A Panorama of Hungarian Mathematics in the Twentieth Century, I (Bolyai Society Mathematical Studies Volume 14)

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finite fact or represe ntations, as later st udied by Elm ar T homa {25}. An interesting fact about FC- groups is th at if th ey are finitely generated , then t hey are virtually abelian gro ups, t ha t is, have an abelian subgro up of finit e ind ex. It was eventually shown by Thoma {26} t hat t he virtua lly abelian groups are pr ecisely the class of typ e I groups, discrete groups whose uni tary representati ons always genera te ty pe I von Neum ann algebras, and in fact the dim ensions of their irredu cible representat ions are bounded . Hence for finitel y generated FC- groups , non-commut ative harmonic analysis in t he sense of Haar is precisely t he decomp ositi on of functi ons into mat rix coefficients of irredu cible representat ions, as in t he situation of t he PeterWeyl Theorem.

2.

SZ.-NAGY A N D P UKAN SZKY

2.1. Bela Sz.-Nagy Bela Sz.-Nagy, one of t he great operator t heorists of t he twent iet h century, made a few interestin g cont ribut ions to non-commu tative harmonic ana lysis, even t hough this was not his primary math emati cal interest .P Here we will just mention four of t hem. The first is a st rengthening of von Neumann's auto matic ana lyticity th eorem for homomorphisms of Lie groups (see Sect ion 1.1) : Nagy {19} proves that measurability is enough to guarantee ana lyticity; one does not need to assume cont inuity from t he start. T he second contribution, on its face, only dea ls with commutative harmonic analysis. In {20}, Nagy pr oved t hat a one-parameter gro up {Tn}nEZ or {Ts } sEIR of invertible linear operators on a Hilb ert space H is similar to a unitary representation (of Z or lR) if and only if it is uniformly bounded, i.e. , th ere is a const ant K > 0 such t hat IITn ll ::; K for all n E Z or I Ts li ::; K for all s E R The connection wit h unit ary representati on t heory is t hat essentially t he sa me t heorem , wit h almost t he same proof, holds for un iformly bound ed represent at ions 1[' of arbitrary amenable locally compact groups G, as was pointed out by J acques Dixmier {4}. In ot her word s, if 1[' is a strongly continuous homomorphism from G to t he inverti ble linear

A.

9T he bulk of Sz.-Nagy 's work is discussed in the companion art icle in this volume by Csaszar and D. Petz.

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operators on H, and if 1111" (g) II ~ K for all 9 E G, then

(~, 17)' =

L(11"(g)~,

11" (9)17)

dg,

Ie

with denoting an invariant mean on G, defines a new inner product on the Hilbert space, equivalent to the original one, with respect to which the representation 11" is unitary. Incidentally, the requirement of amenability is essential here; the group SL(2, IR) was shown in {7} to have uniformly bounded representations that are not unitarizable. The third concerns the following problem. Suppose H is a Hilbert space and a : G - t l3(H) is a map from a discrete group to bounded operators on H. What is the necessary and sufficient condition for a to be the compression of a unitary representation, or in other words, for there to be a Hilbert space H' ~ H and a unitary representation 11" of G on H' such that if P : H' - t H is the orthogonal projection, then (j(g) = P11"(g)P for all 9 E G? Nagy solved this problem in {21}. This can be viewed as an operator-valued analogue of the characterization of matrix coefficients 9 t---t (11"(g)~,~) of unitary representations as the positive-definite functions, which follows from the Gelfand-Naimark-Segal construction. The fourth problem studied by Nagy (in joint work with Ciprian Foias and Laszlo Ceher {8} and in {22}) can be viewed as a postscript to von Neumann's work (cited above in Section 1.3) on uniqueness of representations of the canonical commutation relations (1). Recall that the method of Weyl and von Neumann was to study unitary representations of the Heisenberg Lie group, not representations by unbounded operators of the original relations, which are more numerous and which pose difficult analytic questions . However, Nagy was able to give necessary and sufficient conditions for a representation of the CCR to come from a representation of the group : an irreducible pair of closed symmetric operators Q and P on a Hilbert space H is a "Schrodinger couple" if and only if QP - PQ = if holds on a subspace 'D c 'DQ P - PQ which is large enough so that the restrictions of Q and P to 'D are essentially selfadjoint and that at least one of the eight sets (Q ± iI)(P ± if)'D, (P ± iI)(Q ± -iI)'D is dense in H . 2.2. Lajos Pukanszky

Lajos Pukanszky, who was born in Budapest in 1928, studied with Sz.Nagy in Szeged. His earliest papers (dating from 1951 through 1960) deal

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with von Neumann algebras; all his subsequent publications were on t he uni t ary representation theory of Lie groups . While it was von Neumann who first ant icipated t he importan ce of the t heory of rin gs of ope rators (i.e., von Neumann algebras) to non- commut ati ve harmonic ana lysis, it was Pukan szky who finally achieved a dee p synt hesis of t hese two subjects. Again , we have no roo m here to go into det ails, some of which may be found in {6} and {5}. T he reader interested in Pukanszky's work could also see his posthumous monograph [139J . The subject of this book, ind eed of mu ch of Pukan szky's work, concerns t he following question. For a ty pe I Lie group G , t here is a bijective correspondence between G, the set of uni tar y equivalence classes of irredu cible uni t ar y represent ations 7r, and t he set of "characters" of G, t he generalized functions 9 t--t Tr 7r (g). W hat is t he substit ute for this bijection in t he case of non-typ e I Lie groups? The ans wer , which is qui te rem arkabl e, is t hat for non-typ e I Lie groups, ~

n

one needs to repl ace G by G , t he set of quasi-eq uivalence classes-'' of "normal" representations. T hese are factor representations tt of G of types I or II , for which t he (usua lly unb ounded ) trace Tr on t he facto r 7r( G)" is finite and non-z ero on some ideal in C* (G ), the C*-complet ion of t he convolut ion algebra L1(G). On a type I group, there is no difference between n

~

G and G. But on non- typ e I connected Lie gro ups , Pukan szky showed n

t hat G is big enough to sup port t he canonical (i.e., cent ra l) direct integral decomp osit ion of t he left regular representation of G on L 2 (G). T hus harmonic analysis of L 2 functions on the group som etimes f orces one to consi der normal representations that are not irreduci ble, and the left regul ar representati on of G on L 2 (G) generates a von Neuman n algebra with no ty pe III summand. (T his last fact was proved t hrough t he joint efforts of Pukan szky and Dixmier. ) Not only t his, but every primit ive ideal of C *( G) (t hat is, t he kern el of a n irr educible represent ati on of this algebra) is t he kernel of a uni que qu asi-equivalence class of norm al representations, n

so t hat there is a natural bijection between G and Prim C *(G). Finally, in a stunning gene ralization of Alexander Kirill ov's cha racter formula {12}, Pukan szky [139J was ab le to give a geometric par am etrization of t he nor mal represent ations and a formula for t heir characters, at least for connect ed solvable Lie groups . IOTwo factor rep resenta ti ons 11"1 and 11"2 of G a re ca lled quasi-equivalent if t he re is an isom orphism : 11" 1 (G) /I -+ 11"2 (G )" such t hat 71"2 = 0 71"1 . T his is the na t ur al eq uiva lence rela t ion on factor represen tati ons. The d ifferen ce between t his and unit ary equivalence is tha t need not be given by conj ugation by a un itary op erator.

Non-Commutative Harmoni c Ana lysis

207

REFERENCES

[62]

Haa r, Alfred , Oss zegyujtott Munkrii = Gesammelte Arb eiten, ed . Bela Sz.-Nagy , Akademiai Kiado (Budapest, 1959).

[118J Neumann, Janos (Jo hn von Neumann), Collected Works , 6 volumes, ed. A. H. Ta ub , Pergamon Press (New York, 1961). Of int erest to this sur vey are Vol. I: Logic, theory of sets and quantum mechanics ; Vol. II: Operators, ergodic th eory and almost periodi c functions in a group ; Vol. III : Rings of operators; and Vol. IV : Continuous geomet ry and other t opics. [139] Pukanszky, Lajos, Characte rs of Connected Li e Group s, With a preface by J . Dixmier and M. Duflo, Mathematical Surveys and Monographs , Volume 71, American Mathematical Society (Providence, R1, 1999). [199J

Wigner , J eno (E .P.), Collect ed Works , ed. A. Wightman, Springer-Verlag (BerlinNew York, 1993).

{I}

V. Bargmann , Irr educible unitary representations of th e Lorent z group , Ann. of Math . (2) , 48 (1947), 568-640.

{2} H. Bohr , Zur Theor ie der Fastperiodischen Funktionen, I, Acta Math. , 45 (1925), 29-127; II , ibid., 46 (1925), 101-214. {3} H. Cartan, Sur la mesur e de Haar, C.R . A cad. S ci. Paris, 211 (1940), 759-762. {4} J . Dixmier , Les moyenn es invariantes dans les semi-groups et leur s app lications , Acta S ci. Math . Szeged, 1 2 (1950), "Leopoldo Fejer et Frederico Riesz LXX annos natis dedicatus," Pars A, 213-227. {5} J. Dixmier , M. DuRo, A. Hajnal and A. Koran yi, Pukanszky Lajos (1928-1996 ): About his mathematical work: a personal commemora t ion (Hungarian), Mat . Lapok (N.S.) 4 , no . 2-3 (1994), 71-81 (1998). {6} J . Dixmier , M. DuRo, A. Hajnal, R. Kadi son , A. Koranyi, J . Rosenb erg and M. Vergne, Lajos Pukanszky (1928-1 996), N otices A m er. Math . Soc., 45 , no . 4 (1998), 492-499. {7} L. Ehrenpreis and F . I. Mautner, Uniformly bound ed representations of groups , Proc, Nat . A cad. Sci. U.S. A ., 41 (1955) , 231-233 . {8} C. Foias, L. Geher and B. Sz.-Nagy, On the permutability condition of quantum mechanics , Acta S ci. Math . (Sz eged), 21 (1960), 78- 89. {9} F . G. Frob enius, Gesammelte Abhandlungen , J .-P. Serr e, ed ., 3 vols., SpringerVerlag, Berlin and New York, 1968. {10} J . Glimm , J . Impagliazzo and I. Singer , edit ors, Th e legacy of Joh n von Neumann , Proc. of Symposia in Pure Math ., 50., Amer . Math . Soc. (Providence, RI , 1990). {11} D. Hilbert , Mathematische Probleme: Vortrag, gehalten auf dem int ern ational en Mathematiker-Con gress zu Paris 1900, Gint, Na chr . (1900) , 253-297 , translated and reprinted as Mathematical Problems, Bull. Amer. Mat h. So c., 37 , no. 4 (2000) , 407-436.

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{12} A. A. Kirillov, Unitary representations of nilp otent Lie gro ups, Uspekhi Ma t. Nauk , 17, no. 4 (1962) (106), 57- 110 (Russian ); English tr anslation in Russian Math . Su rveys, 17, no. 4 (1962), 52- 104. {13} G. W . Mackey, Imprimitivity for representations of locally compact groups, I, Proc. Nat . Acad. Sci . U.S.A ., 35 (1949), 537-545. {14} G. W . Mackey, Introduction to th e mathematical pap ers of E. P. Wigner , in : [199], pp . 241-290. {15} J . C. Oxtoby, B. J. Pet tis and G. B. Price (edito rs) , John von Neumann, 19031957, Bull. Amer. Math. S oc., 64 , no. 3, Part 2 (1958); reprinted as a monograph , Amer. Mat h. Soc. (Providence, RI , 1988). {16} F. Pet er and H. Weyl , Der Vollstandingkeit der primitiven Darst ellun gen einer geschlossen kontinuierl ichen Gruppe, Math . Ann., 97 (1927) , 737-755. {17} 1. Schur, Gesammelte Abhandlungen, Alfred Brauer and Han s Rohrbach , eds ., 3 vols., Springer-Verlag (Berlin and New York, 1973). {I8}

J. Szab ados and K. Tandori, eds ., A . Haar Mem orial Conf erence, 2 vols., Janos Bolyai Math. Soc. and Elsevier (Amst erd am and New York, 1987).

{19} B. Szc5kefalvi-Nagy, Uber messbar e Darstellun gen Liescher Gruppen , Math. Ann., 112 (1936) , 286- 296. {20} B. Szc5kefalvi-Nag y, On uniformly bounded linear tr ansformations in Hilbert space, A cta Univ. SzegedSect. S ci. Math ., 11 (1947), 152-1 57. {21} B. Szokefalvi-Nagy, Transformations de I'espace de Hilbert , foncti ons de type positif sur un groupe, A cta S ci. Mat h. (Sz eged), 15 (1954), 104-114. {22} B. Szokefalvi-Nagy, The commutation relation of quantum mechan ics, in : Fun ctional analysis and its applications (Nice, 1986), l CPAM Lecture Notes, World Sci. Publishing (Singap ore, 1988), pp. 323- 337. {23} B. Szc5kefalvi-Nagy, Alfred Haar (1885-1933): invari ant measure of mathematical excellence, in : {I 8}, vol. 1, pp . 17- 24. {24} B. Szc5kefalvi-Nagy (1913-1998), A cta S ci. Math . (S zeged), 65 , no. 1-2 (1999), 318. {25} E. Thoma, Zur harmonischen Analyse klassenfiniter Gruppen , Invent . Math ., 3 (1967),20-42. {26} E. Thoma, Uber unitare Darstellungen abzahlbar er , diskr et er Gruppen , Math . Ann., 15 3 (1964) , 111-138; Ein e Char akterisierung diskreter Gruppen vorn Typ I, In vent . Math. , 6 (1968) , 190-196. {27} A. Weil, L 'integration dans les groupes topologiques et ses applications, Herm ann (Paris, 1940). {28} H. Weyl, S eleeta , Birkhau ser (Bas el, 1956). {29} D. P. Zhelobenko, S. M. Nikol'skii and S. A. Telyakovski i, Bela Szc5kefalvi-Nagy (obituary) , Uspekhi Mat. Nauk, 54 , no. 4 (1999) ,143-146 (Russian); English tr an slation in Russian Math . Surveys, 54 , no. 4 (1999) , 819-822.

