) < 00. On the other hand, is E r = (A. oo)+ = (A. o(@; » + by our assumption, and hence (SUPiEIIS(a/l()(i)ii(i)))SE@; converges to O. At this point,

v(i)/ y(i)j

=

v/I()(i)/ y(i)la/l()(i)ii(i)";; (sup(v/I()(i)! y(i)j)a/l()(i)v(i), iEI

for all i E I, clearly implies y E K o(@;). 0 By the way, the special case @; = lS of Lemma (3.6) yields (ii)~ (ii') in Proposition (1.4) (2), and the equivalence with (iv) in that proposition also is a special case of our next result, which immediately follows from Proposition (3.3) and Lemma (3.6). Proposition (3.7). The following assertions are equivalent: (1). .st1 satisfies condition (M, @;). (2). Ao(@;) = A. oo•

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(3). K o(® ) = «; (4). satisfies condition (M,6). (5). On each bounded subset of K; (or, equivalently, of K o(6 » , the induced topology is given by the system (qV.s)VEr.sE~ of seminorms, where qv,s(x) = sUPiEs(ii(i)lx(i)/) for all x E K""

r

This ends our discussion of the more general form (M, 6) of the second part (M,~) of condition (D). (Note that (M, 6) was formulated in terms of the Kothe matrix .sI1 = (an)n while we had preferred a formulation in terms of the associated decreasing sequence "Jf in the last section.) Let us now turn to the first part (N,~) of condition (D). Again, we begin with a generalization, but, after a few remarks, we will need (two) countability assumptions to establish our main results. Definition (3.8). A Kothe set ~ on 1 is said to satisfy condition (N,6) whenever, for each S E ®, there exists an a E rJ'J with al s strictly positive such that, for each a' E ~ with a';;a: a, infiEs(a(i)/a'(i»? O. If 6 1 and 6 2 denote two directed systems of subsets of 1 and if 6 1 ::;; 6 2 (in the sense of the remark after Definition (3.2», then condition (N, 6 2)

m

implies condition (N, 6J Note that condition (N, is always satisfied while, for 6 = {I}, (N, 6) holds if and only if Aoo(1, ~) is a Banach space. In our next remark, we use the following terminology: If S is a subset of I, the sectional subspace A",(I, rJ'J)ls is the space {xis Ix E A",(I, ~)}, endowed with the l.c. topology induced by the seminorms (q~tE!1" where q~(x) = sup(a(i)lx(i)/)

for all x E A",(1, rJ'J)/s'

iES

The sectional subspace Ao(I, ®; f!I»/s is defined in the same way.

Remark (3.9). The following assertions are equivalent: (1). f!I> satisfies condition (N,6). (2). For each S E 6, there exists an a E f!I> with al s strictly positive such that the function a = a-lIs, defined by a(i) = a(ifl for i E Sand a(i) = 0 elsewhere, belongs to tj) = 11. (1, ~)+. (3). For each S E 6, the sectional subspace A",(1, f!I»ls (or, equivalently, Ao(I, 6; f!I»ls) is norm able. I.e. there exists an a E f!I> with al s strictly 00

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positive such that the norm q~ induces the topology of Aoo(I, 9Jl)ls (resp. Ao(I, (S; 9Jl)1 s). This is quite obvious. By the way, it clearly follows from Remark (3.9), (1)<:> (2) that rIP satisfies condition (N, (S) whenever 9Jl does . . At this point, we again make our first countability assumption. The (strictly positive) Kothe matrix d = (an)nEN on I satisfies condition (N, G) if and only if, for each 5 E (S, there exists an n E N such that, for each m ~ 11, infiEs(an(i)/am(i» > O.By Remark (3.9)(2), this holds if and only if, for each 5 E @S, there exists an n EN such that vn Is E r = r('V). Moreover, similarly to Remark (3.9) (3) (and with analogous terminology), we also have equivalence with the following assertion: For each 5 E 13, the sectional subspace Atl s = At(I, d)ls is normable. Next, by the remark following Remark (3.9), condition (N, (S) for sf implies that satisfies (N, G), too. I.e. for each SEe;, there is a v E with vis strictly positive such that, for every v' E with v' ~ v, infiEs(v(i)/v'(i)) > a or, equivalently, for each 5 E G, the sectional subspace Kooi s (resp. Ko(@S)ls) is normable. However, this can be improved as follows:

r

r

r

Remark (3.10). If the Kothe matrix sf = (an)n on I satisfies condition (N, G), then, for each 5 E G, there exists an n EN such that the sectional Vn Is) subspace K,,,ls (resp. Ko(CS)ls) has the norm topology of 1 (= 1 vn)/s)· 00(5,

00(1,

In fact, this follows directly from Remark (3.9), (1) ~ (2), and this remark even provides a converse to Remark (3.10). Definition (3.11). For the associated decreasing sequence 'V = (v n )nEN ' o; = a~l, we put

n....

I.e. ~ o(G) is the increasing union of the Banach spaces co(I, G; vn ) , endowed with the strongest I.e. topology under which the injection from each of these Banach spaces is continuous. The spaces ~o(G) interpolate between ~o and ~oo since ~oOS) == ~o

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while k O(t5 ) = k", for 8= {I}. If S, and 8 2 denote two directed systems of subsets of I with 8 1 :s:; S2' then ko(SI) is continuously embedded in ko(Sz); in particular, for arbitrary S, the canonical injection of ko(S) into k oo is continuous, and if 1= USEe S, then k O is continuously embedded in ko(S). Moreover, each Banach space coU, ®; Vn) is continuously embedded in co(I, S; v) for arbitrary v E Of, and this induces a continuous injection ko(6)~ K o(6). Putting 'IO(S) = 'IOU, S; =

u

or) = {x E

k x = koo(I,

or)1 for

some S E S, xlI\s

== O}

'IO(I, S; Un),

nEN

we have '10m) = 'IO(I) and 'IO(S) = k oo for S = {I}. Clearly, '10(6) is (sequentially) dense in ko(S) for arbitrary 6, but, in general, kO(S) is not the closure of '10(8) in k oo• (See Theorem (1.5) for the case S = is.) Now we are ready to make our second countability assumption; instead of considering general directed systems (5 of subsets of the index set I, as we have done so far in this section, we will fix an increasing sequence ~ = Um )mEN of sets 1m C I at this point and study the corresponding condition (N,~) (which is nothing else but condition (N,~) of Section 2, reformulated in terms of the Kothe matrix JIi). The following is our main lemma in this direction; it contains the essential idea. Lemma (3.12). If sti = (an )nEN is a Kothe matrix on I which satisfies condition (N, ~) (i.e. for each mEN, there is an nm;?; m such that, for each k ;?; nm, inf iElm(anm(i)/ak(i» > 0), then ko(~) and Ko(~) induce the same topology on '10 (~). Proof. We first remark that, replacing the increasing sequence (am )mEN by (anJmEN' if necessary, we may assume that nm = m in condition (N,~) so that, for each mEN and each k ;?; m, infiEl,.(am(i)/ ak(i» > O. Next, put 8k = inf (ak(i)/ak+I(i»

iElk

> 0,

k

=

1,2, ... ,

and fix a neighbourhood U of 0 in ko(~)' Without loss of generality, we may take

U = absconv ( U

kEN

ekB~),

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where Ck > 0, k = 1,2, ... , B~ = caU, ~; vk ) , denotes the closed unit ball of caU, ~; vk ) , and absconv stands for the absolutely convex hull. Inductively, we can now choose an increasing sequence (ak)kEN of positive numbers with

k = 1,2, .... Then, for each i E lk (k

=

1,2, ...),

and hence

Putting

and observing that (lk )nEN is increasing, we conclude k = 1,2, ....

After these preparations, it suffices to verify

{x E cp (~)I sup(v(i)lx(i)/)";; I} CU. iEI Fixing xEcp(~) with sUPiEI(v(i)lx(i)/).,;;l, there exists a KEN such that xCi) = 0 for all i E I\IK' For each i ElK' however, there is a smallest integer 1.,;; ui,« K with

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Putting In={iEIKlk(i)=n}, n=l, ... ,K, it is obvious that I K= U ~;I In and In n 1m = 0 for n -# m. Now, if n E {I, ... , K} and i E In' then

thus vn(i)!x(i)/.s;;; (~YEn' Therefore, x n = (Xn(i»iEJ' defined by xn(i) = 2nx (i ) for i E In and x n(i) = 0 elsewhere, clearly belongs to EnB~, n = 1, ... , K, and the proof is finished by noting that K

X=

L

Gtxn E absconv

n=l

(u EnB~)

=

U.

0

nEN

Theorem (3.13). If .st1 is a Kothe matrix on I which satisfies condition (N, :S), then ko(:S) is a topological subspace of Ko(:S), and hence

Proof. Since
The proof is immediate from Theorem (3.13); see the following diagram:

k",4K",

1

u top.

ko(:S) c Ko(:S)' top. For 1 = !Ill and :s = (Im )mEN' 1m = {1, ... , m}, any Kothe matrix .st1 on N satisfies (N, :S), while ko(:S) == k o and Ko(:S) == K o, and hence K o == 20 and

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the fact that k o is a topological subspace of k oo follow from Theorem (3.13) and Corollary (3.14). As remarked in Theorem (1.2) (I), (3), these results remain true for an arbitrary (not necessarily countable) index set 1. However, Theorem (3.13) (resp. Corollary (3.14)) clearly extends Theorem (1.2) (I) (resp. (3)) for 1= N considerably. And it is such an extension which we need in the following Proof of Theorem (2.3). In Theorem (2.3), we assume condition (D), hence there exists an increasing sequence ~ = (Im )mEN of subsets of the index set I such that the Kothe matrix .s'i = (an)nEN satisfies both (N,~) and (M, ~). Since k is complete (by Theorem (1.3) (2)), we can infer from Theorem (3.13) and Corollary (3.14) that 00

--------..

Ko(~) == ko(~) == koo-c1os kO<~)

is a topological subspace of k oo' But it follows from Proposition (3.7) that Ko(~) == K oo ' and since the latter space contains k oo (it even equals k oo algebraically), we have the desired conclusion k oo == K oo ' 0 We finish this section with some (simple) remarks on the regularity, completeness and bounded retractivity of koo(G) = indlim,.... co(I, is; vn ) . The following proposition generalizes (1)<:> (4)<:> (5) in Theorem (1.5), and its proof easily follows from that result. Proposition (3.15). For a directed system is of subsets of I with 1=

U SE$ S, the following assertions are equivalent: (1). "fI' is regularly decreasing. (2). k o(is) == indlim,.... co(I, G; vn ) is a boundedly retractive inductive limit

(and hence complete). Proof. First, we note that, since G covers I, we have the following inclusions of topological subspaces:

Next, from Theorem (1.5), we know that "fI' is regularly decreasing if and only if koo==indlimn.... loo(I, vn ) (or, equivalently, ko==indlim n .... co(I, vn )) is a boundedly retractive inductive limit. But, clearly, whenever E == indlim,.... En is an injective boundedly retractive inductive limit of I.e.

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spaces and (Fn)n is an inductive sequence of topological linear subspaces Fn C En' n = 1,2, ... , then F == indlim,.... F n is boundedly retractive, too. Hence indlim,.... I"'(I, Vn) boundedly retractive implies indlim;.... Co (I, ®; Vn) boundedly retractive while, conversely, the bounded retractivity of ~o(®) implies the same property of ~o == indlim,.... Co (I, vn ) . 0 Corollary (3.16). Let r denote a regularly decreasing sequence. For arbitrary n E N, m n ~ n is chosen as in condition (ON). If ~ = (I/)/Er, is all increasing sequence of subsets of I such that,

(*) for each lEN, there exists an n then

~o(~)

=

n(l) EN with inf (vm/i)/vn(i» > 0, iEII

== Ko(~) (algebraically and topologically).

Proof. In view of (ON), (*) implies infiEI,(Vk(i)/Vn(i» > a for all k ~ m n, and hence, obviously, infiE1/an(i)/a k(r) = infiE1/(vk(i)/vn > 0 for all k ~ n, I = 1,2, ... ; i.e. the corresponding Kothe matrix.:!f...:::.. (an)n satisfies condition (N, ~). Now Theorem (3.13) yields Ko(~) == ~o(~); but ~o(~) is complete by Proposition (3.15). 0

(i»

It is evident from Proposition (3.15) that the bounded retractivity of ~o(e) == indlim,.... co(I, ®; vn ) does not depend on the directed system ® of subsets of I (with 1= U SE6 S). However, regularity and completeness for these spaces behave in a completely different way. From Section 1, we know that ~om:) == ~o is complete (or, equivalently, a regular inductive limit) if and only if r is regularly decreasing while, on the other hand, for ® = {I}, ~o(®) == ~'" is always complete (and a regular inductive limit).

Appendix In this appendix, we complement our discussion of distinguished echelon spaces by presenting the example, due to Kothe [9, I] and Grothendieck [6], of a space Al = A\(N x N,.sIl) which is not distinguished, together with a reformulation of the classical proof in our terminology.

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Proposition. Let sd = (a k )kEN denote a (strictly positive) Kothe matrix on an index set 1. If there exist 11 E N and a decreasing sequence (lk subsets of I such that, for each k ~ n,

)kEN

of

then Al = AI(I,.st/.) is not distinguished.

Proof. Omitting a" ... , an-I' if necessary, we may take n = 1. To simplify notation, let us now assume that a l is identically I, this amounts to dividing by al' and also clearly involves no loss of generality. Then our hypothesis reads as follows: There exists a decreasing sequence (lk )kEN of subsets of I such that, for each kEN

while, for some lk ~ k, (2) inf V'k (i)

=

0.

IEJk

With the sequence

(ckhEN

of positive numbers defined by (1), we put

B~ = ckl",,(I, Vk)l' k = 1,2, ... , and

U

= absconv

(u B~) . kEN

Clearly, U is a neighbourhood of 0 in k"" == indlim,.... /",,(1, vk ) , and we would show that U is not a neighbourhood of 0 relative to (AI)~ == Ki; Then, in view of Theorem (1.3) (3), the proof is finished. Let us first derive a property that all elements of U share: For each x E U, there is a kEN with SUP'EJk Ix(i)I::so; 1. In fact, x E U has a representation x = L :=1 'Y,x" where kEN, x, E B: and L ~=l /'Y,I::so; 1. Now, for all / E {I, ... , k} and all i o Elk'

/x,(io)/

=

v,(io)lx,(io)/a,(io)::SO; (sup(v,(i)lx,(i)/»a,(io)::SO; c,o,(io) lEI

=

(inf v,(i»a,(io)::SO; (inf v/(i)a,(io)::SO; v,(io)a/(io) = 1, iEJI

iEJk

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whereby k

2:

k

2:

sup/x(i)/ ~ 11'11 sup Ixl(i)1 ~ 11'11 ~ 1 . iElk 1= I iElk 1=1

For the rest of the proof, fix i5 E 'Y. We can take i5 of the form infw,lll'kvk) with ll'k > 0, k = 1,2, .... We claim that there always exists an x = (X(i))iEI E k oo with SUPiEI(i5(i)lx(i)l):::; 1 such that, for every kEN, there is an ik E Jk with Ix(ik)1 ~ 2. If this claim is established, it follows that {x E kool SUPiEI(V(i)lx(i)i) ~ I} fL U,

i.e. U is not a neighbourhood of 0 relative to K oo, as we wanted to show. Hence it remains to construct a function x = (X(i))iEI with the desired properties. Given kEN, let us choose lk ~ k according to (2). Clearly, there exists an t, E i, with Vlk(ik) < ~ll' ~I, and hence V(ik) ~ ll'hVlk (id ~~. Now, with 10 = {iI' i 2 , • • •}, we put x(i) = 2 for i E 10 and x(i) = 0 elsewhere. Then x = (X(i))iEI E loo(I, 1) = loo(I, VI)' i.e. x E k"" with SUPiEI (v(i)lx(i)1) = SUPkEN 2V(ik) ~ 1, while X(ik) = 2 for each ik E Jk, k = 1,2, .... 0 Example (Kothe-Grothendieck), Letting 1= N x N, we put SI1 where

i ~ k, i~k+l,

=

(ak)kEN'

(i, j EN, k = 1,2, ...).

For J k = {(i, n! i ~ k + I}, (Jk )kEN is a decreasing sequence of subsets of I and, for each k = 1,2, ... ,

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We see that the hypothesis of the preceding proposition is satisfied, hence A!(N x N, d) is not distinguished.

Il. Function Spaces 4. Notation and Preliminaries From [4], we recall some definitions and results. In the sequel, we will let X denote an arbitrary completely regular (Hausdorff) space. For spaces of continuous functions on X, the directed system = C£(X) = all compact subsets of X takes the role that ~ = ~(I) played for sequence spaces.

cr

4.1. Nachbin Families and Weighted Spaces A nonnegative real valued function on X which is upper semicontinuous (u.s.c.) will be called a weight on X. If 611 und "fI are two sets of weights on X, we write 611 ".;; "fI whenever, corresponding to each u E 611, there exist a v E "fI and a A > 0 such that u ".;; AV (pointwise on X). A set "fI of weights on X is said to be directed upward provided that, for every pair VI' Vz E "fI and every A > 0, it is possible to find a V E "fI so that AV! ".;; v and AVz "';; v. Since there will be no loss of generality, we shall hereafter assume that sets of weights are directed upward. If a set of weights on X also satisfies "fI> 0, i.e. if, given any x E X, there is some v E "fI for which v(x) > 0, then "fI will be referred to as a Nachbin family on X. The system JC = JC(X) = {A lKI A> 0, K C X, K compact}, where lK denotes the characteristic function of the set K, will be helpful to us in what follows. In the sequel, ~(X) indicates the collection of all continuous complex (or real) valued functions on X. Corresponding to each Nachbin family "fI, we associate the following weighted (Nachbin) spaces: ~"fI(X) = {fE ~(X)I qv(f) ~"fIo(X) =

= sup(v(x)lf(x)i) < 00 for each

v E "fI},

xEX

{fE ~(X)I for each v E "fI, vf: x~ v(x)f(x) vanishes at infinity} .

vf is said to vanish at infinity if for each e > 0, there is a compact set K = K(e) C X with v(x)lf(x)1 < e for all x E X\K.

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Under the system {qvl v E V} of seminorms, ~'V(X) is a (Hausdorff) I.c. space, while ~'Vo(X) is a closed subspace of ~'V(X) which will be endowed with the induced system of seminorms. We note that ~'V(X) is complete whenever X is a kR-space and J{ ~ 'V. Also, ~'Vo(X) is the closure in C€'V(X) of the space ~c(X) of all continuous functions on X with compact support provided that X is locally compact or 'V consists only of continuous functions on X. Note that, in the terminology of Part I, each Nachbin family 'V on X is a Kothe set on X, and we have ~'V(X) = ~(X) n A",,(X, 'V) and ~'Vo(X) = ~(X) n Ao(X, (£; 'V) with the topology induced by A",,(X, 'V). Also, ~c(X) = ~(X) n q;(I, (£; 'V). More generally, if 'V is a Nachbin family on X and d(X) is some predetermined linear subspace of ~(X), we put d'V(X) = d(X) n cg'V(X) and d'Vo(X) = d(X) n cg'Vo(X) with the induced weighted topology. For a kR-space X and J{ ~ 'V, these spaces are complete whenever d(X) is closed in ~(X) under the compact-open topology co. If 'V = {Avl A > O} for a single strictly positive weight v on X, we will write ~v(X), ~vo(X), dv(X), and dvo(X) instead of ~'V(X), ~'Vo(X), d'V(X), and d'Vo(X), respectively.

4.2. Weighted Inductive Limits and their Projective Hulls For a decreasing sequence 'Yd = (vn )nEN of strictly positive weights V n on the completely regular space X and a subspace d(X) of cg(X) we put

The I.c. inductive limit topologies on these spaces are Hausdorff, and the canonical injection 'Vdod(X)~ 'Ydd(X) is continuous. For the case d(X) = ~(X) of this definition, we should again note that, in the terminology of Part I, we have

and

Clearly, the inclusion mappings continuous.

'Vdcg(X)~ k oo

and

'VdO~(X)~ k O«(£)

are

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199

At this point, we define the Nachbin family 'V = 'VYd on X by - = { i5: X ~ R+ u.s.c.] sup v(x) } "fI - < 00 for each n EN. Yd xEX vn(x)

Note that, taking X as an index set, we had introduced the Kothe set 'V("fId) associated with the decreasing sequence "fId = (vn)n (d. Subsection 1.3). Obviously, 'VYd C 'V("fId) while each i3 E 'V("fId) is dominated by a function in 'Vr, -viz. v is dominated by a function of the form ~ = infnEN(anvn) with an> 0, n = 1,2, ... , and since all weights un are u.s.c., we have ~ E 'V'Yd' Hence we obtain

as well as cg'V(X) = cg(X) n A",(X, 'V) = cg(X) n K"" cg'Vo(X) = cg(X)

n Ao(X,~; 'V) = cg(X) n Ko(~)'

As a consequence, taking 'V = 'V'Yd in the sequel will not give rise to any confusion with the terminology of Part 1. Some attention will be given _ _ to the subset 'Vcont = ('V'Yd )cont consisting of all continuous functions in 'Y = "fI'Yd' If X is locally compact and o-compact and 'Vd = (vn)n is a decreasing sequence of strictly positive continuous functions on X, we can prove 'Vcont = 'V; i.e., not only 'Vcont C 'V, but also 'V ~ 'Vcont (see [4, 0.2, Prop.]). We remark that, according to our conventions, the spaces d'V(X) and d'Vo(X) are defined. It is easy to see that 'Vdd(X) can be continuously injected into d'V(X), as can 'Ydod(X) into d'Vo(X). 4.3. Some General Results

Choosing d(X) = cg(X) in Subsection 4.2, we have the following result [4, Section 1]. Theorem (4.1). Let X be a locally compact space. (1). 'YdO~(X) is a topological subspace of ~'Vo(X). In particular, the l.c. inductive limit topology of 'YdOcg(X) is the weighted topology induced by the system (qi;)ijEf" of seminorms. If, in addition, infxEK vn(x) > holds for each

°

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K.D. Bierstedt, R. Meise / Distinguished Echelon Spaces

compact subset K of X and n = 1,2, ... , then we obtain
t

u top.

'YdO
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201

In comparison with Theorem (1.3) (2), it should be remarked that, due to the lack of a satisfactory duality theory in the case of spaces of continuous functions. it is not known if, say, the hypotheses of the last part of Theorem (4.1) (1) are sufficient for the completeness of 'V/€(X). On the other hand, the following result holds [4. Th. 1.13].

Proposition (4.3). Let X denote a locally compact space and take 'Vd = (vn)n to be a decreasing sequence of weights on X with inf x E K vn(x) > 0 for each compact subset K of X, n = 1,2, .... If the linear subspace d(X) is closed in (cg(X), co) and if (d(X), co) is a semi-Montel space, then 'V~(X) is the barrelled space associated with d'V(X) and hence is complete.

5. Some Additional Results In the sequel. we fix a decreasing sequence 'Vd = (Vn)nEN of strictly positive weights on a completely regular space X, and we assume that infx E K vn(x) > 0 for each compact subset K of X, n = 1,2, ... (our assumption is clearly satisfied if all the weights Vn are continuous). After having surveyed the general situation in the last section, we now turn to conditions on the decreasing sequence 'Vd = (v n )n' First, we would mention results parallel to Proposition (1.4) (1) for spaces of continuous functions, even though they are not as significant as in the case of sequence spaces since, in general. the property involved is much weaker than 'Vdcg(X) being a (DFS)-space. Here is the function space analog of condition (S): (S'). For each n EN, there exists an m > n such that vm/vn vanishes at infinity.

Remark (5.1). If 'Vd = (vn)n is a decreasing sequence of strictly positive continuous weights on X, then the following assertions are equivalent: (1). satisfies condition (S'). (2). 'Vdcg(X) = 'VdOcg(X). (3). For each n E N. there is an m > n such that cgvm(X) induces the co topology on each bounded subset of cgvn(X). Moreover, if X is locally compact, (1}-(3) are also equivalent to: (3'). For each n E N, there is an m > n such that cg(vmMX) induces co on each bounded subset of cg(vnMX).

r,

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In fact, (1)~ (2) is trivial, and for (1)~ (3), see [4,004]; it remains to note that, for a locally compact space X, an obvious modification of the proof of [4,0.4, Prop.] yields (3')~ (1). We briefly remark in passing that, under the conditions of Remark (5.1), taking un = l/vn, n = 1,2, ... , defines a Nachbin family OIl = {AunI A > 0, n EN} on X for which the weighted spaces cgOU (X) and cgOUo(X) are metrizable. We leave it to the reader to reformulate condition (S') and its equivalence to properties (3) and (3') of Remark (5.1) in terms of the increasing sequence (un)n and the generating projective sequences (cgun(X»n and (cg(unMX»n of cgOU(X) and cgOUo(X), respectively. Clearly, (S') implies cgOU(X) == cgOUo(X), but the converse does not hold (d. Proposition (1.4) (2», Now, if 1fd = (vn)n is a decreasing sequence of weights on X with inf xE K vn(x) > 0 for each compact subset K of X, n = 1,2, ... , if 1fd satisfies condition (S') and (d(X), co) is a semi-Montel space, then 1fdd(X) == 1fdod(X) is a (DFS)-space. Using an open mapping lemma due to A. Baernstein, it then follows that~1fdod(X) is a topological subspace of 1fdOcg(X) == 1fdcg(X), and thus we obtain from Theorem (4.1) (1) (cf. [4, Th. 1.6])

Corollary (5.2). Let X denote a locally compact space. If 1fd = (vn)n satisfies condition (S') and (d(X), co) is a semi-Montel space, then 1fdd(X) == 1fdod(X) == d'Vo(X) == d'V(X) (algebraically and topologically).

If, under the hypotheses of Corollary (5.2), all the functions V n are continuous and OIl denotes the Nachbin family {AV~ll A> 0, n EN}, then dOll(X) == dOUo(X) is a Frechet-Schwartz space. 5.1. Condition (M') and its Consequences

Next, we turn to a discussion of the analogue (M') of condition (M) for function spaces: (M'). For each n E N and each subset Y of X which is not relatively compact, there exists an m = m(n, Y»n with infYEy(vm(y)/vn(Y»=O. Comparing with Definition (3.2) (also, see the discussion after Remark (3.5», we realize that (M'), formulated in terms of 1fd = (vn)n' exactly equals condition (M, <£') for the associated increasing sequence (un)n' un = V~l, where <£' again denotes the system of all compact subsets of the completely regular space X.

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It is easy to see (d. the corresponding result for (M) and (S) [5, Prop. 4.8]) that the sequence 'Vd satisfies condition (S') if and only if it is regularly decreasing (in the sense of Subsection 1.6) and satisfies (M'). Our first result concerning (M') is parallel to (part of) Proposition (1.4) (2).

Proposition (5.3). If 'Vd = (vn)n denotes a decreasing sequence of strictly

positive continuous functions on a locally compact space X, the following assertions are equivalent: (1). 'Vd satisfies condition (M'). (2). 'Y = 'Y'Yd satisfies condition (M') = (M, <:£) (I.e. for each v E 'Y and each subset Yof {x E XI v(x) ¢ O} which is not relatively compact, there is a ee 'Y, i3'~v, with inf yEy(v(y)Ii3'(y)) = 0).

(3). ~'Y(X) = ~'Yo(X). (4). ~'Y(X) (or, equivalently, ~'Yo(X)) induces co on each bounded subset.

Proof. Since (M') = (M, <:£) and each v E 'Y('Vd) is dominated by a function in 'Y'Yd' (1)~ (2) follows from Proposition (3.7). Furthermore" in view of ~'Y(X) = ~(X) n K., = ~(X) n K",(X, 'Y) and ~'Yo(X) = ~(X) n K o(<:£ ) = ~(X) n Ko(X, <:£; 'Y), we also obtain (1) ~ (3), (4) from that result. (Note that, by our general assumptions on 'Vd, the topology of ~'Y(X) must be stronger than the compact-open topology.) To show (3) ~ (1), we suppose that 'Vd does not satisfy condition (M'). Then we can find an n E N and a subset Y of X which is not relatively compact such that infyEy(v m (y )Ivn(y)) = Em > 0 for each m > n. Now

defines an element

v E 'Y with for all y E Y.

Since Vn is continuous with (iilvn)/y ~ 1, we have 1/vn E ~vn(X) C 'Vd~(X) C ~'Y(X), but Ilvnfi. ~'Yo(X), i.e. we arrive at a contradiction to (3). Finally, to prove (4)~ (3), we suppose that fE ~'Y(X)\~'Yo(X). Then there exist a v E 'Y and an E > 0 such that, for each K E <:£, we can find

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XK E X\K with v(xK)/f(xK)1 ;a. E. Next, construct a continuous function gK on X with 0 ~ gK ~ 1, gK(XK) = 1 and gKIK == O. (Since X is locally compact, we can assume that gK has compact support.) Defining fK = gKf, we have fK E (~"r(X) (resp. even fK E ~c(X) c ~'Yo(X» and v(xK)lfK(xK)I;a. E, fKIK == 0 and IfKI ~ IfI for each K E~. (fK)KErr is a bounded net in ~'Y(X) (resp. ~'Yo(X» which clearly converges to 0 in the compact-open topology, but does not converge to 0 in the weighted topology, a contradiction to (4). 0 Note that the additional hypotheses of Proposition (5.3) were not used in the proof of (1)~ (3), (4). (In fact, only for (3)~ (1) we have made use of the assumption that all functions Vn are continuous, and the local compactness of X is only necessary to show the implication (4)~ (3) in the case of the space ~'Yo(X).) Hence we obtain Corollary (5.4). Let the sequence 'Vd = (vn)n satisfy condition (M'). If each bounded subset of (d(X), co) is precompact, the same holds for d'Y(X) == d'Vo(X). Here are the corresponding 'dual' results. Proposition (5.5). Let (un)n denote an increasing sequence of strictly positive continuous functions on a locally compact and a-compact space (i.e. un = V~1 for a decreasing sequence 'Vd = (vn)n) and put 6IL = {Aunl A > 0, n EN}. Then the following assertions are equivalent: (1). (un)n satisfies condition (M') = (M,~) (i.e. for each n E N and each subset Y of X which is not relatively compact, there is an m = m (n, Y) > n with inf y E Y (un(y)/um(y» = 0). (2). ~6IL (X) = ~6ILo(X). (3). <€6IL(X) (or, equivalently, ~6ILo(X» induces the compact-open topology on each bounded subset. Proof. Since

= ~(X) n A", = ~(X) n A",(X, (un)n)

and ~6ILo(X) = ~(X) n Ao(~) = ~(X) n Ao(X, ~; (un)n)' (1)~ (2), (3) holds by Proposition (3.3). Furthermore, following exactly the proof of (4) ~ (3) in Proposition (5.3), we obtain (3) ~ (2). (2) ~ (1). Thus, suppose that (un)n does not satisfy condition (M'). Then we find an n E N and a subset Y of X which is not relatively compact with infYEy(un(y)/um(y» = em >0 for each m >n. As in the proof of Proposition (5.3), (3)~ (1), letting 'Vd = (Vn)n' Un = U~I, we can construct ~6IL(X)

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i3 E 'Y =' 'Y'Vd with unV Iy ;;. 1. But since all the functions vn are continuous and X is locally compact and o-compact, i3 is dominated by a continuous function v' E 'Y. At this point, we get the desired contradiction to (2): v' E 'Y clearly implies v' E Cfd6IJ(X), but v' g Cfd6IJ o(X) because of unv'ly;;.1.

o

Again, part of the preceding proposition holds without the additional hypotheses. In fact, for the implication (1) =? (2), (3) of Proposition (5.5), the countability assumption on the Nachbin family 611 can also be dropped since Proposition (3.3) is true for arbitrary Kothe sets PfJ. I CoroUary (5.6). Let 611 denote a Nachbin family on X with JC = JC(X):';:; 611. (1). If6IJ satisfiescondition (M') =' (M, 1J)( in the sense ofDefinition (3.2)), then Cfd6IJ (X):::= Cfd6IJoCX), and Cfd6IJ (X) induces co on each bounded subset. (2). In particular, let the Nachbin family 611 on X satisfy condition (M'). If each bounded subset of (d(X), co) is precompact, the same holds for d6IJ (X):::= d6IJ o(X), and then d6IJ (X):::= d6IJ o(X) is a semi-Montel space whenever it is quasi-complete. 5.2. Regularly Decreasing Sequences 'Yd

In contrast to (S') and (M') which arise from (S) and (M), respectively, by replacing the finite subsets of the index set I by the compact sets in the completely regular space X, it also makes sense for spaces of continuous functions to study the regularly decreasing condition (ON) in the form of Subsection 1.6 (i.e. without any change). There is an analog of Theorem (1.5) (see [4, Section 2]). Theorem (5.7). Let X denote a locally compact space and 'Yd = (vn)n a decreasing sequence of strictly positive continuous functions on X. Then the

following assertions are equivalent: (1). 'Yd is regularly decreasing (i.e. for each n E N, there exists an m ;;. n such that,Jorany subset Yo! X with infYEy(vm(y)/vn(Y))>O, we also have infyEY(vk(y)/vn(y)) > 0 for all u » m). (2). 'YdCfd(X) =' indlim,... Cfdvn(X) is a boundedly retractive inductive limit (and hence complete). (3). 'YdOCfd(X):::= indlim,... Cfd(vn)(X) is a regular (or, equivalently, boundedly refractive) inductive limit. (4). 'YdoCfd(X) is complete. (5). 'YdOCfd(X):::= Cfd'Yo(X).