BOLYAI SOCIETY MAT HEMAT ICAL STUDIES , 14

A P anorama of Hungarian M athematics in the Twentieth Century, pp . 211-244 .

A PANORAMA OF THE HUNGARIAN REAL AND FUNCTIONAL ANALYSIS IN THE 20TH CENTURY

AKOS c s As zAR and DENES PETZ 1

The theory of real functions is a relati vely young cha pte r of mathemat ical analysis. In fact , in t he 19th cent ury, the expression "function t heory" was applied to mean t he t heory of complex-valued analytic funct ions of one or more complex variab les. T he first investigat ions that initiat ed the part of analysis called today "t heory of real functions" or briefly "real analysis" . were const ruct ions of various real-valued funct ions of a real variable whose characteristic prop erties are very far from those of analyt ic functions; as a ty pical example, we can mention the const ruct ion, due to K arl Weiers trass , of a real-valued function cont inuous in an interval but not differenti able at any point of t his int erval. The investigations in this dir ection obtained a very powerful instrument in the t heory of sets discovered and developed by Georg Cantor. As the first impo rtant results of the t heory, we can mention the investigations of Cami lle Jordan on propert ies of the functions of bounded variat ion, or the theory of the area of plane point sets due to t he same aut hor . However, the last discovery that finished the acceptance of real analysis as a well-adopt ed cha pte r of analysis was t he concept and theory of a very general kind of integral, du e to Henri Lebesgue, in t he first years of the 20th cent ur y. This acceptance was perh aps due to the fact t hat Lebesgue not only developed t he t heory of t he integral but also reached amazing applicat ions of his theory so t hat the imp ort an ce of t he new, general theory of t he integ ra l convinced everybo dy interested in mathematical analysis. On t he ot her hand , Lebesgue's integration t heory catalyzed the cristallization of t he ideas of fun ctional analysis in t he setting of infinite dimensional function spaces . IThe authors are grateful to Janos Horv ath , Joz sef Szucs and Laszlo Zsido for many helpful suggest ions.

212

A.

Csaszar and D. Petz

There were Hungari an mathem atician s who joined t he investi gations on real analysis in the earliest period of its exist ence, i.e., in the last decad es of the 19th cent ury. We must first mention G y ula [J ulius] K onig (18491913) who , in his courses on analysis given at the Techni cal University Budapest, presented a definition of the integral that included not only the classical definition of the Riemannian integral but also the Sti eltj es int egral. Despite the fact that he did it in about 1890, he formulated t hese ideas not earlier than 1897 in a pap er in Hungari an language {27} so t ha t the priority evident ly belong s to Th omas Jean S tieltj es who published his discovery in 1894. However , Konig was very probabl y the first researcher who const ruc te d a real -valued function cont inuous in an interval and having an ext remum in every subinterval {26}. This result cert ainly influenced t he disciple of Konig, Zoard G eocze (1873-1916) , who const ru ct ed a continuous function that is not rectifiabl e in any subinte rval {lO}. Many years later , it turned out t hat Ceocze's const ru ct ion yields essentially more ; in fact it is one of t he simplest const ructions furni shing a continuous nowhere differentiable fun ction i.e., it presents t he W eierstrass singularity (see work {21} of S and or K antor ). On t he other hand, t he sa me method of const ruction, under another choice of the par am et ers , furnishes a continuous fun ction that is increasin g and singular (i.e., its derivative is equa l to 0 almost everywhere, see {8}) . In t he last sente nce, the expression "almost everywhere" means , of course, "with the excepti on of a set of points of Lebesgue m easure zero" . It is an imp erishabl e merit of the history of mathematics in Hun gary that we had a resear cher who, a few years afte r its birth, not only mad e him self a master of Lebesgue's t heory of measure and integral, but also added to t his theory essent ial cont ributions. This resear cher is Frigyes [Frederic] R ie s z (18801956) . Frederic Riesz was definitely the first mathematician in Hungary who understood the great importance of th e new theory of integration . He was born in Gyor , a town in t he midw ay between Budap est and Vienna. After a two-year- study at the polytechni c in Zurich, he continued at t he science university in Budapest. From 1904 to 1912 he was a high-school teacher and wrote fund am ental pap ers already in this period. Although he published very good works also in Hun gari an all his life, he was clever enough to understand t ha t Paris was not only the capital of Fran ce bu t t he capital of mod ern analysis as well. His publications in Comptes R endu s, t he journal of the French Academy of Sciences, earne d a world fam e for him very early and

A Panorama of the Hungarian Real and Functional Analysis in the 20th Century

213

he became a professor of the University of Kolozsvar (called Cluj-Napoca now in Roumania) in 1912. Frederic Riesz was one of the fathers of functional analysis. Although functional analysis in the sense of nowadays had several roots in the 19th century, such as Fourier expansion of functions and spectral theory of some differential equations, its genesis could be put at the beginning of 20th century. Even at the end of 1800's linear algebra was very finite dimensional and dealt with n-tuples of real numbers, and Frechet's thesis on metric spaces appeared in 1906. The cristallization of the ideas was catalyzed by Lebesgue's integration theory. Many of the basic concepts of functional analysis were born in the setting of infinite dimensional function spaces . The intimate relation of the Lebesgue integral and infinite dimensional functional analysis is very transparent in the work of Frederic Riesz. The space £2 of square integrable functions on an interval of the real line was the first infinite dimensional space on which functional analysis in the modern sense was studied. The so-called Riesz-Fischer theorem (1907) {59} claims that the space £2 is complete, that is, all Cauchy sequences are convergent in the £2-sense. That time it was shown that if fn E £2 and f I f n(x) - f m (x) 1 2 dx is arbitrarily small when nand m are large enough , then there exists a function f E £2 such that

JI

fn(x) - f(x)!2 dx

--t

0

as

n

--t

00.

This is Fischer's version of the Riesz-Fischer theorem but Riesz was aware of the equivalence . Riesz actually showed that given an orthogonal sequence !I, 12, ... offunctions of unit length and a sequence Cl, C2, ... of scalars such that L:i ICi 12 is finite, there exists a function f in the £2 space such that

(Ii, J)

=

Ci·

Another early result of Riesz, discovered independently by Frechet relying on an earlier paper of Riesz, tells us that any bounded linear functional A of £2 is induced by an element 9 of £2 in the form of integration of the product {55}:

(1)

AU) =

J

f(x)g(x) dx

for some 9 E £2 if there exists a constant MA such that

214

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Csaszar and D. Petz

It is still a pleasure to read t he original works of Riesz. In 1910 he published a paper in Hungarian (Integralhato fiiggvenyek sorozatai {57}) , in which he explains his understanding of L 2 and the above mentioned two results. It is remarkable that abstract functional analysis did not exist at t hat time, nevert heless he understood his own (as well as Lebesg ue's, Frechet 's and Fischer's) result in a very modern and intrinsic way. For example, he wrote that "The extension of the concept of integral due to Lebesgue is an indispensable condition for my theorem, similarly to the fact that validity of certain theorems of algebra or arithm etics require the appropriate extension of the concept of numbers". In this pap er he defined the weak topology on th e spac e L 2 and shows th at a bounded sequence cont ains a weakly convergent sub sequence. In our present language, (1) describes the dual of the L 2 space . T he dual space was cert ainly a concept that shou ld be attributed to Riesz. He defined the du al of L 2 in 1907 and in 1909 he dealt with the dual of th e space of continuous funct ions , Sur les operations fonetionn elles lin eaires. Let C[a, b] denote the set of all cont inuous real-valued functions on the interval [a , b]. In 1903 Had amard want ed to describe all linear functionals U : C[a, b] ---t lR such th at U i« ---t U f whenever I« ---t f uniformly. He took a function F such th at

nl~ n

f(x) =

l

b

f(t)F( n(t - x )) dt

uniformly in x E [a , b] , for example F( x) = exp( -x 2 ) would do, and he showed that U(J)

= J~~

l

b

f(t)Cf?n(t) dt ,

where Cf? n(t) is the value of the functional U at th e function x I-t nF( n(t x)) . Riesz described the continuous linear functionals of C[a, b] by means of the Sti eljes int egral and removed the arbitrariness of th e function F in Hadamard 's theorem {56}. He proved t hat there exists a funct ion a of bounded var iation such that

(2)

U(J) =

l

b

f( x) da( x) ,

moreover a is unique if a (a) = 0 and the left continuity of a are required . For any a < t < b he considered the function a,

if a ::; x ::; t,

t - a,

if t::; x ::; b.

X -

ft( x)

=

{

215

A Panorama of the Hungarian Real and Functional An alysis in the 20th Cent ury

He showed that the function A : t f-+ U(fd is Lipschitz and took -a(t) as one of the derived numb ers of A at the point t. Then it was a st andard procedure to modify a to fulfil th e addit ional requir ement s and to keep (2). Extending his work on the space L 2 , Riesz devoted a fundamental pap er to LP spaces in 1910, Untersuchungen iiber Systeme integrierbarer Funktionen {58}. LP is the set of all complex valued measurable functions such that IfIP is integrable. He restricted himself to th e case p > 1 and extended th e Holder and Minkowski inequali ties where

(t,

lad

1

1 = 1, q

- +P

bkl'fP ~ (t,lakIPt + (t,lbkIPt

to measurable functions. If

f

E LP and 9 E Lq, then

f g is integrable and

Moreover, if f , g E LP, t hen f + g E LP and (

j lf(x )+g(x )IPd X)

IIp

:s

(

j

If(x) IPdx

)

II p

+(

j

!g(x ) IPdx

)

IIp

.

He exte nded several definitions and results from t he theory of L 2 spaces. He defined strong convergence in LP as I« ---7 f if and only if If n(x ) - f (x) IP dx ---7 O. His first definition of weak convergence was different from today's usual one. He said that I« ---7 f weakly if

J

t

it fn( x) dx ---7i f( x) dx for all t in the interval on which th e functions are defined. He showed that this is equivalent to j (J( x) - fn( x)) g(x) dx

---7

0 for all

9 E U.

He proved t he weak compactness of the unit ball of LP and he was particularly interested in the solution of the infinite syst em of linear equations

(3)

A.

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Csaszar and D. Petz

where ~(x) is the unknown and the !i(X)'S belong to L", (The subscript i can run over an arbitrary set, countable, or not.) One cannot give an easy condition for the existence of the solution. He claimed that the condition is the existence of a constant M such that

(4) holds for all finite subsets I of the index set and for all complex numbers fli. Riesz was aware of the importance of the case where the Ji's are all the functions in Lq. Then (4) is exactly the roundedness of the functional L defined as LUi) = Ci and he discovered that the dual of Lq can be identified with LP. He achieved the first example of what we call today reflexive Banach space. The system of equations (3) is related to the moment problem. This means that given the continuous functions Ii on an interval [a, b] and a sequence Ci of real numbers (i = 1,2, . . . ), an increasing function 0: should be found such that

(5)

(i=1,2, ... )

(d. (2)). In the original moment problem !i(t) = t i . A trivial necessary condition for the existence of 0: is the property that imply "£';=1 AjCj ~ O. If this is fulfilled then

'£/]=1 Aj!j

~ 0 should

defines a positive functional on the linear span of the functions Ii and this functional should be extended to all continuous functions. The representation theorem (2) could be used. The moment problem belonged to the circle of ideas Frederic Riesz worked on. His brother, Marcell [Marcel] Riesz (1886-1969), considered the moment problem as the question of extension of a positive functional. His method works in a very general setting, where a linear functional is defined on a subspace and the positivity is determined by a convex cone. His beautiful method is applicable not only to the power moment problem but many related problems in function theory (see Section II.6in{1}).

A Panorama of the Hungaiian Real and Functional Analysis in the 20th Century

217

In 1920 Frederic Riesz published a detailed paper dedicated to an elementary presentation of Lebesgue's integral (B4 in [156]) . The method is based on a completely elementary particular case of Lebesgue measure, namely on the definition and simplest properties of the sets of measure zero (briefly null sets) : A set A c R is a null set if it can be covered, for an arbitrary c > 0, with a sequence of intervals [an , bn] such that 2:~=1 (bn -an) < c. A statement is true a.e. if it is true everywhere with the exception of the points of a null set . The starting point is the, still quite elementary, definition of the integral of simple functions , i.e., functions in [a , b] such that there is a decomposition of [a, b] into finitely many pairwise disjoint subintervals in each of which the function is constant. For simple functions , the (Riemann) integral can be given by a finite sum . Now a function i, bounded in [a , b], is said to be integrable if there exists a bounded sequence of simple functions fn (i.e., Ifni ~ M for some M) such that fn ---. f a.e, in [a , b] . It can be shown that, under these conditions, the integrals fn(x) dx converge to a limit depending only on

f:

f:

f (i.e., independent of the sequence); this limit defines f(x) dx . In order to define the integral of an unbounded function , let us first say that a (bounded or unbounded) function is measurable in [a , b] if it is the pointwise limit of an a.e. convergent sequence of simple functions. The integral f(x) dx of a measurable function f is defined as the limit of the

f: f: i.;»;

sequence (x) dx , where (cn) is an arbitrary sequence tending to -00 and (dn ) is one tending to +00, while f cn ,dn is the truncation of f: it is equal to f(x) if Cn ~ f(x) ~ dn , to Cn if f(x) < Cn, and to dn if f(x) > dn; the function f is said to be integrable if the above limit exists , is finite and independent of the choice of the sequences (cn ) and (dn ) . Based on these definitions, it is not difficult to deduce the usual prop erties of the integral (linearity, theorems on the integration of sequences of functions, etc.) It can be easily shown that, for a function integrable in the sense of Riemann, the new integral exists and is equal to the Riemann integral. Riesz presented his exposition of the theory of the Lebesgue integral in his courses on analysis. A detailed exposition can be found in the monograph [157] which was later translated into several languages. In two papers (B5 and B6 in [156]) Riesz analyzes the role of Egoroff's theorem, which states that a convergent sequence of measurable functions is uniformly convergent eliminating a subset of arbitrarily small measure

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{9}, in th e theory of the Lebesgue integral. In par ticular, he indicates t he modifications necessary for exte nding t he theorem for applicat ions in the theory of the Lebesgue-Stieltjes integral. The St ieltjes integral ap pa rently captured Riesz's attent ion because it played a decisive role in his result concern ing the int egral representation of bou nded linear operations on the function space C(a , b) of cont inuous functi ons (see (2)) . In three short pap ers (B7, B8 and B9 of [156]) and in his letters to G. H. Hardy, Riesz gives simple pro ofs for some int egral inequalities in particular , t he celebrated maximal inequality of Hardy and Lit tlewood ; in general, arguments are based on the use of t he distri bution fun ction m(y ) = m({ x E [a , b] : f (x ) < v} ) associated wit h a funct ion f measurable in [a, b], where m denotes Lebesgue measure. In B9, he uses t he so called Riesz lemma to furni sh an elementary proof of Lebesgue's t heore m: every monotone functi on is almost everywhere differentiable (see BlO, Bll and B12 in [156]). In its simplest form , i.e., for cont inuous functions, the Riesz lemma is so elementary that its proof can be included here.

Riesz lemma. If f is continuous in the interval [a, b], then the set H of points x E [a , b] for which there exists some point x < x' :::; b such that f (x ) < f( x' ) is open: H = Uk(ak, bk ) and f (ak) :::; f (bk) for each k. The set H can be empty; in this case we have not hing to prove. If H =I 0, it is evidently open by the cont inuity of f so that t he represent ation H = U(a k, bk ) is clearly possible. Fix a k and consider ak < x < bk . Let XQ be one of th e points in the int erval [x, bJ where t he value of f is maxim al. Then x :::; XQ < bk is imp ossible since it would imply XQ E H and the existence of an Xo < x' :::; b satisfying f (xo) < f (x' ). T hus bk :::; Xo :::; b and t hen f (bk ) 2: f (xo) as bk ~ H . On the other hand, f( x ) :::; f( xo) by the choice of Xo, hence f (x ) :::; f (bk) and, from t he continuity of I, x - t ak yields f( ak) :::; f (bk ). Using th e Riesz lemm a , the proof of Lebesgue's th eorem becomes almost complete ly elementary. The Riesz lemma quite aut omatically provides coverings of t he exceptional set by systems of intervals of arbitrarily small total length in the pr oof of Lebesgue's t heorem on t he a.e. differentiabi lity of a monotone function. Riesz himself was aware tha t his lemma can be used for proving further int eresting t heorems of measur e theory (B9 and B13 of [1 56]) and ergodic theory (G5, G7 and G8 in [1 56]). Much later, the lemma was generalized to severa l variables (d. {5} and [166]).

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The fact that Lebesgue's th eorem on the differentiability of monotone functions obtained an elementary proof through the Riesz lemma, suggested to Riesz a new approach to Lebesgue's integral theory, based on the different ability of monotone functions. He present s his ideas in two papers (B14 and B15 of [156]) The start ing point is t he following observation: Let f ~ 0 in t he interval [a , bJ and suppose that there exists a function F , increasing in [a , bJ and satisfying F'(x) = f( x) a.e. in (a,b). Then t here exists , among these F , one for which the difference F (b) - F(a ) is the smallest possible. After having proved this , we say that

J:

f

~

0 is integrable in [a , b] if

th ere is an F as above, and define f as the minimum of F(b) - F(a) . A function f of arbitrary sign is said to be integrable if = max (J, 0) and

r

r

J: J:

J:

= - min (J, 0) are integrable and then we define f = f+ j>. From these definitions, one can deduce without any difficulty the usual properties of the integral, e.g., the theorems on the integration of sequences of functions.

In the years after World War II , Riesz wrot e some big expository pap ers on the evolut ion of the concept of the integral (B16, B17 in [156]) and on th e role of null sets in real analysis (B18 in [156]). It is natural that his own ideas played a central role in all th ese summaries. The original proof of th e Riesz lemma due to Frederic Riesz, was slightly more complicated; the idea of appl ying the above point XQ is due to his brother Marcell [Marcel] Riesz (1886-1969). Mar cel Riesz was also an outstanding mathematician , he lived most of his life in Sweden and had a wide scientific int erest , including functional an alysis, partial differenti al equations and algebra. Assume that a linear operator A is defined on a set of measurable functions and its values are also measurable functions on a different space. Assum e that A has a finite norm C(p , q) when it is regarded as a map from LP to Lq (1 ~ p ~ 00, 1 ~ q ~ 00). The Riesz convexity theorem of Marcel Riesz tells us that log C(p , q) is a convex function of the variables (p-l,q-l) E [0 ,1] x [0 ,1]. The convexity theorem became a starting point of abstract int erpolation theorems. The spaces LP and £P' are connected by a path of Banach spaces (namely th e Lq spaces, when q is between p and p'). Und er some condit ions a const ruct ion works for any two Ban ach spaces , t his is a very concise description of t he interpolation theory du e Calderon , Lions and Peetr ewhich has it s root s in t he work of Mar cel Riesz.

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A considerable part of the work of several Hungarian mathematicians in the first part of the 20th century was devoted to an important application of Lebesgue integral, namely to the calculation of the area of surfaces. Surface area is only seemingly an easy two-dimensional analogue of arc length. Since the work of Hermann Amandus Schwarz, we know that the theory of surface area is essentially more complicated. Recall that if, say,

x=cp(t),

y='ljJ(t), z=X(t)

(a~t~b)

is the parametric representation of a continuous curve in R 3 (i.e., ip , 'ljJ, X are continuous in [a , b]) then the length of the curve can be defined as the least upper bound of the lengths of polygons inscribed in the curve, namely obtained with the help of a subdivision of [a ,b] by points a = to < tl < t2 < .. . < t n = b and taking as vertices of the polygon the points of the curve with parameters ti ; the curve is rectifiable iff this least upper bound is finite . Schwarz discovered that the area of a surface cannot be defined in a similar way. Even for very simple surfaces, e.g., for a circular cylinder, it can happen that the areas of all inscribed polyhedra are unbounded from above, and the surface can be uniformly approximated by inscribed polyhedra so that their area tends to an arbitrary limit not less than the usual (elementary) area of the surface . Consider, for the sake of simplicity, a surface having an equation z = f(x , y) where f is continuous in a rectangle R = [a, b] x [c, dJ. In this case , an idea due to Lebesgue again produces a suitable definition of the area of the surface. Consider a subdivision of R into pairwise non-overlapping triangles T l , .. . ,Tn and a function g continuous in R and linear in each of the triangles Ti. The equation z = g(x, y) corresponding to the piecewise linear function 9 can be considered as representing a polyhedron P having an elementary area a(P). Let us consider a sequence of subdivisions having the property that the functions 9n converge uniformly to f and the areas a(Pn ) have a limit l; this limit may depend on the sequence (9n) and then the smallest possible limit can be considered as the area of the surface; we shall call it the Lebesgue area L(1) of z = f(x, y) . In the case of a good function f (e.g. if f has continuous partial derivatives fx and fy in R) it is not difficult to show that the Lebesgue area can be computed with the help of the classical formula

(6)

L(1) =

JJ Jn + fJ + 1 dx dy;

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however, in the general case of a continuous I, the definition of L(j) does not directly involve any method for calculating it. This was the motivation for Zoard Geocze, in one of his first papers on the theory of surfaces, presented as a Thesis to the Sorbonne in Paris in 1908 (Quadrature des surfaces courbes, Ungar. Ber., 26 (1910), 1-88.), to introduce the following expressions:

G1(j,I) =

J:

G2 (j,I) =

1

If(x,5) - f(x,I')1 dx,

8 If (j3, y) - f(a,Y)1 dy,

G(j,1) = (G 1 (j, 1)2 + G2(j, 1)2 + III) 1/2, where I = [a , j3] x [,,5] is a subinterval of R and III denotes the area (j3 - a) (5 - 1') of 1. He considered further the limit that we call nowadays the Burkill integral of the interval function G; in order to define it, let us consider a subdivision I = {II, ... , In} of R into pairwise non-overlapping subintervals Ii , then take the sum n

s(I) =

L G(j, Id 1

and the (always existing) limit H(j, R) of s(I) , i.e., a (finite or infinite) number to which (s(In)) converges whenever the subdivision In is infinitely refining (Le., varies such a manner that the maximum of the diameters of the intervals belonging to In tends to 0). Now Ceocze proposes to consider the value H(j, R) as the area of the surface z = f(x, y). This is motivated by the result that H(j, R) = L(j) whenever the function f satisfies a Lipschitz condition (i.e., there is a constant M such that f(x', y') - f(x, y)1 < M( lx' -xl + Iy' - vl) whenever (x, y), (x', y') E R). This proposal is well-motivated because Tibor Rad6 (1895-1967) proved later that the equality H(j, R) = L(j) is valid for any continuous function f {48}. Thus Ceocze found in fact a method for calculating the Lebesgue area of an arbitrary continuous surface defined by an equation z = f(x , y) ((x , y) E R). Geocze found also a necessary and sufficient condition for the value H(j, R) to be finite , i.e., by Rad6's theorem, for the continuous surface z = f(x, y) to have a finite Lebesgue area. This is the following: let

I

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the funct ion f be of bounded variation as a funct ion of x in the interval [a ,b] for almost every fixed y E [c, d] and as a functi on of y in t he interval [c, d] for almost every fixed x E [a, bJ; let us denote by V1 (y) t he total variation of f (x , y) as a function of x over the inte rval [a , bJ and by V2(X) th e total variation of f (x , y ) as a function of y over the interval [c, dJ; the condition postulat es that VI should be (Lebesgue) integrabl e in [c, dJ and V2 be int egrable in [a, bJ . This condit ion due to Ceocze was rediscovered by Leonida Tonelli {66}; a function satisfying this condition is said to be of bounded variation in the Tonelli sense. Tonelli foun d also a necessary and sufficient condition for the classical formul a (6) to give the Lebesgue area of the cont inuous surface z = f( x , y ). A functi on f satisfying this condit ion is said to be absolutely continuous in the Tonelli sense. This theory fills Chapter V of the brilliant monograph {63} of St anislaw Saks, where works due to Ceocze and Rad6 are often quoted . The problem of calculat ion of the area of sur faces is essentially more complicated if we consider continuous surfaces having a parametr ic representation; suppose t he surface 8 is given in the form

x

= f (u , v),

y

= g(u , v),

z = h(u , v ),

where f , g , h are cont inuous in a rectangle R = [a, bJ x [e, dJ of the uvplane. Ceocze made a few first steps in this direction in {12} and in the works "A rectifiabilis feliiletrol" {14} "A feliilet teriiletenek Peano-fele definitioj arol" {15} written in Hungarian. However, t he t horough discussion of this problem was mainly accomplished by Rad6 who not only published a long series of papers on this sub ject but is also t he aut hor of a great monograph [144J containing a deep analysis of the serious diffi culties of the problem. Besides the Lebesgue area £(8) of th e surface 8 , defined with t he help of sequences of polyhedr a quite similarly as in the case of surfaces represent ed in t he form z = f (x , y), it is convenient to introduce anot her kind of area a(8 ) playing a role similar to the expression H (j,R) in the t heory of surfaces z = f( x , y). This is done, based on ideas of Ceocze {12} by Rad 6 in {49, 50, 51} and in [144J. The concept of a(8) can be used in examining the properties of the sur faces of zero area {13, 53}. The role of t he surface area a(8 ) in calculat ing th e Lebesgue area £ (8 ) is emphasized by the fact t hat, in many cases, we have a(8) = £ (8) and ,

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at the same time, a(8) is often equal to the value of the classical integral formula

fL

(7)

W(u , v) dudv,

where and

8(g , h) J1(u,v) = 8(u ,v) '

8(h, f) J 2(u ,v) = 8(u ,v)'

8(j,g) J3(u ,v) = 8(u, v)

are Jacobians. Rado {52} has shown that the value of (7) is always :S £(8) , whenever the partial derivatives i x, i y, gx, gy, hx , hy exist a.e. in R. In the general case, Rado has shown in {54} that , inst ead of the concept of functions of bounded variation and absolutely continuous in the Tonelli sense, it is possible to introduce th e concept of a surface 8 of essential bounded -uoriatioti and essentially absolutely continuous, respectively, further instead of the ordinary J acobians Ji , essential generalized Jacobians and , with the help of them, a generalized function We(u, v). Now if £(8) is finite, then 8 is of essent ial bounded variation, We(U , v) exists a.e. on R and we have the inequality

fL

We(u,v)dudv:S £(8) .