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(6). 'YdO'e(X) is closed (or, equivalently, stepwise closed) in 'Yd'e(X) (i.e. 'YdO'e(X) = 'Yd'e(X)-clos 'ec(X».

We note that "one half" of Theorem (5.7)--viz. the implication (1):::;> (2), (3), (4), (5) and (6)--holds without the hypothesis that all functions vn are continuous; only the local compactness of X (and infx E K vn(x) > 0 for each compact subset K of X, n = 1,2, ...) is required for this part. We will now add the following dual result. Proposition (5.8). Let X denote a locally compact space and take au to be the Nachbin family {Au; I A >0, n EN}, where (un)n is an increasing sequence of strictly positive continuous functions. Then the following assertions are equivalent: (1). For each n E N, there exists an m ;;:0 n such that, for any subset Y of X with infyEY(un(y (y» > 0, we also have infyEY(un(y )/uk(y» > 0 for all

k se m.

v«;

(2). The Frichet space able.

-eu (X)

(or, equivalently, 'eauo(X» is quasi-norm-

Proof. We start with two well-known remarks. (i). A I.e. space E is quasi-normable if and only if, for each O-neighbourhood Win E, there is a neighbourhood W' of 0 in E such that, for every e > 0, we can find a bounded set B in E with W' c e W + B. (ii). If ZI' Z2 C X are zero sets of continuous functions h), h 2 on X, respectively, with Z) n Z2 = 0, then h = Ih)I/(lh.1 + Ih 2Ddefines a continuous function on X with values in [0, 1] such that h IZI == 0 and h Iq == 1. Moreover, we note that, under the hypotheses of Proposition (5.8), 'Yd = (vn)n' vn = U~I, defines a decreasing sequence of strictly positive continuous functions on X which is regularly decreasing if and only if (un)n satisfies (1). Hence, by the equivalence of (ON) and (wS) (d. Subsection 1.6): (iii). (1) is satisfied if and only if, for each n EN, there exists an m » n such that, given any e > 0, it is possible to find a v E 'V = 'V'Vd (or, equivalently, a e E 'V('Yd» such that

Un (X) --;;:Os:::;> um(x)v(x);;:01 um(x)

To show (1):::;> (2), we may take

for each x E X.

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W

=

207

{IE ~OU(X) (resp. ~OUo(X»1 sup(un(x)lJ(x)J)~p}, xEX

where n E 1'.1, p >0. Given this integer n, we choose m

W'

=

~

n according to (iii) and put

{I E ~OU(X) (resp. ~OUo(X»1 sup(um(x»)f(x)J) ~ I}. xEX

For each e > 0, (iii) allows us to find a

Using the definition of

'Y,

v E 'f = 'YYd

such that

it is easy to check that

B = {IE ~OU(X) (resp. ~OUo(X»llf(x)1 ~ vex) for each x E X}, defines a bounded subset of ~OU(X) (resp. W' C e W + B. To establish this claim, put

~OUo(X».

We claim that

and note that X\YC{xEXlum(x)v(x)~l}. Y and Z are two disjoint zero sets of continuous functions on X, so by (ii) there is a continuous function g on X with values in [0, 1] such that g Iy == 1 and g Iz == O. At this point, fixing fE W', we have gf= e(e-lgf)E eW and (1- g)fE B since sUP(Un (x) .!.Ig(x )f(x )1) xEX

e

1

~-

e

and

pe sup(um (x )/f(x)J) ~ p xEX

1(1-g)(x)f(x)I~lf(x)l~um(x)lf(x)lv(x)~v(x)

for

all xEX\Y

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while (1 - g )(y )f(y ) == a ~ v(y) for each y E Y. Hence

f == gf + (1 - g)f E E W i.e. W' C E W + B, as desired. Conversely, assume that ~OU (X) (resp. Then, for the a-neighbourhood

+B , ~OUo(X»

is quasi-normable.

W == {IE ~OU(X) (resp. ~OUo(X»1 sup(un(x)lf(x)l) ~ I}, xEX

there is a a-neighbourhood W' in ~OU(X) (resp. loss of generality, we can take of the form

~OUo(X»

which, without

W' == {IE ~OU(X)(resp. ~OUo(X»1 sup(e; (x)/f(x)i) ~ 8}, m ~ n, 8> a, xEX

such that, for every ~OUo(X» with

E

> 0, one can find a bounded set B in

~OU (X)

(resp.

W'C~8EW+B.

Let 4 vex) == - sUPlf(x)/ 8E fEB

for each x E X.

v E 'V == 'V(rd ) . We notice now that, given n EN, we have chosen m ~ n so that, for every E;3 0, there are B and v with (*) and (**), respectively. It remains to show that um(x)v(x);31 whenever un(x)/um(x)~ E, and thus let us fix xoE X with un(XO)Jum(XO)~ E. Since X is locally compact, there exists an fE ~c(X) with O~f~ 1, f(x o) == 1, and f(x) == 0 whenever u; (x)/ um(x) ~ ~E. Because of It is easy to see that

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209

we have ~5ef/un E W'. Now (*) implies the existence of agE Wand an hE B with ~5ef/un = ~8eg + h. In particular,

so that

whereby um (xo)v(xo) ~ un (xo)v(xo) ~ 1, as desired. (Note that the local compactness of X was only needed for the implication (2)~ (1).) 0 At this point, we remark that Theorem (5.7) yields a complete characterization of 'YdOC6' (X ) == C6''Vo(X) for decreasing sequences 'Yd ,:" (vn)n of continuous functions on a locally compact space X. On the other hand, it is still an open problem to characterize when the inductive limit topology of 'Yd C6'(X ) coincides with the weighted topology C6''V(X), i.e. to find necessary and sufficient conditions for 'Yd C6'(X) == C6''V(X) topologically. As a step towards the solution of this problem, we will introduce a new, rather weak, condition (D') (analogous to condition (D)) and we will prove in the next section that (D') implies 'Yd C6' (X ) == C6''V(X). In particular, let us note that, in the sequence space case, it was clear from the duality of echelon and co-echelon spaces that condition (M) or condition (ON) implies k oo == K; topologically. (E.g. this follows from A, reflexive or quasi-norm able implies Al distinguished, and a direct argument in the quasi-norm able case is based on a strong form of Theorem (1.5) (4), but involves the fact that K; == (Al)~ is a (DF)-space.) Since similar tools are not available in the case of spaces of continuous functions, Theorem (5.7) does not directly imply 'Yd C6'(X ) == cg'V(X) for regularly decreasing 'Yd, and it is not obvious from Proposition (5.3) that 'Yd C6' (X ) == cg'V(X) topologically whenever 'Yd satisfies (M'). However, as a consequence of our main result in the next section, we will show that, under reasonable additional assumptions, 'Yd C6'(X) == C6''V(X) is indeed true if 'Yd satisfies condition (M') or is regularly decreasing.

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6. The Main Theorem and its Coronaries

For the purposes of this section, we fix a nonvoid system ® of subsets of the completely regular space X which is directed upward with respect to inclusion. Since the case @; = ~ = all compact subsets of X has already been treated, the setting where ® is an increasing sequence ~ = (Xm )mEN will be of special interest in what follows. Definition (6.1). Corresponding to a Nachbin family 'Von X, we associate the spaces ~'Vo(X, @;)= {IE ~'Y(X)' for each v E V, (sup (v(x)lf(x)l)sEe xEX\S

converges to O}

= ~(X) n Ao(X, ®; 'Y) and ~c(X, @;; 'Y) = {IE ~'V(X)) for some S E @;,flx\s

= ~(X) n

== O}

cp(X, ®; 'Y).

cg'Vo(X, @;) is a closed subspace of cg'V(X) which will be endowed with the induced system (qJvEV of seminorms. If 'Y = {Avl A > O} for a single strictly positive weight v on X, we will sometimes write ~vo(X, @;) instead of cg'Yo(x, ®) and cgc(X, ®; v) instead of ~c(X, @;; 'Y). Note that we have ~'Yo(X,~) = ~'Yo(X) and cgc(X,~; 'V) = ~c(X) while, for @; = {X}, ~'Yo(X, @;) = ~'Y(X) = cgC<X, @;; 'V); hence the spaces ~'Vo(X, ®) 'interpolate' between cg'Vo(X) and ~'Y(X) in all interesting cases. Since ~c(X, ®; 'Y)C
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Proof. We fix fE ~'Vo(X, 6), v E 'V, and e >0. (1). By definition, there exists an S E 6 with sUPxEX\s(v(x)lf(x)/) < e, and by hypothesis we find an S' E 6 with S C int S'. The closed sets ZI = {x E xl v(x)lf(x)1 ~ s} and Z2 = X\int S' are disjoint, and since X is normal, we can choose g E ~(X) with 0 ~ g ~ 1, glZt == 1 and glq == O. Clearly, h = gfE ~c(X, 8; 'V) satisfies

sup(v(x)lf(x) - h(x)/) = sup(v(x)(l- g(x))lf(x)/) ~ sup (v (x)lf(x)i) ~ e. xEX xEX\Z,

xEX

(2). If the weight v is continuous, Z, = {x E Xl v(x )If(x)1 ~ s ] and Z2 = {x E Xl v(x)lf(x)/ ~ ~e} are disjoint zero sets of continuous functions on X, and by remark (ii) at the beginning of the proof of Proposition (5.8), we can choose g E C(X) with 0 ~ g ~ 1, glZt == 1 and glq == O. Since there is an SE8 with sUPXEX\s(v(x)lf(x)i)<~£, Z2 contains a set of the form X\S so that h = gf defines an element of ~c(X, ®; 'V) satisfying sUPxEx(v(x)lf(x)-h(x)I)~e. 0 We briefly mention a consequence of Proposition (3.3) which, for the case 6 = Q:, was (essentially) Corollary (5.6) (1): If the Nachbin family 'Von X satisfies condition (M, ®) (cf. Definition (3.2)) then ~'V(X) = ~'Vo(X, 6) and, on each bounded subset of ~'V(X), the induced topology is given by the system (qv,S)VE'Y.SE6 of seminorms, where

qv,s(f) = sup(v(x)lf(x)l)

for all f E

~'V(X)

.

xES

Definition (6.3). For a decreasing sequence 'Vd = (un)n of strictly positive weights on the completely regular space X, we put 'VdO~(X, ®) == indlim,.... ~(vnMX, 6) (endowed with the I.e. inductive limit topology) and ~c(X,6;'Vd)={fE'Vd~(X)lflx\s==O for some SE6}= U nEN ~c(X, ®; vn)' Obviously ~c(X, ®; 'Vd) C 'VdO~(X, ®), and it follows from Remark (6.2) that ~c(X, 6; 'Vd) is (sequentially) dense in 'VdO~(X, 6) in each of the following cases: (1). X is normal and, for each S E ®, there exists an S' E ® with S Cint S'. (2). All weights vn are continuous.

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=

We note 'VdO~(X; ~) 'VdO~(X) and ~c(X;~; 'Vd) = ~c(X) while 'VdO~(X, ®) == 'Vd~(X) = ~c(X; ®; 'Vd) for @) = {X}. If @)I and @)2 are two directed systems with @)I :,;;; @)2 (in the sense explained in Section 3), then 'VdO~(X, @),) is continuously embedded in 'Vd~(X, @)2); in particular, the canonical injection of 'VdO~(X, @) into 'Vd~(X) is always continuous, while 'VdO~(X) is continuously embedded in 'VdO~(X, @) whenever each compact set K in X is contained in some S = S(K) E @). Clearly, 'VdO~(X, ®) = ~(X) n k o(@) = ~(X) n ko(X, @); 'Vd), and the canonical injection of 'YdO~(X, ®) into k o(@) is continuous. Finally, there is also a canonical injection 'VdO~(X, ®)~ ~'fo(X, @), where 'f = 'fYd' and it is continuous with respect to the I.c. inductive limit topology of 'VdO~(X, @) and the weighted topology of ~'fo(X, @). The following corollary to Proposition (3.7) can now be stated. Corollary (6.4). If 'Vd = (vn)n satisfies condition (M, @) (i.e. if, for each n EN and each subset Y of X with Y n (X\S) ~ 0 for all S E @), there is an m ;;;. n such that inf EY(vm(y)Jvn(y» = 0), then we have: y (1). 'V = 'VYd satisfies condition (M, ®) ti.e. for each v E 'V and each subset Yof [x E XI vex) ~ O} with Y n (X\S) ~ 0 for all S E ®, there exists a v' E 'f with v';;;. v such that infyEY(v(y)/v'(y» = 0). (2). ~'fo(X, @) = ~'f(X). (3). On each bounded subset of ~'f(X), the induced topology is given by the system (qu,s )iiE'Y,sEe of seminorms, where qii,S (f) = SUPXES(V(X )If(x)1) for all f E ~'f(X), From the side of condition (M, ®), the preceding corollary is already sufficient for the proof of the main theorem; concerning condition (N, ®), however, more preparations are necessary. For the next results, we again need a countability assumption. Thus, at this point, let us fix an increasing sequence ~ = (Xm )mEN of subsets of a completely regular space X and a decreasing sequence 'Vd = (Vn)nEN of strictly positive weights on X. Lemma (6.5). If the sequence 'Vd = (vn)n on the normal space X satisfies condition (N,~) (i.e. for each mEN, there is an nm;;;' m such that inf x E x ", (vk(x)/vnJx» > 0 for all k;;;. nm) then 'VdO~(X;~) and ~'fo(X,~) induce the same topology on ~c(X,~; 'Vd ) . Proof. (The first part of the proof exactly proceeds as in the case of

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Lemma (3.12).) Replacing the decreasing sequence (vm)mEN by (vnm)mEN' we assume that nm = m in condition (N, ~) and put l)k = inf xE x k (Vk+l(X)/V k(x » > 0, k = 1,2, .... In the sequel, we fix a neighbourhood U of 0 in 'VdOcg(X ~) which, without loss of generality, we may take of the form

U= absconv (u

ck

kEN

B~)

,

where E k > 0 and B~ denotes the closed unit ball of C(vkMX ~). Inductively, we can choose an increasing sequence (adkEN of positive numbers with

k

and

1,2, ....

=

Then, defining ak

V = inf -

kEN E k

Vk

-

E 'V ,

we conclude (as in the proof of Lemma (3.12» V1Xk = min {;- Vj I

I

j = 1, ... k },

k

=

1,2, ... ,

and claim:

{f E cgc(X~; 'Vd)1 sup(v(x)lf(x)J) < I} cU. xEX To establish this claim, we fix fE cgc(X,~; 'Vd ) with sUPxEx(V(x)lf(x)1) < 1. There is an N E N with flx\XN == O. For each x E X N ' however, we can find an n = n(x)~N with v(x) = E~lanvn(x). Hence, putting n

=

1, .. . ,N,

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we have obtained a finite open cover (Yn ):= 1 of X, and since X is normal, it is possible to choose a continuous partition of unity (c,on):=1 on X which is subordinate to the covering (Yn):=1 (i.e. c,on E cg(X), O:s:;: e, ~ 1, s, Ixw. == a (n = 1, ... , N), and L :=1 c,on == 1 on X). At this point, we note that gn = 2nc,onf belongs to enB~, n = 1, ... , N, since gn(x) = 0 if x E X\ Y n while x E Yn implies

If all the weights V n are continuous, a refinement of the proof of Lemma (6.5) works without the assumption that X is normal. In fact, assuming V n E ~(X), n = 1,2, ... , we will let the first part of the preceding proof remain unchanged. Choosing now (¥k?; 2 . 2 k (instead of (¥k?; 2 k ), k = 1, 2, . .. and fixing f E '€C (X, ~; 'Vd ) with sUPxEx(i3(x)lf(x)/)< 1 and N EN with flxlXN == 0, we conclude that (Zn:=l'

is a finite closed covering of X. At this point, taking

Yn =

{x E XI an vn(x)lf(x)1 < 2} 10"

and

Z~ = X\ Y n ,

and using our additional hypothesis, we see that Z~ and Z~ are two disjoint zero sets of continuous functions on X, and hence, by remark (ii) in the proof of Proposition (5.8), we can choose gn E '€(X) with O:s:;: B; ~ 1, gnlz~==1 and gnlz2==0, n=I, ... ,N. Clearly, L:=lgk is strictly positive on X so that the functions

gn c,on = - - satisfy c,on E '€(X), O:s:;: e, :s:;: 1, c,on IXI r, == a and L :=1 c,on n = 2 c,o"t again belongs to enB~, n = 1, ... , N, since

gn

== 1 on X. Now,

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Finally, f= ~:=I gil E U, as desired. Furthermore, still assuming that all weights v" are continuous, the first part of the proof of Lemma (6.5) shows that the restriction of v to each set Xm of the sequence :J is a finite minimum of continuous functions, and hence itself is continuous on X m • If, additionally, a function f: X -H~+ with fix continuous for all mEN must already be continuous on X, the preceding argument actually yields a v E C€(X), i.e. v E 'ii'cont " (E.g. this is the case if :J = (Xm )mEN covers X and if, for each mEN, there exists a k m ;;.: m with X m C int X k m . ) We have now proved the following: Corollary (6.6). (1). The conclusion of Lemma (6.5) remains true for an arbitrary completely regular space X whenever 'Vd = (v"),, is a sequence of continuous weights on X (satisfying condition (N, :J)). (2). If all weights v" are continuous and if a function f: X ~ R+ with fixm continuous for all mEN must necessarily be continuous on X, then the Nachbin family 'ii' may be replaced by its subset 'ii'cont of continuous weights in Lemma (6.5) (and in part (I) of this corollary).

To illuminate the background of Corollary (6.6) (2), we would add another fact which also follows from the method of proof of Corollary (6.5) (and clearly implies Corollary (6.6) (2)). Corollary (6.7). Let 'Vd = (v"),, denote a sequence of continuous functions on a completely regular space X which satisfies condition (N,:J) for the increasing sequence ~ = (Xm )mEN of subsets of X. If a function f : X ~ R+ with flxm continuous for all mEN must already be continuous on X (this clearly holds whenever ~ covers X and x; C int X m + 1 for each mEN), then 'ii' = 'ii'conI i.e. we not only have 'ii'cont C 'ii', but also 'ii':s;; 'ii'cont ' In this case, in particular, 'VconI is a Nachbin family on X, and ee'ii'o(X, ~) ee'ii'contO(X,~) holds as well as ee'ii'(X) ee'ii'c ont(X) and ee,(x, :J; 'ii') = ee,(x, ~; 'ii'cont)'

=

=

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Proof. As in the beginning of the proof of Lemma (6.5), we assume that nm = m _ in (N,~) and put Ok _= inf x E x k (Vk+1(X)/Vk(x)) > 0, k = 1,2, .... To _ prove 'Y ~ 'YconI' fixing i3 E 'Y, we may assume v = infkEN(akvk)' a k > 0 (k = 1,2, ...). Inductively, we can now choose positive numbers ak with

v

r,

Then, defining = inf kEN (akvk) E it is clear that i3:::; ii; and we claim v E CC(X). But, for each x E X k (k = 1,2, ...), a kvk(x)/Vk+1(x):::; a k5// :::; ak+l and hence akvk(x)::::; ak+1vk+l(X), Since (Xm)mEN is increasing, we conclude

By 'Yd C CC(X), vl X k is continuous for each kEN and, by hypothesis, E CC(X), as we had claimed. 0

v

Proposition (6.8). (1). Let 'Yd = (vn)n denote a decreasing sequence of strictly positive weights on a completely regular space X which satisfies condition (N,~) for the increasing sequence ~ = (Xm )mEN of subsets of X. Assume that one of the following conditions (i) or (ii) holds: (i) All weights vn are continuous. (ii) X is normal and X m C int X m + 1 for m = 1,2, .... Then 'YdOCC(X, ~) is a topological subspace of ccrofx, ~). (2). Moreover, under these hypotheses, 'YdOCC (X, ~) is also a topological subspace of each of the following spaces: 'YdCC(X), kO(~) = ko(X,~; 'Yd) and e; = koo(X, 'Yd ) . (3). In case (i), if additionally, (i') A function f: X ~ R+ with flxm continuous for all mEN must already be continuous on X, then CCro(x, ~) = ccrconto(X, ~). ~ (4). Finally, CC'Yo(X,~) = 'YdOCC(X,~) holds under the following conditions: 'Yd = (vn)n is a decreasing sequence of strictly positive weights on a kR-space X, 'Yd satisfies condition (N, ~), and (i) and (i') hold, or X is normal, X m C int X m + 1 for all mEN and inf x E K vn(x) > 0 for each compact subset K of X, n = 1,2, ....

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Proof. In view of the remark made after Definition (6.3), Lemma (6.5) and Corollary (6.6) (1), assertion (1) follows from [4, Lemma 1.2]. Now, (2) is a consequence of the following diagram:

And (3) is implied by Corollary (6.7). It remains to prove (4). From Corollary (4.2) (1), we get 'V/6'(X) = ce'ii'(X) algebraically so that ceAx, ~; 'Vd) = cec(X, ~; 'ii'). (In fact, if (i) and (i') hold, the latter space also equals cec(X,~; 'Ycont ) ') Hence we can apply Proposition (6.2) (1) to ce'ii'o(X,~) (resp. Remark 6.2 (2) to ce'ii'contO(x, ~» to obtain the density of 'YdOce(X,~) =:J cec(X,~; 'Yd) in ce'ii'o(X, ~). But since X is a kR-space and x « 'ii' holds under our assumptions on 'Yd, ce'ii'o(X, ~) is complete (as a closed subspace of ce'ii'(X». Hence, the desired conclusion follows from (1).0 If X is a locally compact and o-compact space and 'Vd = (Vn)nEN denotes a decreasing sequence of weights on X with infxE K Vn (x) > 0 for each ~ compact subset K of X and n = 1,2, ... , then cero(X)== 'VdOce(X) holds. This follows from Proposition (6.8) (4) by taking ~ = (Xm)mEN to be a fundamental sequence for the compact subsets of X, which satisfies X mCint Xm+l for all mEN, and observing 'VdOce(X)== 'VdOce(X, II)== 'VdOce(X, ~), ce'ii'o(X) == ce'ii'o(X, II) == ce'ii'o(X, ~). However, as we have seen in Theorem (4.1) (1), the result remains true for arbitrary locally compact spaces X. On the other hand, if X is a hemicompact kR-space and all weights vn are continuous, 'YdOce(X) == ce'ii'o(X) == ce'ii'conto(X) again follows from Proposition (6.8) (4). At this point, all preparations are finished, and we can proceed to state (and prove) our main result.

-

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Theorem (6.9). Let 'Vd = (vn)n denote a decreasing sequence of strictly positive weights on a completely regular space X. We assume the following condition (0'): (0'). There exists an increasing sequence ~ = (Xm)mEN of subsets of X such that, (N, ~). for each mEN, there is an n m ~ m with inf x E X.. (vk (x)/vn)x» > 0, k = nm + 1, n m + 2, ... , while, (M, ~). for each n E N and for each subset Y of X with Y n (X\Xm ) of- 0 for all mEN, there is an n'> n such that infYEy(vn,(y)lvn(y» = 0; and (1) Xis normal and, (*)foreach mEN, thereisak; ;a. m with X; C int X km, or

(2) all weights vn are continuous, and a function f: X ~ R + with fix continuous for all mEN must already be continuous (which is im~ plied by part (* ) of (i». Then we have 'V/€(X) == «6'Y(X) topologically; in case (2), there is also equality (as a I.e. space) with «6'Ycont(X), In particular, if condition (0') holds, if X is a kR-space. and if, in case (1), we also suppose that inf xE K vn(x»O for each compact subset K of X, n = 1,2, ... , then 'Vd«6(X) is complete.

Proof. To prove the desired topological equality 'V/€(X) == «6'Y(X), we consider various completions, but no injectivity problems will arise. Since 'Vd satisfies condition (N,~) as well as (i) or (ii) of Proposition (6.8) (1), it follows from that proposition that 'VdO«6(X,~) is a topological subspace of «6'Yo(X, ~). Moreover, as in the proof of Proposition (6.8) (4), using Remark (6.2) (and Corollary (6.7) in the case (2», here we need the second part of hypothesis (2). Also note that condition (M,~) already implies X = U mEN X m , cf. the remark made after Definition (3.2)).. we obtain density of 'VdO«6 (X, ~) in «6'Yo(x, ~). As a consequence, -----=-----.: 'VdO«6 (X, ~) == «6'Vo(X,~) holds. But since 'Vd also satisfies condition (M, ~), ~e «6~~) == «6'Y(X) from Corollary (6.4) so that, finally, 'VdO«6 (X, ~) == «6'V(X). On the other hand, the canonical injection of 'VdO«6 (X, ~) into 'Vd«6 (X) is a topological isomorphism into by Proposition (6.8) (2), whence

----

'VdO«6 (X, ~)

i.e.

~)

----

== 'Vd«6(X)-C!os V dO«6 (X, ~) ,

is a topological subspace of

'V~).

At this point, it is

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219

obvious that not only 'Vd'e(X) = 'e'V(X), but that the two spaces must also have the same topologies (under our hypotheses). The rest of the assertions is clear from Proposition (6.8) (3) and the remark on completeness of weighted spaces at the beginning of Section 4.

o

The corollaries which we had already announced at the end of the last section now follow. Corollary (6.10). Let V, = (vn)n denote a decreasing sequence of weights on a locally compact and a-compact space X which satisfies infx E K vn(x) > 0

for each compact subset K of X, n = 1,2, . . .. (Or, alternatively, let 'Vd = (vn)n denote a decreasing sequence of strictly positive continuous weights on a hemicompact kR-space.) If 'Vd satisfies condition (M'), then 'Vd'e(X) == 'e'V(X) holds topologically (and, in the second case, we also have equality with 'e'Vcont(X». In particular, 'Vd'e(X) then induces the compact-open topology on each bounded subset. Proof. Proceeding as in the remark after Proposition (6.8), we see that condition (D') is satisfied, whence the first part of the result. The last assertion then follows from Proposition (5.3) (also, see the remark after Proposition (5.3». 0 Corollary (6.11). Let 'Vd = (vn)n denote a decreasing sequence of strictly positive continuous weights on a completely regular space X. Then each of the following conditions (1), (2) implies 'Vd'e(X) == 'e'V(X) == 'e'Vcont(X): (1). There exists an increasing sequence 5:5 = (Xm )mEN of subsets of X such that, (*) for each mEN, there is a s;> m with X m C int X k m and, (N,5:5). holds as well as: (5, 5:5). For each n EN, there exists an n' > n with limm -+ oo SUPXEX\Xm (vAx)lvn(x» = O. (2). 'Vd is regularly decreasing. Proof. Since (S,5:5) clearly implies (M,5:5), the assertion for (1) directly follows from Theorem (6.9). It is now sufficient to show (2) ~ (1). Essentially, this was already accomplished in [5, Prop. 3.9, Cor. 3.10], and we closely follow that proof. First, 'Vd regularly decreasing implies condition (G) of Section (1.6), where I: = {x E Xl vn,(x)lvn(x);?; 11m} (with n' the integer m of the regularly decreasing condition), n, m = 1,2, .... Note

220

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that here I~ C int I~+l holds by the continuity of the functions Vn' Next, (G), in its turn, implies condition (wG) (d. Section (1.6)), where now 1m = U n+/"'m I~. Observe that 1m C int Im+1 is true for each mEN, hence this clearly proves (1) for X m == 1m • m == 1, 2, .... D The next two results are parallel to Proposition (3.15) and Corollary (3.16) for sequence spaces, with identical proofs (which we therefore omit). Proposition (6.12). For a decreasing sequence "lid == (Vn) of strictly positive continuous functions on a locally compact space X and for a directed system G of subsets of X with the property that, for each compact set K in X, there is an S E G with K C S, the following assertions are equivalent: (1). "lid is regularly decreasing. (2). "lidO ce(X, 6) =: indlim;.... ce(vnMX; ~) is a boundedly retractive inductive limit (and hence complete). Corollary (6.13). Let "lid = (vn)n denote a regularly decreasing sequence of strictly positive continuous functions on a locally compact space X and let, for arbitrary n EN, m n;;;' n be chosen as in the regularly decreasing condition. If ~ == (Xm)mEN is an increasing sequence of subsets of X such that, for each compact set K in X, there is an m = m (K) E N with K C X m and, for each mEN, there exists an n = n(m);;;. m with infxEx)vm,<x)/vn(x)) > 0, then "lidO ce(x, ~) =: ce'fi'o(X, ~) (algebraically and topologically). Finally, we mention a consequence for function spaces on a topological product (which is analogous to Lemma (2.4) and Corollary (2.5)). Here, we do not present the most general version which could be derived by a careful analysis of the proof. Let "IId 1 = (v~)n and "Ifd2 = (v~)n denote two decreasing sequences of strictly positive continuous functions on the locally compact spaces XI and X 2 , respectively. Our notation in the sequel is similar to the one explained in the case of sequence spaces. Corollary (6.14). If there exist increasing sequences ~I = (X~)m and ~2 = (X~)m of subsets of XI and X 2 , respectively, with the properties (1). for each compact set K, in X;, there is an m j == mj(K;) EN with K, C X~i (i = 1,2) and (2). "IId l satisfies both (N, ~1) and (M, ~I) and "IId2 satisfies both (N, ~2) and

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221

(M, ~z),

then 'Vd10 'VdZ= (v~ 0 v~)n satisfies condition (0'), and hence

In particular, this topological isomorphism holds if XI is a locally compact and a-compact space on which the sequence 'Vd1 satisfies condition (M') and if the sequence 'VdZ is regularly decreasing on a locally compact space

x,

The proof of Corollary (6.14) proceeds along the same lines as the proof of Lemma (2.4) and Corollary (2.5) for sequence spaces. Again, it is not known whether 'Vdl,,€(XI) == "€'Vydl(X2 and 'VdZ"€(XZ) == "€'V'Yd2(Xz) already implies ('VdI 0 'Vdz) "€(XI x X z) = "€'V'YdII8iYd2(XI x X z). And, clearly, in the case of function spaces, it is not possible to interpret this question in terms of a duality between the 1T- and s-tensor products (as it was the case at the end of Section 2). .

Appendix

Up to this moment, the parallels between the sequence space and the function space cases have always been emphasized. As a balance, it should be pointed out that there are also some differences between the two cases and that, in fact, spaces of continuous functions can behave in a better way than sequence spaces. To this purpose, we will now give an example of a completely regular space X and a decreasing sequence 'Vd of strictly positive continuous functions on X such that 'Vd"€(X) == "€'V(X) holds topologically, but the topologies of the corresponding sequence spaces k oo = koo(X, 'Vd ) and K; = Koo(X, 'V) (on X as an uncountable index set) are different. That is to say, the continuity of the functions in 'V/6'(X) helps to establish the desired projective description of the weighted inductive limit topology while such a projective description breaks down for the larger space k oo , where arbitrary functions are admitted. Moreover, in our example, 'Vd does not satisfy condition (D); a fortiori, (the first part of) (0') is not satisfied. Hence condition (0') is only sufficient, but not necessary for the topological equality 'Vd"€(X) == "€'V(X), and the converse of Theorem (6.9) does not hold. Let us start with two easily established, but quite useful remarks.

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Lemma. (1). If a decreasing sequence "II = (vn)n of strictly positive functions on an index set I satisfies condition (D) and if J denotes a nonvoid subset of I, then (D) is satisfied for 'Y/ J = (e, IJ)n as well. (2). For a nonvoid subset J of I, k oo(], 'VIJ) == indlim,...,loo(J, vnII) (resp. Aoo(J, r('Y)I/) == Koo(J, r('YI I ») is topologically isomorphic to the sectional subspace kool J (resp. KooI /), and hence to a complemented topological subspace, of k oo (resp. K oo). If koo(I, "II) == Koo(I, r('Y», we also have koo(J, 'VII) == Koo(J, r('VI I

».