The sign of equality holds if S is essent ially absolutely continuous. Moreover, if the partial derivatives i- ,..., h y exist a.e. in R, then We(u, v) can be replaced here by W(u , v). As to the equality a(8) = £(8), it holds whenever £(8) is finite and also if a(8) = O. Rad6's results in the theory of surface area play, of course, an important role in his monograph on a famous question in differential geomet ry [142] . He also published a monogr aph together with Reichelderfer [145], where the methods developed in the theory of surface area play an essential role. In his last papers, he combines the methods of this theory with methods of general measure theory {52} and of three papers written in collaborat ion with E. J. Mickle {31, 32, 33}. Geocze and Rado were decisive personalities in the theory of surface area and their works are quoted everywhere in the literature of this important chapter of Analysis.

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G yorgy [George] Alexits (1899- 1978) became later a famous researcher in th e theory of orthogonal series; however, one of his early papers {2} is an essential cont ribution to an important chapt er of real anal ysis, namely to the theory of Baire functions . A paper of P al [Paul] Veress (1893-1945) {67} is concerned with the same theory. Both papers are quoted in the monograph of Hans Hahn (Reelle Funktionen, Leipzig, 1932). Veress was th e author of th e first textbook on real analysi s in Hungarian .

At th e beginnings of functional analysis integral equations enjoyed a lot of attention. They are of th e form

ep(s ) = f(s)

+

Alb

K(s , t)f(t) dt ,

where ep is a continuous function on [a, b], A is some complex parameter, K(s , t) is continuous on [a , b] x [a , b] and f(t) is th e unknown function . For example, the Dirichlet problem could be reduced to such an integral equation. David Hilbert, in his very famous and fundamental paper of 1906, replaced th e integral equation by an older concept of an infinite syst em of linear equations. Let U n be a complete orthonormal sequence of continuous functions on [a, b] . If f is a solution of the equation, then we can consider the generalized Fourier coeffi cients

ll b

kij = Xi =

l

b

k(s, t)Ui(S)Uj(t) dsdt,

b

ep(S)Ui (S) ds and Yi =

l

b

f (S)Ui(S) ds.

In this way we arrive at th e infinit e syste m of linear equations 00

Yi

+ AL

kijYj = Xi,

j=l

where the sequences Xi and Yi are square summ able and kij is an infinite matrix (of a certain bilinear form). Hilbert himself worked with square integrable sequen ces and introduced the import ant concepts of continuity and complete cont inuity, mostly for symmetric bilinear forms. It is not our aim to give more details about Hilbert 's work on integral operators, we want to turn to the work of Riesz on completely continuous operators. His lecture delivered in a session of t he Hungarian Academy of Sciences in 1916 appeared in the journal Mathematikai es Term eszettudomanyi

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Ertesito with the title Linearis fuggvenyegyenletekrol in Hungarian {60} in 1917, and the German translation Uber luieare Funktionalgleichungen {61} was published in 1918. The subject of this paper is the invertibility of certain transformations and Riesz gave the definition and spectral th eory of completely continuous transformations. He works on the space of all continuous functions on an interval, but he notes that similar methods work on other function spaces, i.e. on £2, where they are even simpler. He uses the norm of a function f: 11111, which is the maximal value of the function f(x)l · The same concept and the same notation is standard today. He carried over Hilbert's definition of a completely continuous bilinear form (based on the weak topology) to the new situation. He defined a linear mapping as completely continuous if the image of a bounded sequence is compact. (In today's language one would replace "compact" by "precompact" or by "relative compact" .) The novelty of his paper is that he realized that Frechet's concept of compactness is the proper tool to deal with completely continuous operators and he uses only the axiomatic definition of norm years before it was introduced under the name of "Banach space". He gives in an entirely geometric language what is known nowadays as the Riesz-Fredholm theory of compact operators.

I

In 1918 Riesz left Kolozsvar , after the World War I the town became part of Roumania. For two years Riesz lived in Budapest and in 1922 he became professor of the newly founded university at Szeged. A partially ordered real linear space L has an order structure which is compatible with the linear structure. This means, that for any pair of elements f and 9 in L satisfying f :::; 9 it follows that f + h :::; 9 + h holds for all h E L and af :::; ag holds for all real numbers a 2: O. If, in addition, the order structure in L is a lattice structure, then L is called a Riesz space. In the present language Frederic Riesz was interested in the ordered dual of an ordered vector space and the basic example was the space of continuous functions. His lecture at the International Mathematical Congress at Bologna in 1928 was devoted to this subject and he returned to it in an 1940 Annals of Mathematics paper (which was the translation of his 1937 inaugural lecture at the Hungarian Academy of Science). Riesz put emphasis on the following decomposition property: If !I + 12 = gl + g2, then there are elements f11, !I2, 121 and 122 such that !I = f11 + !I2, 12 = 121 + 122 , gl = f11 + 121 and g2 = !I2 + 122. In the space of continuous functions one can choose 111 := min (!I , gd and 122 = !I + 12 - max (!I , gd · Between 1938 and 1948 Riesz dealt in eight papers with ergodic theorems. The ergodic and quasi-ergodic hypothesis were born in statistical mechanics

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and von Neum ann gave th e following mathematical formulation. Let H be a Hilbert space and T be a bound ed linear transformation on 'H, Accordin g to von Neumann's m ean ergodic theorem the averages

converge to a T-invariant vector for every vector 1 E H , when T is a unitary operator. Riesz gave a very elegant proof for von Neumann 's result. Riesz showed that the orthogonal complement of the set

{1-

Sn

(1) : n EN,

1 E ti}

is the fixed point set of T. From this fact one can prove the convergence of the averages and the proof requir es only the hypothesis liTIII ~ IIJII for every 1 E H , that is T is a contraction . Th e Hilbert space version of th e mean ergodic theorem corresponds to £2 spaces and Riesz considered oth er £P spaces as well. Much lat er ergodic theory app eared again in Hung arian functional analysis in the context of operator algebr as: Istvan Kovacs and J6zsef Szucs obtained the first mean ergodic theorem in von Neumann algebras {25}. Their result implies that if a is an automorphism of a von Neumann algebra admitting a faithful normal invariant state, then the averages 1

Sn

:= - ( I

n

+ a + ... + a n - 1 )

converge to a condit ional expect at ion onto the fixed point algebra, pointwi se in the st rong operator top ology. John von Neumann was born Neumann Janos in 1903 in Budapest . He was a child prodigy, a prodigious student and he left his mark not only on pure mathematics but on theoretical physics, on meteorology, on economics, on digital computers and on more. He was th e mathematician admired by most scholars outside of his own discipline. In its December 24 issue in 1999, The Fin ancial Times has declared John von Neumann to be "Man 01 the Century ". In the years 1914-21 von Neumann st udied in Budapest 's Lutheran Gymnasium. In 1921 he went to become a chemical engineer first to Berlin University and then in 1923 he took the entrance examin ation for the prestigious course in the chemical engineering department of the famous

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Eidgenossische Technische Hochschule in Zurich. When Hermann Weyl was absent from Zurich, the undergraduate chemist von Neumann took over the teaching of some of his classes. During his university years at the ETH, von Neumann was passing courses in Budapest University (which he never attended) from where received his Ph.D. with highest honors in 1925. In early autumn of 1926 von Neumann arrived in Gottingen. He immediately learnt quantum theory from Heisenberg's lectures. Von Neumann became an axiomatizer of quantum mechanics on behalf of the so-called Copenhagen school (which did not include Schrodinger.) To Hilbert's delight, von Neumann's mathematical exposition made much use of Hilbert's own concept of Hilbert space. However, it is not sure that axiomatization of the Hilbert space and its linear operators (as a substitute for infinite matrices) by the twenty-three-year-old von Neumann was to Hilbert 's delight . Our present concept of Hilbert space , infinite dimensional complex vector space endowed with an inner product whose metric is complet e and separable , was formulated by von Neumann. The rigorous quantum mechanics required the use of unbounded operators defined only on a subspace of a Hilbert space . Von Neumann developped several technicalities concerning such operators. The role of the graph, the difference between symmetric and selfadjoint operators, the spectral decomposition of unbounded selfadjoint operators were discovered by him. In his excellent textbook {29} Peter Lax makes the following historical comment: In the 1960s Friedrichs m et Heisenberg, and used the occasion to express to him the deep gratitude of the community of mathematicians for having created quantum m echani cs, which gave birth to the beautiful theory of operators in a Hilbert space. Heisenberg allowed that this was so; Friedrichs then added that the mathematicians have, in some m easure, returned the favor. Heisenberg looked noncommittal, so Friedrichs pointed out that it was a mathematician, von Neumann, who clarified the difference between a selfadjoint operator and one is m erely symmetric. "What's the difference, " said Heis enberg. After some earlier work on single operators, von Neumann turned to families of operators. He initiated the study of rings of operators, which are commonly called von Neumann algebras today. The papers which constitute the series "R ings of operators" opened a new field in mathematics and influenced research for half a century (or even longer). In the standard theory of modern operator algebras, many concepts and ideas have their origin in von Neumann's work. A von Neumann algebra consists of bounded linear Hilbert space operators. The characteristi c feature of t he concept of von Neumann algebra is

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it s very rich st ructure. A von Neum ann algebra contains the spectral pr ojecti ons of all selfadjoint oper ators belonging to the algebra . In particular, there are many orthogonal projections in the algebra itself. Roughly speaking, the point in the concept of von Neum ann algebra is that formation of pro du ct and spectral diagonalization of selfadjoint elements are posssible within the algebra. It t urns out t hat the projections of a von Neumann algebra form a lat tice in the sense that any two of t hem have a least upper bound and a greatest lower bound with respect to an appropriate and natural ord ering. The lattice of proj ections is t he st arting point in t he classificat ion of von Neumann algebras and a ground for quant um logic. Von Neum ann algebras are classified in terms of the range of a dimension function defined on th e lat t ice of projectio ns. T he dimension function is the ext ension of the simple concept of rank (for matrices) and the peculiarity of t he subject begins with the observation t hat in nontrivial cases t his "rank" can be noninteger. Below t he classification of von Neumann algebr as is described. Also, t he influence of measure theory on early operator algebra t heory is demonstrated by a comparison of a measure-theoretic const ruction of Alfred Haar with the dimension function of Murray and von Neum ann. T his exa mple shows th at the connection with measure th eory and ergodic th eory has been very imp ort ant for operator algebras since t he very beginn ing. We denot e by B (7t ) the set of all bounded opera tors acting on the Hilbert sp ace 'H. For a subset S ~ B (7t), its commutant S' is defined as the set of opera tors commuting with S:

S'={KEB(H ): K S=SKfor allSES} . Note that S ~ (S' )' holds obviously for any S ~ B (H ). A family of operators acting on a Hilbert space and containing t he identi ty operator is called a von Neum ann algebra if it contains the adjoint , the linear combinat ions and t he products of its elements and forms a closed subspace of the space of all bounded operators with respect to the topology of pointwi se convergence. A von Neum an n algebra is linearly spanned by its selfadjoint elements and the spectral resolution of the latter ones lies conveniently in the algebr a. One of t he first results of von Neum ann, the von Neumann's double com mu tan t theorem, was an equivalent algebraic definition of von Neumman algebr as. Von Neum ann's double commutant t heorem asserts t hat a fam ily of opera tors is a von Neum ann algebra if and only if it contains t he adjoint of its elements and coincides wit h its second commut ant (t hat is, the commut ant of its commutant ). T he remarkable point in t he double commutant

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theorem is the lack of any topological requirement. In the concept of von Neumann algebra, topology and pure algebra are in great harmony. The selfadjoint idempotents, called (orthogonal) projections, of a von Neumann algebra form an orthomodular, complete lattice with respect to the lattice operations 1\, V, .1 and the partial ordering K. Below we describe how these operations are defined in terms of the algebaric operations. The projections are in natural correspondence with the closed subspaces of the underlying Hilbert space and the set theoretical inclusion of subspaces induces a partial ordering on the projections. This ordering is equivalently defined as

(8)

p :S q if pq = p.