Example. Let X denote the set ([-2,2]X[-2,2])\{(1In,0)lnEN}, endowed with the topology induced by [R2. Clearly, X is a normal Hausdorff space, but not locally compact. For x E R 2 and E > 0, Ue(x) denotes {y E R 21IIy - xlb < s}. Inductively, we can choose a decreasing sequence (En)nEN of positive numbers such that En < lin and clos U; n clos Um = 0 for n;j. m, where U; = U (lin, 0) (n = 1,2, ...). Next, we take continuous functions gk on clos U, with e, IaUk == 1, gk(l/k, 0) = 0, 0 < s, < 1 on Vk = u,\{(1Ik, O)} (k = 1,2, ...), and put vn == gk on clos Vk = (c1os U k)\{(1Ik, O)} (k = 1, ... '. n), with vn == 1 on X\ U ~=1 clos Vk (n = 1,2, ...). Letting V o == 1, "lid = (Vn)nENo is a decreasing sequence of strictly positive continuous functions on X. By construction, for each kEN, it is possible to find points YkJ E clos Vk such that gk(Ykt) = 1// (l = 1,2, ...); we note Ykt;j. Yk'l' for (k, l) ;j. (k ', I'). Since E

k

•

~n,

(k, lEN, n = 1,2, ' ..) , k~n+1,

setting Y = {Ykllk, lEN} defined a subset Y of X for which (after the canonical identification Y~ N x N) (vn IY )nEN coincides with the decreasing sequence "II associated with the Kothe matrix SI1 in the example (due to Kothe and Grothendieck) of a nondistinguished echelon space (cf. the appendix of Part I). Hence koo(Y, 'Vdly) and KooCY. r('Ydly» have different topologies, and 'Vdl y does not satisfy condition (D) . From the preceding lemma, we conclude that neither (D) nor (D') is satisfied for "lid itself and that k oo = koo(X, 'Vd ) and K; = Koo(X, r) do not agree topologically. It remains to prove that, nevertheless, the topological equality

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223

°

'V/€(X) == C€'Y(X) holds. Thus, for the rest of the proof, let us fix a neighbourhood U of in 'VdC€(X); without loss of generality, U

= absconv

(u

kENo

°

ekB~),

where ek > and B~ denotes the closed unit ball of C€vk(X), k = 0, 1, .... By induction, we can select an increasing sequence (an )nENO of positive numbers with a o = 2, a k ~ 2· 2k and ak+l!ek+l > ak/ek (k = 0,1, ...). For

we claim that the O-neighbourhood W = {fE C€r(X)1 Pi!(f) = sup(v(x)lf(x)I):so;n xEX

in C€'Y(X) is contained in U. To establish this claim, fix f E C€'Y(X) with Pi! (f) :so;~. First, we note that 1

,

2~ v(x) f(x)1

a

2

eo

eo

o = -If(x)/ = -If(x)/,

°

and hence If(x)1 :so; ~eo, for all x E X\ U kEN clos Vk • Since (0, 0) is an element of the last set, there exists a 8> such that If(x)1 < ~eo for each x E X n Us(O, 0). We choose N E N with clos ti, C Us(O, 0) for all k > N. Next, we remark that, for each k = 1, ... , N,

since, for each x E clos Vk ,

Clearly, Wk is closed in X and contained in Vk • Moreover,

vl(c1os Vk)\Wk ==

224

K.D. Bierstedt, R. Meise / Distinguished Echelon Spaces

aol Eo SO that, in view of (*), If(x)1 ,.;; ~ Eo also holds for all x E (c1os Vk ) \ W k (k = 1, ... , N). At this point, we notice that v is continuous on clos V k so that putting

defines an open neighbourhood of Wk in Vk (and hence in X). Since U ~=1 W~ U (X\ U ~=I Wk ) is a finite open covering of the normal space X, there exists a continuous partition of unity (CPk )~=o on X such that supp CPk C W~ (k = 1, ... , N) and supp CPo C X\ U ~=l Wk' Now, for each k = 1, ... ,N,

~~~(vo(x)I£l'oCPo(x )f(x)l),.;; 2 sup {If(x )/1 x E X\~I W ,.;;2max(su p {lf (x )/lx E

01

k }

(tclos Vk)\Wk ) }

,

sup{lf(x) II x E X n U6(O, O)}, sup{lf(x)11 x E X\

k~N clos Vk })

,.;:: 2 max{ III}_ ..., 4 EO' iEo, 4 EO - EO' and hence £l'oCPof E EB~. Finally, N

f = 2:

k=O

cpd =

N

1

k=O

£l'k

2: -

£l'kCPd

E absconv (

U kENo

EkB~),

i.e. We U, as desired. 0

Acknowledgment The first author, K.D. Bierstedt, gratefully acknowledges support under

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225

a travel grant from DFG for his visit to the Universidade Federal do Rio de Janeiro to attend the 1982 Analysis Conference honouring L. Nachbin. The authors would like to thank Bill Summers (University of Arkansas at Fayetteville) for several helpful conversations on the subject of this article.

Note added in proof (May 1985). Since late 1983 (when the present article was written), several new results related to the material presented here have been obtained. (1). S. Heinrich, Ultrapowers of locally convex spaces and applications I, Math. Nachr. 118 (1984) 285-315, introduces the 'density condition' for I.e. spaces E and (in his Prop. 1.7) gives an example of an echelon space Al without this density condition.-Now, dualizing the definition, it is easy to see that a Frechet space E with density condition must be distinguished. And, using the Proposition in the Appendix to Part I of the present article, it is clear that Heinrich's example Al is not distinguished. (In fact, Heinrich's construction is only a slightly more general reformulation of the classical example of Kothe and Grothendieck.) (2). J. Bonet and A. Defant, Projective tensor products of distinguished Frechet spaces (manuscript, Valencia and Oldenburg, 1985) put our results at the end of Section 2 (in particular, Proposition (2.7» in a broader context: They prove that an echelon space Al is Montel if and only if AI01rF is distinguished for each distinguished Frechet space F. (3). K. Reiher, Kothesche Folgenraume mit veranderlichen Stufen (manuscript, Paderborn, 1984, to be included in his Ph.D. thesis) generalizes part of our study on quasi-normable and distinguished Kothe echelon spaces to the setting of 'Dubinsky echelon spaces' where 'steps' I more general than II (or Ip' 1.,.:;; v< (0) are allowed. (4). J. Bonet, A projective description of weighted inductive limits of spaces of vector valued continuous functions, Collect. Math. 34 (1983) 117-124 extends some of the projective description results to functions with values in I.e. spaces satisfying the countable neighbourhood property or in complete (gDF)-spaces. These investigations are continued (in a different direction) in J. Bonet, On weighted inductive limits of spaces of continuous functions, Math. Z. (to appear).

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References (1] K.D. Bierstedt, Gewichtete Raume stetiger vektorwertiger Funktionen und das injektive Tensorprodukt I, II, J. Reine Angew, Math. 259 (1973) 186-210; 260 (1973) 133-146. [2] K.D. Bierstedt, Injektive Tensorprodukte und Slice-Produkte gewichteter Raurne stetiger Funktionen, J. Reine Angew. Math. 266 (1974) 121-131. [3] KD. Bierstedt and R Meise, Induktive Limites gewichteter Raume stetiger und holomorpher Funktionen, J. Reine Angew. Math. 282 (1976) 186-220. [4] K.D. Bierstedt, R Meise and W.H. Summers, A projective description of weighted inductive limits, Transact. Amer. Math. Soc. 272 (1982) 107-160. [5] K.D. Bierstedt, RG. Meise and W.H. Summers, Kothe sets and Kothe sequence spaces, In: Functional Analysis, Holomorphy and Approximation Theory, NorthHolland Math. Stud. 71 (North-Holland, Amsterdam, 1982) 27-91. [6] A. Grothendieck, Sur les espaces (F) et (DF). Summa Brasil. Math. 3 (1954) 57-122. £7] A. Grothendieck, Produits tensoriels topologiques et espaces nucleaires, Mem. Amer. Math. Soc. 16 (1955), reprint 1966. [8] R Hollstein, Inductive limits and s-tcnsor products, J. Reine Angew. Math. 319 (1980) 38-62. [9] G. Kothe, Topological vector spaces I, II, Grundlehren Math. Wiss. 159; 237 (Springer, Berlin and New York, 1969, 1979). [10] L. Nachbin, Elements of approximation theory, Van Nostrand Math. Stud. 14 (Van Nostrand, Princeton, 1967); reprinted by (Krieger, Huntington, N.Y., 1976).

l.A. BARROSO editor, Aspects of Mathematics and its Applications © Elsevier Science Publishers B.V. (1986)

227

SOME NUCLEAR HANKEL OPERATORS

F.F. BONSALL Department of Mathematics, University of Edinburgh, Edinburgh, Scotland, United Kingdom Dedicated to Leopoldo Nachbin on his sixtieth birthday

1. Introduction

Let H be a Hilbert space with orthonormal basis (e)nEZ+' A Hankel operator is a bounded linear operator Ton H with matrix (Tej , e») of the form (a j +j ) with {an} a complex sequence called the coefficient sequence of the operator. An operator T is nuclear if it belongs to the trace class, that is if the positive operator (T* Ty/2 has finite trace (see [5]). The nuclear Hankel operators have been identified by Peller [3] as the Hankel operators S", with their symbols

2. The Nuclear Norms of Certain Elementary Hankel Operators

Let Tn with n E Z + denote the elementary Hankel operator for which the coefficient sequence {ad has an = 1 and a k = 0 (k 0/- n). Let

We shall obtain estimates for

IISnlll and IIRnill'

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F.F. Bonsall I Nuclear Hankel Operators

1 Theorem (2.1). - n log n - n ,,;; IISn III";; n log n + n 27T

(n E N).

Proof. With n EN, let D; denote the operator with matrix (d;) given by d;j = 1 (0";; j ,,;; i ,,;; n - 1), d;j = 0 for all other i, j, and observe that

(2.1) Since IITII I = Tr«T*T)II2), this follows from the fact that S:Sn = D:Dn, or, alternatively, it is easy to find a unitary operator U with USn = D n. Let

Inspection of the matrix shows that X n is a linear combination with positive coefficients of two projections, and so X; ~ O. Thus (2.2) With the aid of (2.1) and (2.2), Theorem (2.1) will have been proved when we have proved the following: 1 Lemma (2.2). 27T n log n ::s:;; II Ynll l

,,;;

n log n

(n E N).

Proof. Let .in (A) denote the determinant of the n x n matrix in the top left-hand corner of the matrix of AId + D; with Id the identity. It has long been known that

D:,

This follows at once from the recurrence relation

or see [1, p. 346]. The eigenvalues of 2 Yn are the zeros AI' ... , An of the polynomial equation .in(-iA) = 0, that is of the equation (-ii\

-ly + (-iA + ly

=

o.

229

F.F. Bonsall / Nuclear Hankel Operators

With A = cotan 0, this equation becomes cos nO = 0, so that

1T

Ak = cotan(2k - 1)2n

(k = 1, ... , n).

If n = 2m, the eigenvalues are ±A 1, ±A2 , ••. ,±Am , while if n = 2m + 1 they are ±A 1, ±A 2••••• ±Am , O. Thus in both cases

and so

1T

31T

IIYnll = cotan -+ cotan -+ .. , + cotan(2m 2n 2n

(2.3)

1T

1)-, 2n

with m = [nI2]. To estimate (2.3) we use the relation cotan 0 + cotan( 1T12 - 0) = 2 cosec 20, and find it convenient to write am = 1 + 1/3 + ... + 1/(2m - 1). If n = 4p,

IIY III = (cotan ~+ cotan(4p-l)~) + (cotan 31T + cotan(4p- 3)~) ~

n

~

~

~

+ ... + (cotan(2p - 1) ~ + cotan(2p + 1)~) 8p

= 2(cosec ~ + cosec 31T + ... + cosec(2p -

4p

4p

and since 0- 1 :s;; cosec O:S;; ~1TO-I

8p

1)~) , 4p

,

(2.4) Similarly, if n = 4p + 2, we have

IIYnll. = 1 + 2 (cosec _1T_ + ... + cosec(2p _ 4p+2

and so

1) _1T_) , 4p+2

F.F. Bonsall I Nuclear Hankel Operators

230

(2.5) Now suppose that n = 2m + 1. From (2.3), we have ~

3~

2n

2n

IIYnll l ~cosec-+cosec-+'"

~

+cosec(2m -1)-, 2n

and so (2.6) Also, since cotan decreases, tr

2~

2n

2n

211 Y n III ;?l: cotan - + cot an -

Zmtr + ... + cotan - -

= 2 ( cosec -tr + cosec -2~ + n

n

2n

m7T)

... + cosec - . n

Since ~log(2p + 1) ~ up ~ 1 + ~log(2p - 1), (2.4) and (2.5) give

n -log(2p + 1) ~ IlYnlll ~ n(1 + ~log(2p - 1))

(n = 4p),

~

and

n

1 + -log(2p + 1) ~ llYn III ~ n(1 + 210g(2p - 1)) I

(n = 4p+2).

7T

Here we have n ;?l: 4, and so 1 + ~log(2p - 1) ~ 10g(eGn)ll2) ~ log n , and log(2p + 1);?l: log ~n ;?l: ~Iog n .

F.F. Bonsall I Nuclear Hankel Operators

231

Thus the required inequalities are proved for even n except perhaps for n = 2 for which (2.3) gives IIY2111 = cot an ~7T, again satisfying the required inequalities. When n = 2m + 1, (2.6) gives

II Y n III ~ n(1 + ~log(2m - 1)) = n log(e(2m - 1)112). The inequality e 2(2m - 1) ~ (2m + Ii is obvious when m ~ 4 and easily verified when m = 1, 2, 3, so that llYn 1/1 ~ n log n. Finally (2.7) gives

n

IlYnlll~-log(m 7T

n

+ 1)~-log(2m + 1), 27T

and the proof is complete. 0 (n E 1\1).

Proof. The coefficient sequence of R; is (n, n - 1, ... ,2,1,0,0 ... ) so that its matrix is (/3;) with (i+j~n-l),

={n-(i+j) 0

/3;j

(i+j~n).

With D; as before, D:Dn has the matrix (a;) with

a .. = {n - max{i, j) "

0

(max{i, j} ~ n - 1), (max{i, j} ;a. n) .

With 1 ~ P ~ [n/2], let Fp be the finite rank operator with matrix (IJ- ~)), where IJ-~) = I for p ~ i, j ~ n - p and IJ-~) = 0 for all other i, j. Then the matrix of F 1 + F 2 + ... + F[n/21 is (IJ-ij)' with IJ-;j the cardinality of the set of integers p with

1 ~ P ~ min{i, i).

n- p

~max{i,n.

n}

(0 ~ i, j

Thus II ..

,-"

= {min{min{i, i). n - max{i,

0

~

n - 1),

(all other i, j) .

232

F.F. Bonsall / Nuclear Hankel Operators

For 0,,;;; i, j ,,;;; n - 1, we have min{i, j}

a jj

-

{3jj

= { n - max{i, j}

(i + j,,;;; n - 1) , (i+j;;;:'n).

Noting that min{i, j} < n - max{i, j} if and only if i + j < n, we see that aij- {3jj = /-Lij' and so D:Dn - R n = F I + F z+ ' " + F[n/2j' Since (n - 2p+ 1rl~ is a projection, it follows that D:Dn - R; ;;;:. 0, and so (n even) , (n odd).

Since also IID:Dn ll l = Tr(D:Dn ) = ~n(n + 1), we have ~n(n+1)-jn2

";;;IIRnlll,,;;;~n(n+1)+jn2 z ~n(n + 1) - j(n - 1)";;; IIRnll l ,,;;; ~n(n + 1) + j(n 2- 1)

(n even), (n odd),

and the theorem is proved. 0

3. Sufficient Conditions for Nuclearity of Hankel Operators Throughout this section, {an} is a sequence of complex numbers, and we obtain sufficient conditions that it is the coefficient sequence of a nuclear Hankel operator. Theorem (3.1). Let limn--+oo an = 0 and L :=z/an-I - an In log n < 00. Then {an} is the coefficient sequence of a nuclear Hankel operator. Proof. Let An be the Hankel operator with coefficient sequence (a Q, ai' ... , an-I' 0, 0, ...), that is, in terms of the elementary Hankel operators Tn'

Let bn = an-I- an (n E N) and B; = b.S, + bzSz + ... + bnSn. Since S; = To+ T, + ... + Tn-I' we have

233

F.F. Bonsall I Nuclear Hankel Operators

B n = (a o- al)SI + (a l - a2)S2 + ... + (an-I - an )Sn

= aoS I + a l(S2- 5 1) + " , + an_.(Sn - Sn_I)- anSn =

An - anSn,

and so (3.1)

(n

> j + k).

By Theorem (2.1), we have

L IIbn Sn ll 1 < 00,

n=1

and, since the space of all nuclear operators is a Banach space with respect to the norm 11·111' there exists a nuclear operator B with

Since 11-11:0:::; 11-11. and Iimn --+oo an = 0, it now follows from (3.1) that (Bej, ek ) = lim (Anej, ek ) = aj+k , n_oo

thus B is the Hankel operator with coefficient sequence {an}' 0

:L:=1 n 2lan_ 1 -

2a n + an+11 < {an} is the coefficient sequence of a nuclear Hankel operator. Theorem (3.2). Let limn_ooa n = 0 and

Proof. Let b;

= an-I - an' c; = b; - bn+ l, let An be as above, and let

Since R; - R n-I = Sn' we have

en

= = =

00.

(b, - b2)R I + (b2- b3)R z + ... + (bn - bn+I)Rn blR I + b2(R2- R I) + ... + bn(Rn - R n-I) - bn+IRn

b l SI + b2S2+ ... + bnSn - bn+IRn

Then

F.F. Bonsall / Nuclear Hankel Operators

234

= (a o-

al)SI + (a l

-

a~S2 +

... + (a n_ 1 - an)Sn - bn+IRn

= aoS I + a l T I + ... + an_ 1Tn-I -

anSn - bn+IRn

= An - anSn - bn+tRn·

Thus (3.2)

(n

> j + k).

We prove next that (3.3) With p > q, we have cq + cq +1 + ... + cp _ 1 = bq - bp , and so 2 q 1bq - bpi"';

p-t

2: n 2Jenl·

n=q

Given e > 0, there exists an N such that L ~:~ n 2 Jen I< e (p > q '2!= N Since limn..."" bn = limn....""(an_ l - an) = 0, we therefore have E

and (3.3) is proved. It now follows from (3.2) that, for all j, k, (3.4)

lim«A n - Cn)ej , ek ) OO n....

=

By Theorem (2.3),

2: IlcnRn III <

00 ,

n=1

and so there exists a nuclear operator C with

°.

E

) .

F.F. Bonsall / Nuclear Hankel Operators

235

From (3.4) it follows that (Cej , ek ) = aj+k' and so C is the Hankel operator with coefficient sequence {an}' Corollary (3.3). For n ;?; 0, let an satisfy

(i). an;?; 0,

(ii). an - an+ 1 ;?; 0,

(iii). an - 2a n+ 1 + an+z;?; 0.

Then {an} is the coefficient sequence of a nuclear Hankel operator if and only if (iv).

2: an <

00 •

n~O

Proof. Let A be the Hankel operator with coefficient sequence {an}' It is almost obvious that (iv) holds if A is nuclear, for

Suppose on the other hand that (iv) holds, and let bn = an-l - an' an = ~n(n + 1). We have b l + 2b z + ... + nb; = a o+ a 1 + ... + an-l - nan' and so lim nan = a n_ oo

Since

a

>

= 2: an n=O

2: nb; ;?;

n=l

°.

°would contradict (iv), we have lim n....'" nan 2: nb; = 2: an <00.

n=l

n=O

Since an - a n-1 = n, we have

and so

n=l

n=l

n=l

=

0, and

236

F.F. Bonsall / Nuclear Hankel Operators

Since lan-I - Za; + an+11 = b; - bn+ l , Theorem (3.2) now shows that {an} is the coefficient sequence of a nuclear Hankel operator. 0

Remark (3.4). An alternative proof of Corollary (3.3) can be based on a theorem of Howland [2] as follows. With p > q, we have

and, since {b n } decreases, (p > q).

Taking q

= nand p = 2n + 1, 2n + 2, ... , we have

and so

Likewise

Thus

and, by Howland [2, Th. 1.5] (with a misprint corrected), this implies that {an} is the coefficient sequence of a nuclear Hankel operator. Example (3.5). In the light of Corollary (3.3), it is natural to ask for a sequence {aJ satisfying (i), (ii) and (iv) that is not the coefficient sequence of a nuclear Hankel operator.

237

P.P. Bonsall I Nuclear Hankel Operators

Take k o = 1 and choose positive integers k l , k 2 ,

(n

(3.5)

~

•••

in turn with

0),

so that {an} is a decreasing sequence with an - an+1 > ~an > O. Let Cn be the operator with matrix ('Yij), where

I

(i + j OS; k; - 1) , (0 OS; i OS; k; - 1,0 os;j OS; k; -1, i + j (all other i, j) ;

an

'Yij =

~n+l

~

k n ),

and let En be the operator with matrix (Sj), where S.. = IJ

{I0

(i, j

os;

k; - 1) ,

(all other i. j) .

Then Cn = (an - an+l)Sk n + an+1En, and, since k~l En is a projection, we have

IICnll

1

~ (an - an+I)IISkn III - an+IIIEnIII

~ (an - an+l){2~ k n log k; -

k; } - an+l k;

Plainly IICnll l -+ 00 as n -+ 00. Now let {an} be the sequence with (2k n

Then {an} satisfies (i) and (ii) and

os; j

< 2k n + l , n

= 0, 1,2, ...)

238

F.F. Bonsall / Nuclear Hankel Operators

i Lan = n=O

Uo+ u1(k 1- k o)+ u 2(k2 - k 1) + ' " < 1 +

L

unkn <00.

n~l

Let A be the Hankel operator with coefficient sequence {an}' Since k n;;;': 2k n_ 1, the k n x k n matrix (Yii) occurs as a submatrix of the matrix of A, and so there exist orthogonal projections Pn, On with PnAQn unitarily equivalent to en' If A were nuclear, we would therefore obtain the impossible inequality

Acknowledgments I am indebted to T.A. Gillespie for the crucial idea in the above example and to the Science and Engineering Research Council for their Special Replacement Scheme.

References [1] 1.W. Archbold, Algebra (Pitman, London. 1958). [2] 1.S. Howland, Trace class Hankel operators, Quart. 1. Math. Oxford Ser. (2) (1971) 147-159. [3] V.V. Peller, Nuclearity of Hankel operators, LOMI Preprint E-I-79, Leningrad, 1979. [4] S.C. Power, Hankel operators on Hilbert space, Res. Notes in Math. 64 (Pitman, London, 1982). [5] 1.R. Ringrose, Compact non-self-adjoint operators, Van Nostrand Reinhold Math. Stud. 35 (Van Nostrand Reinhold, New York, 1971).

l.A. BARROSO editor, Aspects of Mathematics and its Applications © Elsevier Science Publishers B.V. (1986)

239

UNIQUENESS OF THE MAPPING DEGREE FOR ELLIPTIC OPERATORS WITH STRONG ZERO ORDER NONLINEARITIES Felix E. BROWDER Department of Mathematics, University of Chicago, Chicago, Ill., U.S.A.

Dedicated to Leopoldo Nachbin

In a number of recent papers, [1], [2], [3], [4], [5] the writer has constructed degree theories of the classical type (i.e. with values in the integers rather than in some general algebraic structure) for operators of monotone type, i.e. mappings f with domain in a reflexive Banach space X and values in its conjugate space X*. One of the most important examples of such mappings from the point of view of applications is that in which the Banach space X is the Sobolev space W;,P(fl) where fl is an open subset of the Euclidean space IR n with n;;;' 1, m ;;;. 1, 1 < P < +00. If Wm,P(fl) is the usual Sobolev space of functions u on fl such that u and all its partial derivatives Dtu for lal"s; m lie in U(fl), where the latter space is the LP space corresponding to Lebesgue measure on Il, then W;;"P(fl) is the closed subspace of Wm,P(fl) generated by the testing functions with support in fl. Its conjugate space X* is the space of distributions W-m'P'(fl) with p' = p(p _1)-1, the conjugate exponent to p. The mappings f we wish to consider for our present discussion are those of the form f(u) = A(u) + g(x, u),

where A is a regular elliptic operator mapping W;;"P(fl) into W-m'P'(fl) of the form A(ll)=

2:

lalEm

(-l)la ID aA a (x, u, ... ,D m ll) ,

and g is a function satisfying the Caratheodory condition from fl x R 1 into IR 1. If the coefficient functions A a satisfy a polynomial growth

240

F.E. Browder I Uniqueness of Mapping Degree

condition of the form

with h in U'(fl), as well as ellipticity and semi-coerciveness conditions, then A defines a continuous mapping of the whole space X = 'W~,P(fl) into X* = 'W-m,p'(fl) which maps bounded sets of X into bounded sets of X * and falls (depending upon the precise assumptions) into one of the two following classes of mappings: (PM). If uj is a sequence in X which converges weakly to w

uj ~ u) and if limsup(A (uj

),

uj

-

U

(we write

u) ,;:;; 0, then

A(u) converges weakly to A(u) in X*

(A(u),

and

u)~(A(u), u).

(These are the pseudo-monotone mappings.)

(SL. If uj

w

~ U

in X and if limsup(A(u), uj

-

uj converges strongly to

u) ,;:;; 0, then II

in X.

For the function g. we assume the following: (1). g(x, r) is continuous in r for fixed x, measurable in x for fixed r. For each integer s, there exists a function h, in L'oc(fl) such that for all r with /rl,;:;; S, Ig(x, r)/,;:;; h,(x). (2). g satisfies the sign condition: i.e. g(x, r) has the same sign as r. Before we proceed to specify the class of mappings in more detail, let us remark that by a degree theory on a class F of mappings f, corresponding to a class of permissible homotopies H, we mean an integer-valued function which for every mapping f in the class and each bounded open set G in X with f:clos

G~X*

and for each target point Yo in X * such that Yo ~ f(JG), (where JG is the boundary of G), we have an integer-valued function d(f, G, Yo) defined having three basic properties: (1). Normalization. If d(f, G, Yo) 01- 0, then yoEf(G). For a suitable canonical mapping fo (in this case, the duality mapping J), dUo, G, Yo) = +1 if Yo Efo(G).

F.E. Browder / Uniqueness of Mapping Degree

241

(2). Additivity on the domain. (3). Invariance under homotopies {/,} in H In [1]' a degree function was defined on the class (S)+ and was shown to be unique in [2] if the class of homotopies included affine homotopies. In [3], (see also [5]) these conclusions were extended to maps of the form T + f, with T maximal monotone and f a map of class (5)+ which maps bounded sets into bounded sets. With a slight generalization of the concept of degree, similar conclusions were obtained in the same papers if f were only pseudo-monotone. Finally, in [4], a degree function was constructed for the class of maps of the form f + N g with f in the class (S)+ and N g the Niemytskii operator corresponding to g in the following sense: If Ng(u) is given by Ng(u)(x) = g(x, u(x», then N g maps dom(Ng) into X'" where dom(Ng ) = {ul u E X, N/u) E X'" n LI~}' Theorem. Suppose that d and d I are two degree functions defined on the class F of maps of the form A + N g, with A in (S)+ and N g the family of Niemytskii operators corresponding to functions g satisfying the sign condition as above. Suppose that the class of homotopies H for both degree functions includes all affine homotopies between A + N g and maps Al of class (S)+. Then d and d, must coincide. Proof. We let A k = A + N k , where N, is the Niemytskii operator corresponding to the truncation of g at level k, multiplied by the characteristic function of a compact subset G, of G. The mapping N, is then compact from X to X"', and since the class (S)+ is invariant under compact perturbations, A k must lie in (S)+ for each k. We assume that as k goes to infinity, G k expands to fill out G. Consider the homotopy g, = (1- t)Ng + tNk + A .

By hypothesis, the homotopy g, lies in the class H. On the class (S)+, by the uniqueness theorem of [2] (see also [5]), d and d, must coincide. Hence the two degrees will coincide on g, for t = 0, i.e, if Yo g go(JG), then

We assume that Yo g (A + Ng)(JG). We propose to show that for k

242

F.E. Browder / Uniqueness of Mapping Degree

sufficiently large and for all t in the unit interval [0, 1], we shall have Yo~ g,(aG).

Suppose the contrary. Then we should have a sequence {Uk} in aG for k going to infinity, and a sequence {td in [0,1] such that

Since G is bounded in the reflexive Banach space X, we may assume without loss of generality that Uk converges weakly to some U in X, and that tk converges to t in [0,1]. Using the compactness of the Sobolevimbedding on compact subdomains of G, we may also assume that uk(x) converges almost everywhere in fJ to u(x) and that Nk(uk)(X) converges almost everywhere to Ng(u)(x). By the sign condition on g and the definition of Nk(Uk)' the function Nk(Uk)Uk is non-negative and converges a.e. to Ng(u)u. Hence Ng(u)u is integrable and Ng(u) lies in L:oc(fJ). Let

By the same argument as above, WkUk ;;'0, and Wk converges a.e. to Ng(u) in a. By assumption,

We now apply the result of Brezis and Browder [6], based on the results of Hedberg [7], which asserts that if an element T of X * n Lt.ehas the property that for u in X, T(x )u(x) ~ 0, then (T, u)=

J T(x)u(x)dx,

where the wedge pairing indicates the distribution pairing between X and X*. Hence

F.E. Browder I Uniqueness of Mapping Degree

243

By Fatou's lemma,

while by the result of [6], since we may assume that wk = Yo - A(uk ) converges weakly in X* and its weak limit must coincide with its a.e. limit,

Hence, we know that

On the other hand

since A(uk ) Finally,

=

Yo- wk . : Yo - Ng(u).

limsup (A(u k ) ,

Uk -

u):so; a.

Since A is assumed to lie in the class (S)+, it follows that Uk converges strongly to u. By the continuity of A, A(uk ) converges to A(u). Hence u lies in aG and (A + Ng)(u) = Yo' This contradicts our initial assumption. Thus for k sufficiently large and all t E [0, 1], dig; G, Yo) and d1(gl' G, Yo) are independent of t in [0,1]. Since they coincide for t = 0, they coincide for all t. Hence d(A + N g , G, Yo) = d1(A + N g , G, Yo). 0 References [1] F. Browder, Degree of mapping for nonlinear mappings of monotone type, Proc. Nat. Acad. Sci. U.S.A. 80 (March 1983) 1771-1773. [2] F. Browder, l'Unicite du degre topologique pour des applications du type monotone, C.R. Acad. Sci. Paris ser, I 296 (24 janvier 1983) 145-148.

244

F.E. Browder / Uniqueness of Mapping Degree

[3] F. Browder, Degree of mapping for nonlinear mappings of monotone type: Densely defined mapping, Proc. Nat. Acad. Sci. U.S.A. 80 (April 1983) 2405-2407. [4] F. Browder, Degree of mapping for nonlinear mappings of monotone type: Strongly nonlinear mapping, Proc. Nat. Acad. Sci. U.S.A. (April 1983) 2408-2409. [5] F. Browder, Fixed Point Theory and nonlinear problems, Bull. Amer. Math. Soc. (9) 1 (1983) 1-39. . [6] F. Browder and H. Brezis, Some properties of higher order Sobolev spaces, J. Math. Pures Appl. 61 (1982) 245-259. [7] L.I. Hedberg, Spectral synthesis in Sobolev spaces and uniqueness of solutions of the Dirichlet problem, Acta Math. 147 (1981) 237-264.

J.A. BARROSO editor, Aspects of Mathematics and its Applications © Elsevier Science Publishers B.V. (1986)

245

ON A CONFORMAL INVARIANT OF THREE-DIMENSIONAL MANIFOLDS Shiing-shen CHERN t Mathematical Sciences Research Institute. Berkeley. California 94720, U.S.A. Dedicated to Leopoldo Nachbin

1. Introduction In [3] James Simons and I introduced an invariant for a compact orientable three-dimensional Riemannian manifold as follows: Let M denote the manifold, oriented, and let P be the bundle of its orthonormal frames, so that we have (1.1)

where 7T is the projection, mapping a frame xe teZe 3 to its ongm x. A section s : M 4 P of the bundle satisfies the condition 7T 0 S = identity and can be viewed as a field of frames. It is well known that in our case such a section always exists; we say that M is para lieliz able. (All our manifolds and maps are ~oo.) To such a frame field the Leoi-Cioita connection of the metric is given by an antisymmetric matrix of one-forms: (1.2)

and its curvature by an antisymmetric matrix of two-forms: (1.3)

Throughout this paper our small Latin indices will run from 1 to 3. We have I Work done under partial support of National Science Foundation grants MCS 77-23579 and MCS 8120790.

246

(1.4)

S.-s. Chern I Conformal Invariant dWik

=

L W ij

A W jk

+ n ik

•

j

We introduce the three-form (1.5)

and consider the integral (1.6)

cP(s) =

fiT. M

It will be proved below that for another section t: M ~ P, cP(t) - cP(s) is an integer, so that cP(s) mod 1is independent of s. This defines an invariant cP(M) ERa, where cP(M) = cP(s) mod 1. In [3] we proved the theorems: Theorem (1.1). cP(M) is a conformal invariant, i.e. it remains unchanged under a conformal transformation of the metric.

Theorem (1.2). cP(M) has a critical value at M if and only if M is locally conform ally flat.

In this paper we will give direct proofs of these theorems, both because the three-dimensional case has special features and because the invariant cP(M) has come up in various other connections. In fact, up to an additive constant it is the eta invariant of M, which was introduced by Atiyah, Patodi, and Singer via spectral theory [1]. It has also been used by W. Thurston in his theory of hyperbolic manifolds [6], and Robert Meyerhoff has shown that, for certain hyperbolic manifolds it takes values which are dense on the unit circle [5]. It is also related to the concept of 'anomaly' in two-dimensional gauge field theory in physics.

2. Family of Connections We shall use arbitrary frame fields to develop the Riemannian geometry on M. Let xe,e 2e3 be a frame, and w\ w 2 , w 3 , its dual coframe.

247

S.-s. Chern I Conformal Invariant

Let the inner product be (2.1)

We introduce gij through the equations (2.2)

and use the g's to raise or lower indices, as in classical tensor analysis. Then the connection forms w~ or W ij are determined, uniquely, by the equations (2.3)

W;j

+ Wj;

=

dg jj

•

The curvature forms are defined by (2.4)

'''''k n ~.= dw {LJ W;

1\ W

.

~

•

By exterior differentiation of (2.4) we get the Bianchi identity (2.5) To avoid confusion notice our convention that the upper index in is the second index. Thus :L j W{gjk = w ik (¥ W ki in general). We introduce the cubic form

wi, n~

On our three-manifold M, T is of course closed. But the basic reason for its importance and interesting properties is that formally by (2.4), (2.5), we have (2.7)

which is the first Pontrjagin form. Consider a family of connections on M, depending on a parameter Then w~, ni, T all involve T, and we have the fundamental formula

T.

248

(2.8)

S.-s. Chern I Conformal Invariant

«t= - d {"" j aw~ "" j aWl} 8 1T 2 L.W.II--L.W,aT 'aT 'aT aw : . aw{ .) +22": ( - II a l - 1\ a '. . aT 1 aT 1

The proof of (2.8) is straightforward. It follows by differentiation of (2.6), and using the formulas obtained by differentiation of (2.3), (2.4) with respect to T. It is useful to observe that the last term is a polarization of the Pontrjagin form. Let P' be the bundle of all frames of M, so that we have j

P - P'

~/.r.

(2.9)

,

M where i is the inclusion. Then T in (2.6) can be considered as a form in P' and its pull-back i*T is the T given by (1.5). We will make such identifications when there is no danger of confusion. We consider C/l(s) defined by (1.6). When t is another section, then t(M) - s(M), as a three-dimensional cycle is homologous, modulo torsion, to an integral multiple of the fiber Px ' x E M. But P, is topologically SO(3) and w ij reduces on it to its Mauer-Cartan forms. If j: P, = SO(3)~ P is the inclusion, '*T-

+1 -'iJ

1 -2

81T

W12 II W23

II

W31'

whose integral over P, is 1. Hence C/l(s) mod 1 is independent of s. We wish to clarify the relation between orthonormal frame fields and arbitrary frame fields. By the Schmidt orthogonalization process P is a retract of P', under which the origin of the frame is fixed. The retraction we denote by r : P' ~ P. We consider the form T defined in (2.6) to be in P'. If s' : M ~ P' is a section, then s = r 0 s' is also a section and they are homotopic through sections. Since d T = 0, we have (2.10)

J T= J i*T. s'M

sM

249

Si-s. Chern / Conformal Invariant

Hence C/J(M) can be computed through an arbitrary frame field by the left-hand side of the last equation.

3. Proofs of the Theorems

In view of the above remark we can, for local considerations, use a local coordinate system u' and the resulting natural frame field a/Bu'. We shall summarize the well-known formulas, which are Wij --

'" LJ

r!ik du k '

(3.1 a)

The

R ijkl

satisfy the symmetry relations R;jkl = R klij,

R ijkl = - R jikl = - R ijlk '

(3.1 b)

0,

R;jkl+ R iklj+ R ilik =

Rijkl.h

+ R;jlh.k + Rijhk.1 = 0,

where the comma denotes covariant differentiation. They imply (3.2)

and (3.3)

We also introduce the (3.4)

' " L..J

" g ik{lik = {l ii = 'L..J

Ricci curvature i ' " Rij R k -- L..J kj' j

0.

and the

scalar curvature

by

250

S.-s. Chern / Conformal Invariant

For treatment of the conformal geometry we define

c:u :-Rik.l- n!i.e>:l(dR dR) Uk .I-u l .k' (3.5)

Then the Bianchi identities give

C i kl + C kli + C li k

(3.6)

=

O.

These relations imply that the matrix

C=

(3.7)

is symmetric and that the matrix GC, where G = (g i), has trace zero. Schouten proved [4, p. 92] that the three-dimensional: manifold M is conformally flat, if and only if C = 0, i.e. C;ik = O. By integrating (2.8), we get

(3.8)

87T

2

:T J T = 2 J L1, M

M

where (3.9)

~ (aw: -/I

L1=LJ

aT

.) = - L ~Jaw{ . -/l{l'·, I aT I

. aw{ I aT

[]l--/l{l'.

by (3.3). We consider a family of metrics gii(T) and put (3.10)

ag

V .. = _ ' I 'I

aT

To prove Theorem (1.1) we suppose this is a conformal family of metrics, i.e.

251

S.-s. Chern / Conformal Invariant

(3.11)

V ij =

ug ij

•

From (3.1a) we find (3.12) where Uk = Ju/Jx k • By the second equation of (3.2) and the equation (3.3) we find L1 = O. This proves that JM T is independent of T, and hence Theorem (Ll). To prove Theorem (1.2) we consider Vii such that the trace 2: v; = o. Geometrically this means that we consider the tangent space of the space of conformal structures on M. From (3.1a) we find (3.13) It follows that

(3.14) The term in the middle is zero, because n i is antisymmetric and dV ij is symmetric in i, j. To the integral of the last term we apply Stokes theorem to reduce it to an integral involving only the Vii' and not their derivatives. We will omit the details of this computation. The result is that the condition

is equivalent to (3.15)

JTr(VC) du

l

1\

du 2

1\

du 3 = O.

M

If the metric is conformally flat, we have C

the above integral.

= 0 and

hence the vanishing of

252

S.-s. Chern / Conformal Invariant

Conversely, at a critical point of rp we must have Tr(VC) = a for all symmetric V satisfying Tr(VG- 1) = O. Hence C is a multiple of G- 1 or GC is a multiple of the unit matrix. But GC has trace zero. Hence it must itself be zero and we have C = O. This proves Theorem (1.2).

References (1J M.F. Atiyah, V.K. Patodi and I.M. Singer, Spectral asymmetry and Riemannian geometry II, Math. Proc. Cambridge Philos. Soc. 78 (1975) 405-432. [2] S.S. Chern, Complex manifolds without potential theory, 2nd Edition (Springer, Berlin, 1979) Appendix 97-150. [3J S.S. Chern and J. Simons, Characteristic forms and geometrical invariants, Ann. of Math 99 (1974) 48-69 or S.S. Chern, Selected papers (Springer, Berlin, 1978) 444-465. [4J L.P. Eisenhart, Riemannian Geometry (Princeton Univ. Press, Princeton, 1949). [5J R. Meyerhoff, The Chern-Simons invariant for hyperbolic 3-manifolds, thesis (Princeton University, Princeton, 1981). [6J W. Thurston, Three-dimensional manifolds, Kleinian groups, and hyperbolic geometry, The Mathematical Heritage of Henri Poincare (Amer. Math. Soc., Providence, RI, 1983) 87-111.

1.A. BARROSO editor, Aspects of Mathematics and its Applications

253

© Elsevier Science Publishers B.V. (1986)

SOME ASPECTS OF INFINITE-DIMENSIONAL HOLOMORPHY IN MATHEMATICAL PHYSICS J.F. COLOMBEAU u.E.R. de Mathematiques et d' Informatique, Uniuersite de Bordeaux I, 33405 Talence, France Dedicated to Leopoldo Nachbin who has been developing and inspiring infinite-dimensional holomorphy for the past fifteen years

O. Introduction Infinite-dimensional holomorphy originated at the beginning of the century in works of Hilbert [21], 1909, and others. In the thirties the theory had been developed in the setting of normed spaces: Michal [26] and others. Recently the theory has been very much developed not only in Banach spaces but also in general locally convex spaces: Nachbin [28], [29], [30], Barroso [1], Chae [3], Coeure [4], Colombeau [5], FranzoniVesentini [20], Dineen [16], Mujica [27], Ramis [34], Noverraz [33], just to quote a few books or survey articles. The purpose of this article is to illustrate the interconnections between this theory and applications by recalling two aspects of its influence in mathematical physics. The first one, known as the 'method of analytic functionals', is quite old (Fock [19] 1934). It is based on the remark that the Fock spaces of Boson fields may be considered as spaces of holomorphic functions over the space L 2([R 3) (or similar spaces). It gave rise to a large amount of literature in mathematical physics (see [2], [32]) and mathematics. Besides holomorphic functions on L 2(1R. 3) it also uses holomorphic functions on the spaces 9'([R 3) and ,9"(1R 3). The second one is quite recent. Using holomorphic functions and Cf5'" functions over the spaces jg'(fl) and f»(fl) (11 open subset of IR n) the author defined a general multiplication of distributions which has all the needed computational properties to explain heuristic computations where distributions are treated like usual Cf5'" functions (see [9]). Some other motivations for infinite-dimensional holomorphy can be found in [31], [22], [38].

J.F. Colombeau / Aspects of Infinite-Dimensional Holomorphy

254

l. Fock Spaces of Boson Fields and Entire Functions on the Hilbert Space

L

Z(1R3

),

on the Frechet Space 9'(1R 3 ) and on the (DF)-Space 9"(R 3)

The Fock spaces of Boson fields are constructed on the following simple model: F

+'"

= EB (L z(1R 3»0;.$ , n=O

where ®~$ denotes the completed symmetric n-fold tensor product and where EB::o denotes the Hilbertian direct sum. Therefore an arbitrary element K of F may be written as an infinite sequence

IIK/I; = IKol 2+ ~

n=1

J!Kn(x., ... , xnW dx •... dXn <+00.

The more fundamental operators on the Fock space are the creation operators a+(ip) and the annihilation operators a-(ip) defined if ip E L 2(1R 3) by:

and

a-(ip )(Kn)::o

=

(n + lyl2

J

Kn+M,· )ip(~) d~)

i'ER3

+oc

n=O

(in the two above second members we use abbreviated formulas for the functions (Xl" .. , Xn)~ n l/ZSym(ip(x1)Kn_l(xZ' ... , xn and (Xl' .•. , xn)~ (n + 1)IIZ Ji'ER3 Kn+M, Xl' ... , Xn)ip(~) d~). The domains of these operators contain the space of the states K such that K; = 0 for n large enough (i.e. the states with a finite number of particles). The introduction of infinitedimensional holomorphy in this setting may be traced back as far as the beginnings of the theory [19]. To K we associate the function f/J from

»

J.F. Colombeau / Aspects of Infinite-Dimensional Holomorphy

l'/>(a)=Ko+

255