The smallest projection with respect to this ordering is 0 and the largest one is the identity. For projections p and q, their meet (that is, greatest lower bound) p 1\ q is the orthogonal projection onto the intersection of the range spaces of p and q. The projection pl\q may be obtained as the strong limit of (pqt as n ----t 00. The projection p V q is defined as the smallest upper bound in the lattice of projections. (p V q projects onto the closed subspace spanned by the range spaces of p and q.) The orthocomplementation .1 is defined as pi- = I - p. The orthomodularity of the lattice of projections means that the following so-called orthomodularity condition is fulfilled in the lattice:

(9)

q = p V (pi- 1\ q) for P:S q.

This relation is a weakening of the distributivity condition and is an essential property of the lattice of projections. Let p and q be two projections in a von Neumann algebra M . The projections p and q are called equivalent (with respect to M), p rv q in notation, if there is an operator x in M such that p = x*x and q = xx*. In terms of the underlying Hilbert space, the equivalence of p and q means that there exists a partial isometry x in the given von Neumann algebra which sends the range space of p isometrically onto the range of q. An extended positive-valued function D : P(M) ----t [0,00] on the set P(M) of all projections of M is called a dimension function if it satisfies the following requirements: (a) D(p) > 0 if p =1= 0 and D(O) = O. (b) D(p) = D(q) if p and q are equivalent projections.

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It is fundamental in the theory of von Neumann algebras that the dimension function is determined up to a positive multiple if the center of the algebra is trivial, that is, the algebra is a [actor.

We sketch how the dimension function was obtained in {34}. A nonzero projection is called finite if it is not equivalent to a smaller projection. "Smaller" is understood here in the sense of the partial ordering (8). Murray and von Neumann proved in {34} that if I is a finite and e is an arbitrary proj ection in a factor then there exists a unique integer k such that

I

= ql

+ q2 + ...+ qk + P,

where ql , q2, . .. , qk are pairwise orthogonal projections equivalent to e, p is a projection orthogonal to all qi and equivalent to a subprojection of f. This integer k is denoted by

[~]

(10)

and this is the number of projections equivalent to e which may be packed into I in a pairwise orthogonal way. (10) is an integer and is only an approximate measure of the ratio of the subspaces corresponding to I and e. Now we fix a finite non-zero projection eo and a sequence en of non-zero finite projections converging to O. The limit lim

(11)

n->oo

[tJ

[~J

=

(L) eo

forms a quantitative ratio of relative dimensionality, when the sequence en converges to 0 strongly. (Heuristically, the projection eo will have dimension 1, first we estimate the dimension en by comparison with eo and then the dimension of I is estimated by comparison with In.) The relative dimension was defined in {34} as

D(e) =

0

if e = 0,

(:0)

if e is finite,

+00

if e is not finite.

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The use of the relative dimension in the classification of factors will be discussed below. Now we make a detour and compare the construction of the dimension function with that of the Haar measure on a locally compact topological group. The existence of a measure on an abstract locally compact group which is invariant under right translations was proven in 1932 by the Hungarian mathematician Alfred Haar {17}. Von Neumann was in contact with Haar and knew his celebrated result before it was published. It is instructive to trace back the dimension function of a ring of operators to Haar's beautiful idea for the construction of the invariant measure. Let G be a locally compact topological group and for a precompact BeG and an open U C G denote by h(B ; U) the number which gives at least how many right-translates of the set U are needed to cover the set B . This is an integer showing the size of B compared to U. h(B; U)

is translation invariant by construction. Of course, the smaller the U, the larger the h(B ; U) . The latter one may increase to infinity when U runs over the neighbourhoods of a point. We need a normalization of h(B; U) . A compact set S of nonempty interior is chosen to normalize the measure. (S will be a set of unit measure.) (12) gives the measure of a compact set B if (Rn) is the filter of neighbourhoods of a point. The set function Jl is right-translation invariant and additive on disjoint compact sets . After the measure Jl of compact sets is obtained, measure-theoretic arguments are used to extend Jl to a larger class of sets. It is difficult to refrain from comparing Haar's idea with the construction of dimension function of projections in a von Neumann algebra: the similarity between the formulas (12) and (11) is striking. (12) yields the right-translation invariant size of subsets of a group G and (11) defines an invariant under partial isometries for projections in a von Neumann algebra. This example demonstrates how measure-theoretic arguments can survive in the apparently different discipline of operator algebras . Von Neumann devoted two papers to Haar measure. In {39}, he gave another proof for the existence and uniqueness in the compact case and in {40} he obtained uniqueness in the general locally compact case. Von Neumann presented several courses on measure theory and invariant measures at the Institute for Advanced Study. His lecture notes were published in 1999 by the American

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Math ematical Society {43}. For him operator algebra t heory was a non commutative outgrowth of measure th eory. Rosenberg's article (in thi s volume) is a complementary readin g about noncommut ative harm onic analysis {62}. Now we cont inue t he comparison of the relative dimension and Haar measure. The obj ective of integrat ion theory is to const ruct a linear functio nal, called integral, from a certain measur e. Murray and von Neum an n ext ended the relative dimension functional to arbitrary selfadjoint elements of the given von Neumann algebra M . Let A = A* EM and let )"dE()") be its spectra l resolution with a projection-valued measure E on the rea l line. Then by property (c) of the relat ive dimension, D (E ) is an ordinary measure and

J

(13)

TrM(A) =

J

)"dD(E) ()")

determines a real numb er when th e integral on the right-h and side exists. The inconveniency of definition (13) is in the fact t hat for noncommu ti ng self-adjoint operators A and B one cannot say much about the spectra l resolution of A + B in terms of the spectral resolution s of A and B. Mur ray and von Neum ann expected that

but t his was proven in {34} only for commuting A and B. T he general case was postponed to t he subsequent pap er {35}. It was established there that th e abst ract trace functional TrM is linear. TrM yields an analogue of an integral. (This analogy has developed into an operator-algebraic int egration t heory, including LP spaces, measurable operators and so on. For this Segal proposed t he term "noncom m utative integration" in {64} since a commutative von Neumann algebra admits represent ations by functions.) In {37} von Neumann established th e st ructure of commutative von Neum ann algebras: The selfadjoin t part of a commutative von Neum ann algebra consist s of all bounded measurable functions of a certain selfadjoint operator. The classification of nonab elian algebras was carried out in {34}. Murray and von Neum ann recognized t hat t he cente r of the algebra plays an import ant role in t he st ruct ure problem. T he cente r of a von Neum ann algebra M is a von Neumann algebra again and if it contains a project ion z , then M becomes the direct sum of zM and (I - z)M. Hence to decrease the complexity of an algebr a, one may assume that its center does not contain a nontri vial proj ecti on. A von Neumann algebra is called a fa ctor if its center is trivial, that is, if it cont ains t he mult iples of the identity operator only.

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On a von Neumann factor, the dimension function is unique up to a scalar multiple. Murray and von Neumann proved that there are the following possibilities for the range of the dimension function of projections:

(I n) {a, 1, (Ioo)

{O,l ,

,n}, where n is a natural number. ,n, .. . ,oo}.

(II d The interval [0,1]. (II 00) (III)

The interval [0, +00] . The two-element set {O, +oo}.

In this classification all von Neumann factors were found to belong to the classes type I, type II or type III. (However, it is worth mentioning that at the time of the discovery of the classification it was not known whether type III factors exist.) Factors are the building blocks of von Neumann algebras, hence the understanding of their structure has primary interest. According to the range of the dimension function of projections, a factor might be "trivial" , "regular" or "singular" . The trivial or type I is characterized by integer dimension, in th e regular or type II case the dimension function has a continuous range and the singular or type III case is free of finite nonzero projections. To investigate the type I and type II cases Murray and von Neumann could utilize the dimension function; however , that tool was insufficient for type III factors . To have a feeling about the "singularity" of type III factors , one can think of a measure space in which all nonempty measurable sets have infinite measure. The full understanding of the type III case needed half a cent ury. Ergodic theory was the first source of factors. Classification of von Neumann algebras is strongly related to conjugacy classes of transformations of measure spaces. The Tomita- Takesaki theory provided the new tools and revolutionized operator algebras in the 1970's. (The book {55} by Serban Stratila and Laszlo Zsido is a suggested introductory reading about von Neumann algebras.) Factors of type I are characterized by the existence of minimal projections . If a maximal pairwise orthogonal family of minimal projections has cardinality n, then the factor is isomorphic to B(1t), where H is a Hilbert space of dimension n. In particular, for every sEN U { +00}, there exists only one factor of type Is up to isomorphism. The existence of factors of

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type II and type III is not at all apparent, however. Murray and von Neumann constructed factors of type III and type II 00 by means of ergodic theory in {34}. Below we describe a method called "group measure space construction". This construction yields factors of different type. Let (X, B, f.L) be a measure space and let G be a countable group of measure-preserving transformations of X . The group measure space construction yields a von Neumann algebra acting on the Hilbert space L 2(f.L) i2l l 2(G), which is regarded as a set of functions ~ defined on G and with values in £2(f.L). (In this identification 8g i2l f corresponds to 8g x f for 9 E G and f E L 2(f.L ).) For every f E £00(f.L) define a bounded operator M] acting on L 2 (f.L) i2l z2 (G) as

(14)

((MJO(g)) (x) = f(g-lx) (~(g)(x))

(~ E £2(f.L) i2l l 2(G), 9 E G)

and for every 9 E G we define a unitary Vg by the formula

(15) Let M(f.L, G) be the von Neumann algebra generated by the operators

Then the choice of the unit circle with Lebesgue measure and (the powers of) an irrational rotation yields a factor of type II 1 . The real line with Lebesgue measure and the rational translations give a factor of type II 00 ' A factor of type III was const ruct ed only in the third paper of the "Rings of Operators" series {42}. Von Neumann modified the above measure theoretic procedure by allowing measurable transformations preserving measure 0, nowadays they are called nonsingular transformations. In this way he produced a factor of type III from the Lebesgue measure of the real line and the group of all rational linear transformations. (Although Murray and von Neumann used the group measure space construction for th e production of factors, which are called Krieger factors nowadays, the difficult question of isomorphism of factors that arose from different actions was clarified only 40 years lat er {28}. Krieger proved that two ergodi c nonsingular transformations of a Lebesgue space give rise to isomorphic factors if and only if the two transformations are orbit equivalent.) Von Neumann believed that among all factors the case II 1 has the strongest interest and expected that not all factors of type III are isomorphic to each other. Von Neumann preferred th e typ e III case for two main reasons. One of these is the nice behavior of the unbounded operators

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affiliated with a type II 1 factor. It is well-known that addition and multiplication of such operators are particularly troublesome. The crux of the difficulty lies in the unrelatedness of the domain and range of such an operator with the domain of another one. Much of the difficulties evaporates, however, if one considers selfadjoint operators with spectral resolution in a factor of type 111 , The other reason why von Neumann attributed great importance to continuous finite factors is that he interpreted this lattice as the proper logic of a quantum system. The lattice of projections of such a factor is modular, that is, in addition to the orthomodularity property (9), the stronger condition

p V (p' A q) = (p V p') A q for p::; q holds for every p' (and not only p' = pl.). (Non-modularity of the projection lattice of an infinite dimensional factor of type I was considered by von Neumann as a pathology of the usual Hilbert space quantum mechanics as a non commutative probability theory.) The paper "Rings of Operators IV" {36} has two important achievements concerning type II 1 factors. It is proved that there exist nonisomorphic type II 1 factors, and that there is only one hyperfinite type III factor. A von Neumann factor is called hyperjinite if it is generated by an increasing sequence of finite dimensional subalgebras. (Nowadays such algebras are preferably called approximately finite dimensional, or AFD for short.) The hyperfinite type III factor R may be produced in many different ways; for example, the above group measure space construction yields R . The uniqueness of R reminds us of the uniqueness of a finite, atomless separable measure space. Factors of type II 1 did not play much role in the theory of von Neumann algebras until recent years. After Jones founded his index theory {19}, the study of subfactors of type II 1 factors has received much interest. Even a concise review of index would require a lot of space (d. {23}) but its flavour is given below. Let N be a von Neumann algebra acting on a Hilbert space 1{ and having commutant N'. Assume that both Nand N' are type II 1 factors and let TrN and TrN' be the canonical normalized traces. For any vector ~ E 1{ the projection [N~J onto the closure of N~ belongs to N' and similarly [N'~J EN. The quotient

(16)

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is known to be independ ent of t he vector ~ and is called the coupling constant since the work of Murray and von Neum ann . In a certain sense the coupling constant is the dimension of the Hilbert space 'H with respect to the von Neumann algebra N. (When N == ceI, the coupling constant is the usual dimension of H , hence the not ation dimN (1t ).) V. J ones used the coupling constant to define t he size of a subfacto r of a finite factor. He was inspir ed by the notion of the index of a subgroup of a group, he therefore called this the relative size index. Let N be a subfacto r of a ty pe II 1 von Neum ann factor M possessing a unique canonical normalized trace TrM . T he index is obtained as the quotient (17)

[M : N]

= dimN(1t) .

dimM (1t)

The number [M : N] is not always an integer, and t he possible values of t he ind ex form t he following set:

(18)

{t E lR : t 2:: 4} U {4cos2( 1r/p) : p EN, P 2::

3}.