JK1(x)a(x)dx+'"

+~ (n!)

J Kn(x

l , •••

,xn)a(x l )

"

•

a (xn) dx l

· · .

dx, + ... ,

if a E L 2(1R 3). This series is bounded above in absolute value by

Therefore it is immediate to check that I'/> is an entire function on L 2(1R 3 ) which has a lot of further properties, e.g. it is bounded on bounded sets. Further since ,n(n)(o) K n -- _1_ (n !)1/2 '¥ ,

the map K ~ I'/> is injective from r into .re(L2(1R 3) ) , the space of all entire functions on L 2(R 3). This representation of the Fock space as a space of holomorphic functions on L 2(R 3) is very nice since basic operators on r have very simple formulas: for instance the formulas for the creation and annihilation operators are: (a+(
J(a) ,

(a-(

(J)(a)
if a, tp E L 2(1R 3) . In this framework it was proved in Berezin [2, Th. 1.5] that any linear bounded operator on r admits some 'kernel' called normal form in Berezin [1]. Then this result was extended to most unbounded operators in [10], [11], [12], [13], [14], [23], [24], [25]. A glimpse at the situation is as follows: Let us denote by [) the algebraic direct sum +00

[) =

ffi (9'(1R 3))~~'S . n=O

256

1.P. Colombeau / Aspects of Infinite-Dimensional Holomorphy

D is a dense subspace of IF . Then considering as above IF as a subspace of Ye(L 2(R 3 ) ) , we have

where 9l(9"(R 3) ) denotes the space of all continuous polynomials on the (DF)-space 9"(R 3) and where Exp(9"(R 3) ) denotes the space of all entire functions on the (DF)-space 9"(R 3 ) which are of the exponential type (if E is a complex locally convex space we say that a holomorphic function r:/J : E ~ 0 such that 1r:/J(x)1 ~ c exp(p(x)) for all x E E). The basic mathematical tools for these 'kernel' theorems are the following general results of infinite-dimensional holomorphy [5, Chap. 6, 7]. Let E be a dual nuclear complete complex locally convex space and let E' denote its strong dual (i.e. E' is the topological dual of E and E' is equipped with the topology of uniform convergence on the bounded subsets of E). The space Ye(E) of all the holomorphic functions on E is equipped with the topology of uniform convergence on the compact subsets of E and Ye'(E) denotes its strong dual. The Fourier-Borel transform is the linear map from Ye'(E) into Ye(E') defined by $(T) = rp(e T ) if rp E Ye'(E) and TEE'. Then we have Theorem (1.1). In the above conditions for E the Fourier-Borel transform is a topological isomorphism from Ye'(E) onto Exp(E').

Now let E and F be nuclear complete complex locally convex spaces. If G is a locally convex space, 0' its strong dual and if H is another locally convex space we denote by 2(0', H) the linear space of the continuous linear maps from 0' into H, equipped with the topology of the uniform convergence on the equicontinuous subsets of 0'. Theorem (1.2). In the above conditions for E and F and if fl, fl' are respectively open subsets of E and F then we have the natural topological isomorphism

2(Ye'(fl'), Yes(fl)) ~Yes(fl x fl'), L~[(a, T)~L(DT)(a)].

The notations are: a E fl, TEfl', DT E Ye'(fl') is the Dirac measure at

J.F. Colombeau / Aspects of Infinite-Dimensional Holomorphy

257

the point T: 8T ( cP) = cP(T) if cP E :le(n') and :les(n), :les(n x fl') denote the spaces of the Silva-holomorphic functions on n, n x fl' (equipped with the topology of uniform convergence on the compact subsets). Therefore one has the isomorphisms: Exp(9"(R 3» == Jt"(9'(1R. 3» A, where Jt"(9'(1R. 3)f is the image of Jt"(9'(1R. 3» by the Fourier-Borel transform, and

which give, since the Fourier-Borel transform is an isomorphism,

Thus the linear continuous operators from Exp(9"(1R. 3» into Jt'(9'(R 3»-which includes all usual unbounded operators on. the Fock space f -are characterized by a 'kernel' which is a holomorphic function on (9'(R 3)l As a remark this shows that even if in this method one considers a-priori holomorphic functions on a Hilbert space, one is led directly to the consideration of holomorphic functions on Frechet and (DF)-spaces. A detailed study of these results of infinite-dimensional holomorphy is in the book [5]. 2. General Multiplication of Distributions and Entire Functions over the (DF)-Space '(;'(n) and the (LF)-Space g;(n)

The basic heuristic computations of quantum field theory, which are dating back as far as the thirties, are based on computations on vectorvalued distributions (the free field operators) which are treated as usual err functions mainly concerning derivation, multiplication, integration and exponentials (these last ones are of the kind exp(iH) with H some symmetric operator-valued 'object'). This is a classical motivation to look for a theory of multiplication of distributions. Starting from the consideration of holomorphic functions over '(;'(n) and g;(n) the author developed such a theory, see [6], [7], [8], [9], [40]. The idea which is both natural and original is as follows: At first one considers the map

258

i.F. Colombeau

I Aspects of Infinite-Dimensional Holomorphy

(2.1)

where x Efland 8x denotes the Dirac measure at the point x; it is easy to check that AR E 'lJ(fl) and that the quotient algebra .?'t'('lJ'(fl»/Ker A is algebraically isomorphic to the algebra ~(fl): in short this quotient algebra is a very good algebra of functions on fl but does not bring anything new. However this suggests the following idea: In order to obtain in this way something new it should be necessary to consider a space larger than .?'t'(~'(fl». .?'t'(~'(fl» is a subalgebra of .?'t'(f!lJ(fl» (via the restriction map R ~ RI!'lJ(ll): use the well-known result that f!lJ(fl) is everywhere dense in 'lJ'(fl». Then one may look for an ideal .H of .?'t'(f!lJ(fl» such that .H n .?'t'('lJ'(fl» = Ker A. This is done in [6], [7], [8], [9], with the minor modification that .H is only an ideal of a (large enough) subalgebra .?'t'M(f!lJ(fl» of .?'t'(f!lJ(fl». Then we set (2.2) which is our algebra of 'new generalized functions' on fl. Let us give the definitions. For q = 1,2, ..., we set

R"

R"

If IpEdq, E>O and xElR n we set lpe.AA)=E-nlp(E-1(A-X».It is immediate to check that Ipe E d q if and only if Ip E d q and that Ip E,X ~ 8x in 'lJ'(1R n) when E ~ O. If R E .?'t'('lJ'(fl» then REKer d if and only if R(8x) = 0 for every x E fl. Now in the more general case when R is in .?'t'(f!lJ(fl», the expression R(8x) does not make sense. Since IpE,x~8x we are led to characterize Ker d by means of some properties of the set of values R (Ip e, x). It is clear that REKer d if and only if R (Ip E, x) ~ 0 when E ~ 0, for any x Efland any tp E d j • However, to seek for the ideal .H

we shall use the following more subtle characterization [9, Prop. 3.3.3]. Proposition (2.1). R E .?'t'(~'(fl» is in Ker d if and only if for any q EN, Ip E «: and for any compact subset K of fl there are constants c > 0, TJ > 0

such that

J.F. Colombeau / Aspects of Infinite-Dimensional Holomorphy

if 0 <

E

259

< 1/ and x E K.

This property can be extended to elements of :It'(@(fl)) and it is the starting point of the following definitions. Before we state them we need to introduce a natural concept of partial derivatives (in the numerical variable x E fl) for the holomorphic functions on @(fl). Definition (2.2). If R E :It'(@(fl)), fl eRn open, and if 1 <s i <S n we define an element ajax; R of :It'(@(fl)) by:

aR ) (cp)= -R'(cp)acp (ax; ax; when cp ranges over @(fl) (above R'(cp) : @(fl)-+ C is the derivative of R at the point cp E @(fl); it is an element of @'(fl)). This definition generalizes obviously the derivation of distributions; i.e. the particular case R is linear. If D = alkljax~l . . . ax~n one defines DR E :It'(@(fl)) by an obvious induction. We may check that (DR)(cp c.x) = D(x -+ R(c,oc' x)) [9, 4.4.1]. Now we may define :It'M(@(fl)) and X. The subalgebra :It'M(f1iJ(fl)) of :It'(f1iJ(IJ)) is by definition made of the elements R of :It'(@(fl)) such that R(c,oc,x) (and all its partial x-derivatives DR (CPc' x)) have a 'moderate' growth when E -+0. Definition (2.3). :It'M (@(fl)) is the set of the elements R of :It'(f1iJ(fl)) such that for every compact subset K of fl and every partial derivation D

there is an N E fill such that for every

if x E K and 0 <

E

cp E SI1N

there are c, 1/ > 0 such that

< 1/.

:It'M (@(fl)) since, if R E :It'(~'(fl)), then DR E (DR)(cp c,J-+ (DR)(~x) when E -+ O. It is easy to check that f1iJ'(IJ)C :It'M(f1iJ(fl)). Now we define the ideal X of :It'M(f1iJ(fl)). Clearly

:It'(~'(fl)) C

:It'(~'(IJ)) and

Definition (2.4).

j{

is the set of the elements R of :It'(f1iJ(fl)) such that for

260

1.P. Colombeau / Aspects of Infinite-Dimensional Holomorphy

every compact subset K of fl and every partial derivation D there is an N E N such that if q ;;;. Nand cp E d q there are c, TJ > 0 such that

if x E K and 0 < e < TJ. We note that the definition of N is a straightforward consequence of the above characterization of Ker d (if REKer d one checks easily that DR E Ker .91). Our algebra ,§(fl) is now defined. In [9, Prop. 3.4.7] we prove: Proposition (2.5). ~'(fl) may be considered as contained in ,§(fl) through the map T ~ (class of R T ) = R T +}( where R T (cp) = (T, cp). Therefore we have the inclusions: 'if) (fl) C

~'(fl)

C ,§(fl) .

Therefore, since 'if)(fl) is a subalgebra of ,§(fl), the multiplication in ,§(fl) is a natural candidate for a multiplication of distributions. This multiplication has all usual computational properties. However it is well known [39] that a 'reasonable' general multiplication of distributions is impossible, even in a setting larger than distribution theory. Let us explain this apparent contradiction. It is an indispensible requirement that the new multiplication should generalize the multiplication of continuous functions. For this, Schwartz assumes very precisely that the new product should be exactly equal to the classical product in the new algebra (possibly larger than ~'(n». In our case this last mentioned requirement is satisfied for cgoo functions. If f and g are two continuous functions on fl then their new product in ,§(fl) denoted here by f0 g is in general different from their classical product fg. But we have: 10 g is an 'abstract object' (i.e. an element of ,§(fl) which is not in ~'(fl» which admits in a natural and non ambiguous sense a projection in ~'(fl) which is the classical product fg. More precisely this coherence of the two products of the continuous functions is obtained as follows: In our theory of generalized functions we construct an algebra cg containing C such that if G E ,§(fl) and x E fl then G(x) E cg (note that cg depends on n if fl is an open subset of R "), As an example 8(0) E cg\C (i.e. 8(0) E cg and 8(0) ~ C) if 8 is the Dirac

l.F. Colombeau / Aspects of Infinite-Dimensional Holomorphy

261

measure on IR ". The construction of ~ is modelled on the construction of C9(fl): ~ is obtained as a quotient. Then we define naturally the integral of a generalized function. If K is a compact subset of fl and if G E C9(fl) then 1K G(x) dx is naturally defined as an element of~: this is defined by means of an arbitrary representative R of G and the usual Lebesgue integral; then one proves that the class in ~ of the result does not depend on the choice of R. This integration generalizes exactly the classical duality brackets of distribution theory: if T E gg'(fl) and 1JI' E gg (fl), if 1JI' G T = T G 1JI' denotes their new product in C9(fl), then

J(TO 1JI')(x) dx

= (T,

1JI') E C C

e.

Now if f and g are two continuous functions on fl then (from Schwartz's impossibility result) their new product f8 g in C9(fl) is not always equal to the classical product fg (which is also an element of C9(fl». Fortunately this lack of equality takes place in C9(fl) which is some 'abstract' space; we are going to see that from a 'concrete' (i.e. a classical) viewpoint this lack of equality is insignificant. Indeed if 1JI' E ~(fl) the new integral I (f0 g)(x)1JI'(x) dx gives as a result the classical integral I f(x )g(x) 1JI'(x) dx. Therefore, although different in the abstract space C9(fl), f8 g and fg give back the same numerical result when smeared out with a test function 1JI' E gg(fl) (indeed two different elements G!. G 2 of C9(fl) may be such that I G J(x )1JI'(x ) dx = J G z(x )1JI'(x ) dx for all 1JI'Egg(fl), a fact which-by definition-is impossible in distribution theory). This reconciles our new product with the impossibility result of Schwartz: the new product of continuous functions, when looked at within classical analysis, gives back the classical product. When working in the new theory one is easily convinced that this weakening of the concept of equality does not lead to trouble. Anyway Schwartz's impossibility result shows that this phenomenon is unavoidable. Following this idea a 'New Generalized Analysis' is developed in [9] and [40]. In this generalized analysis one may very often treat distributions exactly as if they were usual ~'" functions: if 8 denotes the Dirac measure on IR, 8 2 is a generalized function with derivative 288', the value 8(0) is a generalized number (and 8(x) = 0 if x 01- 0); 10+'" 8(x) dx is a generalized number and C: 8(x) dx = 1 .... One has even much more than the multiplication of distributions: for instance e i8, cos 8, sin 8 are elements of C9(R). It appears that this new theory may be considered as some original model of Nonstandard Analysis.

262

i.F. Colombeau / Aspects of Infinite-Dimensional Holomorphy

References [1) J.A. Barroso, Introduction to Holomorphy (North-Holland, Amsterdam, 1984). [2) EA. Berezin, The Method of Second Quantization, Pure Appl. Physics 24 (Academic Press, New York and London, 1966). [3] S.B. Chae, Holomorphy and Calculus in Normed Spaces (Marcel Dekker, New York, 1984). [4] G. Coeure, Analytic Functions and Manifolds in Infinite Dimensional Spaces, NorthHolland Math. Stud. 11 (North-Holland, Amsterdam, 1974). [5) J.F. Colombeau, Differential Calculus and Holomorphy, Real and complex Anlaysis in locally convex spaces, North-Holland Math. Stud. 64 (North-Holland, Amsterdam, 1982). [6) J.F. Colombeau, New Generalized Functions, Multiplication of Distributions, Physical Applications, Contribution of J. Sebastiao e Silva, Portugal. Math. 41 (1982) 57--69. [7) J.F. Colombeau, A Multiplication of Distributions, J. of Math. Anal. Appl. 94, 1 (1983) 96-115. [8] J.F. Colombeau, Une multiplication generale des distributions, C.R. Acad. Sci. Paris Ser, A 296 (1983) 357-360. [9] J.E Colombeau, New Generalized Functions and Multiplication of Distributions, North-Holland Math. Stud. 84 (North-Holland, Amsterdam, 1984). [10) J.F. Colombeau and B. Perrot, Theoreme de Noyaux analytiques en dimension infinie, c.n. Acad. Sci. Paris Ser. A 284 (1977) 759-762. [11] J.F. Colombeau and B. Perrot, Transformation de Fourier-Borel et Noyaux en dimension infinie, C.R. Acad. Sci. Paris se-. A 284 (1977) 963-966. [12) J.F. Colombeau and B. Perrot, Infinite dimensional normal forms of operators on the Fock spaces of Boson fields and an extension of the concept of Wick product, In: Advances in Holomorphy, Ed. J.A. Barroso, North-Holland Math. Stud. 34 (NorthHolland, Amsterdam, 1979) 249-274. [13] J.E Colombeau and B. Perrot, The Fourier-Borel transform in infinitely many dimensions and applications, In: Functional Analysis, Holomorphy and Approximation Theory, Ed. S. Machado, Lecture Notes in Math. 843 (Springer, Berlin, 1981) 163-186. [14] J.F. Colombeau and B. Perrot, Reflexivity and kernels in infinite dimensional Holomorphy, Portugal. Math. 36, 3-4 (1977) 291-300. [15] J.F. Colombeau and B. Perrot, Convolution equations in spaces of polynomials on locally convex spaces, In: Functional Analysis, Holomorphy and Approximation Theory, Ed. G. Zapata, Lecture Notes in Pure App!. Math. 83 (Dekker, New York, 1983) 21-32. [16] S. Dineen, Complex Analysis in locally convex spaces, North-Holland Math. Stud. 57 (North-Holland, Amsterdam, 1981). [17] Th.A.W. Dwyer III, Partial differential equations in Fischer-Fock spaces for the Hilbert-Schmidt Holomorphy type, Bull. Amer. Math. Soc. 77, 5 (1971) 725-730. [18] Th.A.W. Dwyer III, Holomorphic representation of tempered distributions and weighted Fock spaces, Analyse Fonctionelle et Applications (Hermann, Paris, 1975) 95-118. [l9] V.A. Fock, Zur Quantenelektrodynamik, Physikalische Zeitschrift der Sowjet Union 6 (1934) 425-469. [20] T. Franzoni and E. Vesentini, Holomorphic Maps and Invariant Distances, NorthHolland Math. Stud. 40 (North-Holland, Amsterdam, 1980).

J.F. Colombeau I Aspects of Infinite-Dimensional Holomorphy

[21]

[22J

[23] [24J

[25J [26J [27J

[28J [29J [30J [31J [32J

[33J

[34] [35J [36J [37J [38J [39J

[4OJ

263

D. Hilbert, Wesen und Ziele einer Analysis der unendliehvielen unabhagigen variablen, Rend. del Circ, Mat. Palermo 27 (1909) 59-74 or Gesammelte Abhandlungen III 56-72. C.O. Kiselman, How to recognize supports from growth of functional transforms in real and complex analysis, In: Functional Analysis, Holomorphy and Approximation Theory, Ed. S. Machado, Lecture Notes in Math. 843 (Springer, Berlin, 1981) 366-372. P. Kree, Calcul symbolique et seconde quantification des fonctions sesqui-holomorphes, C.R. Acad. Sci. Paris Ser. A 284 (1977) 25-28. P. Kree, Methodes holomorphes et methodes nucleaires en Analyse de dimension infinie et en Theorie Quantique des champs, In: Vector Spaces Measures and Appl. I, Lecture Notes in Math. 644 (Springer, Berlin, 1978) 212-254. P. Kree and R. Raczka, Kernels and Symbols of operators in Quantum Field Theory, Ann. Inst. H. Poincare 28, 1 (1978) 41-73. A.D. Michal, Le calcul Differentiel dans les espaces de Banach I (Gauthier-Villars, Paris, 1958). J. Mujica, Holomorphy in finite and infinite dimension (North-Holland, Amsterdam, 1984). L. Nachbin, Recent developments in Infinite Dimensional Holomorphy, Bull. Amer. Math. Soc. 79 (1973) 615-640. L. Nachbin, A glimpse at Infinite Dimensional Holomorphy, In: Proc. Infinite Dimensional Holomorphy, Lecture Notes in Math. 364 (Springer, Berlin, 1974) 69-79. L. Nachbin, Topology on Spaces of Holomorphic Mappings, Ergeb, Math. Grenzgeb. 47 (Springer, Berlin, 1969). L. Nachbin, Why Holomorphy in Infinite Dimension?, Ens Math. 26, 3-4 (1980) 257-269. Yu.V. Novozhilov and A.V. Tulub, The method of functionals in the quantum theory of fields, Russian Tracts Adv. Math. and Phys. 5 (Gordon and Breach, New York, 1961). Ph. Noverraz, Pseudo convexite, Convexite Polynomiale et Domaines d'Holomorphie en dimension infinie, North-Holland Math. Stud. 3 (North-Holland, Amsterdam, 1973). J.P. Ramis, Sous ensembles analytiques d'une variete Banachique Complexe, Ergebn. Math. Grenzgeb. 53 (Springer, Berlin, 1970). J. Rzewuski, On a Hilbert space of entire functionals, Bull. Acad. Polon. Sci. Ser, Sci. Math. XVII (1969) 453-458, 459-466, 571-578. J. Rzewuski, Hilbert spaces of Functional Power Series, Rep. Math. Phys. 1 (1971) 195-210. D. Sarasvati and J. Valatin, An Analytic representation of Quantum Field Theory, Comm. Math. Phys. 12 (1969) 253-268. M. Schottenloher, Michael problem and algebras of holomorphic functions, Arch. Math. 37 (1981) 241-247. L. Schwartz, Sur l'irnpossibilite de la multiplication des distributions, CiR, Acad. Sci. Paris 239 (1954) 847-848. J.F. Colombeau, Elementary Introduction to New Generalized Functions, NorthHolland Math. Stud. 113 (North-Holland, Amsterdam, 1985).

1.A. BARROSO editor, Aspects of Mathematics and its Applications © Elsevier Science Publishers B.V. (1986)

265

ULTRACONTRAC~SEMIGROUPSANDSOME

PROBLEMS IN ANALYSIS

E.B. DAVIES Department of Mathematics, King's College, London, England

B. SIMONI Division of Physics, Mathematics and Astronomy, Caltech, Pasadena, CA 91125, U.S.A. Dedicated to Leopoldo Nachbin in recognition of his contributions to mathematic. We study L 2 to L properties of exp(-t/l), where if is the Dirichlet form associated to a Schrodinger operator or to a Dirichlet semigroup. We use this study to obtain results about boundary behaviour of functions in suitable Sobolev spaces, and to obtain information of Brownian paths. OO

1. Introduction One of the central themes of Leopoldo Nachbiri's career has been the interplay of various aspects of abstract analysis with problems in concrete analysis. In this note, we want to sketch some results involving the relation of some abstract theory of LP properties of semigroups and some concrete problems involving Schrodinger operators and Dirichlet Laplacians; complete details, refinements, etc. will appear elsewhere [8]. Here are three concrete problems we will address:

1.1. Sobolev Estimates up to the Boundary Let [1 be a bounded open region in IR n and let H n denote the Dirichlet Laplacian on L 2([1, dx) which has compact resolvent. Let En be its smallest eigenvalue and l/Jn the corresponding eigenfunction, so l/Jn is determined by

Let Wp = Dom(IHn lp/2) be the usual Sobolev space. It can be proved [8] I

Research partially supported by USNSF grant MCS-81-20833.

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that if t.p E Wp and p > ~ n and if an obeys a weak condition (each boundary point regular in the potential theory sense), then t.p is a continuous function on clos n vanishing on an. We want to ask how fast t.p vanishes, where, if necessary, we are willing to take p very large. If an is smooth, it is easy to prove that any t.p E ~ with p sufficiently large vanishes at least linearily in dist(x, an) =- d(x) and !/In vanishes exactly that fast (see e.g. [10]). That the situation for general n is more complicated is seen by the study of polyhedral regions (see e.g. [12]). The situation is especially easy to describe if n = 2 so n is a polygon: If X o E an is a vertex of interior opening angle a and if x ~ X o along the bisector of that vertex, then !/In vanishes as dist(x, xot with m = 1T/a. Some thought suggests that the correct rate of vanishing should be precisely that of !/In' Thus: Problem (1.1). For what nand p It.p(x)/:s;;; ell IHnl p/2t.plh!/Jn(x)?

IS

there an estimate of the form

Surprisingly, we know of no previous work, for general natural question.

n,

on this

1.2. Conditioned Brownian Paths Given x, y E IR nand t > 0, let P x, y; t be the probability measure on Brownian paths conditioned to begin at x and end at y at time t. Explicitly, P is a measure on continuous functions b(s), O:s;;; s :s;;; t, with b(O) = x, b(s) = y; the components of b(s) are jointly Gaussian random variables with mean m(s) = Ex,y;t(b(s)) = (1- t)x + ty and covariance Ex,y;t«bj(s)-mj(s))(bj(u)-mj(u)))=d;js(l-rlu) if O:S;;;s:S;;;u:s;;;t (see e.g. [16], [21]). Let n C IR n be open and bounded and define Fn(x, y; t)

= P x,y;t({bl b(s) En, for all s, O:s;;; s:S;;; t}),

the fraction of paths that stay in n. Fn should go to zero as either x or y approach an (and will if an has weak regularity; see [16]). Paths that don't leave n stay 'away' from an, so most paths that leave n should do so when s is near 0 or t, i.e. we expect that Fn(x, y; t) should go to zero as Fn(x, x; t) l12Fn (y, y; tl12 . Thus: Problem (1.2). Let Dn(x; t) = Fn(x, x; t)1/2. When is it true that for some

E.B. Davies, B. Simon / Ultracontractioe Semigroups

267

a, f3 > 0 and all x, yEn, we have that

We note that the upper bound is easy, since (27Ttf nl2 exp(- (x - yi/2t)· Fn(x, y; t) is the integral kernel of the positive operator exp(-~tHn)' 1.3. Ultracontractivity of Schrodinger Semigroups

Consider a semigroup e -tA, t ~ 0, of selfadjoint operators on L 2(X, dJ.l) with X a probability measure space and so that A obeys lie -tA cp lip';;;; IIcplip for all t ~ 0, 1,;;;; p ,;;;; 00. (Such semigroups arise naturally as follows: Given H, a selfadjoint operator on L 2( y; dv), so that e:" is positivity preserving and so that HI/1 = EI/1 for some 1/1, a strictly positive, normalized vector in L 2( y , dz/), then one can pick X = Y; dJ.l = 1/12 dv and define U:L2(Y;dv)~L2(Y,dJ.l) by Ucp=I/1-1cp. U is unitary and A= U(H - E)U- 1 obeys e- IH , = 1 and e- tA is positively preserving. Such a semigroup is a contraction on all LP spaces (see e.g. [17]). We will occasionally write A = H and refer to the H construction). Given an LP contractive semigroup, we say that it is hypercontractioe if lIe- tA cp Il4';;;; cllcplb for some t> a and supercontractive if lIe- tA cpI14';;;; c(t)llcplb for all t> O. We introduce here the notion ultracontractive to mean that lIe- tAcpll",,;;;; c(t)lIcplb for all t > O. (We note that there is no point in replacing 00 by some p'i- 4 in (2,00); all p < 00 yield a definition equivalent to supercontractivity.) If H is given of the form discussed parenthetically above, so that the corresponding e -IH (note the _) is #-contractive, we say that e -IH is intrinsically #-contractive. Problem (1.3). Are any Schrodinger operators, -.1

+ V, intrinsically

ultracontractive? Here is some background on this problem. Since their introduction as tools in constructive quantum field theory (see e.g. [11], [21]), and especially after Gross' paper [13] on logarithmic Sobolev inequalities, hypercontractive estimates have provoked a large mathematical literature. In a recent bibliography on the subject, Gross [14] lists 51 papers! Nelson [15] showed that -.1 + cx 2 is intrinsically hypercontractive in the initial paper on the subject, and he later showed that it is not intrinsically supercontractive. Eckmann [9], Rosen [19] and Carmona [4]

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E.B. Davies, B. Simon I Ultracontractiie Semigroups

studied the intrinsic #-contractive properties of general Schrodinger operators. They show that if V(x) roughly goes to Ixl a , then one doesn't even have intrinsic hypercontractivity if a < 2 and one has intrinsic supercontractivity if a> 2. Apparently no one studied ultracontractivity because there was a belief that it couldn't hold for (-..1 + V). This belief, which we originally shared, seems to come from the fact that intrinsic ultracontractivity implies that for any eigenfunction ({! of H, ({!ifi -1 is bounded. This is false for the harmonic oscillator, so it was uncritically assumed false in general. Indeed, a simple WKB argument shows that, in one dimension, if V -Ixl a with a> 2, then each ({!ifi- 1 is bounded. We will see below that if V -Ixl a with a> 2, then -..1 + V is intrinsically ultracontractive. Indeed, ultracontractivity is the rule: If V(x) = Ixla(log(lxl + 2)l, then one has no intrinsic contractivity if a < 2, b » 0; intrinsic hypercontractivity but not intrinsic supercontractivity if a = 2, b = 0; intrinsic supercontractivity but not intrinsic ultracontractivity if a = 2, 0< b :s;;; 2 and intrinsic ultracontractivity if a > 2, b ~ a or a = 2, b > 2. We should mention that one of us in [6], which was one motivation for us here, showed that very general one-dimensional Schrodinger operators on a finite interval with Dirichlet boundary conditions are intrinsically ultracontractive. ~

2. Some Abstract Theory Let X be a locally compact, second countable Hausdorff space with regular Borel measure, v, and let H be a semibounded self-adjoint operator on L 2(X, dz-) so that e- tH has a jointly continuous integral kernel at(x, y). Suppose also that (i). at(x, y) > a for all x, y and (ii). Tr(e- tH ) < 00 for all t > O. Because of (ii), H has purely discrete spectrum {En} with Eo:S;;; E 1 :s;;; E 2:S;;; •• " where Eo < E 1 follows from (i) [18, XIII.12l, and normalized eigenfunctions ifin (x) with ifio(x) > 0 for all x. It is not hard to see that

:=0

(2.1)

at(x, y) = ~ e -tEn ifin (x )ifin (y) n=O

converges uniformly on compact subsets of X. Define Mx) = Yat(x, x)

E.B. Davies, B. Simon I Ultracontractioe Semigroups

269

(not to be confused with the function d(x) = dist(x, an) appearing in Problem (1.1». Two estimates automatically hold (2.2) (2.3)

for (2.2) (which one of us has used extensively elsewhere [22]) is a consequence of setting x = y in (2.1), and (2.3) is a consequence of the positivity of the operator e". Theorem (2.1). Under the above conditions, the following are equivalent: (i). e- tH is intrinsically ultracontractive. (ii). For all 0 < t < 00, there exists a c, < 00 with

where

II·Jb is the L 2(X, dv)

norm.

(iii). For all 0 < t < 00, there exists a c~ such that

(2.4)

(iv). For all t > 0, there exists a c'; <

00

such that

(v). For all t > 0, there exists a c'; <

00

such that

Before turning to a sketch of the proof, we note two things: First, that by (2.2), any (and hence all) of (i)-(v) imply (2.5)

Second, the remarkable fact that an upper bound on at like (2.4) implies a lower bound like (2.5). (Sketch of) Proof. (i)~ (ii). Just involves disentangling the definition of intrinsic ultracontractivity.

270

B.B. Davies, B. Simon I Ultracontractive Semigroups

(i)~ (iii). The integral kernel, at(x, y), of e- tH is easily seen to be a,(x, y) = etEol/lo(xfll/lo(yfla,(x, y). Thus (iii) is equivalent to saying that a, is bounded. By the Dunford-Pettis theorem, this says that (iii) is equivalent to the assertion that e -IH is bounded from L i to L'" for all t. Given that e -tEl is a contraction on each L P, duality, interpolation and the semigroup property show that e- IH is bounded from L 1 to L'" for all t if and only if it is bounded from L 2 to L'" for all t. (iii)~ (iv). A triviality. (iv) ~ (iii). Follows from (2.3). (iii)~ (v). Given t, pick a compact K, so that

J

(2.6)

I/Io(X)2 dv(x) 0;;;; ~(C;/3rl exp(-jtEo).

X\K

Then, using (iii), I/Io(x) exp(-jtEo) =

Jat/ix, Y)I/Io(Y) dv(y) x

K

X\K

Using (2.6) (2.7)

J at/3(x, y)l/Io(y) dv(y) ~ ~I/Io(x) exp(-jtEo) . K

Since K is compact and at(x, y)l/Io(xf11/lo(yf l is continuous and non-zero on all of X x X, it has a strictly positive infimum, 'Y, on K x K. Thus, by the semigroup property and (2.7)

J dv(z)dv(w)at/3(x, z)at/iz, w)a,/3(w, y)

at(x, y)~

KxK

~ 'Y

J dv(z)dv(w)at/3(x, z)l/Io(z)at/3(w, Y)I/Io(w) KXK

B.B. Davies, B. Simon I Ultracontractive Semigroups

271

(v) ~ (iv). We have that, using (2.2) and (v) %(x)

= e Eo'

f

[a,(x, y)]%(y) dv(y)

x

~ e Eo,

f [c'~

exp(-;tEo)%(y)b,(x)]%(y) dv(y)

x

With this theorem, we can reduce the solution of Problems (1.1) and (1.2) to statements about intrinsic ultracontractivity of Dirichlet semigroups. 2.1. Problem (1.2), revisited

Let a, be the integral kernel of e-'Hn /2. Then, the Feynman-Kac formula [21] says that Fn(x, y; t) = [(27rtfn/2 exp(-(x - y )2/2t)fl a,(x, y) , Dn(x; t)

= [(21Tttn/4fl b,(x).

Since (x - y)2 is bounded on n X n, we see that Fn(x, y)/Dn(x)Dn(y) is bounded above and below if and only if a, (x, y)/b,(x)b,(y) is bounded above and below. Thus: Corollary (2.2). For fixed n, Problem (1.2) has a positive solution for all t if and only if exp(-tHn) is intrinsically ultracontractive. 2.2. Problem (1.7), revisited

The estimate (2.8) is equivalent (since H n is invertible) to

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E.B. Davies, B. Simon / Ultracontractite Semigroups

Thus, since lIe-1H lb decays exponentially for large t and H-p/2 = cp f tP/2-1 e -IH dt, we see that Corollary (2.3). (2.8) holds for all p > a if for 0 < t < 1: (2.9) We remark that if (2.8) holds for some p, then (2.9) holds for all a > p, so Corollary (2.3) is 'almost' if and only if.

3. General Theory of Ultracontractivity of Dirichlet Forms In this section, we combine known results of Gross [13] and Rosen [19] to reduce ultracontractive estimates to a single family of operator inequalities. We restate their results carefully, because there seems to be a tradition in the subject to misstate them. Eckmann [9] and Carmona [4] both misstate Gross' estimate because they copy this inequality exactly although they have changed the convention on one of his constants. Rosen [19] isn't explicit about his constants; when Carmona tries to be explicit, he makes two errors! Fortunately, these errors don't affect the main conclusions of those papers. For us, the behaviour of the constants is critical. In the results below, we will not always state conditions on domains explicitly; these are discussed in detail in [4], [9], [19]. Gross' first important idea in [13] is the following: Theorem (3.1). (Gross [13].) Let Ji be a probability measure. Let A be an operator on L 2 (il, du ), Let I, = ItIP-1 sign (f) and suppose that for some r E (2, 00] and all p E (2, r), we have that, for all f E Dom(A):

(3.1)

JItIP loglfl:o;;; c(p) Re(Af,fp) + r(p)IIt1I~ + 1It11~ 10g:ltll

p '

Suppose that

(3.2)

t=

J 2

dp c(p)p'

M=

J 2

dp F(p)p

E.B. Davies, B. Simon / Ultracontractive Semigroups

273

are both finite. Then (Gross' estimate): (3.3)

Remark (3.2). The proof goes by letting pes) be defined by s = If(S) C(p)p-I dp and differentiating IIe-sHfll~l:~. See Gross for details. Remark (3.3). Gross' y is related to our r by y = Flc. He defines M by M = f; y(p(s» ds which can be seen to be equivalent to (3.2) by a change of variables. Remark (3.4). Gross only states his result for r < 00. Using 1111100 = Iim~", II11lp , the proof extends to r = 00; the conditions t, M < 00 are nontrivial if r = 00. Remark (3.5). (3.1) is called a logarithmic Sobolev inequality. In our examples below, one has that (3.1) holds for any p and c, i.e. there is {3(p, c) with (3.1')

J Ifl

P

log If I ~ c Re(Af, Ifl p ) + {3(p, c)llfll~ + IIfll~ log Ilfll p •

The second result in Gross [13J (quoted in a more explicit form due to Eckmann [9]) deals with a situation where n is an open subset of IR nand where (3.4)

(Af, g)

=

JVf(x)' Vg(x) dJL(x) , n

on suitable f, g and with suitable domain hypotheses. A is called a Dirichlet form. If H = -.1 + V and A = iI, then A is a Dirichlet form.

Theorem (3.6). (Gross [13], Eckmann [9].) Let A obey (3.4) and suitable domain hypotheses. Suppose that (3.1') holds for p = 2 and all c with {3(2, c) == b(c). Then (3.1') holds for all 2 ~ Po < 00 where 2 (2(Po-1) ) {3(Po,c)=-b c. Po

Po

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E.B. Davies, B. Simon I Ultracontractive Semigroups

Remark (3.7). The basic idea is to replace Ifl by IfIPo/2 in (3.1') for Po = 2. The left-hand side of (3.1') becomes ~Po f IfIPo log IfIPo. The first term on the right using (Vlflpo/2)2= (Vlflpo-l)VIfIGPof/(Po-1), becomes (AI, I/l p) times (~Po)2/(po - 1). See [13] for details. If we note that 2(p - l)lp varies from 1 to 2 as p varies from 2 to 00 and use the fact that without loss we can suppose b(2c) ~ b(c), we have by combining the last two theorems:

Theorem (3.8). Suppose that A obeys (3.4) and suitable domain conditions and that (3.1') holds lor p = 2 with f3 (2, c) == b(c). Given t, suppose we can choose c(p) so that

t=

J

dp

c(p)p'

2

M =

dp 2b(c(p» p2 .

J 2

Example (3.9). b(c) = Ac- k at least for c small. We take (for t small) c(p) = t(log 2)/(log p)2 and find M = dkAr k. This can be used to show suitable fractional powers of H generate supercontractive semigroups. Example (3.10). b(c) = exptc'"') at least for c small. Pick c(p) = td(a )/(log p with a> 1. Then M < 00 if aa < 1. Thus, if a < 1, we have ultracontractivity. It is interesting that the borderline is related to the borderline in Trudinger-type estimates; see [3].

t

Example (3.11). b(c) = A o+ A1log(c- 1) . Since f; 2 dp/p2 = 1, we have

Take

and

This is relevant for Problem (1.1) as we shall see.

c(p) = t(log 2)/(log p)2.

E.B. Davies, B. Simon / Ultracontractive Semigroups

275

One needs to ask when L 2 logarithmic Sobolev inequalities hold for Dirichlet forms. This is answered by an argument of Rosen [19] (extended by Carmona [4]). Theorem (3.12). (Rosen's lemma [19].) Let A = H where H = -.1 + Von LZ(Rn) or H = -.1 n with n C IR ". Suppose that one has the operator inequality -logil/Jol ~ ~oH + g(o).

(3.5)

Then (3.1') holds for p

=

2 with

f3 (2, c) =

n 4

g (0) + an - - logo,

for a universal (n-dependent) constant an'

Remark (3.13). The proof just makes the constant explicit (and .correct) in Carmona's version [4] of Rosen's argument [19]. an depends on the constant in the classical Sobolev inequality.

4. Getting our Act Together The net result of the last section is that ultracontractivity of Schrodinger and Dirichlet semigroups is reduced to upper bounds on -log l/Jo' i.e. lower bounds on l/Jo' and lower bounds on H = -.1 + V. It is remarkable that (as we shall see) rather crude lower bounds on l/Jo suffice; we say remarkable because ultracontractivity says that l/Jo/b, is bounded above and also away from zero. Thus, for example, for an x 4 oscillator in one dimension where one knows that b, - cx- 1exp(-dx 3 ) , a lower bound l/Jo ~ C1 exp(-c2X 4- e ) , e > 0, plugged into our machinery bootstraps to a lower bound by c'x- 1 exp(-dx\ 4.1. Schriidinger Semigroups (Solution of Problems (1.3)) If one looks at the argument of Carmona-Simon [5], one sees that the following is true:

Lemma (4.1). If V(x) ~ c1lxl a + C2 for some

C1

> 0, Cz real, then

E.B. Davies, B. Simon / Ultracontractive Semigroups

276

for some d., d 2 > O.

Indeed, Lemma (4.1) follows from an almost trivial path space estimate [8]. Given this lemma and Theorem (3.12), we have

Theorem (4.2). Suppose that for some c I ' c3 > 0, c z, c4 we have that

where ~ a + 1 < b. Then H

Proof. Let a

=

= - L1 + V

is intrinsically ultracontractive.

b/aa + 1). By Lemma (4.1),

By Example (3.9), and by Theorems (3.8) and (3.12), we obtain ultracontractivity. 0 Note that ~a + 1 < band b ~ a imply b > 2. More refined estimates [8] and b > 2, then show that if czxZlog(lxl + 2)b ~ V(x) ~ clxZ[log(lxl + one has ultracontractivity. More results on the Schrodinger case will appear in [8]. We emphasize that these results are multi-dimensional.

2W

4.2. Dirichlet Semigroups (Problems (1.1) and (1.2))

We will prove ultracontractivity under suitable geometric hypotheses. Lest the reader think such hypotheses are unnecessary, we mention that there are examples of regions in IR Z for which intrinsic ultracontractivity fails; for one can show that btl/l~1 is unbounded (see [8]). To verify (2.10), one needs lower bounds on 1/10' i.e. upper bounds on - log 1/10' and lower bounds on H n by functions of x. The latter problem is solved by a recent estimate of Davies [7]. Given n and x E n, and given a unit vector wE sr', let d(x, w) be defined by: d(x, w) = inf{)rll x

and the quasidistance q(x) by

+ reo g n},

E.B. Davies, B. Simon I Ultracontractive Semigroups

1 q(X)2 =

f S·-1

dw d(x, w?'

where dw is the normalized invariant measure on

Theorem (4.3). (Davies [7].) For any

277

s:'. Then

a

Remark (4.4). This is an elementary consequence of the inequality (f, (4x 2 1f ) ~ (/''/') for f E Cff~(O, 00). Combined with Agmon's method [2], this is useful for proving upper bounds on 1/1, and critical in the example mentioned above where 1/1-ld, is unbounded. To use this, we need

r

Definition (4.5). We say that n obeys an exterior cone condition if and only if there exists E > 0, a > 0 so that for each x E an, there is a unit vector e(x) with {yl 0 < Ix - yl <

E,

e- (y - x) > aly - xl} C Rn\n.

A simple geometric argument shows that if n obeys an exterior cone condition, then q(x) ~ ad(x) where a depends on a, E and diam(fl) and thus Corollary (4.6). (Davies [7].) If n obeys an exterior cone condition, then for a suitable constant c

To get a lower bound on t/Jo, it is useful to define special cones as follows: Let A C sr' be an open set; then given x E IR n and E, we define C(x, A, E) = {yl 0 <

Ix -

yl < E, y - x/ly - x] E A}.

An elementary comparison argument [8] leads to:

Definition (4.7). Let A be an open subset of s:'. We say that n obeys an A-interior cone condition, if and only if there exists an E, a 8> 0, a f3 > 0,

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E.B. Davies, B. Simon / Ultracontractioe Semigroups

and for each x E that

n with d(x) < S, a point

y(x) E

x E C(y(x), RAA), s ) C

an and a rotation R, so

n,

dist(x - y(x)/Ix - y(x)l, sn-I\RAA» > {3.

Definition (4.8). Given A C s:', let A(A) define the lowest eigenvalue of the Laplace-Beltrami operator on L 2(A) with Dirichlet boundary conditions. Define a = a(A) > 0 by: a(a + v - 2) = A. Theorem (4.9). [8] Let n obey an A-interior cone condition. Then

Remark (4.10). If v = 2 and A = {(cos 0, sin 0) I0 < 0 <
n obeys an exterior cone condition and an A-interior cone condition, then e -tHn is intrinsically ultracontractive and

Theorem (4.11). If

lie -tlinll"", 2 :s; cr 1n/4+a (A l/2j • Remark (4.12). For cubes, one can use the method of images to compute exactly the divergence of lie -tHnll"" I and to get a lower bound on lie -tHnll",, 2 which has the same power behaviour as Theorem (4.11) if one takes A to be a quadrant. See [8] for a discussion of 'trumpet' conditions replacing cone conditions.

.

Given the discussion at the end of Section 2, we have solved Problems (1.1) and (1.2) under suitable cone conditions.

E.B. Davies, B. Simon / Ultracontractive Semigroups

279

Acknowledgments

This work was done while we were both vrsrtmg The Australian National University (E.B.D. at the Research School of Physical Sciences, and B.S. at the Centre for Mathematical Analysis). We would like to thank Derek Robinson and Neil Trudinger for their hospitality, and we would like to thank Derek for his suggestion of the term 'ultracontractive'.

References [1] S. Agmon, Lectures on elliptic boundary value problems (Van Nostrand, New York, 1965). [2] S. Agmon, Lectures on exponential decay. of solutions of second-order elliptic equations: Bounds on eigenfunctions of N-body Schrodinger operators (Princeton Univ. Press, Princeton, 1982). [3] M. Aizenman and B. Simon, Brownian motion and Harnack's inequality for Schrodinger operators, Comm. Pure Appl. Math. 35 (1982) 209-273. [4] R. Carmona, Regularity properties of Schrodinger and Dirichlet semigroups, J. Funct. Anal. 33 (1979) 259-296. [5] R. Carmona and B. Simon, Pointwise bounds on eigenfunctions and wave packets in N-body quantum systems, V. Lower bounds and path integrals, Comm. Math. Phys. 80 (1981) 59-98. [6] E.B. Davies, Hypercontractive and related bounds for double well Schrodinger operators, Quart. J. Math. (2) 34 (1983) 407-421. [7] E.B. Davies, Some norm bounds and quadratic form inequalities for Schrodinger operators II, J. Operator Theory 12 (1984) 177-196. [8] E.B. Davies and B. Simon, Ultracontractivity and the heat kernel for Schrodinger operators and Dirichlet Laplacians, J. Funct. Anal. 59 (1984) 335-395. [9] J.P. Eckmann, Hypercontractivity for anharmonic oscillators, J. Funct. Anal. 16 (1974) 388-406. [10] D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order (Springer, Berlin, 1977). [11] J. Glimm and A. Jaffe, Quantum physics (Springer, Berlin, 1982). [12] P. Grisvard, Singularities for the problem of limits in polyhedrons, Sem. GoulaoucSchwartz 1981/82, Ex. 8. [13] L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975) 1061-1083. [14] L. Gross, Papers on logarithmic Sobolev inequalities and hypercontractivity, unpublished, Aug. 1982. [15] E. Nelson, A quartic interaction in two dimensions, In: Mathematical theory of elementary particles, eds. R. Goodman and I. Segal (MIT Press, Cambridge, MA, 1966) 69-73. [16] S. Port and C. Stone, Brownian motion and classical potential theory (Academic Press, New York, 1978).

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[17] M. Reed and B. Simon, Methods of modern mathematical physics, II. Fourier analysis, self-adjointness (Academic Press, New York, 1975). [18] M. Reed and B. Simon, Methods of modern mathematical physics, IV. Analysis of operators (Academic Press, New York, 1977). [19] J. Rosen, Sobolev inequalities for weight spaces and supercontractivity, Trans. Amer. Math. Soc. 222 (1976) 367-376. (20] B. Simon, The P('h Euclidean (quantum) field theory, Princeton Ser. Physics (Princeton Univ. Press, Princeton, 1974). (21] B. Simon, Functional integration and quantum physics (Academic Press, New York, 1979). (22] B. Simon, Semiclassical analysis of low lying eigenvalues, II. Tunneling, Caltech preprint.

l.A. BARROSO editor, Aspects of Mathematics and its Applications © Elsevier Science Publishers B.V. (1986)

281

LA NOTION DE RIGUEUR EN MATHEMATIQUES

Jean DIEUDONNE 10, Rue du General Camou, 75007, Paris, France Amon vieil ami Leopoldo Nachbin

Les mots de 'rigueur' ou de 'raisonnement rigoureux' qualifiant les demonstrations rnathematiques sont utilises des Ie XVIIIe siecle (par exemple par d'Alembert en 1748), et plus frequemrnent a partir du debut du XIx< siecle: par reaction contre la pratique de leurs predecesseurs qui se bornaient a invoquer 'Ia generalite de I'Analyse' pour justifier des affirmations douteuses, Gauss et Cauchy entre autres leur opposent 'Ia rigueur qu'on exige en geometric' (Cauchy). Apres eux, les analystes du XIX e siecle se targueront a qui mieux mieux de leurs dernontrations faites 'de Iacon complete et rigoureuse' (Poincare), ce qui n'ernpechera pas leurs successeurs d'y trouver souvent des lacunes! L'abus de ces malencontreuses marques d'autosatisfaction a fini par tendre a devaluer Ie concept merne de 'rigueur', a en souligner Ie caractere changeant avec les epoques, voire a finir par en faire une sorte d'ideal dont on peut s'approcher mais qui demeure toujours inaccessible; contraste frappant avec la tradition heritee de Descartes et Pascal qui voyaient les demonstrations mathematiques comme seules capables de mener a la 'verite'. Ce scepticisme a l'egard de la rigueur est encore exprime plus ou moins implicitement par certains mathematiciens, comme par exemple R. Thorn; et il a trouve recemment une expression fracassante dans I'ouvrage Proofs and refutations de 1. Lakatos, qui semble avoir eu un grand succes aupres des philosophes, et dont nous aurons a reparler. Je me propose de chercher a avoir des vues plus nuancees sur cette question, sans passion ni exageration; je pense que pour cela il faut avoir une idee precise de l'evolution historique des diverses branches des mathematiques, ce qui fait souvent defaut dans les affirmations peremptoires sur Ie sujet. Comme il s'agit de controverses anterieures a 1920, done etant ap-

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parues a une epoque ou l'unification actuelle de la mathernatique n'etait pas realisee, iI convient d'examiner separement chacune des disciplines traditionnelles. Si distinctes qu'elles soient dans I'esprit de la plupart des mathernaticiens, elles ont toutefois en commun la demarche des demonstrations, telle qu'elle est deja decrite par Platon dans la Republique com me un dialogue avec un interlocuteur fictif, ou, 'd'un commun accord', on part de 'choses connues, ... , dont on n'a plus a rendre compte', sur lesquelles, 'procedant par ordre', on parvient 'au but que l'on s'etait propose'. Comme avant Brouwer il n'y a jamais eu de divergences entre mathematiciens sur ce qui constitue une operation logique correcte, les controverses sur la 'rigueur' ne peuvent venir que de deux sources: ou bien on ne s'entend pas sur la definition des objets mathernatiques, ou bien on ne leur attribue pas les memes proprietes de base ('axiomes' ou 'postulats' selon la terminologie traditionnelle).

1. Arithmetique

Avant Dedekind et Peano, il n'y a jamais de besoin de definir les entiers naturels; ce sont des notions communes sur les proprietes desquelles tout Ie monde est d'accord. Contrairement a la Iacon dont il precede en geometric, Euclide ne dresse pas non plus de liste des proprietes sur lesquelles il base ses demonstrations d' Arithmetique; il est facile d'en detecter quelques-unes, comme Ie fait qu'une suite decroissante d'entiers est stationnaire, ou la commutativite de l'addition, ou l'existence de la difference a - b pour b < a; au XIII" siecle, Campanus tentera d'en faire une Iiste (4 axiornes et 10 postulats) qui reste incomplete (i1 ne mentionne pas la commutativite de l'addition); en tout cas, les demonstrations d'Euclide en Arithmetique n'ont jamais ete contestees, et sont restees des modeles d'ingeniosite mathematique, aussi convaincants aujourd'hui qu'il y a 2000 ans.

2. Algebre

II faut rappeler qu'avant Viete (debut du XVII" siecle) on ne peut guere parler d'algebre au sens actuel; mais sous leur forme geometrique, les operations algebriques sur les nombres reels positifs etaient en usage depuis les Grecs, et bien que non codifiees explicitement, leurs proprietes

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ont toujours ete considerees comme 'evidentes' et, ici encore, n'ont jamais donne lieu a des controverses. L'extension de ces proprietes aux nombres negatifs a pu soulever des doutes hors du milieu des mathematiciens professionnels, beaucoup, dans Ie public cultive, ayant du mal a admettre par exemple que Ie produit de deux nombres negatifs soit positif! Mais comme Ie dit d'Alembert, a la fin de I'article Negatif de l'Encyclopedie ou il a essaye de faire comprendre la question a tous ses lecteurs (et n'y arrive guere) '... les regles des operations algebriques sur les quantites negatives sont admises generalement par tout Ie monde et recues generalernent comme exactes, quelque idee qu'on attache d'ailleurs aces quantites.' C'est seulement lorsque les algebristes italiens du XVl" siecle ont commence a manipuler des nombres complexes que les mathematiciens, jusqu'au debut du XIX e siecle, se sont sentis mal a l'aise dans les calculs sur ces nombres 'impossibles' dont ils ne pouvaient donner de definition; mais tout est rentre dans l'ordre des que leur representation geometrique par les points du plan a ete generalement acceptee,

3. Geometrie

L'attitude des rnathematiciens vis-a-vis des demonstrations en geometric est curieusement ambigiie. D'une part, ils les considerent comme des modeles de 'rigueur', comme en temoignent les expressions employees par Gauss et Cauchy citees plus haut; la raison en est sans doute que c'est alors la seule branche des rnathematiques ou definitions et axiomes sont enonces explicitement, Mais d'autre part, depuis Ie XVI e siecle, on avait observe que les preuves d'Euclide ne reposaient pas exclusivement sur les axiomes enonces par lui: par exemple Clavius note que l'existence de la quatrieme proportionnelle est utilisee sans qu'aucun axiome la garantisse, et Leibniz remarque que Ie fait que deux cercles se coupent si chacun passe par Ie centre de l'autre est affirme sans aucune demonstration. Des controverses sur la valeur probante des demonstrations geometriques auraient done ete possibles, et il est facile de donner des exemples ou l'absence de certaines notions chez Euclide pourrait conduire a des conclusions erronees (par exemple la 'preuve' que tout triangle est isocele, en prenant l'intersection de la bissectrice d'un angle avec la mediatrice du cote oppose en placant cette intersection a l'interieur du triangle, notion non definie par Euclide). Mais en fait, il ne

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semble jamais y avoir eu de telles controverses en geometric euclidienne classique (en entendant par la la partie ou n'intervient aucune consideration topologique). Peut-etre faut-il en voir la raison dans l'equivalence, etablie par la methode des coordonnees de Descartes et Fermat, entre les relations de cette theorie et leurs traductions algebriques, celles-ci ne soulevant aucune difficulte. Quant aux geometries non euclidiennes, ce n'est pas la correction des raisonnements de leurs fondateurs qui a ete mise en cause, mais leur possibilite logique, et les doutes a cet egard ont cesse lorsqu'on a obtenu des modeles de ces geometries en algebre.

4. Analyse

On sait que, contrairement aux theories mathematiques precedentes, c'est ici que surgissent de toute part les contestations et les controverses, a partir du XVII" siecle. La raison en est apparente: il s'agit de nouvelles notions ou de nouveaux precedes de calcul, dont aucune definition ne peut etre trouvee dans les ecrits anterieurs; chacun s'en remet donc a son 'intuition' pour mener ses calculs et ses raisonnements, sans se soucier en general de tenter de rattacher ce dont il parle a des concepts deja acquis et ne soulevant pas de discussion. Un exemple typique de la confusion qui en resulte est la fameuse controverse sur les logarithmes des nombres complexes au debut du XVIII" siecle: il s'agit de prolonger la fonction x ~ log x lorsque x prend des valeurs complexes ¥- O. Or cette fonction jouit de nombreuses proprietes connues a l'epoque: elle est injective, satisfait a l'equation fonctionnelle log(xy) = log x + log y, a pour differentielle dx/x, et on a I'expression de log(l + x) par une serie entiere. Mais en admettant sans justification qu'il est possible de prolonger la fonction aux nombres complexes (ou merne seulement aux nombres reels negatifs) en conservant I'une ou I'autre de ces proprietes, on tombe aussitot dans des contradictions. Par exemple, en admettant que la fonction prolongee est encore injective, Leibniz soutient que les logarithmes des nombres negatifs doivent etre imaginaires, puisque log x prend to utes les valeurs reelles pour x> 0; par contre, Jean Bernoulli pretend que I'on doit toujours avoir log(-x) = log x, puisque (-X)2=X 2, d'ou 210g(-x)=210gx par l'equation fonctionnelle! C'est seulement lorsqu'Euler, en 1749, revient a la definition de y = log x comme solution de l'equation e " = x, et qu'il examine quelles solutions peut avoir cette equation lorsqu'on donne a x

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une valeur complexe, qu'il decouvre l'existence d'une infinite de determinations du logarithme; encore n'ose-t-il pas utiliser I'expression des parties reelle et imaginaire de e a +ib , et prefere-t-il definir e" pour y complexe comme limite de (1 + yIn)" lorsque n tend vers l'infini, en resolvant done d'abord l'equation (1 + yin)" = x avant de faire tendre n vers l'infini, de Iacon a ne faire que des calculs algebriques sur les nombres complexes. Mais lorsqu'il s'agit de concevoir, par exemple, ce qu'est la 'somme' d'une serie, Euler n'a pas d'idees moins confuses que ses contemporains. Tant qu'il s'agit d'une serie convergente a notre sens, if ne se pose pas de problerne au XVfl]" siecle: on sait que Ie terme general tend vers 0, mais que cette condition n'est pas suffisante pour assurer la convergence (la divergence de la serie harmonique est connue depuis Jacques Bernoulli). Mais lorsqu'il s'agit de series de fonctions :L:=lf,,(x), dont la somme au sens precedent existe pour certaines valeurs de x et est egale a une fonction connue g(x), il se peut que cette fonction soit definie pour des valeurs de x telles que la serie des f,,(x) ne converge pas; la tentation est alors grande de lui attribuer tout de merne une 'somme' egale a g(x), de considerer que la 'somme' des derivees f~(x) est la derivee g'(x), etc. Un exemple extreme est la facon dont Euler attribue une 'somme' a la serie (4.1)

1- I! + 2! - ... + (-1 )"n! + ....

Euler lui associe la serie de fonctions (4.2) et voudrait prendre pour 'somme' de (4.1) la valeur y(l). Malheureusement, Ie terme general de (4.2) ne tend vers 0 pour aucune valeur x ;: 0; mais cela n'arrete pas Euler, qui en derivant terme a terme la serie (4.2), montre que I'on a (4.3)

Or cette equation a effectivement une seule solution telle que yeO) = 0, don nee par x

(4.4)

y(x)=e ll x

f~e-lIldt o

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et Euler considere que cette fonction est la 'somme' de (4.2), done y(l) la 'somme' de la serie (4.1). On peut encore citer comme exemple de calculs sur des objets non definis les 'changements de variables' ou l'on remplace dans une integrale une variable reelle par une variable imaginaire, sans savoir au juste ce que cela signifie. Sans doute pouvons-nous voir dans ces hardiesses une premonition de theories qui ne seront developpees que bien plus tardivement, comme la theorie de Cauchy, les precedes de sommation ou les series formelles, mais les mathernaticiens du XVIII" siecle qui les utilisent se rendent bien compte qu'ils ne peuvent les faire reposer sur rien de solide, merne lorsque les resultats qu'ils obtiennent de cette facon sont exacts; Euler, en substituant la valeur x = -1 dans la serie entiere developpant 1/(1 - x) autour de 0 ne trouve pas choquant d'obtenir la 'somme' (4.5)

~=

1-1 + 1-1 + ... + (-ly-l + ... ,

mais il reste perplexe lorsqu'il fait de merne sur la serie dormant 1/(1 - 2x) et obtient pour x = 1 (4.6)

-1

=

1 + 2 + 4 + ... + 2n - 1 + ... ,

ne comprenant pas comment une somme de termes positifs peut etre negative! De merne, apres avoir obtenu les valeurs de certaines integrales definies en passant a une variable imaginaire, Laplace insiste sur Ie fait qu'il vaudrait mieux verifier Ie resultat par d'autres methodes. On est done bien loin des 'raisons certaines et evidentes' ou Descartes voyait I'essence des mathernatiques! Comme historiquement c'est la premiere fois depuis l'Antiquite que de tels doutes s'elevent dans la rnathematique, la reaction qu'ils declenchent au debut du XIX" siecle est aussi Ie premier exempIe du scenario connu sous Ie nom de 'retour a la rigueur', qui va se repeter ensuite plusieurs fois dans d'autres domaines. II est assez caracteristique que ce processus ne soit pas du tout instantane, mais s'etale sur de nombreuses annees. On peut meme y distinguer deux etapes assez differentes. Dans la premiere, illustree par les noms de Gauss, Balzano, Cauchy et Abel, on donne pour la premiere Iois les definitions precises des entites utilisees jusque la en Analyse infinitesimale: limite, somme d'une serie, fonction continue, integrale definie, en les ramenant essentiellement ades egalites et inegalites sur

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des nombres reels. Mais au debut les choses se gatent lorsqu'il s'agit d'appliquer ces definitions a des theorernes tels que l'existence de l'integrale d'une fonction continue ou la continuite d'une limite de fonctions continues, et il est assez surprenant de voir des mathematiciens tels que Cauchy et Abel trebucher sur l'utilisation de leurs propres definitions! En fait, tout comme Euclide, ils melent quelquefois, a des raisonnements appliquant strictement les definitions, des assertions qui ne reposent que sur 'I'intuition' qu'ils ont heritee de leurs predecesseurs sans penser a devoir la justifier. Nous retrouverons d'autres exemples de paresse dans la verification de propositions simples mais ennuyeuses; s'ils l'avaient faite avec soin, elle les aurait conduits tout naturellement aux notions de convergence uniforme et de continuite uniforme que vont degager leurs successeurs. C'est en effet dans une seconde etape, avec Riemann, Heine et surtout Weierstrass, que l'on acquiert la patience de suivre pas a pas la marche des calculs d'inegalites et l'ordre des quantificateurs. C'est cette peri ode de flottement, entre 1820 et 1860, que cite Lakatos a l'appui de sa these sur les fluctuations de la notion de rigueur, qu'il pretend etre ineluctables et permanentes dans tous les raisonnements mathematiques, Ce qu'il ne dit pas, et qui rend cet exemple sans valeur, c'est qu'apree Weierstrass il n'y a jamais plus eu la moindre controverse sur Ie canon d'une demonstration d' Analyse tel qu'il I'avait fixe. ce que l'on dit II faut peut-etre signaler ici que, contrairement souvent, l'absence d'une 'construction' des nombres reels a partir des nombres ration nels n'a rien a voir avec les tatonnements dont nous venons de parler. Si on ne cherche pas avant 1860 a donner une definition explicite des nombres reels, tous les mathematiciens sont cependant implicitement d'accord sur leurs proprietes, exactement com me les Grecs n'avaient jamais eleve de doutes sur les proprietes des en tiers naturels. C'est ce que montrent leurs raisonnements, ou l'on voit que les seules proprietes qu'ils utilisent sont que les nombres reels forment (dans notre langage) un corps ordonne, archimedien et complet. On sait que ces proprietes les caracterisent effectivement a isomorphie pres et qu'on les emploie aujourd'hui comme systeme d'axiomes lorsqu'on ne veut pas s'assujettir aux penibles 'constructions' classiques, qui n'acquierent leur importance que lorsqu'on veut prouver la non contradiction (relative) de ces axiomes. On peut done resumer l'evolution de la 'rigueur' en Analyse comme une rnontee continue a partir de notions vagues et confuses, qui, a travers des vicissitudes diverses, se clarifient peu a peu, jusqu'a ce qu'on arrive a

a

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un palier stable, a partir duquel il n'y a plus aucune contestation parmi les mathernaticiens sur ce qui constitue une demonstration correcte en Analyse. En realite, on peut considerer que c'est la merne evolution qui s'est produite pour I'Arithmetique, I' Algebre des nombres reels et la Geometric euclidienne, sauf que l'on est parti de points differents de la courbe: dans les deux premiers cas, on etait deja au 'palier' stable des Ie debut de la theorie (ou tout au moins dans les premiers textes que nous connaissons); tandis qu'on peut dire qu'avec la Geometric euclidienne, on part a mi-chemin de la rigueur, Ie 'palier' n'etant atteint qu'avec les axiomatisations completes de Pasch et de Hilbert. Nous allons voir qu'une evolution tout a fait semblable caracterise deux des theories modernes, la Topologie algebrique et la Geometric algebrique.

S. Topologie AIgebrique Cette theorie n'a vraiment debute qu'en 1900 avec H. Poincare, mais a eu une 'prehistoire' aux XVIII" et XIXe siecles; jusque vers 1850, les questions que nous y rattachons a present sont considerees comme faisant partie de la geometric ou de la combinatoire. Vne des plus discutees est la fameuse formule d'Euler (1751) sur les 'polyedres' (5.1)

s-a+f=2,

OU s, a, f sont les nombres de sommets, d'aretes et de faces du polyedre. Or, bien qu'on soit en Geometric (Ie temple de la 'rigueur' pour Gauss et Cauchy!) on chercherait en vain dans Euclide des definitions precises des mots 'polyedre', 'arete' ou 'face'; il ne parle que de 'solides', ou il comprend entre autres les cones et les spheres, et les seuls polyedres qu'il definit explicitement sont les pyramides, les prismes et les polyedres reguliers, Euler n'est pas plus explicite, et il n'est done pas etonnant que selon les interpretations donnees aux nombres a et f dans (5.1), sa demonstration, comme eelles de la douzaine de rnathematiciens qui, dans Ie siecle qui la suit, s'occupent de cette formule, aient donne lieu a des eontroverses, et que pour certains 'polyedres', on ait merne obtenu des valeurs autres que 2 pour le premier membre de (5.1); on est done tout a fait dans la meme situation qu'en Analyse pour la notion de logarithme

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d'un numbre complexe. En fait, il est vraisemblable qu'Euler n'ait pense qu'aux polyedres convexes, sans d'ailleurs les definir explicitement, car dans un travail de la merne epoque, il decompose un polyedre en pyramides ayant un merne sommet interieur au polyedre, ce qui ne se conceit guere que si Ie polyedre est convexe. En fait, ce n'est qu'au XIX e siecle que l'on trouve une definition des polyedres convexes, et encore n'est -elle pas la definition que nous donnons maintenant des ensembles con vexes par la condition de contenir tout segment dont les extremites sont dans I'ensemble; c'est la surface du polyedre qu'on appelle convexe lorsque Ie plan de chaque face est un plan d'appui. De toutes facons, il s'agit la d'une propriete qui n'est pas conservee par deformation et ce n'est pas avant 1860 qu'on a vu que la formule d'Euler restait valable lorsque les faces et les aretes sont courbes, moyennant des conditions qui ne sont plus de nature geornetrique, mais topologiques; en fait c'est seulement von Staudt en 1847 qui formula ces conditions (qui reviennent a dire, dans Ie langage actuel, que Ie premier 'nombre de Betti' du polyedre est nul). Ce sont ces controverses autour de la formule d'Euler dont I'histoire est developpee avec prolixite par Lakatos sur environ 100 pages, et par lesquelles il pretend prouver l'irnpossibilite d'arriver a la rigueur en rnathematiques, II oublie simplement que l'histoire ne s'arrete pas en 1850. Let mot de 'Topologie' est a peine mentionne dans son ouvrage, et il semble qu'en raison de son ignorance de l'histoire recente des mathematiques, il n'ait pas soupconne que l'episode auquel il s'attache faisait partie de toute une serie de problemes qui allaient marquer la naissance de cette science. C'est Riemann Ie premier qui, pousse par la necessite de classifier les 'surfaces' compactes connexes qu'iI introduit en Analyse compIexe, songe ales differencier les unes des autres par un caractere numerique invariant par deformation (1851): une des deux definitions qu'il donne de ce qui sera plus tard Ie 'premier nombre de Betti' est Ie plus grand nombre hi de circuits (courbes homeomorphes au cercle) qu'on peut tracer sur la surface de facon que leur reunion ne 'forme pas frontiere' d'un ouvert; ce terme n'est pas defini precisernent, mais le contexte montre qu'une reunion de circuits F 'forme frontiere' d'un ouvert U si tout point d'un de ces circuits appartient a la fois a l'adherence de U et a celle du cornplementaire de U U F; sur Ie tore, Ia reunion d'un meridien et d'un parallele 'ne forme pas frontiere'. Riemann lui-rneme, puis Betti et plus tard Poincare dans son premier memoire sur 'l'Analysis situs' (1895)

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tenteront de batir une doctrine topologique en generalisant cette idee a un nombre quelconque de dimensions; mais ils n'aboutiront jamais ainsi qu'a des definitions vagues et a des resultats rendus simplement plausibles pour l"intuition' (pour autant qu'on puisse parler d"intuition' dans des espaces de dimension ~4). Les choses ne vont changer que lorsque Poincare, en 1900, s'etant sans doute rendu compte que cette voie ne rnenerait a rien de solide, introduit les methodes 'combinatoires' qui marqueront Ie debut de la Topologie algebrique en tant que veritable discipline mathernatique. II commence par donner une definition generate d'un 'polyedre' dans un espace euclidien de dimension arbitraire. C'est un cas particulier de ce qu'on appellera apres lui un 'complexe' rectiligne, dont la definition est plus simple: c'est une reunion d'un ensemble T d'un nombre fini d'ensembles disjoints, dont chacun est l'interieur d'un polyedre convexe (appele aussi 'cellule') dans la variete lineaire qu'il engendre; en outre, sa frontiere (dans cette variete lineaire) est reunion de polyedres con vexes de I'ensemble T. Un espace admet une triangulation s'il est homeomorphe a un complexe rectiligne. Si a j est Ie nombre des cellules .de T dont la dimension (egale par definition a celie de la variete Iineaire qu'elle engendre) est j, Poincare attacha a T, par un precede purement algebrique, n + 1 'nombres de Betti' bo, b., ... , b; (n etant la dimension maxima des cellules de T) et demontra la 'formule d'Euler-Poincare' (5.2)

a o - a 1 + a 2 - ... + (-l)n a n

= b0 - b1 + b2 - ... + (-l)nbn'

II chercha ensuite a prouver que les b, sont des invariants topologiques, c'est-a-dire que si un espace admet deux triangulations differentes T, T', les nombres de Betti de T et T' sont les memes et ne dependent done que de la topologie de l'espace; par exemple, pour la sphere, on a bo = b2 = 1 et b, = 0, et (5.2) n'est autre que la formule d'Euler (5.1). Mais les raisonnements de Poincare sont tout a fait insuffisants, et il fallut les efforts de plusieurs mathematiciens durant une trentaine d'annees pour parvenir a une demonstration correcte, et fonder ainsi ce qu'on appelle la theorie de l' homologie. On voit done que l'evolution de cette theorie s'est deroulee de facon tout a fait analogue a celie de I'Analyse, bien que plus rapidement: on part de notions vagues dont on n'est pas capable de donner une definition precise, puis une premiere etape permet de cerner de veritables objets mathematiques, sans qu'on sache encore bien les manier, et on finit par

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aboutir au 'palier' a partir duquel il n'y a jamais plus de contestation sur les resultats de la theorie, On ne peut manquer de remarquer a propos de cette theorie de l'homologie que les premiers roles dans son elaboration, Riemann et Poincare, ant ete des mathematiciens universels s'il en fut jamais. En particulier, ils etaient rompus a la pratique de l'Analyse infinitesimale rigoureuse et en ont donne mainte preuve dans leurs ecrits: comment, dans leurs memoires sur la Topologie algebrique, ont-ils pu se contenter de raisonnements reposant sur des affirmations sans aucune valeur probante, qu'ils auraient severement condamnes dans d'autres domaines? A la decharge de Riemann, on peut dire que les notions de variete differentielle ou analytique n'etaient pas forrnulees explicitement de son temps; mais Poincare en donne des definitions precises equivalentes aux notres; apres quoi, il n'hesite pas, par exemple, a raisonner comme si l'intersection de deux varietes analytiques etait toujours une variete, alors qu'il connait fort bien (par exemple pour les varietes algebriques) des exempIes du contraire! II y a la comme un dedoublement de la personnalite qu'il est difficile de comprendre.

6. Geometrie Algebrique

S'agissant d'une theorie necessitant beaucoup de connaissances techniques prealables, je ne pourrai que donner une breve esquisse de son histoire. Jusque vers 1925, la theorie est essentiellement l'etude des courbes algebriques et des surfaces algebriques dans les espaces projectifs complexes; on notera que du point de vue topologique, une courbe algebrique est un espace a 2 dimensions et une surface algebrique un espace a 4 dimensions. II faut examiner separernent l'histoire des courbes algebriques et celle des surfaces algebriques, qui sont bien differentes. La premiere deb ute avec Riemann en 1857, chez qui il n'est guere question de geometrie, mais de fonctions algebriques d'une variable complexe et de leurs integrales, et c'est pour leur etude qu'il developpe les premiers rudiments de l'homologie, dont nous avons parle plus haut; ses resultats sont extremement originaux et suggestifs, mais aucune de ses demonstrations ne peut etre consideree comme correcte. A partir de 1863, Clebsch, puis Brill et Max Noether developpent une theorie dont les objets sont les courbes algebriques dans Ie plan projectif complexe et les systemes de

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points sur ces courbes, au moyen de laquelle ils veulent donner des preuves des theorernes de Riemann en ne faisant pas usage de I'Analyse. Pour Riemann, I'objet fondamental etait Ie corps des fonctions merornorphes (ou rationnelles, ce qui revient au meme ici) sur la courbe, et il ne faisait done pas de distinction entre deux courbes algebriques si elles ont merne corps de fonctions rationnelles; or on voit sans peine que cela signifie qu'il y a une correspondance birationnelle entre les deux courbes (qui peuvent etre situees dans des espaces projectifs de dimensions differentes); cela veut dire que chaque coordonnee d'un point de I'une des courbes est fonction rationnelle des coordonnees de I'autre, chaque point de I'une des courbes, a I'exception possible d'un nombre fini d'entre eux, correspondant a un seul point de I'autre. On dit que chacune de ces courbes est un modele du corps de fonctions considere. Un des premiers buts de l'ecole 'geometrique' de Brill et M. Noether fut d'obtenir par des transformations birationnelles des modeles sans singularite d'une courbe algebrique plane quelconque (qui en general a des points singuliers); des 1870 ce but etait atteint, toute courbe algebrique irreductible ayant un modele sans singularite dans l'espace projectif a 3 dimensions, et en outre un tel modele est unique a isomorphisme pres. Un autre modele etait une courbe algebrique plane n'ayant que des points doubles a tangentes distinctes. Tout ceci etait obtenu par des raisonnements essentiellement corrects, encore qu'au debut les geometres de cette ecole aient eu tendance a negliger certains types de points singuliers cornpliques dont l'etude leur paraissait fastidieuse et dont ils etaient convaincus qu'ils ne creeraient pas de difficulte serieuse; effectivement, une etude exhaustive de toutes les singularites possibles d'une courbe algebrique fut rapidement faite et etablit l'existence de modeles sans signularite de facon incontestable; on avait donc certainement fait de cette maniere un progres decisif par rapport a Riemann, dont les intuitions geniales de nature topologique ou analytique restaient alors sans preuve. Un peu plus tard, en 1882, la theorie purement algebrique developpee par Dedekind et Weber mettait definitivement hors de doute les theorernes de Riemann; la theorie avait done evolue comme dans les exemples anterieurs, mais d'une Iacon beaucoup plus resserree dans le temps, puisqu'on etait tres rapidement arrive a la stabilite qui finit toujours par succeder aux tatonnements du debut. Vers 1870, Cayley, Clebsch, M. Noether et Zeuthen entreprennent de fonder la theorie des surfaces algebriques sur Ie modele de celle des courbes algebriques, en remplacant les systemes de points sur la courbe