This is t he fund amental result of Jones which influenced a huge amount of subseq uent resear ch and renewed the almost forgot ten coupling constant. Vaughan F. R. Jones was awarde d t he Fields Medal in 1992 for discovering a surprising relat ionship between von Neum ann algebras and geometric topology (see {4} for a review). T he index th eorem was the first step towards his discovery. Construct ion of factors was t he main act ivity in t he field of operator algebras after the pap ers "Rings of operators" for many years . It is out of t he scope of this sur vey to summa rize t he constructions that were used to get more and more fact ors. Instead , we turn to the very end of the st ory. By th e time t he pap er "Rings of Operat ors IV" was published (year 1943) it was known t hat t he classes of ty pe I n, II 1 contain a unique (up to algebraic isomorphism) hyp erfinite von Neumann fact or. However , t he typ es II 00 and III remained unclear for many years. In 1956 Lajos Pukanszky const ructed two different factors of type III {47, 24}. After his breakt hrough infinitely many factors were const ructe d but the final classificat ion was not achieved until th e discovery of new invariants. Opera tor algebras achieved a revolutionary development in th e late 60's after a relati ve isolation of 30 years.

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Type III factors may be produced by means of infinite tensor product. Let M2(C) be the algebra of 2-by-2 matrices. Fixing 0 < A < 1 we can define a state rp on this algebra as follows .

rp(A) = Tr (AD) ,

where

D=

()'~1

o ), ) . ),+1

(The matrix D is called the density matrix inducing 'P .) A representation of the inductive limit of the n-fold tensor product of copies of M2 (C) can be constructed by means of tensor product states of copies of .p, (Th e socalled Gelfand-Naimark-Segal construction is involved here, but we do not give more details.) The generated von Neumann algebra is a hyperfinite factor. For A = 1, the type IiI factor shows up, for A = 0 we obtain a type l oo factor and for 0 < A < 1 a typ e III), factor R), app ears. In fact , R), is the only hyperfinite type III), factor. Confined to hyperfinit e typ e III), factors with 0 < A < 1 th e Connes spectrum is a complete invariant due to the results of Alain Cann es. He received th e Fields Medal in 1983 for his work on von Neumann algebras including the classification of type III factors , approximately finite dimension al factors and automorphisms of the hyperfinite type III factor {3}. After the work of Connes, the uniqueness of the hyperfinite type III 1 factor remained und ecided. This question was answered positively somewhat lat er by Uffe Haagerup {16}. (In the case of typ e III o, there are infinit ely many nonisomorphic hyperfinit e factors.) Quantum mechani cs influenced von Neumann to develop several ideas. He was the first person who summarized quantum theory in a comprehensive and mathematical form, his monograph [119] has been a standard reference in mathematical physics . Operator algebras consist of bounded operators but quantum mechanics needs unbounded ones. Von Neumann und erstood the importance of maximal symmetric operators on Hilbert spaces and introduced the ent ropy of statistical operators ([119] and {46}. The von Neumann entropy got a new information theoretic interpret ation recently. Bela Sz8kefalvi-Nagy was born in Kolozsvar in Transylvania on July 29, 1913. His father was also a mathematician and his mother was a high school teacher. In his scientific pap ers he did not use his full nam e but the abbreviat ion B. Sz.-Nagy. Native Hungari ans have always been surprised about the strange pronounciation of his name by foreigners . After World War I, the family moved to Szeged (Hung ary) . During his university studies Sz.-Nagy was deeply influenced by fugye s Riesz, B ela K erekjdrt6 and Alfred Haar. Von Neumann's book on the foun-

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dations of qu an tum mechan ics [119] and van der Waerd en 's book on gro up th eory and qu antum mechan ics were his favorite readings. This was the time when qu antum t heory revoluti onized both phy sics and mathemati cs. Between 1937 and 1939 Sz.-N agy sp ent some time in Leipzig, Grenobl e and Paris. From 1939 he worked for the University of Szeged; he became full professor in 1948. Sz.-Nagy had rather wide mathematical interest s. In one of his first pap ers he gave a new proof for Stone's t heorem about t he spectra l represent ati on of a strongly cont inuous one-pa ra mete r semigro up of uni t ary Hilb ert space operators . Such a semigroup U(t) is obtained in t he form

U(t) =

1:

iAt

e

dE (A ),

by means of a projection-valued measure E on th e real line. Lat er he extended t his result to semigroups of normal operators. He also wrote a very concise book on spectra l t heory. Generations learn t the spectra l t heorem from [1 77] published in 1942 by Springer Verlag. Alth ough Sz.-Nagy cont ribute d to t he t heory of Fouri er series and to approximation t heory, t he cente r of his interest was t he Hilb ert space and its linear opera tors . The basic exa mple of a Hilbert space is t he L 2 space, th e space of square integrabl e functions over a measure space . On top of th e st andard Hilb erti an structure L 2 has an order structure which is det ermined by t he cone of positive functions. In an early pap er Sz.-Nagy gave an abstract characterizat ion of t he positive cone. In ot her words, he listed t he requirements of a cone of an abstract Hilb ert space unde r which an isomorphism of the space wit h an L 2 space exists , such that t he positi ve cones correspond to each ot her. He also proved t hat an invertible Hilb ert space operator whose pos it ive and negative powers are uniformly boun ded is similar to a unitar y operator . The analysis of Hilb ert space operators mostly concerns some particular classes of operat ors such as self-a djoint, unit ary etc . The highlight of t he scient ific activity of Sz.-Nagy was t he th eory of cont ractions. It started with t he uni t ary dilation t heore m obtained in 1953. Let H be a Hilb ert space and let T be a genera l bounded linear operator. Hence IITII is finit e an d mult iplying T by some constant we can achieve that IITII ::; 1. (Such a T is called a cont raction.) T he dilation t heorem says that t here exist a Hilb ert space K ::J 'H and a unit ar y opera tor U on K such t hat

T"]

= Pll"]

and

(T*tf

= PU - nf

(J E 1i)

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for any n E N, where P denotes the orthogonal projection from K onto 'H. The space K could be the direct sum of infinitely many copies of 'H and U can be written in the form of an infinite matrix (with operator entries) as follows : I

U=

o o o

0 D* -T*

0

o

0

T 0 D 0

o

I

where D = (I - T*T)1/2 and D* = (I - TT*)1/2 . Since the structure of a unitary operator is rather well-understood, the contractions could be investigated through the dilation.

In the study of contractions Sz.-Nagy had a longstanding cooperation with Ciprian Foias from 1956 to the end of his life. They wrote together 50 papers and the monograph [44]. An interesting class of operators is formed by the completely non-unitary contractions, they do not act unitarily on any subspace. The class Co is formed by the completely non-unitary contractions T for which there exists a function 0 i= w E H'" such that w(T) = O. An operator T E Co has the following remarkable properties: (1) For every vector

l , T"] --t 0 and

(T*)n

--t

0 as n

--t 00.

(2) T has a nontrivial invariant subspace. Recall that a linear operator T of a finite dimensional space always admits a polynomial p such that p(T) = O. The definition and several properties of the class Co resemble the finite dimensional scenario . Sz.-Nagy and Foias found a quasisimilarity model for the Co-contractions and a unitary equivalence model for arbitrary completely non-unitary contractions. Their lifting theorem is connected with the minimal isometric dilation of a contraction T. Let T; be a contraction acting on a Hilbert space 'Hi and let Vi be the minimal isometric dilation of T; acting on the space Ki , i = 1,2. If a bounded linear operator X from the space 'H 1 to 'H2 has the property T2X = XT1 , then there exist an operator Y from the space J(l to J(2 such that V2Y = YV1 . The lifting theorem of Sz.-Nagy and Foias extends earlier results of T. Ando and D. Samson. Many applications are known, in particular to interpolation problems.

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[44]

Foias, Cipri an - Sz.-Nagy, Bela, Harmonic Analysis of Operators on Hilbert Space, North-Holland Publishing Co. (Amsterdam, 1970).

[62]

Haar, Alfred , OsszegyujtOtt Munkai = Gesammelte Arbeiten, ed. Bela Sz.-Nagy, Akademiai Kiado (Budapest, 1959).

[118J

Neumann, Janos (John von Neumann) , Collected Works, 6 volumes, ed. A. H. Taub, Pergamon Press (New York, 1961).

[119]

Neumann, Johann von, Mathematische Grundlagen der Quantenmechanik, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Band XXXVIII, J. Springer (Berlin , 1932).

[142]

Rado, Tibor, On the Problem of Plateau, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 2, No.2, Springer-Verlag (Berlin , 1933).

[144]

Rado , Tibor, Length and Area, Amercian Mathematical Society Colloquium Publications, Vol. XXX. (1948).

[145]

Rado , Tibor - Reichelderfer, P. V., Continuous Transformations in Analysis with an introduction to algebraic topology, Die Grundlehren der Mathematischen Wissenschaften in Einzedarstellungen, Band LXXV, Springer-Verlag (Berlin-Cottingen-Heidelberg, 1955).

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[157]

Riesz, Frigyes -Sz.-Nagy, Bela, Lecons d'Analyse Fonctionnelle, Akademiai Kiado (Budapest, 1952). English translation: Functional Analysis, Ungar (New York, 1955).

[166]

Stein, Elias M., Harmonic Analysis: Real Variable Methods , Orthogonality and Oscillatory Integrals, Princeton Mathematical Series, 43, Princeton Unversity Press (Princeton , NJ, 1993).

[177]

Sz.-Nagy, Bela, Spektraldarstellung linearen Transformationen des Hilbertschen Raumes, Ergebniss e der Mathematik und ihrer Grenzgebiete, Band 5, No.5 . Springer-Verlag (Berlin , 1942).

= Gesammelte Arbeiten,

{1} N. 1. Akhiezer, The classical moment problem, Hafner Publishing Company, 1965. {2} G. Alexits , tiber die Erweiterung einer Baireschen Funktion, Fund. Math ., 15 (1930), 51-56. {3} H. Araki , The work of Alain Connes , Proceedings of the Int . Congress of Mathematicians, Warszawa 1983 (ed.: Z. Ciesielski, C. Olech) , PWN - Polish Scientific Publisher, 1984. {4} J. S. Birman, The work of Vaughan F. R. Jones , Proceedings of the Int. Congress of Mathematicians, Kyoto 1992 (ed.: 1. Satake) , Springer, 1992,9-18. {5} A. P. Calderon and A. Zygmund , On the exist ence of certain singular integrals, Acta Math ., 88 (1952),85-139 .

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{6} A. Connes, Une classification des facteurs de type III, Ann. Sci . Ecole Norm. Sup ., (4) 6 (1973), 133-252 . {7} A. Connes , Classification of injective factors, Ann. Math., 104 (1976), 73-115. {8} Csaszar A., Megjegyzes Geocze Zoard fiiggvenyehez [Remark to Z. Ceocze's function], Mat. Lapok, 8 (1957), 268-271. {9} D . Th . Egoroff, Sur les suites de fonctions mesurables, C.R . Acad. Sci . Paris , 152 (1911),244-246. {1O}

Ceficze Z., Folytonos rendszert kepezo sikgorbek ivhosszar6l [On the arc length of plane curves forming a continuous system] , Az Ungvari Realiskola 1904/05. evi ertesitoje, 32 pp .

{11} Z. Geocze, Recherches generales sur la quadrature des surfaces courbes, Ungar. Ber ., 27 (1911), 1-21 and 131-163, 30 (1914), 1-29. {12} Z. Ceocze , Sur la quadrature des surfaces courbes, C.R . Acad. Sci. Paris , 154 (1912),1211-1213. {13} Geocze Z., A zerus teriiletfi feliiletrol [On the surface of zero area], Mathematikai es Termeszetiudoiruiruji Ertesito, 33 (1915), 730-748 . {14} Geocze Z., A rectifiabilis feliiletrol [On the rectifiable surface], Mathematikai es Termeszetiudottuitun Ert esito, 34 (1916), 337-354, 587. {15} Geocze Z., A feliilet teniletenek Peano-fele definitiojarol [On Peano's definition for the area of a surface], Mathematikai es Termeszetiudonuinui Ertesito, 35 (1917), 325-358. {16} U. Haagerup, Connes' bicentralizer problem and the uniqueness of the injective factor of type Lll i , Acta Math ., 158 (1987), 95-147. {17} A. Haar, Der MaBbegriff in der Theorie der kontinuierlichen Gruppen, Annals of Math ., 34 (1933), 147-169; [62] H2, pp . 600-622. {18} H. Hahn, Reelle Funktionen, Akademische Verlagsgesellschaft, Leipzig, 1932. {19} V. Jones, Index for subfactors, Invent. Math., 72 (1983), 1-25 . {20} P. Jordan , E. Wigner and J . von Neumann, On an algebraic generalization of the quantum mechanical formalism, Ann. of Math., 35 (1934), 29-64; [118] . {21} Kantor S., Ceocze Zoard fiiggvenye mindeniitt folytonos, de sehol sem differencialhato [Z. Geocze's function is everywhere continuous but nowhere differentiable], Mat . Lapok, 8 (1957), 264-267 . {22} L. Kerchy and H. Langer, Bela Szokefalvi-Nagy, in Recent Advances in Operator Theory and Related Topics . The Bela Szokefalvi-Nagy Memorial Volume, 11-38, eds . L. Kerchy et al, Birkhauser, 2001. {23} H. Kosaki, Index theory for operator algebras, Sugaku Expositions, 4 (1991), 177197. {24} Kovacs 1., Pukanszky Lajos es a faktorok elmelete [Lajos Pukanszky and the theory of factors], Mat. Lapok., 6 (1996), 28-39 . {25} 1. Kovacs and J . Szucs, Ergodic type theorems in von Neumann algebras, Acta Sci . Math . Szeged, 27 (1966), 233-246 .

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{26} Gy. Konig, Uber ste tige Funktionen, die innerh alb jedes IntervaIIes ext reme Werte besitz en, Monat shefte f. Math ., 1 (1890),7-12. {27} Konig Gy., A hat aroz ott int egralok elme!etehez [On th e th eory of definite int egrals], Mathemat ikai es Terme szett udotruinui Eriesiiii, 15 (1897), 380-384 . {28} W . Krieger , On ergodic flows and the isomor phism of fact ors, Math . Ann., 223 (1976), 19-70. {29} P. Lax , Functi onal analysis, J ohn Wiley, 2002. {30} N. Macrae , John von Neumann. The Scientifi c Genius Who Pion eered the Modern Comput er, Gam e Th eory, Nu clear Deterrence and Much More, American Mathematical Society, 1992. {31} E. J. Mickle and T. Rado, Density t heorems for outer measures in n-space, Proc. Amer. Math. Soc., 9 (1958), 433-439. {32} E. J . Mickle and T . Rado, On redu ced Caratheodory outer measures, Rend. Circ. Mat . Palermo (2), 7 (1958), 5- 33. {33} E. J . Mickle and T. Rado, A uniqu eness theorem for Haar measure, Trans. Amer. Math. Soc., 93 (1959), 492- 508. {34} F . J . Mur ray and J . von Neum ann , On rings of opera tors , Ann. of Math., 37 (1936), 116-229 ; [118J Vol. III ., No.2, 6-119. {35} F. J. Murray and J. von Neumann, On rings of operators II , Trans. Amer. Math . Soc., 41 (1937), 208-248; [118] Vol. III ., No.3, 120-160. {36} F. J. Murray and J . von Neum ann, On rings of operators IV, Ann. of Math ., 44 (1943), 716-808; [118J Vol. III ., No.5 ., 229-32l. {37} J . von Neumann , Zur Algebr a der Funk tionaloperatoren und T heorie der norm alen Op eratoren , Math. Ann ., 102 (1929), 370-427; [11 8] Vol. 11. , No.2, 86- 143. {38} J . von Neumann, Die Eind eutigkeit des Schrodingerschen Op eratoren, Math . Ann., 104 (1931), 570-578; [118] Vol. 11., No.7, 221-229. {39} J. von Neumann , Zum Haarschen MaB in Topologischen Gruppen, Compos. Math. , 1 (1934) , 101-114 ; [118J Vol. 11. , No. 22, 220-229. {40} J . von Neumann , The un iqueness of Haar's measure, Matem. Sbor., 1 (1936), 721734; [118] Vol. IV., No. 6, 91-104. {41} J. von Neumann , On algebraic generalizat ion of th e quantum mechani cal form alism (Part I) , Matem . Sbor., 1 (1936), 415-484; [118] Vol. III. , No.9, 409-444. {42} J . von Neumann, On rings of operators III , Ann. Math ., 41 (1940), 94-161 ; [118] Vol. III ., No.4, 161-228. {43} J. von Neumann, Invariant m easures, American Mathemat ical Society (Providence, 1999). {44} M. Ohya and D. Petz, Quantum Entropy and Its Use, Springer-Verlag, Heidelb erg, 1993. {45} D. Petz and M. Redel, John von Neum ann and the th eory of operat or algebr as , in: Th e Neumann Compendum, eds. F . Brodi , T . Vamos, World Scientific Series in 20th Cent ury Math. vol. 1, World Scientific, Singapore, 1995, 163-1 85.

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{46} D. Petz, Entropy, von Neumann and the von Neumann entropy, in: John von Neumann and the Foundations of Quantum Physics, eds. M. Redei and M. Stoltzner, Kluwer, 2001. {47} L. Pukanszky, Some examples of factors, Pub!. Math. Debrecen, 4,135-156 (1956). {48} T . Rado , Sur le caicul de I'aire des surfaces courbes, Fund. Math., 10 (1927), 197210. {49} Rado T ., A felszinmeres elmeletehez [On the theory of measuring surface area], Mathematikai es Termeszettudonuinui Ertesito, 45 (1928), 225-244. {50} T . Rado , Sur I' aire des surfaces continues, Atti. Congr. Internaz. Bologna, 6 (1928), 355-360. {51} T . Rado, Uber das Flachenrnass rektifizierbarer Flachen, Math. Ann., 100 (1928), 445-479. {52} T . Rado , On th e derivative of the Lebesgue area of continuous surfaces, Fund. Math., 30 (1938), 34-39. {53} T . Rado, On continuous path-surfaces of zero area, Ann. of Math., 44 (1943), 173-191. {54} T . Rad6 , On surface area, Proc. Nat. Ac. Sci. U.S.A ., 31 (1945), 102-106. {55} F. Riesz, Sur une espece de geometrie analytique des systemes de fonctions sommables, Comptes Rendus, Acad. Sci. Paris , 144 (1907), 1409-1411 ; [156] C4, pp . 386-388. {56} F . Riesz, Sur les operations fonctionnelles lineaires, Comptes Rendus, Acad. Sci. Paris , 149 (1909), 947-977; [156] C7, pp. 400-402. {57} Riesz F ., Integralhato fiiggvenyek sorozatai [Sequences of int egrable functions], Math . es Phys . Lapok, 19 (1910), 165-182; [156] C9, pp. 407-440. {58} F . Riesz, Untersuchungen iiber Systeme integrierbarer Funktionen, Math. Annalen, 69 (1910), 449-497; [156] ClO, pp. 441-489. {59} F. Riesz, Sur Ie systemes orthogonaux de fonctions , Comptes Rendus, Acad. Sci. Paris, 144 (1907), 615-619; [156] C2, pp. 378-381. {60} Riesz F., Linearis fiiggvenyegyenletekrol , Mathematikai es Termeszettudonuiruii Ertesito, 35 (1917),544-579; [156J F5 , pp . 1017-1052 . {61} F . Riesz, Uber lineare Funktionalgleichungen, Acta . Math., 41 (1918),71-98; [156] F6, pp, 1053-1080 . {62} J . Rosenberg, A Panorama of Hungarian Mathematics in the Twentieth Century: Non -Commutative Harmonic Analysis, this volume . {63} S. Saks , Theory of the integral, Nak!. Pols. Towarzystwa Mat emarycznego, Warszawa , 1937. {64} I. Segal, On non-commutative extension of abstract integration, Ann. of Math., 57 (1953),401-457. {65} $. Stratila and L. Zsido, Lectures on von Neumann algebras, Abacuss Press, Tunbridge Wells, 1979.

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{66} L. Tonelli, Sulla quadratura delle superficie, Atti Accad. Naz. Linc ei (6) 3, 357-363; (1926), 445-450 and 633-658. {67} P. Veress, Uber kompakte Funktionenmengen und Bairesche Klassen, Fund. Maih ., 7 (1925), 244-249.

Akos Csaszar

Denes Petz

Alfred Renyi Institute of Mathematics P.G .B . 127 Budapest H-1364 Hungary

Budapest University of Technology and Economics Department of Analysis Budapest H-1111 Egry J. u. 1. Hungary

csaszar~renyi .hu petz~math .bme .hu

A Panorama of Hungarian Mathematics in the Twentieth Century, pp . 245-294.

BOlYAI SOCIETY MATHEMATICAL STUDIES, 14

DIFFERENTIAL EQUATIONS: HUNGARY, THE EXTENDED FIRST HALF OF THE 20TH CENTURY

I

!ARPAD ELBERT and BARNABAs M. GARAY

Introduction. It is well known that the theory of differential equations does not belong to the most important chapters in the history of Hungarian mathematics. Yet, when making preparations for this paper, both of us were astonished to realize how many prominent Hungarian scholars had been concerned with the theory of differential equations, even if marginally, and how much they had been aware of the relations of their primary fields to our topic.

Summary. We give a detailed account on • the relation between Fejer 's summation theorem and Dirichlet's problem on the unit disc; • Fejer's work in stability theory (in connection to his habilitation lecture); • F. Riesz' subharmonic functions ; • Haar's inequality for partial differential equations of the first order; • the Haar-Rado results in the calculus of variations (with a particular emphasis on the minimal surface problem) • what is called 'von Neumann's stability analysis' and the underlying Lax equivalence "consistency & stability ¢:} convergence"; • Lax's contribution to the theory of conservation laws (a field of research he entered under the influence of Neumann's interest in shock waves); • M. Riesz' theory of fractional potentials; • the work of Polya and Szego on isoperimetric inequalities.

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The concluding pages are devoted to differential equations in Hungar y afte r t he second world war. As before, t he emphas is is placed on •

resul ts pr oviding a significant cont ribution to classica l problems (like Bihari's 1956 inequality, the first nonlinear version of Gronwall 's Lemma) ;

•

resul ts which pave the way to modern theories (like the 1950-60 contributions by Renyi an d Barna to t he eme rging t heory of int erval maps). In connection to some statist ical data from t he

•

first decad e of t he twe nt iet h cent ury;

• years 1928 and 1953 (re present ing t he period before and afte r t he second world war ) a couple of general rem arks are also mad e. At the turn of the century. At t he beginning of t he twent ieth cent ury t h.e general t heory of differenti al equations was closer to physics, and within t his, to mechani cs t han it is nowad ays. Let us mention here that even t he famous pap er on linear inequalities by Gyul a Far kas had its roots in t he onesided const raints of mechan ics. In t he volumes of Mathemati sche Annalen between 1900 and 1910 34 art icles (among approx. 450) had Hungari an aut hors . Ou t of them 16 are concerne d with differential equat ions in a wider sense, including 8 pap ers on physi cal applications, nam ely: 4 papers by Mar Rethy discuss the variational principles of mechan ics, 1 pap er by Gyozo Zemplen t reats hydrodyn ami cs, another is on elect rody na mics, 1 paper by Gyul a Far kas is on shock waves and, finally, 1 pap er by Lipot Fejer is concerned wit h t he variational princip les of mechanics. The last one will be discussed in detail later. Three pap ers on linear ord inary different ial equations by Lajos Schlesinger come under pure mathematics, definit ely. One pap er by Kar oly Goldziher and 4 pap ers by J ozsef Kiir schak , who is known mainly as an algebraist, are on partial differenti al equations. E. g. Kiirschak provides a new proof to a theorem of Lie according to which, under certain compatibility condit ions, Monge-Ampere equations are t ra nsferred into Mongo-Ampere equations by contact tran sformations {44}. Although t he authors received t heir most important professional stimulations from abroad, even, while staying abroad, t hese resul ts could not have been achieved wit hout certain Hungarian scholarly antecedents. Gyul a Konig taught courses on differenti al equat ions regularly at t he Budap est University of Technol ogy, so did Gyul a Valyi at t he University of Kolozsvar . Their most important results obtain ed in the field of par ti al differenti al

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247

equations were highly appreciated by the international scientific community. The studies in partial equations of second order by Valyi, Konig and J6zsef Kiirschak, a follower of Konig, belong to the theory of formal integrability. Their main aims were to set up compatibility criteria on the reduction of equations (i.e. to a system of partial equations of the first order or that of ordinary ones). At the Budapest University of Technology Cusztav Kondor, who belonged to an earlier generation and had much more modest abilities than Valyi and Konig, also read lectures on differential equations. Among the predecessors, the name of J6zsef Petzval, a pioneer of photography, who left the chair of the University of Budapest for that of Vienna University in 1837, should also be mentioned. His two-volume work on differential equations was highly appreciated all over the Austro-Hungarian Monarchy. The papers mentioned, as well as the preliminaries discussed, represent demonstratively the state of the theory of differential equations in the early twentieth century: it had a very close relation to various branches of physics and there was a great effort to solve equations in a closed form. Even in our days it seems to be surprising how many of them could be solved by integral representations and series expansions in terms of elementary and special (higher transcendental) functions. By 1900 the development of the general theory of linear ordinary differential equations was already at a fairly high level. This can be exemplified, primarily, by the generalization of Galois theory, widely known in the field of algebraic equations, to linear ordinary differential equations (whose coefficients are meromorphic functions) . On the other hand, at that time only the germs of a general theory of nonlinear ordinary differential equations, namely, the existence theorems of Peano and Picard, and elements of the qualitative theory began to emerge. But the number of known nonlinear types of equation which could be solved by quadrature became several dozens. As far as partial differential equations are concerned, we cannot speak about a general theory at all, except for the equations of first order. Here emphasis was laid on the concrete solutions of initial-boundary value problems for linear and, to a much lesser extent, for nonlinear equations closely connected with physical applications.