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par des systemes de courbes sur la surface; mais ils se heurtent rapidement a des difficultes totalement imprevues: 'la theorie des courbes algebriques' disait M. Noether 'est une invention de Dieu, mais celie des surfaces algebriques est une invention du diable!' Un des premiers theoremes qu'ils chercherent a prouver est l'existence, pour une surface algebrique irreductible quelconque dans I'espace projectif a 3 dimensions, d'un modele sans singularite dans I'espace projectif a 5 dimensions; mais la methode calquee sur celie qui avait reussi pour les courbes et utilisant une suite de transformations birationnelles bien choisies rencontrait de formidables obstacles dans Ie cas des surfaces, dus a la complexite beaucoup plus grande des singularites; par exemple un point singulier isole peut se transformer en une courbe de points singuliers. Aussi, malgre les efforts d'une demi-douzaine de mathematiciens allemands, italiens et francais, c'est seulement en 1935 qu'une preuve entierement correcte de l'existence d'un modele non singulier fut obtenue par l'Arnericain Walker. En outre, des exemples tres simples (Ie plan projectif et une quadrique non degeneree) montrent qu'un modele sans singularite n'est pas unique a isomorphisme pres en general, contrairement a ce qui se passe pour les courbes. La geometric sur les surfaces algebriques doit done distinguer entre deux types de proprietes: celles qui sont invariantes par toute transformation birationnelle et celles qui ne sont invariantes que par isomorphisme (transformation birationnelle bijective). Comme il est souvent necessaire, dans les raisonnements, d'utiliser un modele sans singularite, il semble que la theorie aurait dfi etre bloquee des Ie debut. Mais les geometres francais et italiens prefererent passer outre; convaincus de I'existence d'un modele sans singularite sans pouvoir Ie demontrer rigoureusement, ils n'hesiterent pas a I'utiliser. Comme pour Riemann et Poincare en Topologie, Ie remarquable sens geometrique des geometres italiens (Castelnuovo et Enriques tout specialernent) leur permit de decouvrir toute une serie de proprietes souvent tres profondes, dont la plupart ont pu etre dernontrees rigoureusement plus tard; mais ils se rendaient compte du caractere precaire de ces resultats: 'on ne reussit pas, pour Ie moment, a exposer des demonstrations de maniere a eviter toute difficulte' avouent Castelnuovo et Enriques en 1897. D'ou de nombreuses controverses entre les specialistes des surfaces algebriques; on verra encore, par exemple, a la date tardive de 1943, Enriques et Severi publier deux articles OU chacun con teste les theoremes de l'autre! A partir de 1900, certaines parties des resultats obtenus anterieurernent

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furent partiellement justifies par Severi et Poincare, puis par l'usage que fit Lefschetz de resultats de Topologie algebrique; mais ce n'est qu'a partir de 1927 avec les methodes purement algebriques de van der Waerden, puis de Zariski et A. Weil, qu'on aboutira a une theorie entierement rigoureuse (et valable par surcroit sur un corps de base quelconque). Et comme toujours, une fois ce palier atteint, toutes les controverses ont completement cesse. Pour conclure, que peut-on dire de la rigueur dans la mathernatique actuelle? Si on se borne aux mathematiciens qui utilisent la logique classique (et qui constituent la tres grande majorite), est rigoureux pour eux un raisonnement comportant une chaine de deductions faites suivant les regles de cette logique, et partant d'un systeme d'axiomes explicite; comme ces regles logiques sont entierernent codifiees, la verification d'une demonstration est en principe un travail mecanique, pourvu qu'elle soit suffisamment explicite; ce travail peut etre long et penible et exiger d'etre fait independamment par plusieurs specialistes lorsqu'une demonstration comporte plusieurs centaines de pages, ce qui n'est plus rare aujourd'hui; et pour certaines demonstrations, la verification exige I'usage d'ordinateurs, ce qui introduit un certain element d'incertitude. Bien entendu, chez Ie mathematicien qui imagine une demonstration nouvelle, Ie role de son 'intuition' demeure preponderant, la 'mise en forme' ne venant qu'ensuite. Certains repugnent a une tache aussi ingrate et malheureusement indispensable; ils esperent alors que certains de leurs collegues ou eleves s'y attelleront a leur place, ce qui en fait se produit en general au bout de quelques mois ou de quelques annees, surtout si Ie theoreme est juge tres important. Mais jusqu'a ce que la demonstration ait ete assez explicitement ecrite, puis verifiee, Ie resultat doit demeurer en doute; on en connait a present plusieurs exemples.

l.A. BARROSO editor, Aspects of Mathematics and its Applications © Elsevier Science Publishers B.V. (1986)

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POLAR SUBSETS OF LOCALLY CONVEX SPACES Sean DINEEN Department of Mathematics, University College Dublin, Belfield, Dublin 4, Ireland

Reinhold MEISE Mathematisches lnstitut, Universitiit DUsseldorf, D-4000 Dusseldorf, Fed. Rep. Germany

Dietmar VOGT Fachbereich Mathematik, Universitiit Wuppenal, D-5600 Wuppertal, Fed. Rep. Germany Dedicated to Leopoldo Nachbin on his sixtieth birthday

o.

Introduction

Lelong [8], [9] and Kiselman [7] have remarked that in certain locally convex spaces all bounded subsets are polar. Using completely different arguments we have shown in [6] that a nuclear Frechet space in which every pseudoconvex domain is a domain of existence, contains a bounded non-polar subset if and only if it has the linear topological invariant (Ji) introduced by Vogt [21]. Moreover, we proved that for a nuclear Frechet space E property (Ji) is equivalent to the existence of a bounded subset A which is not uniformly polar, i.e. every plurisubharmonic function f on E which factors by a continuous linear map through some Banach space is not identically -00 on A. In the present note we refine the basic ideas in our previous article [6] to a sufficient condition on a locally convex space so that all absolutely convex compact polar subsets are uniformly polar. This condition is satisfied by all (DFN)-spaces and by all Frechet spaces in which the Levi-problem has a positive solution. Then we characterize the bounded Hilbert balls in nuclear I.e. spaces E which are not uniformly polar in E and the bounded Hilbert balls in (DFN)-spaces E which are non-polar in

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E. In these characterizations the nuclearity is needed only to show that the conditions are sufficient. In both cases the proofs are based on two essential lemmata. In the first lemma we construct plurisubharmonic functions with certain properties, while the second one provides a condition on a compact Hilbert ball A in a Hilbert space H such that for a given point w in H there exists a holomorphic path through w defined on a complex half plane with values in A on some smaller half plane. Using arguments due to Vogt [18], [19] the characterizations stated above give extensions of the main results of [6] with new proofs. Moreover, they show that for a (DFN)-space E the existence of a bounded subset which is not (uniformly) polar in E is equivalent to the fact that E~, the strong dual of E, has the property (ON). A strongly nuclear Frechet space has (ON) iff it is isomorphic to a subspace of some nuclear power series space of finite type, as Vogt [20] has shown. Our result has the interpretation that there is a large class of nuclear Frechet spaces for which the strong dual contains non-polar bounded sets. We also give examples of nuclear Frechet spaces admitting a continuous norm for which all bounded sets in the strong dual are polar.

1. Preliminaries

We shall use the standard notation from the theory of locally convex spaces as presented in the books of Pietsch [14] or Schaefer [15). A I.e. space always denotes a complex vector space with a locally convex Hausdorff topology. If E is a I.e. space and if U is an absolutely convex zero neighbourhood, then E u denotes the canonically normed space EIKer Po- where Pu is the gauge functional of U. By ·Eu we denote the completion of E u and by 7TU : E ~ Eu the canonical map. If B is an absolutely convex bounded subset of E then E B , the linear hull of B in E, becomes a normed space under the gauge functional of B. If the normed space E B is a Hilbert space, then we call B a bounded Hilbert bal/. For an absolutely convex subset M of E we define 11·lltt: E' ~ [0, 00] by I/Ylitt = sup xEM Iy(x )1, where E' denotes the topological dual of E. Obviously I/·lltt is the gauge functional of MO = {y E E'/ SUPXEM/y(x)l:s; I}. By E~ we denote the strong dual of E, while E; is E' endowed with the topology 11. (E', E) of uniform convergence on the precompact subsets of E. For ACE we denote by Int A the interior of A.

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A I.e. space E is called a (DFN)-space, if it is the strong dual of a nuclear Frechet space. For a Frechet space E we always assume that its I.e. structure is generated by an increasing system (11'lln)nEN of seminorms. 1.1. Sequence Spaces Let PfJ be a family of non-negative sequences (Pn)nEN with the following properties: (1.1)

For every P, q E PfJ there exists an r E PfJ and a c > 0 with

p+q (1.2)

~cr.

For each j E N there exists apE PfJ with Pj > O.

For 1 ~ s ~ 00 (resp. s

= 0) we

define the sequence space AS (PfJ) by I/s

cc

AS ( PfJ) =

{x E c N IlIxll = (~ (Ixj IPj)' ) < 00 for all P E PfJ} p

A""(PfJ)

= {x

E (;Nlllxlip = sup (Ixjlpj) < JEN

00

(1 ~ s <

00) ,

for all P E PfJ}

and Ao(PfJ) = {x E A""(PfJ)llim (xjP) = 0 for all P E PfJ}. j--

AS (PfJ) is a I.e. space under the natural I.e. topology induced by the seminorms (11'llp )PE9" ' We write A(PfJ) instead of AI(PfJ). We recall that AS(PfJ) is Schwartz (resp. nuclear) iff for every P E PfJ there exists a q E PfJ and a JL E Co (resp. JL E [I) such that Pj ~ JLjqj for all jEN. Consequently AO(PfJ)=A""(PfJ) if A""(PfJ) is Schwartz and AS(PfJ) = A(PfJ) for all s;;:.1 if AS(PfJ) is nuclear. If AS ( PfJ) is a Frechet space then we can always assume that AS ( PfJ) = AS(d), where .91 = (aj.k)U.k)EN2 is a matrix satisfying (1). O~aj.k~aj.k+lforallj, kEN. (2). For each j E N there exists a kEN with a t: k > O. From Bierstedt, Meise and Summers [1] we recall that AS(d)~, the strong dual of AS (d), can be identified under the canonical bilinear form (x, Y)~ ~7=1 xjYj with A'(A( .91)+) where r = 1 for s = 0 and l/r + l/s = 1 for 1 < s <00, and A(d)+ = {x E A(d)1 x ~O}.

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Particular examples of Frechet sequence spaces are the so-called power series spaces, which are defined in the following way: Let a be an increasing unbounded sequence of positive real numbers (called an exponent sequence) and let R = 1 or R = 00. Then we define for I:$; s :$; 00 the spaces A ~(a) = AS(d(R, a)),

where

A ~ (a) is called a power series space of finite (resp. infinite) type if R = 1 (resp. R = 00). Well-known examples of power series spaces are s = A",,((Iog(n + rj), EN) =:; 't?""(sl), .rt'(Ck ) =:; A ",«\Y;Z)n EN) and .rt'(B1(O/) =:; Al«(~r,i\EN)' where B1(O) stands for the open unit disk in C and where .rt'(fJ) denotes the space of all holomorphic functions on fJ endowed with the compact open topology.

1.2. Polar Sets

Let E be a I.e. space, f: E ~ [-00,00) is called plurisubharmonic (psh.) if f is not identically -00, if f is upper semicontinuous (i.e. 1([-00, c)) is open for every c E R) and if z ~ f(a + zb) is subharmonic or identically -00 for every a, bEE. f is called uniformly plurisubharmonic, if there exists an absolutely convex zero-neighbourhood U in E and a plurisubharmonic function g on the canonical Banach space Eu such that f = go 1TU is not identically -00. A subset B of E is called (uniformly) polar, if there exists a (uniformly) psh. function f on E with fiB = -00. For more information on psh. functions on I.e. spaces we refer to [12].

r

1.3. Holomorphic Functions

Let E be a I.e. space and let fJ be a non-empty open subset of E. A function f: fJ ~ C is called Gateaux-analytic if for every a E fJ and every bEE the function z ~ f(a + zb) is a holomorphic function on its natural domain of definition in C. f is called holomorphic, if f is Gateauxanalytic and continuous. By .rt'(fJ) we denote the vector space of all holomorphic functions on Il. For details concerning holomorphic functions on l.c, spaces we refer to [2], [4] and [12].

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2. Locally Convex Spaces in which Every Absolutely Convex Compact Polar Subset is Uniformly Polar In [6, Thm. 10], we have shown that for a nuclear Frechet space, in which every pseudo-convex domain is a domain of existence, there exists a compact non-polar subset if and only if there exists a compact subset which is not uniformly polar. Here we prove that in certain I.e. spaces E every absolutely convex compact subset of E is polar in E iff it is uniformly polar in E. Lemma (2.1). Let F be a Banach space and B a bounded absolutely convex subset of F. Let (nj ) JEN be a strictly increasing sequence in N and let (p) JEN be a sequence of homogeneous polynomials with the following properties: (2.1)

Pj is n-homogeneous for every j EN.

(2.2)

sup sup Ipj(x)lllnj ~ ~. JEN

sup IPj (x)1 Iinj ~ e -j2 for every j EN.

(2.3) (2.4)

Ilxll'" I

xEB

There exists a z E F with

Ilzll < 1 and

inf Ipj(z)lllnj =

S

> O.

JEN

Then the function

is psh. on F, satisfies g(z) >

-00

and glB =

-00.

Proof. For every x E F, x of- 0, we have by (2.1) and (2.2) (2.5)

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This implies that the series

converges either to -00 or to a real number. Hence it defines a function g : F ~ [-00, 00). Let U denote the open unit ball in F. From (2.2) it follows that glu is the limit of a decreasing sequence of psh. functions and hence psh. on U, provided glu ¥- -00. In order to see this, note that by (2.4) we have

=

=1

1

g(z) = L -;z log /pj(z)l;;:. L -:2 log s > -00. j=I} nj j=I} To see that g is psh. on F we first remark that for R > 1 and every x ERU we have

I

cc 1 g(x)= L-;zlog R't p, (X)/ =g (X) - + = 1 lo g. R . j=I} nj R R j=I}

L-:2

This shows that glRU is obtained as the pull-back of a psh. function on U by a continuous linear map. Hence glRU is psh. on RU for every R > 1. Consequently g is psh. on F. For every x E B we have by (2.3) cc

g(x) = L

1

-.2j=IJ j

n

cc

1

cc

log Ipj(x)! ~ L -:2 (-l) = L (-1) = -00. j=I}

j=1

This completes the proof. 0 Theorem (2.2). Let E be a l.c. space having the following properties: (a). Every pseudo-convex open subset of E is a domain of existence. (b). Every

Gateaux-analytic function f on E which is continuous on some non-empty open subset of E is continuous on E. Then every absolutely convex compact subset B of E is polar in E if and only if B is uniformly polar in E. Proof. Since every uniformly polar subset of E is obviously polar, it suffices to show that any given absolutely convex compact polar subset B

of E is already uniformly polar. Since B is polar in E iff E B is polar in E it follows from [13, Prop. 2],

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n

n

that there exists a pseudo-convex open set with E B c ¥ E. By hypothesis (a) n is the domain of existence of some IE H(n). For n E No let qn denote the nth derivative of I at zero. Since I is continuous on there exists an absolutely convex zero-neighbourhood V in E such that supxE v I/(x)1 < 00. From the integral representation

n

1 qn(x)=-Z'

7Tl

J I(zx) ~dz Izl=1

z

it follows that we may assume without loss of generality that SUPnENSuPxEVlqn(x)I~~. Let 7Tv:E~Bv denote the canonical map. By standard arguments it follows that there exist continuous n-homogeneous polynomials Pn on B v satisfying qn = Pn 0 TTV for all n E Nand sup sup IPn(y)1 ~~. nEN 11Y1I"'1 By the compactness of B it follows that SUpxERB I/(x)1 < 00 for all R > O. This implies limn-+.,(suPxEB Iqn(x)/)I/n = 0 and consequently IPn(y)l)l/n = O.

lime sup n-+'" yE ".v(B)

Next we claim that there exists awE E with limsup,....cc Iqn(w)11/n > O. Otherwise x ~ L:=o qn(x) defines a Gateaux-analytic function on E which is continuous on Il. By hypothesis (b) this function is an entire function on E extending I to all of E. Since n E is the domain of existence of I this is a contradiction. Now let z = 7T v( w) and choose a strictly increasing sequence (n j ) jEN in N such that for a suitable s > 0 we have, for all j EN,

s:

and j2 sup IPn.(y)llInj ~ e- .

yE"'v(B)

I

Since all the hypotheses of Lemma (Z.l) are satisfied, there exists a psh.

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function g on Bv with g(z»-oo and gl".v(B)=-oo. Since g01Tv(w)= g(z) > -00, the function h = go 1Tv is uniformly psh. on E and satisfies h 18 = -00. Hence B is uniformly polar in E. D By the extensions of Zorn's theorem (see [12, Thm. 1.3.1]), Frechet spaces and strong duals of Frechet Schwartz spaces satisfy hypothesis (b) of Theorem (2.1). Since Colombeau and Mujica [3] (see also Schottenloher [17]) have shown that every pseudo-convex open subset n of the strong dual of a nuclear Frechet space is a domain of existence, we have the following corollaries of Theorem (2.2): Corollary (2.3). Let E be a (DFN)-space. Then every absolutely convex compact subset B of E is polar in E iff B is uniformly polar in E. Corollary (2.4). Let E be a Frechet space in which every pseudo-convex open subset is a domain of existence. Then every absolutely convex compact subset B of E is polar in E iff B is uniformly polar in E. Remark (2.5). By Schottenloher [16, Cor. 3.4], every Frechet space E with the bounded approximation property satisfies the hypothesis of Corollary (2.4).

3. Essential Lemmata In this section we prove two lemmata which contain the basic ideas for the results presented subsequently. The first lemma has its origin in the construction of the proof of [6, Thm. 7], while the second one has been used implicitly in the proof of [6, Thm. 9]. Lemma (3.1). Let E be a l.c. space and let A be an absolutely convex subset. Let (Bn)nEN be an increasing sequence of absolutely convex subsets of E with the following properties: (3.1)

U B; nEN

(3.2)

=

E and there exists an mEN with A C B m •

Iff:E ~ [-00, 00) is upper semicontinuous on each B n, thenf is upper semicontinuous on E.

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303

If E satisfies

(P)

there exists an X oE E such that for all n EN, all C > 0 and all d > 0 there exists ayE E' with ly(xOW+ d ~ Cllyll; ~lIyll~

then

(3.3)

A is polar in E, if each B; is closed

(3.4)

A is uniformly polar in E, if each B; is open.

Proof. If E A is not dense in E then the Hahn-Banach theorem gives ayE E' with y ¥ 0 and ylEA = O. Then g:x ~ log jy(x)j is a uniformly psh. function on E with glA = -00. Hence we may and shall assume that A is a total subset of E. Moreover, we can assume that we have (3.1) with

m=1. By (P), for every n EN, Cn = e n 2- 1 and d; = n 2 - 1, there exists a 2 Yn E E' with IIYnll;. = lie such that /Yn(x o)ln ~ IIYnll~. Since A is total in E (3.1) implies that for an = log IlYnll~ we have

and hence (3.5)

For xEE let '"

1

g(x) = ~ ~ log IYn(x)/, and note that for x E B k and all n ~ k we have, since B, C B n ,

Hence glnt is the decreasing limit of a sequence of psh. functions. This shows that glnt is upper semicontinuous for all kEN and consequently upper semicontinuous by (3.2). By (3.1) this also implies that for every

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finite-dimensional linear subspace Eo of E the function glEo is either identically -00 or psh. Hence g is psh. on E provided g ~ -00. However, by (3.5), we have

By (3.1) and since m

=

1 we have for all x E A

and hence 00

1

00

g(x) = ~ lan/log IYn(x)1 ~ ~ (-1) = - 0 0 . This shows glA = -00 and completes the proof in case (3.3). In case (3.4) the same arguments work, but then it is easy to see that the function g factors by a continuous linear map through the Banach space E Bl' Hence g is uniformly psh. and consequently A is uniformly polar. 0 Lemma (3.2). Let H be a Hilbert space and let A be a bounded Hilbert ball in H such that H A is dense in H. Assume that the canonical inclusion T: H A ~ H is of type lp and that II Til ~~. Suppose also that for some wE H, C;;;' 1, d » 1 and all Y E H' we have 1

P< - 1+

s

Then there exists a holomorphic function h : G ~ H with h (1) = wand Int h -l(HA ) ~ 0, where G = {z E CI Re(z) > ~}. Proof. If dim H < 00 then H A = H and the lemma holds trivially. Hence we may assume dim H = 00. Then the spectral representation theorem implies that

Tx

=

2: Aj(X I e)A~' j=l

x E HA ,

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where {ej Ij EN} (resp. {~I j EN}) are complete orthonormal systems in H A (resp. H) and where 0:00.;; Aj'~ ~ for all n EN and (Aj)jEN E [p' Next we define 'Pk E H' by 'Pk (x) = (x Ifk) for kEN and note that for all kEN we have

lI'Pkll* =

1 and

lI'Pkll~ = sup I(x / fk)1 _A

= sup I(Tx Ifk)/ = Ak sup I(x I ek )AI = _A _A

Hence our hypotheses imply, that for all kEN, with a

Ak ·

= 1/(1 + d),

(3.6)

For kEN we define the functions hk : G ~ C by

Iwkl:oo.;; A~ or Iwkl > 1.

if

Let M = {k E all z E G

L

Nllwkl > I}. Then

Ihk(z)/2:oo.;;

k~l

by (3.6) and p < 1/(1 + d) = a we have for

L Iwkl + L Iwkl2 :00.; c- L kEM

k~l

k=l

A~ +

L Iwkl2<

kEM

00.

This implies that h: G ~ H defined by h(z) = ~;=1 hk(z)fk is holomorphic on G. By definition we have h(l) = w. For each z E G with Re(z) > 2/ a = 2(1 + d) we have Ihk(z)1 :00.; c;2 Ai for all kEN \M. This implies that

i: hk(z) e A

k~l

k

k

E

HA

for all z E C with Re(z) > 2(1 + d),

and consequently h -l(HA ) :J {z E

C/ Re(z) > 2(1 + d)}. 0

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S. Dineen et aJ. I Polar Subsets of I.e. Spaces

4. Bounded Sets which are Not Uniformly Polar

In this section we use the two essential lemmata to derive a necessary and a sufficient condition for the existence of bounded subsets of a I.e. space E which are not uniformly polar in E. Proposition (4.1). If the locally convex space E contains a bounded absolutely convex subset A which is not uniformly polar in E, then E satisfies the following equivalent conditions:

(4.1)

For every x E E and every absolutely convex zero-neighbourhood U there exist a C> 0 and ad> 0 such that for all y E E' ly(x)l1+d ~ CJIyllt/llyll~.

(4.2)

For every x E E and every absolutely convex zero-neighbourhood U there exist a C > 0 and ad> 0 such that for all r > 0

Proof. Assume (4.1) does not hold. Then there exist an X o and an absolutely convex zero-neighbourhood U such that for any C> 0, d > 0 there exists ayE E' with ly(xoW+ d > Cllyll;;dllyll~. With B; = nU this implies that the hypotheses of Lemma (3.1) are satisfied and hence A is uniformly polar by (3.4). By contraposition this proves the first part of the theorem. The equivalence of (4.1) and (4.2) is proved by the same method as in [19, Lemma 1.4] (see also [22, Lemma 2.1]). 0 Proposition (4.2). Let E be a nuclear l.c. space. A bounded Hilbert ball A in E is not uniformly polar in E if and only if the equivalent conditions (4.1) and (4.2) are satisfied. Proof. By Proposition (4.1) it suffices to show that (4.1) implies that A is not uniformly polar in E. To prove this, let f be a psh. function on E which is uniformly psh. with respect to the absolutely convex zeroneighbourhood U, i.e. f = go 'Tru, where g is psh. on Bu. Since E is nuclear, we may suppose that H = Eu is a Hilbert space. Moreover, we

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307

may assume that 2A C U. We let A' = 7Tu (A ) and note that A' is a bounded Hilbert ball in H. Let i, : E A ~ E resp. T : HA'~ H denote the canonical inclusion maps. Since E is nuclear, TTU 0 i, is of type s. This implies that T is of type s. Since H' is canonically isometric to (E')Int u it follows easily from (4.1) that for every x E H there exist a C> 0, d ~ 1 such that for all y E H' we have

ly(x)1 +

1 d

~

C1lyll;/llyll.1·

Now we choose wE H with g(w) > -00 and apply Lemma (3.2) to get a holomorphic function h: G~ H with h(l) = wand Int h- 1(HA ,) # 0. Since foh is subharmonic, there exists an a'EH A , with g(a'»-oo. Hence if a E E A and 7Tu (a) = a' then f(a) = g(7TU (a» = g(a') > -00. This shows that flEA # -00 and consequently flA # -00. Hence A is not uniformly polar. 0 Next we use an argument of Vogt [18] to reformulate the conditions given in Proposition (4.1). Lemma (4.3). Let A be a bounded absolutely convex subset of the l.c. space E. (1). If E is sequentially complete, then condition (4.2) is equivalent to: (4.2')

For every absolutely convex bounded subset B and every absolutely convex zero-neighbourhood U in E there exist a C> a and ad> 0 such that for all r > 0:

(2). If E is a Frechet space, then condition (4.2) is equivalent to the following condition:

(nA )

For every absolutely convex zero-neighbourhood U in E there exist a zero-neighbourhood V, a C> 0 and ad> 0 such that for all r>O

308

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Proof. Obviously the conditions in (1) and (2) imply (4.2). To show the converse let U be given and put for n EN. Then L; is a closed absolutely convex subset of E satisfying (4.3) for all r > 0 and all n EN. From (4.2) it follows easily that for every x E E there exists an n E N with x E Ln. Hence E = U n E N Ln' In case (2) Baire's theorem implies that for some mEN the set L m is a zero-neighbourhood. By (4.3) this proves (nA ) . In case (1) let B be an arbitrary bounded absolutely convex subset of E. Then the Banach space E clOSB is continuously embedded in E and

Hence Baire's theorem implies that for some e > 0 and some mEN we have

eB C e clos Be L m

•

This implies that for all r > 0 1 Be (me -(m+lJ/m) - U + r A, r

and this completes the proof. D From Proposition (4.1) and Lemma (4.3) we obtain by an argument of Vogt [19, Lemma l.4J the following: Proposition (4.4). Let E be a sequentially complete I.e. space. If E contains a bounded absolutely convex subset A which is not uniformly polar, then E satisfies the following equivalent conditions:

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(4.4)

For every bounded absolutely convex subset B and every absolutely convex zero-neighbourhood U of E there exist a C> 0 and ad> 0 such that for all y E E'

(4.5)

For every bounded absolutely convex subset B and every absolutely convex zero-neighbourhood U of E there exist a C> 0 and ad> 0 such that for all r > 0

Corollary (4.5). Let E be a sequentially complete l.c. space which contains a bounded subset which is not precompact. Then every precompact subset ofE is uniformly polar in E. Proof. Suppose E contains a precompact subset A which is not. uniformly polar. Without loss of generality we may assume that A is absolutely convex. It then follows easily from (4.5) that every bounded set B in E is precompact. Hence the result follows by contraposition. 0 Combining Proposition (4.2) with Proposition (4.4) we get: Theorem (4.6). Let E be a sequentially complete nuclear l.c. space. For a

bounded Hilbert ball A in E the following conditions are equivalent: (4.6)

A is not uniformly polar in E.

(4.7)

Forevery bounded subset B ofE and every zero-neighbourhood U in E there exist a C> 0 and ad> 0 such that for all y E E'

Corollary (4.7). Let E be a sequentially complete nuclear l.c. space which

has a fundamental system of bounded sets consisting of Hilbert balls. Then E contains a bounded subset which is not uniformly polar if and only if E contains a bounded subset A for which (4.7) holds.

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To state another useful corollary of Theorem (4.6) we introduce the following notation. Let A(~) be a nuclear sequence space and let a E A(~) with a> 0 (i.e, a j > 0 for all j EN) be given. Then we put

It is easy to check that N~ eN:, that N~ and N: are absolutely convex bounded subsets of .A (~) and that {N;I b E .A (g}l), b > O} is a fundamental system of the bounded subsets of .A (g}l). This holds also for the system {N~I b E .A (~), b > O} if .A (~) is dual nuclear. Corollary (4.8). Let .A(~) be nuclear and dual nuclear and let a E .A(~), a> 0, be given. Then the bounded sets N~ and N: are not uniformly polar in A( g}l) if and only if the following condition (4.8) holds: .

(4.8)

For every bE .A (~) and every p E ~ there exists a C> 0 and a d > 0 such that for all j EN we have Ibl+ d pf';;; Caj •

Proof. Obviously N~ is a bounded Hilbert ball in .A ( ~). Hence we get from Theorem (4.6) that N~ is not uniformly polar in A( ~) if and only if (4.7) holds. Thus it suffices to show that (4.7) and (4.8) are equivalent. To prove this, we note that A( ~)' = {y E eNI there exists aD> 0, pEP with lyA.;;; Dp.; for all j E fill}. Moreover it is easy to check that for b E A(~) with b > 0 we have co

11'II~r y ~ ( ~ IYjbl) )=1

while for pEP and Up

= {x

E A(~)I

1/2

,

L7=1 IXjpj12 .;;; 1} we have

To show that (4.7) implies (4.8) let bE.A (~) and p E ~ be given and choose c E .A(~) with Ibl.;;; c and c > O. Then we apply (4.7) with B = N;

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311

and U = Up to e; : x -'; xj to get that for some C> a and d > a we have for all j EN 1

C

~+d = lie ~II~ l+d :,;;; C/Ie ~Wle *1I~2 = C (-) I I I Pj

I

0

d

a., I

which implies (4.8), since Ibjl:,;;; c j. To prove the converse implication let b E A(g» with b > a and pEg> be given. Then we get from (4.8) and HOlder's inequality for every Y E A(g»': 00

IIY/I~~ = L

2/(I+d)

j~1

j~1 2

00

:,;;; C

2/(1+d)

(~jlLj) I~I

=

L IXLI 00

IYjbl :,;;; C

2d/(I+d)

Pj

d/(1+d)

(L

Iyj a

j /2!(1+d)

1/(1+d)

00

IYjal)

I~I

PI

C2!(1+d)lIyll* 2d/(I+d)/Iyll* i!(l+d) . Up

No

This implies (4.7), since the sets {N~I bE A(g», b > a} form a fundamental system for the bounded subsets of A(g». The assertion for N: is obtained in the following way. Since N::J N~ (4.8) implies that N~ and consequently N: is not uniformly polar. The converse implication is obtained from Proposition (4.4) by arguments which are similar to those used in the preceding part of the proof. 0

5. Bounded Sets which are Non-Polar In this section we apply the essential lemmata of Section 3 to derive a necessary and a sufficient condition for the existence of non-polar bounded subsets in certain l.c, spaces.

= F~, where F is a Frechet space. If A is an absolutely convex bounded non-polar subset of E then the following equivalent conditions hold:

Proposition (5.1). Let E

(5.1)

For every x E E there exist a bounded absolutely convex set B in E, a C> 0 and ad> 0 such that for all Y E E' = F

312

(5.2)

S. Dineen et al. / Polar Subsets of l.c. Spaces

For every x E E there exist a bounded absolutely convex set B in E, a C> 0 and ad> 0 such that for all r > 0

Proof. The equivalence of (5.1) and (5.2) is shown as in the proof of Proposition (4.1). Hence it suffices to show that (5.1) holds if A is non-polar. To prove this let (Bk)kEN denote an increasing fundamental system of a(E, F)-closed absolutely convex bounded subsets in E with E = U kEN B k. By the Banach-Dieudonne theorem the topology of E is the finest topology which coincides on each B; with a(E, F). Hence (3.1) and (3.2) are satisfied. Since A is non-polar, it follows that (P) in Lemma (3.1) is false and hence (5.1) holds. 0 Proposition (5.2). Let E be a complete dual nuclear l.c. space. A bounded

Hilbert ball A in E is non-polar in E if the (equivalent) conditions (5.1) and (5.2) are satisfied. Proof. Let f be an arbitrary psh. function on E. Then there exists awE E with f(w) > -00. Since E is complete and dual nuclear and since (5.1) remains true if B is replaced by a larger bounded absolutely convex set, we can find a bounded Hilbert ball B in E with w E E B which satisfies (5.1) and has the following properties: The canonical inclusion T: B A ~ E B is of type lp with p < 1/(1 + d) and IITII ~~. Hence the hypotheses of Lemma (3.2) with H = BB are satisfied, provided that for all y E (BBY we have

By (5.1) this holds for all y E TjB«EB)'), where l» : E B ~ E denotes the canonical injection. Since i, is injective and since E B is a Hilbert space, TjB has dense range, which implies the desired property. Hence we get by Lemma (3.2) the existence of a holomorphic function h : G ~ E B with h (1) = wand Int h -l(BA) =I- 0. Since f h is subharmonic on G it follows that flEA =I- -00 and hence flA =I- -00. This proves that A is non-polar in E, since f was an arbitrary psh. function on E. 0 0

Using Baire's theorem as in Section 4, we can reformulate Proposition (5.1) in the following way:

S. Dineen et al. I Polar Subsets of l.c. Spaces

313

Proposition (5.3). Let F be a Frechet space with a fundamental system of seminorms and let

(1/·lln)nEN

E = F;

and

B; = {y E EI sUPllxlln"lly(x)1 ~ I}.

If the absolutely convex bounded subset A of E is non-polar in E then the following equivalent conditions hold: (5.3)

For every kEN there exist n EN, C> 0 and d > 0 such that for all r> 0:

C

d

B, C-Bn+r A. r

(5.4)

For every kEN there exist n EN, C> 0 and d > 0 such that for all x E F we have

Proof. The equivalence of (5.3) and (5.4) has been shown by Vogt [20, Lemma 2.4]. Hence it suffices to show that (5.3) holds if A is non-polar. To prove this we remark that since E' = (Fd' = F we have 1/·1/;1 = I/·I/k' The sets B, are u(E, F)-closed and equicontinuous and hence compact in E. Since we may assume that A is closed, the sets (n/r)Bn + rnA are absolutely convex and compact in E for all n E N and all r > O. By (5.2) it follows that

Hence Baire's theorem implies (as in the proof of Lemma (4.3) (2» that (5.3) holds. 0 Condition (5.4) is well known and has been introduced by Vogt [20] to characterize the linear subspaces of nuclear power series spaces of finite type. From [20] we recall: Definition (5.4). (Vogt [20, 2.1]) A Frechet space F has property (DN) if the following is satisfied:

314

(DN)

S. Dineen et al. / Polar Subsets of I.e. Spaces

There exists a continuous norm 11·11 on E such that for every kEN there exist an nEN, C>O and £>0 with 11'111+'~

q. II'II· lin .

The norm 11·11 on E will be called a (DN)-norm. Using this notation Proposition (5.3) can be rephrased as: Corollary (5.5). Let F be a Frechet space. If F~ contains a bounded non-polar subset A, then II'II~: x ~ SUPyEA ly(x)1 is a (DN)-norm on F. In particular F has property (DN). Proof. We may assume that A is absolutely convex. As a non-polar set A

is total in F~ and hence II'II~ is a continuous norm on F. By (5.4) II'II~ is a (DN)-norm on F. From Corollary (5.5) and Proposition (5.2) we get immediately: Theorem (5.6). Let E be a (DFN)-space. A bounded Hilbert ball A in E is non-polar in E if and only if II'II~ is a (ON)-norm on E~. Corollary (5.7). A (DFN)-space E contains a bounded non-polar subset if

and only if E~ has property

(!?}~)

Remark (5.8). It is easy to check that property (DN) is inherited by linear

subspaces. Remark (5.9). By Vogt [20, 4.1], the sequence space '\(d) has (ON) iff there exists a k o E N such that for every kEN there exists an n EN, C> 0 and e > 0 such that for all j EN: a }.+k' ~ Ca;' ko a i. n • Example (5.10). Many examples of nuclear Frechet spaces with (DN) can

be obtained from the following result of Vogt [20, Sect. 5]: Let X be a connected, N-dimensional, real-analytic manifold (N) 1). Let fJ' be a subsheaf of vector spaces of the sheaf d of germs of all complex-valued real-analytic functions on X for which fJ'(U) is complete with the compact-open topology for every open subset U of X. Then 9'(X) is isomorphic to a subspace of H(B1(Ot- 1) = AI«~/~)nEN) and consequently 9'(X) has (ON).

S. Dineen et al. / Polar Subsets of I.e. Spaces

315

Example (5.11). If a Frechet space has (DN) then it admits a continuous norm. Example (5.12). Infinite-dimensional Frechet spaces F with continuous norm which have property (n) introduced by Vogt [21,4.1], do not have property (DN). This is noted by Vogt [21, p. 192]. An example of such a sequence space A(d) is given in [11, 2.Th] namely d = (aj,k)U,k)EN2, where k

a j• k = exp(

2: exp(v])) . n=l

Examples of weighted Frechet spaces of holomorphic functions with continuous norm and property (iJ) are given in [10]. For s> 1 we get examples by letting

F

=

{fE H(C)I sup(lf(z)/ exp(-(log(l + IzI 2) )')) zEC

<00

for all r> s}.

6. Non-Polar Bounded Sets in (FN)- and (DFN)-spaces In this section we combine the main results of the previous sections to obtain a characterization of certain nuclear Frechet spaces and of all (DFN)-spaces containing bounded non-polar subsets. Let us begin with the nuclear Frechet spaces. In this case we get new proofs of the main results of our article [6]. To state them we recall from [6] that a Frechet-Schwartz space E satisfies condition (.a) (introduced by Vogt [21, Sect. 5]) if it contains a bounded absolutely convex set A such that E has property (.oA ) (see [6, Def. 2 or Lemma 4.3(b)]). For further information and examples of (nuclear) Frechet spaces with property (Ji) we refer to [6], [11]. Theorem (6.1). ([6, Thm. 9, 10]) Let E be a nuclear Frechet space. Then the following are equivalent: (1). E contains a bounded subset which is not uniformly polar in E. (2). E has property (Ji). If every pseudoconvex domain in E is a domain of existence, then (1) and (2) are equivalent to (3). E contains a bounded non-polar subset.

316

S. Dineen et al. I Polar Subsets of l.c. Spaces

Proof. Since E is a nuclear Frechet space it satisfies the hypotheses of Corollary (4.7). Hence the equivalence of (1) and (2) follows from Corollary (4.7) and Lemma (4.3) (2). The equivalence of (1) and (3) follows from Corollary (2.4). 0 For (DFN)-spaces E we have: Theorem (6.2). For a (DFN)-space E the following assertions are equivalent: (1). E contains a bounded non-polar subset. (2). E contains a bounded set which is not uniformly polar in E. (3). E~ has (DN). (4). For every fundamental system (Bn)nEN of absolutely convex bounded subsets of E the following holds: There exists a ko E N such that for every kEN there exist an n EN, C > 0 and d > 0 such that for all r > 0

Proof. (1) and (2) are equivalent by Corollary (2.3), while (3) and (4) are equivalent by [20, 2.4]. By Corollary (5.5) (1) implies (3). In order to show that (3) implies (2) we remark that E satisfies the hypothesis of Corollary (4.7) and that (3) implies that (4.7) holds. Hence (3) implies (2). D Concluding this section we show that for (DFN)-spaces E with basis the equivalence of Theorem (6.2) (1) and Theorem (6.2) (2) can be obtained in a direct way which does not use Corollary (2.3). To see this, we first prove the following lemma. Lemma (6.3). Let A(d) be a nuclear Frechet space with a ;.1> 0 for all j E N and let a E A(d)' with a > O. Then the following conditions are

equivalent: (1). For every kEN there exist an n EN, C> 0 and E > 0 such that for all jEN:

(2). For every kEN and every b E A(d) with b ;;:. 0 there exist a C> 0 and an E > 0 such that for all j EN: a~+k£b.:::;;Ca~. J.

J

J

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317

=> (2). b E A(d) implies that sup jEN (bja j.n) = D; < 00 for every n EN. Hence we get from (1) that for every kEN there exist a C> 0 and an e > 0 such that for every j E N we have

Proof. (1)

D

a t.1+. b.";;; a ~+. _ n ,,;;; CD a: , k / I. k n / a j•n

which proves (2). (2) => (1). This is shown by contraposition. If (1) does not hold then there exists a kEN such that for all n E N and all e > 0 we have SUpjEN(aJ.~·/(ajaj.n»= 00. Without loss of generality we may assume that lim,....""(aj. klaj ) = oo and aj.klaj~I for all jEN. Then we have for 0< 8 < e, all pEN, all q E N with q > p, and all j E N a f.k 1+8

,,::::

a j.k 1+8

_

aJa j. k+q '""'aJa j. k+p -

(a ~ a j)

8 (

a j.k 1+. a i.k: ,,:;:: (a ~ )' (a ~) _ aj.k+P) '"'" a j aj.k+p - aja j. k+p·

For e > 0 and pEN we put N(e,p) = {jENI

a 1+. k • i.

a j aj.k+p

~I}.

Then Nte, p) is an infinite set and we have for 0 < 8 <

E

and p < q

N(8,.q) C N(e, q) C N(e, p).

Hence we have for all j EN; N(I/U + 1), j + 1) C N(II j, j). Consequently we can find a strictly increasing sequence (n) jEN in N with nj E N(11 j, j). Next we define the sequence bEeN by

otherwise. It is easy to check that bE A(d). Let e > 0 be arbitrary. Since n j E N (1/ j, j) we get for all j sufficiently large with E j = II j

318

S. Dineen et al. / Polar Subsets of I.e. Spaces

Since limi ....ee (a t. k/a) = 00 this shows that for every e > a we have b.a

1~£

sup~=oo, iEN

ai

and hence (2) does not hold. 0 Proposition (6.4). Let A(d) be a nuclear Frechet space and let a E A(d)~ with a » a be given. Then the sets N: and N; are non-polar in A (d)~ if and only if they are not uniformly polar in A(d)~. Proof. If ai = 0 for some j E N then N: and N; are contained in a closed hyperplane and hence they are uniformly polar. Thus we may assume a > O. This implies that A(d) has a continuous norm and hence we may assume a i.1 > a for all j EN. It follows easily from Corollary (4.8) that N: resp. N: is not uniformly polar in A(d)~ iff Lemma (6.3)(2) is satisfied, while it follows from Theorem (5.6) (similarly as in the proof of Corollary (4.7» that N: resp. N: is non-polar in A(d)~ iff Lemma (6.3)(1) is satisfied. Hence the result follows from Lemma (6.3).

References [1] K.-D. Bierstedt, RG. Meise and W.H. Summers, Kothe sets and Kothe sequence spaces, In: Functional Analysis, Holomorphy and Approximation Theory, ed. J.A. Barroso, North-Holland Math. Stud. 71 (North-Holland, Amsterdam, 1982) 27-91. [2] J.F. Colombeau, Differential calculus and holomorphy, North-Holland Math. Stud. 64 (North-Holland, Amsterdam, 1982). [3] J.F. Colombeau and J. Mujica, The Levi problem in nuclear Silva spaces, Ark. Mat. 18 (1980) 117-123. [4] S. Dineen, Complex analysis in locally convex spaces, North-Holland Math. Stud. 57 (North-Holland, Amsterdam, 1981). [5] S. Dineen, R Meise and D. Vogt, Caracterisation des espaces de Frechet nucleaires dans lesquels tous les bomes sont pluripolaires, CR. Acad. Sci. Paris 295 (1982) 385-388. [6] S. Dineen, R Meise and D. Vogt, Characterization of nuclear Frechet spaces in which every bounded set is polar, Bull. Soc. Math. France 112 (1984) 41-68. [7] C.O. Kiselman, Croissance des fonctions plurisousharmoniques en dimension infinie, Ann. Inst, Fourier 34 (1984) 155-183.

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[8] P. Lelong, Ensembles de controle de croissance pour l'analyse complexe dans les espaces de Frechet, C.R. Acad. Sci. Paris 287 (1978) 1097-1100. [9] P. Lelong, A class of Frechet spaces in which the bounded sets are Cvpolar, In: Functional Analysis, Holomorphy and Approximation Theory, ed. J.A. Barroso, North-Holland Math. Stud. 71 (North-Holland, Amsterdam, 1982) 255-272. [10] R. Meise, Sequence space representations for (DFN)-algebras of entire functions modulo closed ideals, to appear in J. Reine Angew. Math. [11] R. Meise and D. Vogt, Holomorphic functions of uniformly bounded type on nuclear Frechet spaces, to appear in Studia Math. [12] P. Noverraz, Pseudoconvexite, convexite polynomial et domains d'holomorphie en dimension infinie, North-Holland Math. Stud. 3 (North-Holland, Amsterdam, 1973). [13J P. Noverraz, Pseudoconvex completions of locally convex topological vector spaces, Math. Ann. 208 (1974) 59-69. [14] A. Pietsch, Nuclear locally convex spaces, Ergebnisse der Math. 66 (Springer Verlag, Berlin, 1972). [15] H.H. Schaefer, Topological vector spaces (Springer, Berlin, 1971). [16J M. Schottenloher, The Levi-problem for domains spread over locally convex spaces with a finite dimensional Schauder decomposition, Ann. Inst. Fourier 26 (1976) 207-237. [17] M. Schottenloher, A Cartan-Thullen theorem for domains spread over (DFM)-spaces, J. Reine Angew, Math. 345 (1983) 201-220. [18] D. Vogt, Vektorwertige Distributionen als Randverteilungen holomorpher Funktionen, Manuscripta Math. 17 (1975) 267-290. [19] D. Vogt, Charakterisierung der Unterraume von s, Math. Z. 155 (1977) 109-117. [20J D. Vogt, Charakterisierung der Unterraume eines nuklearen stabilen Potenzreihenraumes von endlichem Typ, Studia Math. 71, (1982) 251-270. [21] D. Vogt, Frechetraume, zwischen denen jede stetige lineare Abbildung beschrankt ist, J. Reine Angew. Math. 345 (1983) 182-200. [22] D. Vogt and M.J. Wagner, Charakterisierung der Ouotientenraume von s und eine Vermutung von Martineau, Studia Math. 66 (1980) 225-240.

J.A. BARROSO editor, Aspects of Mathematics and its Applications

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© Elsevier Science Publishers B.V. (1986)

SUR LES FORMES BINAIRES DONT CERTAINS COYARlANTS SONT NULS

J. DIXMIER Unioersite Paris VI, Analyse, Probabilites et Applications, U.E.R. 48, 75230 Paris-Cedex OS, France

En hommage

a Leopoldo Nachbin

Pour d = 1,2, ... , considerons les polynomes homogenes de degre d en x et y a coefficients complexes, encore appeles formes binaires de degre d. Leur ensemble V d est un espace vectoriel complexe de dimension d + 1, dans lequel Ie groups SL(2, C) opere naturellement. La representation ainsi obtenue de SL(2, C) est, a equivalence pres, l'unique repesentation lineaire irreductible de dimension d + 1. Plus intrinsequement, si West un espace vectoriel complexe de dimension 2, et si C[W]d est l'espace vectoriel des fonctions polynomiales homogenes de degre d sur W; Ie groupe SL(W) opere de rnaniere naturelle dans qW]d' Soient IE V d, g EVe' Parmi les covariants de I et g, les plus simples sont ceux qui sont bilineaires en I et g. Ce sont les combinaisons lineaires des transvectants de I et g. Pour i = 0,1,2, ... , Ie i eme transvectant (f, g)i de I et g est l'element de V d+e-2i defini par

(f,g);(x,y)=

. (d - i)! (e - i)! (a 1 aig a1 aig d'. ' a i-a i-G) a i-l a e. x y x ay x ay i-I " +G)

a1 aig 2 2 '2 ax' ay ax ay'"2

••• )

.

On a (f, g)o = Ig, (f, g)i = 0 pour i > inf{d, e}, et (g'!)i = (-l)i(f, g)i' Plus intrinsequement, on sait que Ie SL(W)-module qW]d®qW]e est somme directe de sous-modules simples, identifiables aux C [W]r avec r = d + e, d + e - 2, d + e - 4, ... , jd - e]; si IE qW]d et g E C[W]e' les transvectants de I et g sont, a des facteurs scalaires pres, les composantes de I ® g dans les sous-modules simples de C [W]d ® C [W]e' Si IE Vd , il resulte de ce qui precede que les conditions 'invariantes' les

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plus simples qu'on peut imposer it f sont les conditions (j,f); = 0 pour i = 0, 2, 4, ... (rappelons que, pour i impair, (f, f); = 0 quel que soit f). La condition (f, f)o signifie que f = O. La condition (f, /)2 = 0 signifie, comme on Ie voit aisement (cf. par exemple [4, p. 235]) que fa une racine d'ordre d ou est nulle. (Lorsque nous parlerons de racines d'une forme f(x, y), il s'agira des valeurs de x/y qui annulent cette forme; ces racines appartiennent it C U {co}.) II est presque immediat (ct. par exemple [1, p. 43, exerc.2]) que, si f E Vd et si f admet une racine d'ordre ~ d - i + 1, on a (j, f)2; = O. D'apres ce qui precede, la reciproque est vraie si i = 1. La reciproque est-elle vraie pour i > I? Pour l:so; i :so; d - 1, I'ensemble des f E Vd qui admettent une racine d'ordre ~ d - i + 1 est un cone algebrique ferrne irreductible de dimension i + 1; la reciproque envisagee ne peut done etre vraie que si la forme (j, /)2i admet au moins d - i coefficients, c'est-a-dire si 2i :so; d et 2d - 4i + 1 ~ d - i, autrement dit si d ~ 3i - 1. Le premier cas a etudier est celui ou i = 2. On doit a Gordan ([2, p. 204-212]) Ie resultat suivant: Proposition (1.1). Soit f E V d avec d ~ 5. Les conditions suivantes sont equiualentes: (1). (f, f)4 = O. (2). fa une racine d' ordre ~ d - 1, ou bien fest GL(2, C )-equivaLente La 'forme de l' octaedre' on a La 'forme de l' icosaedre' .

a

Ces formes exceptionnelles sont de degres 6 et 12. Leurs racines sont simples: ce sont les nombres complexes correspondant aux sommets d'un octaedre regulier ou d'un icosaedre regulier inscrit dans la sphere de Riemann (a l'action de GL(2, C) pres). (En fait, dans son enonce, Gordan analyse en meme temps les formes de degre 4, ce qui Ie conduit facilement a la 'forme du tetraedre") Le but de ce memoire est l'analyse du cas ou i = 3. Nous n'obtiendrons pas un resultat aussi complet que celui de Gordan. Toutefois, remarquons que si fest la forme de l'octaedre ou de l'icosaedre, on a (f, /)6 oj. 0 (ct. [3, p. 148-149]). Une consequence de la Proposition (1.1) est done: Corollaire (1.2). Soit f E V d avec d equiualentes: (1). (j, f)4 = (f, f)6 = O. (2). fa une racine d' ordre ~ d - 1.

~

5. Les conditions suivantes sont

J. Dixmier / Formes Binaires

323

Nous etablirons l'analogue suivant (d. Proposition (1.5)): Soit IE Vd avec d » 8. Les conditions suivantes sont equivalentes: (1). (f, 1)6= (f, I)s = O. (2). I a une racine d'ordre ;a. d - 2. Rappelons la methode cIassique pour exprimer que 1 a une racine d'ordre ;a. d - 2: on ecrit que les d - 2 derivees partieIIes d'ordre d - 3 10' II' ... ,ld-3 de 1 ont une racine commune; pour cela, on introduit des indeterminees AO' AI' ... , Ad-3' !-Lo, !-LI' ... ,!-Ld-3; on considere

qui sont des polynornes hornogenes en x et y de degre 3; on calcule leur resultant, qui est un polynorne de degre 6 par rapport aux coefficients de I, de degre 3 par rapport a AO"'" Ad - 3 , de degre 3 par rapport a J.to, ,J.td-3; on annule les coefficients de ce polynome en AO' ... , Ad-3' J.to, , J.td-3' d'oir (~i equations de degre 6 par rapport aux coefficients de I. La proposition ci-dessus fournit 2d - 12 + 1 + 2d - 16 + 1 = 4d - 26 equations de degre 2 par rapport aux- coefficients de f, ce qui est beaucoup plus economique. En ce qui concerne les formes verifiant seulement (1,1)6 = 0, nous n'avons que des resultats partiels. Nous verrons apparaitre encore les polyedres reguliers dans des cas exceptionnels (d. Remarques (1.7), (1.8), (1.9)). Un element de V d s'ecrit

Si d < 6, on a (1,1)6 = O. Si d » 6, on a: (1.1)

~(f,1)6 = (aoX d-6+ (d~6)alxd-7y

+ (d;6)a zx d-Syz+ ) x (a 6xd-6+ (d~6)a7xd-7y + (d;6)a sx d-Syz+ ) d-S - 6(a lx d-6+ (d~6)azxd-7 y + (d;6)a 3x yz + ) x (a 5xd-6+ (d~6)a6xd-7y + (d;6)a 7xd-Syz+ )

+ 15(azx d-6+ (d~6)a3xd-7 y + (d;6)a 4xd-Syz + ) x (a4xd-6+ (d~6)a5xd-7y + (d;6)a 6xd-Syz+ ) - 1O(a 3xd-6+ (d~6)a4xd-7 y + (d;6)a 5xd-Syz + i

=

(a Oa6- 6a la5+ 15aza4 - lOa~)xZd-IZ

+ (d - 6)(a Oa7- 5a la6+ 9a Za5- 5a 3a4)x2d- 13y + ....

324

J. Dixmier I Formes Binaires

On voit de meme que

Proposition (1.3). Soil f E V d • On suppose que f a une racine d' orde ;;. 3. Si (f, /)6 = 0, a/ors f a une racine d'ordre ;;. d - 2. C'est clair si d ~ 5. Supposons d > 6. Par action de GL(2, C), on peut supposer que, avec aa' ... , ai_I E C, f( X, Y) =

aoX d + a1x d-IY + ...

+ a,_l . Xd-i+lyi-I + Xd-iyi

avec d - i ;;. 3. Si i ~ 2, la proposition est demontree. Supposons i;;. 3 et aboutissons a une contradiction. Soit .A E C\{O}. On a

La forme (x,y)~g(X,y)=.Ai-df(Ax,y) verifie (g,g)6=O. Quand .A~O, on voit a la limite que la forme (x, y) ~ h (x, y) = xd-iy i verifie (h, h)6 = O. Rappelons que i > 3 et d - i ;;. 3. lei et dans la suite, ecrivons A - B si A = pB avec p E C\{O}. On a (h, h)6 = AX2d-2i-6li-6 avec A - (d - i)(d - i - 1)(d - i - 2) (d - i - 3)(d - i - 4)(d - i - 5)

x i(i - 1)(i - 2)(i - 3)(i - 4)(i - 5) - 6(d - i)(d - i - 1)(d - i - 2)(d - i - 3)(d - i - 4)

x i(d - i)i(i - 1)(i - 2)(i - 3)(i - 4)

+ 15(d -

i)(d - i - 1-)(d - i - 2)(d - i - 3)i(i - 1)

x (d - i)(d - i - l)i(i - 1)(i - 2)(i - 3) - lO«d - i)(d - i - 1)(d - i - 2)i(i - 1)(i - 2»2 - (d - i - 3)(d - i - 4)(d - i - 5)(i - 3)(i - 4)(i - 5) - 6(d - i)(d - i - 3)(d - i - 4)i(i - 3)(i - 4)

+ 15(d - i)(d - i - 1)(d - i - 3)i(i - 1)(i - 3) - lO(d - i)(d - i - 1)(d - i - 2)i(i - 1)(i - 2).

J. Dixmier / Formes Binaires

325

Cette derniere expression est un polynome en d de degre :s;; 3; elle est egale a - 6O(d - 3)(d - 4)(d - 5) . En effet, iI suffit de verifier l'egalite pouf 4 valeurs distinctes de d; or c'est facile pour d = i, i + 1, i + 3, i + 4. Ainsi, (h, h)6 ¥- 0, ce qui est impossible. Proposition (1.4). Soit f E V d avec d » 7. On suppose que f a une racine

d' ordre 2. Les conditions suivantes sont equioalerues: (1). (f, /)6 = 0. (2). fa une racine d' ordre ;;:: d - 2, ou bien fest GL(2, C)-equivalente l' une des formes suivantes:

a

(a E C).

(II est clair que ces formes exceptionnelles n'ont pas de racine d'ordre ;;::d-2.) L'implication (2) =? (1) resulte de ce qu'on a dit dans l'introduction et de calculs directs faciles (ou bien d. la fin de la demonstration). Supposons (f, /)6 = 0, et prouvons (2). Soit f = L :~O (~)a;xd-i/. Puisque f a une racine double, on peut supposer, par action de SL(2, C), que a o = at = a 3 = 0, a z = 1. D'apres (1.1), on a aIors a 4 = as = 0. Supposons preuve que (i;;:: 2).

Le coefficient de x 2d-t2-i / C-lineaire de

dans (f'/)6 est, d'apres (1.1), combinaison

(1.2)

Comme ao = a,

= a3 = a 4 = as = ... = ai +3 = 0, il nous suffit de chercher

J. Dixmier I Formes Binaires

326

les coefficients des termes en a Zai+4' On trouve:

(d - 6)(d - 7) (d - 6)' (d - 6)! 2 (i - 2)!(d - i - 4)! - 6(d - 6) (i - 1)!(d - i - 5)!

+ 15

(d-6)! ;!(d -; - 6)!

- (d - 6)(d -7);(; - 1)- 12(d - 6);(d - i - 4)

+ 3O(d
== 8(d - 6)(d - 10) ,

-60d + 420

== -60(d -7),

-6d 2+ 6d + 360

== -6(d 2 - d - 60),

-lOd z+ 70d + 300

== -lO(d + 3)(d -10),

-12d z + 132d + 240 == -12(d 2 - lld - 20), -12d z+ 192d + 180 == -12(d 2-16d -15), -lOd z + 250d + 120 == -10(d 2 - 25d - 12), -6d 2+ 306d + 60

== -6(d2 - SId -10).

Comme d est entier ;;= 7, cela ne peut etre nul que pour d == 7 ou d == 10. Si d == 7,on aao == a I == a 3 == a 4 == a s == a 6 == 0, a z == l,doncf== G)xsl+ a7y7. Si a7 == 0, f a une racine d'ordre 5 == 7 - 2. Si a7 ¥- 0, fest GL(2, C)equivalente a XSy2 + y7. Supposons desormais d > 8, et laissons de cote pour l'instant le cas d == 10. II nous reste a etudier (1.3) pour i > 10. Posons i - 10 == Ad au 0.,;; A < 1. Alors (1.3)

== dZ(Ad + 7)Ad + d(-A Zd2 - 20Ad - 100 + 73Ad + 730 - 270)- 60Ad == Azd 4+ (7A - A2)d3+ 53Ad 2+ (360 - 6OA)d > O.

Par recurrence, on a done aj == 0 pour j;;= 3, et d-2.

f a une racine d'ordre

J. Dixmier / Formes Binaires

Supposons d

=

10. Alors les coefficients de (j, /)6 sont,

327

a des

facteurs

:I- 0 pres:

a Oa 6 - 6a 1aS + 15a 2a4 - lOa;, a Oa7 - 5a 1a6 + 9a 2aS - 5a3a4 , 3aoas - lOa 1a7 + 42a3aS - 35a~, aOa g - 15a 2a7 + 35a3a6 - 2Ia 4a s , aOalO + IOa1a9 - 45a 2aS + 20a 3a7 + 140a 4a6 - 126a;, a1a lO - 15a 3a8 + 35a4a7 - 2Ia sa 6 , 3a 2a lO - lOa 3a g + 42a sa 7 - 35a~ , a 3a lO - 5a 4a9 + 9a Sa8 - 5a 6a 7 , a 4a lO - 6a Sa g + 15a 6a s - IOa~ .

L'annulation de ces coefficients, avec les conditions ao= a1 = a3 = 0, a2 = 1, equivaut a:

D'ou

Si a 6 = 0, fest GL(2, C)-equivalente a x 8 l + xyg, ou bien a une racine d'ordre 8. Si a 6 :1- 0, fest GL(2, Cj-equivalente a C~)xsl+ (~)x4l+ lOagxyg + ¥ylo, d'ou la proposition. Proposition (1.5). Soit f E V d avec d ~ 8. Les conditions suivantes sont

equioaierues: (1). (f, /)6 = (f, /)8 = O.

(2). fa une racine d' ordre ~ d - 2.

Nous divisons la demonstration en plusieurs parties. Soit d

f = 2: (:)aix d - i/ . i=O

(A). L'implication (2)::;' (1) resulte de l'introduction.

J. Dixmier I Formes Binaires

328

Supposons (f, f\ = (f, /)8 = 0, et prouvons (2). Si f a une racine d'ordre ;;;.: 3, cela resulte de la Proposition (1.3). Si fa une racine double, on peut supposer, d'apres la Proposition (1.4), que f = X8 y 2 + xy9 ou f = 135x8l + 630X 4 y6 + axy9 + 35y lO. Les deux cas sont compris, a l'action de GL(2, C) pres, dans le cas

(ct. demonstration de la Proposition (1.4»). Alors

d'ou a 6 = a9 = 0, et f a une racine d'ordre 8 = d - 2. (B). Supposons desormais que f admette une racine simple. On peut supposer, par action de SL(2, C), que a o = a 2 = 0, at = 1. Alors, d'apres (1.1), (1.4) Le coefficient de x 2d - t4y 2 dans ~(f,f)6 est (d;6)aoas + (d - 6ia\a 7 + (d;6)a2a6 - 6«d;6)a\a 7 + (d - 6ia 2a6+ (d;6)a3aS )

+ 15«d;6)a 2a6 + (d -

6ia3aS + (d;6)a~) - 10(2(d;6)a3aS + (d - 6)2a~),

ou, comme a o = a 2 = 0 et a\ = 1, «d - 6)2 - 6(d;6»a 7 + (15(d - 6)2 - 26(d;6»a 3aS + (-lO(d - 6)2 + 15(d;6)a~ - (d - 6- 3(d -7»a7 + (15(d - 6)-13(d -7)a3aS

+ (-lO(d - 6) + ¥-(d -7»a~

= (-2d + 15)a, + (2d + l)a 3a s + ~(- 5d + 15)a~. Done, compte tenu de (1.4), (1.5)

S

a = -7

3

2d + 1 3 5 d - 3 2 a a 2d _ 15 3 22d _ 15 4'

J. Dixmier / Formes Binaires

329

(C). Comme (f,f)g = 0, on a, d'apres (1.1), 0= aOag - 8a la? + 28a 2a6 - 56a 3aS + 35a~ = -8a? - 56a 3aS + 35a~ 40

="3

2d + 1 3 d - 3 2 280 3 2 a 3+20 a 4+Ta 3+35a 4 2d -15 2d-15

soit, apres calcuIs, 5(2d -13) 3 2 3(2d _ 15) (64a 3 + 27a 4 )

·

Done (1.6) (D). Supposons d ~ 10, 15, 30. Nous allons montrer que Ies coefficients ag, a 9 , ••• de f sont, comme as, a 6 , a?, determines de maniere unique par a 3 , a 4 et Ia condition (I, /)6 = O. Supposons cela preuve jusqu'a ai+4' Le

coefficient de X2d-12-iyi dans (I. /)6 est une combinaison lineaire de produits apk donnes par Ie tableau (1.2). Comme a o = a 2 = 0, a i +S sera determine de maniere unique pourvu que Ie coefficient de alai +S soit non nul. Ce coefficient est: d-6

(d - 6)(;_1) - 6(

d - 6- i + 1. . . - id - 6z - 6d + 36 + 6i - 6 z = id-6d+30.

d-6 i ) -

d - 6- 6

Cela est >0 pour i ~ 6. Le cas i = 2 a deja ete examine. Si i = 3, 4, 5, on a id - 6d + 30 = -3d + 30, -2d + 30, -d + 30. Or on a suppose d ~ 10, 15, 30, d'ou notre assertion. Soit a E C, et posons: g(x, y) = dy(x - (d - 2)ay)(x + ay)d-2 = d(xy - (d - 2)al)

J. Dixmier / Formes Binaires

330

= dXd-1y + d(d - 2)ax d-2y2 + ~d(d - 2)(d - 3)a 2xd-3 y 3 + ~d(d - 2)(d - 3)(d - 4)a Jxd-4y4 + ... - d(d - 2)ax d-2/- d(d- 2?a 2xd-Jy 3 - ~d(d - 2)2(d - 3)a Jxd-4l - ...

= dXd-1y -

~d(d - 1)(d- 2)a 2xd-3

l

- jd(d -1)(d - 2)(d - 3)a 3x d-4y4_ ... = (~)Xd-Iy - (~)3a2xd-Jl- (~)8aJxd-4y4 - ....

D'apres (1.6), on peut choisir a de telle sorte que - 3a 2 = a 3 , -8a 3 = a4. Comme (g, g)6 = 0 d'apres I'introduction, la propriete d'unicite vue plus haut prouve que g = f. Done / admet une racine d'ordre ~ d - 2. (E). Supposons d = 10. On a encore les formules (1.4), et (1.5) donne a7 = -7a~-~7a~. Ici, I'annulation du coefficient de x 5l dans (f'/)6 ne permet pas de calculer ag. Mais, si I'on annule Ie coefficient de x 3 y dans (f, /)g, on trouve:

done ag = ~a~a4' Les coefficients a9, alO sont determines de maniere unique par I'annulation de (f, /)6' et l'on peut raisonner comme dans (D). (F). Supposons d = 15. On a encore les formules (1.4), et (1.5) donne a7 = -~a;-2a;. L'annulation du terme en Xl5l dans (f,/)6 donne a g en fonction de a J, a 4. L'annulation du terme en Xl4l dans (f'/)6 ne permet pas de calculer a g • Mais si I'on annule Ie terme en X 12 / dans (f,/)g, on trouve:

d'ou a 9 en fonction de a J , a4 • Les coefficients alO' all' ... ,a 15 sont determines par I'annulation de (/'/)6' et I'on peut raisonner comme dans (D). (G). Supposons d = 30. L'annulation de (f, /)6 donne d'abord a 5 , a 6 , a7 , ag, a g en fonction de a3 , a 4. L'annulation du terme en X41y 3 dans (f, /)g donne une relation de la forme

331

J. Dixmier f Formes Binaires

ou a,

13, ... , (E Z,

avec en particulier

13 =

22(Z;) - 8(2J) ~ 0 ,

d'ou a lO en fonction de a 3, a 4. Les coefficients all' a 1Z' ••• ,a30 sont determines par l'annulation de (f, /)6' et I'on peut raisonner comme dans (D). Soit N d (resp. N~) l'ensemble des f E Vd telles que (f, /)6 = 0 (resp, admettant une racine d'ordre ;:a.: d - 2). Alors N~ eNd' et N~ est un cone ferme irreductible de dimension 4. Proposition (1.6). Soit d > 7. On a dim N d

~

6. Et dim N d

= 6<:> d = 10.

Soit f E Yd' Soit g Ie polynome x ~ f(x, 1). Alors la condition (f, /)6 traduit par l'equation differentielle:

=

0 se

d(d - 1)(d - 2)gg(6l_ 6(d - 1)(d - 2)(d - 5)g'g(Sl

+ 15(d -

2)(d - 4)(d - 5)g"g(4l - lO(d - 3)(d - 4)(d - 5)(g"'i = 0

(cela resulte facilement de [1, p. 201, no. 160]). D'ou dim N d ~ 6. Supposons dim N d = 6. Soit M d l'ensemble des f E Vd admettant une racine multiple; c'est une hypersurface dans Yd' Done dim(Md n Nd);:a.: 5 et par suite M d n N d rt N~. D'apres les Propositions (1.3) et (1.4), on a d= lO. Supposons d = lO. On a dim(Md n N d ) = 5 d'apres les Propositions (1.3) et (1.4). Les indications donnees dans la demonstration de la Proposition (1.4) permettent de trouver facilement toutes les fENd qui admettent une racine simple. On trouve que ce sont les formes GL(2, Cj-equivalentes aux formes lOx 9y + C~)a3x7l + C~)a4x6y4- CsoHa;xsl- (~C)a3a4x4l -(~)(7a~+~a~)x3y7+ C~asxzl+ lO(21a3a~+ 49a~)xy9

+ (15a 3aS + 280a;a 4+ ~s a~)ylO

332

J. Dixmier / Formes Binaires

ou a3 , a4 , as E C. Soit fo une telle forme, avec a3 oJ O. Soit H I'ensemble des 'Y E GL(2, C) tels que 'Y' 10 soit I'une des formes precedentes, Alors I'ensemble des vecteurs tangents a H en e est un sous-espaee de dimension 1 de I'ensemble des matrices diagonales. Done dim(H' 10 ) = 3 + (4 - 1) = 6. On raisonne de meme si a3 = 0, a4 oJ 0, puis si a 3 = a4 = 0, as oJ O. Done dim N d = 6. 0 Concernant la comparaison de N d et N~ pour d seulement faire les remarques suivantes:

;:?;

8 et d oJ 10, je peux

Remarque (1.7). Soit G C SL(2, C) Ie groupe de l'octaedre, L'algebre des fonctions polynomiales G-invariantes est engendree par 3 polynomes homogenes ({)S' ({)12' ({)IS de degres 8, 12. 18 (d. par exemple [5, 4.5.4]). Alors «({)S, ({)S)6 est de degre 4 et G-invariant, done nul. Pourtant, les racines de ({)s' qui correspondent aux sommets d'un cube inscrit dans la sphere de Riemann, sont simples. Un calcul facile montre que N s est la reunion de N~ et du cone N~ constitue par les formes GL(2, C)equivalentes a la forme ({)s ci-dessus. On a dim N~ = 4, et N s est reductible. Remarque (1.8). Le merne raisonnement, applique au groupe de l'icosaedre, prouve qu'il existe une forme binaire ({)20 de degre 20 telle que «({)20' ({)20)6 = 0, et dont les raeines sont simples: elles correspondent aux sommets d'un dodecaedre regulier inserit dans la sphere de Riemann. Remarque (1.9). Des calculs directs assez penibles montrent que d = 9, 11, 12, 13, 14, 15, on a N d = N~.

Sl

Conjecture (1.10). Soit f E Vd • (a). Si (f, /)4; = (f, /)4;+2 = (f, /)4i+4 = ... = (f,/)6; = 0 (et si d » 6i - 1), alors 1 a une racine d'ordre ;:?; d - 2i + 1. (Le cas i = 1 est celui du Corollaire (1.2).) (b). Si (f, /)4i+2 = (f, /)4i+4 = (f, /)4;+6 = ... = (f, /)6i+2 = 0 (et si d » 6i + 2), 1 a une racine d'ordre ;:?; d - 2i. (Le cas i = 1 est eelui de la Proposition (1.5).) A ce sujet, notons que, d'apres [3, p. 149], la forme

1 de

l'icosaedre

333

J. Dixmier / Formes Binaires

(forme qui est de degre 12) verifie (f, /)8 = (f, /)10 = 0, mais n'a que des racines simples. A defaut de prouver la Conjecture (1.10), on peut prouver ceci: Proposition (1.11). Soient f E V d et i un entier tel que i > 1, 4i conditions suivantes sont equicalentes:

~

d. Les

(1). fa une racine d' ordre ;;. d - i + 1. (2). (j, /)u = (f, /)U+2 = (f, /)2i+4 = ... = En particulier, le cone des formes binaires de degre d qui admettent une racine d' ordre ;;. d - i + 1 peut etre defini par des equations quadratiques dans C d +1•

°.

(1) ~ (2) resulte de l'introduction. Faisons l'hypothese (2). Pour construire explicitement un nombre fini de covariants engendrant l'algebre des covariants de f, on a la methode de Gordan (cf. par exemple [4, p. 118-120]). On considere successivement les formes (j,J)2' (j, /)4' (f, /)6' ... , (f, /)2j, ... , et, a chaque etape, on construit certains transvectants. Si 2j < ~ d, l'etape correspondante fournit des covariants qui ne sont jamais des invariants ([4, p. 126]). Les invariants apparaissent seulement aux etapes suivantes. Comme, sous nos presentes hypotheses, (j, f)2j = 0 pour 2j ;;. ~ d, on voit que f est instable. Par action de SL(2, C), on peut done supposer que (si f ¥- 0):

f

= aox d + (~)alxd-ly

+ ... + (~)a.xd-eye ,

avec 2e < d, ae ¥- 0. Si e ;;. i, on a: 0= !(f f) = (a 0 Xd-2e + (d-2e)a Xd-2e-Iy + ... + ae yd-2e) 2 '2e I I d2 X (a 2ox d-2e + (d~2e)a2o+IXd-2e+ly

_ (2Ie)(a I Xd-2o

+

+ (d-2e)a Xd-2e-Iy + I 2

+ adyd-2o)

+ ad-2e+!y d-2e)

x (a 20-1 x d-2e + (d-2e)a Xd-2o+!y + ... + ad-I yd-2e) I 20

+ ... + (_I)e !(2e)(a x d-2o + (d-2e)ae+ I x d-2o- Iy + ... + adyd-2o) 2 e e l -e' On a ae +! = ae +2 = ... = a 20 = 0, et l'annulation du terme en x 2d - 4e donne a, = 0, ce qui est contradictoire. Done e ~ i-I, et par suite f a une racine d'ordre d - e ;;. d - i + 1.

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Bibliographie [1] E.B. Elliott, An introduction to the algebra of quantics, 2nd ed. (Oxford Univ. Press, Oxford, 1913). [2] P. Gordan, Vorlesungen iiber Invariantentheorie (Teubner, Leipzig, 1885). [3] P. Gordan, Biniire Formen mit verschwindenden Covarianten, Math. Ann. 12 (1877) 147-166. [4] J.H. Grace and A. Young, The algebra of invariants (Cambridge Univ. Press, Cambridge, 1903). [5] T.A. Springer, Invariant theory, Lecture Notes in Math. 585 (Springer, Berlin, 1977).

l.A. BARROSO editor, Aspects of Mathematics and its Applications © Elsevier Science Publishers B.V. (1986)

335

INFINITE-DIMENSIONAL BILINEAR REALIZATIONS OF NONLINEAR DYNAMICAL SYSTEMS

Thomas A.W. DWYER,

m

Department of Aeronautical and Astronautical Engineering, University of Illinois, Urbana, /L 6/801, U.S.A.

Dedicated to Leopoldo Nachbin in commemoration of his sixtieth birthday

O. Preface

Dynamical systems representing a large number of physical phenomena are modelled by evolution equations where the state of the system enters analytically, but the control enters linearly (or rather affinely). Such systems have been called 'linear-analytic'. When the state also evolves linearly and the control enters additively, a large body of results on input-output properties has been developed in the 1960's. The intermediate case of systems where the dynamics are separately linear in the state and (affine) in the control was studied in the 1970's, in particular with regard to recursively defined analytic input-output representations. Such systems have been called 'bilinear'. It has been found as a result of work at the end of the decade that all linear analytic systems have equivalent bilinear realizations, wherein the control is unchanged but the new state space is defined to consist of analytic functions defined on the original state space or its dual. It follows that the study of systems evolving analytically and with inputs entering affinely is subsumed in the study of infinite-dimensional bilinear systems. In fact, even if the original state space is infinite-dimensional (e.g. stochastic systems) the construction of bilinear realizations still goes through. In particular, recursive formulas to generate the Volterra kernels of the input-output map of a linear-analytic system, even if evolving in a Banach space, and convergence bounds thereof, can be generated from the corresponding Volterra series for the associated bilinear realization. The approach described above has been dormant since the end of the 70's, but it is expected that the renewed interest in general properties of linear-analytic systems, arising from robotics and spacecraft maneuvers,

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will warrant a deeper study of bilinear realization theory. The present article is therefore intended to provide an overview of the state of knowledge in the field, from the viewpoint of the Nachbin school of holomorphy, as well as to provide suggestions for further research along the same lines. Two other approaches, albeit developed only for finite-dimensional nonlinear systems, deserve the attention of the reader: one is the technique of associating iterated integrals of input signals with indeterminates of non-commutative formal series ala Schiitzenberger, whereby the structure of systems with bilinear realizations has been found by M. Fliess, in [63] and elsewhere. The other is the technqiue of constructing bilinear realizations from Laplace-transformed Volterra series, developed by A. Frazho and discussed in Chapter 4 of W.J. Rugh's book [64]. The flavour of these approaches is completely different from the one discussed here, and they are therefore omitted.

I. Introduction

This investigation is concerned with (linear) function space realizations of a dynamical system characterized by a state vector XI governed by the state evolution equation (1.1)

where 'PI is for each t an analytic vector field on (a manifold modelled on) a real or complex, finite- or infinite-dimensional vector space E (one may take the manifold to be a subset of E to concentrate on the local study). A function space realization is obtained by considering the function ~ ~ exp (XI' ~>, where ~ lies in the dual space E' and (z, D = ~(x), as a new 'state vector' for the system, now regarded as evolving in a space of entire functions defined on E', in the sense of Nachbin [47]. It turns out that this new state vector = state function is governed by the linear state equation (1.2)

with the initial condition fT(g) = exp (x T, g> if the state of the system

Th.A. W Dwyer III/Infinite-Dimensional Bilinear Realizations

337

realization (1.1) at t = T is xT : here l{Jt (atat) denotes the E-valued hyperdifferential operator obtained when the coordinates of x in l{Jt(x) for any basis in E are replaced by the corresponding partial derivative operators in E' for a dual basis. Moreover, if 'P is a scalar-valued analytic function on the domain of l{Jt and

y(t) = 'P(xt)

(1.3)

is a corresponding (nonlinear) output measurement of the state in the realization (1.1) then the same output can be obtained from the realization (1.2) via the (linear) output equation (1.4) A second functional realization is given by the solution 'P at time". of the 'backward' gradient equation T

(1.5) with the 'initial' state 'Pt(x) = 'P(x) (from (1.3» at T = t: that is, the output can also be obtained from the realization (1.5) via the output equation (1.6) Here 'P is analytic on a domain in E depending on the radius of convergence of the Taylor series of l{Jt and on It- TI, to be specified below. The realization (1.2), (1.4) is related to recent work by BaiIIieul [1], Brockett [9], [10], [11] and Krener [44], who consider instead the sequence of tensor powers x, 0 ... 0 x, of the state vector Xt' evolving in the tensor algebra of E (cf. [34]). The realization (1.5), (1.6) is related to the theory of characteristics of first order partial differential equations (Hamilton-Jacobi Theory) (cf. [13], [61, Chap. 19]). When E is finitedimensional the relationship between (1.1), (1.3) and (1.5), (1.6) is well known, although not normally derived from (1.2), (1.4) as done here (d. [4, pp. 167-168]). The present approach links (1.2), (1.4) and (1.5), (1.6) as 'Fourier-Borel duals' of each other, in the sense of Nachbin [47], [48], T

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Th.A. W. Dwyer III/Infinite-Dimensional Bilinear Realizations

[49], [50], [51], [52], Dineen [15], [16], Dwyer [29], [31] and Gupta [39], [40], [41]. Moreover, the treatment is rendered coordinate-free and infinite-dimensional by a natural extension to variable coefficients, carried out in [34], [35], [36], of the differential operators of infinite order on infinite-dimensional domains introduced in [30], [32], [33]. These results are summarized in the next three sections, and new approaches outlined in the remaining sections.

2. Fourier-Borel Duality and Bilinear Realizations of Control Systems The work with this title is found in [34]. Here the state space E is supposed to be a complex Banach space, and the vector field tfJ, of the form tfJ, = tfJI + U(t)tfJ2' with tfJl and tfJ2 holomorphic on a disc with radius a centered at the initial state of (1.1), and u(t) a bounded continuous scalar control. The output map cp in (1.3) is supposed to be holomorphic on the same disc in E. By use of the Cauchy estimates from [47], growth estimates of 'Ovcyannikov type' in the sense of Treves [60], [61] were obtained for the operator in (1.5), which leads to the following result: Theorem (2.1). The equation (1.5) has a unique eel solution CP7' holomorphic on Ilx - x 7 11 < p if cP, = cp and IT - tl is bounded by (a - p )/C e, where C is an upper bound for ItfJ,(x)! for Ilx - x 11 < a. Moreover, CP7 depends continuously on cpo 7

By observing that the spaces EXPN,p(E'), of entire functions f on E' with nuclear Frechet derivatives f(nl(o) (regarded as polynomials) such that IlfllN,p = sup, (lip tllt< n l(O)/IN is finite (nuclear norms in the sense of Gupta [39], [40], [41], Matos [45], [46], Nachbin [48], [49], [50], [51], [52] and Boland [5], [6]) have their duals imbedded in spaces of holomorphic functions bounded on the discs IIxll < p via Fourier-Borel transformation, it is possible to use the 'Fredholm alternative' type of argument of Gelfand-Shilov [38, p. 33], to get: Theorem (2.2). The equation (1.2) has a unique eel solution I, in Exp N, o (E') if fo = exp (x, -), Ilxll:s; p < a and Itl is bounded as in the case of equation (1.5). By setting x = Xo (the initial state of the system realization (1.1», using

Th.A. J.v. Dwyer III/Infinite-Dimensional Bilinear Realizations

339

Fourier-Borel duality and a 'reproducing kernel' property of exp (x, .) with respect to this duality a 'Duhamel principle' ('variation of constants') approach yields the following: Theorem (2.3). The input-output map u ~ y has a convergent Volterra series expansion for /tl bounded by (a - p )/2C e, if Ilull:o;; Rand C = sup{ll~l(x)1I + RII~2(x)lllllxll < a}, with kernels given by iterations of the exponential of the operator ta/a~l(x) following a/a~2(X), applied to