Lajos Schlesinger and his work. In the history of differential equations in Hungary Lajos Schlesinger holds a special and distinguished place for two reasons: he was the first mathematician in Hungary whose prime field of activity was the study of abstract differential equations, namely that of the linear ordinary differential equations, during most of his life and who gained an international fame and reputation for this. His major work, a

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two-volume monograph on linear ordinary differential equations {73} was published again by the Johnson Reprint Corporation in 1968. While the activities and lives of Gyula Konig and Gyula Valyi are discussed by Barna Szenassy in his excellent book on the history of Hungarian mathematics of early times, (i.e. before the twentieth century) in detail, the life and work of Lajos Schlesinger are not treated at all since most of his activity took place after 1900. Lajos Schlesinger was born in Nagyszombat in 1864. He started studying mathematics at Heidelberg, finished his studies in Berlin , and took his doctor's degree with Lazarus Fuchs whose daughter he married (and, thus, he confirmed a peculiar law of genetics according to which the genes carrying mathematical abilities are passed from a father-in-law to a son-in-law. Another example of this is Aurel Wintner born and educated in Hungary, who married Otto Holder's daughter) . Lajos Schlesinger spent most of his life in Germany and died as Professor of the University of Giessen in 1933. His activities are attached closely to his father-in-law's. Similar to the latter, he used methods of the theory of analytic functions to a greater extent and those of group theory to a lesser extent in the theory of ordinary differential equations. Further on, he wrote a great number of papers on Fuchsian equations and Fuchsian groups and published the correspondence of Fuchs and Weierstrass. By the evidence of his references, based on studying and processing almost 2000 professional articles, he wrote his survey {74} on the development of the theory of linear ordinary differential equations from 1865 on, the publication year of his father-in-law's famous paper. Together with Abraham Plessner he wrote a remarkable book on Lebesgue integrals and Fourier series and took part in processing Gauss's unpublished manuscripts. Undoubtedly, he had a very rich career in mathematics. In a late paper related to the solution of the differential equation i = C(t)z where z E eN and C is a complex-valued matrix function , he became the forerunner of the theory of product integrals (i.e. of generalizations of Lie's matrix formula e A +B = limn~oo (eA/neB/nf). This activity of Schlesinger is mentioned with admiration by Felix Browder, too, in the preface to the Mathematical Encyclopedia volume 'J . D. Dollard & C. N. Friedman, Product Integration with Applications to Differential Equations, Addison-Wesley, Reading, Mass., 1979'. Lajos Schlesinger was Professor at the University of Kolozsvar between 1897 and 1911. Among Hungarian mathematicians he stimulated Mario Beke and his impact can be felt on a paper by Lipot Fejer. Beke investigated the irreducibility of homogeneous linear ordinary differential equa-

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249

t ions the coefficient s of whi ch ar e rational fun ctions. Also , for the equat ions with such coefficients the conce pt of irr educibility it self was introduced by him {3}. (An equation is irreducible if it has no solution common with a homogeneous linear ordinary differenti al equation which is of a lower order and has coefficients of the sam e class .) With a highly effective application of Cauchy 's majorant method , Fejer {21} gave a new proof t o Fu chs 's t heorem on the singularities of the solut ions of homogeneous lin ear differential equat ions. For Hungarians, Schlesinger 's reputation is indicat ed by the fact that he becam e the subject of the well-known anecdote attached t o Henri Poincare's visit to Budapest. In 1905 Poincar e came to Budapest to receive the Bolyai Prize. At his arrival his reply to the greetings of the notables who were meeting him was as follows: 'Thank you! But where is Fejer?' Similar to the legends about the lives of medi eval saints, this story has become extant in another version in which the name of Schlesinger replaces that of Fejer 's, Indeed , it is conce ivable t ha t Poincare want ed to meet Schlesinger sin ce he himself had been concerned with homogeneous linear ordinary differential equat ions with rational coefficients . It was he who gave the nam e of Fu chsian fun ctions, in honour of Immanuel Lazarus Fuchs , t o a certain class of au tomorphic fun ctions which played an important role in t he int egration of t he afor em entioned equations.

Fejer summation theorem and the Dirichlet problem on the unit disc. In the first decade of t he twent iet h cent ury Lipot Fejer was concerned with the theory of ordinary differential equat ions in a wider sense in sever al papers. Pal Turan mentions in t he Introduction t o Lipo t Fejer 's Collected Works that the discovery of the famous Fejer's summation t heorem is closely related to the Dirichlet problem on the unit disc B = { (x, Y) E JR2 I x 2 + y 2 ~ I} . Given a continuous fun ction 9 : 8B -+ JR, do es there exist a cont inuous fun ction u : B -+ JR with the properties that u is harmonic on B \ 8B (i.e. u is twice cont inuously differentiable on B \ 8B and satisfies 6.u = 0 on B \ 8B ) and ulaB = g? The positive answer , together with t he form of the solution function

ao/2 (1) u (r costp , r sintp) =

+

f

(ak cos kip + bk sin k tp )r k if r < 1

k=l

{

g(tp)

if r = 1

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(where ao, al ,b 1 ,a2,b 2, .. . are th e corresponding Fourier coefficients) had already been guessed earlier but only H. A. Schwarz succeeded in proving it with th e help of t he so-called Poisson int egral representation of the series expa nsion

L

1 ao/2+ (ak cos kep+bk sin ksp )r k = k=l 271" 00

1 2

0

11"

1- r 2 ( 2 g('ljJ) d'ljJ 1 - 2r cos ep - 'ljJ ) + r

valid for r < 1. Fejer spent the aca demic year 1899/1900 in Berlin where he was influenced greatly by the lectures of H. A. Schwarz. Wh en discussing t he Dirichlet problem on the uni t disc Schwarz stated that it would be exped ient to give an existe nce proof by the theory of series exclusively, moreover he spoke about the unsuccessful at te mpts made in tha t direction. The problem was th at du e to th e possible divergence of t he Fourier series of functi on 9 the Abel summation t heorem could not be applied . The way out of t he sit uat ion was t hat - at t he points of BE - cont inuity of t he solut ion function defined by formula (1) followed from a new summation procedure. Fejer was given the decisive imp etus to frame a new summation pro cedure by t he theorem of Frobeniu s according to which Abel's convergence assumpt ion on 2:: Ck for the existe nce of limr-+l- 2:: Ckrk can be weakened to assuming the convergence of L Sn where Sn = (co+Cl + . .. + cn)/(n + 1), n E N. All these considerat ions led Fejer t o prove that the arithmetic means of the par tial sums of the Fouri er series of cont inuous 271"-periodic functions are uniformly convergent . Compar ed to his revolutionary discovery the origina l questi on posed, i.e. to prove the solvability of t he Dirichlet problem on t he uni t disc by the theory of series exclusively, remained ent irely in the background.

In Fejer 's famous Comptes Rendus note {13} t here is only one sente nce which indicates that his summa tion theorem is also applicable to the theory of Poisson integrals. He worked out the det ails in two sepa rate pap ers published in Hungary. The first one {14} discusses the Dirichl et problem on the unit disc. The second one {15} tr eats the heat equation Ut = U x x equipped wit h t he initi al condit ion u(O, x) = g(x) where 9 : IR ---t IR is a 271"-periodic conti nuous functi on. The results of these two short pap ers are included in Section 3/ a of an extensive pap er on his sum mation theorem he publi shed in Mathematische Annalen in 1904 {16}. The work of Fejer in mechanics and his habilitation lecture. Between 1905 and 1911 Lipot Fejer worked at the Depar tment of Mathematics

Differential Equ ations: Hun gary, th e Extend ed First Half of th e 20th Cent ury

251

and Physics of the University of Kolozsvar . This must explain the fact that he chose the topic of his "habilitation" lecture not from th e theory of Fourier series but from the stability th eory of ordinary differential equations. First , he gave an outline of the definitions of stability used in those days (st rangely enough, including the recurrence prop erty as well): 'T he concept of st ability carries highly different contents even within the framework of mass poin t systems. It is no use arguing which one of them is the best since, except for some inherent features of it , st ability as a popular concept is so indefinite and so relative that, owing to th e variety of existing relations, stability definitions highly differing from one another may be formulat ed without getting into cont radict ion with the popular concept '. Among th e definitions of stability listed by him we can find the one accepted in general nowadays but it is considered too narrow by Fejer, joining Felix Klein's opinion. Then, he discusses some simpler asp ects of th e three-body probl em and finishes his habilitation lecture with th e discussion of the Lagrange-Dirichlet theorem.

In connect ion with his habilitation lecture {19} Fejer published a finding of his own in the problem of equilibrium instability {IS} , {20}. (Ref. {20} is a word-for-word German translation of the Hung arian original {IS}.) Slightly changing notation s, Fejer investigated equation

(2)

f.. = grad 1f(r) - iJ( Id)l id

t: = (x , y , z) E 1R 3

under t he conditions below: The potenti al function tt : 1R 3 ---t IR is an alytic in a small vicinity of the origin Q and its Taylor series about Q begins with a negative definit e 3-variable homogeneous polynomi al of order 2n for some int eger n 2: 1 (in particular , th e potential function tt has an isolated maximum at Q) , f : IR ---t IR is a cont inuous increasing function, f(O) = 0, f (v) > 0 for v > 0 and lim sup f( v)lv <

(3)

00 .

V'- O+

Starting from Jacobi's identity

22 (~lrI2)

:t

=

lil + (grad 1f(r),r) -

(t ,r)f( lrl) lid

Fejer proved via elementary ad hoc inequalities that the equilibrium state

Q is unstable. He raised the question of whether t he result of instability will remain true without t he condition (3), e.g. for the function f (v ) = vet , 0:

E (0,1). The positive answer is a consequence of La Salle's invariance

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Elbert and B. M. Garay

principle elaborated more than 50 years later: A 1968 theorem of Luigi Salvadori (see Thm . III. 5.8 as well as the accompanying discussion) in 'N. Rouche, P. Habets & M. Laloy, Stability Theory by Liapunov's Second Method, Springer, Berlin, 1977' can be directly applied. Another paper of Fejer's {17} published in 1906 is related to the socalled Ostwald principle. In his work entitled 'Lehrbuch der allgemeinen Chemie' published in 1893 Ostwald stated that 'von allen moglichen Energieumwandlungen wird diejenige eintreten, welche in gegebener Zeit den grosstrnoglichen Umsatz ergibt' (from among all possible conversions of energy the one which produces the greatest increment in the given time will happen). For a given planar potential field U (r) and starting point [0 Fejer studied such equations of motion

£. =

- grad U(r)

for which Case 1 U(ro) - U(r(t)) is maximal for a single and fixed t = To> aor, alternatively, Case 2 U(ro)-U(r(t)) is maximalfor all t E [0, To]. He stated that in Case 1 the initial velocity t(O) has to meet some conditions of compatibility and the motion itself will be brachistochronal, and in Case 2 the orthogonal semitrajectory through ro should be a half-line and t(0) should be parallel to the direction vector of this half-line. Otherwise, Ostwald wanted to demonstrate the general validity of his principle by the motions along half-lines obtained in Case 2 . In spite of the great prestige and merits of Ostwald - he was awarded a Nobel Prize in chemistry in 1909 - this energy principle had already been highly criticized. All parties in the debate emphasized that the original phrase 'von allen . .. ' provided several opportunities for different mathematical interpretations. Most of them, including Mar Rethy {66}, investigated which modifications and improved versions of the Ostwald principle were in harmony with the general laws of mechanics and other physical sciences. On partial differential equations, more precisely on the equation of the form n

(4)

L i,k=l

n

aik(x)uX i X k

+ L br(x)ux r + c(x)u =

0,

aik = aki

r=l

Lipot Fejer wrote only a single article {22}. He assumed that the coefficients were analytic functions and c(Q) < O. By using the Cauchy-Kowalevskaya theorem he proved that (4) had a solution having a positive maximum at

253

Differential Equations: Hun gary, the Extend ed First Half of the 20th Century

Q if and only if matrix { aik(Q) }7,k=l is not positive semidefinite. This is a cont ribution to t he t heory of maximum principles. F. Riesz' subharmonic functions. Out of the papers of Frigyes Riesz those on subharmonic functions are related to the theory of differential equat ions dir ectly. The concept of subha rmonic functions was int roduced by Riesz in a lecture in Sto ckholm in 1924 {68}. The definition is as follows: Let n c jRn be open and connected, and f : n ---t [-00, (0) an upper semicont inuous fun ct ion , f (x) =I - 00 for at least one x E n. The function f is subharmonic if for any pair (n', u), t he inequ ality f lan, ~ ulan'

implies t hat fi n' ~ u. Here n' is a bounded domain , n' = cl (n') c n, u : n' ---t jR is a cont inuous function which is harmonic on n'. T he definiti on of subharmonic functions was inspired , partl y, by some charac te ristics of analyt ic functions discovered by Hard y and Landau and, par tly, by Perron's method, which was elaborated for proving the existe nce of classical C(n) n C 2 (n ) solutions to t he Diri chlet boundar y valu e pr oblem Dou = 0, ulan = 9 (where 9 : ---t jR is a given cont inuous functi on and c jRn is a bounded domain having a 'nice' boundary.) Prior to his lecture in Sto ckholm Riesz had generalized {67} t he result s of Hardy and Landau , and in a joint pap er wit h Tibor Rad6, he gave a simplificati on of Perr on 's method {72} . The core of P erron's method as conceived by Riesz and Rad6 is t he following assert ion: If n c jRn is a bounded domain and 9 : an ---t 1R is a continuous functi on , t hen

an

n

u(x ) = sup { v(x) I v E C(n) , vlan ~ 9 and vln is subharmo nic} defines a harmonic function on n. (If an is nice e.g. if it is of class C 2 or satisfies t he outer sphere condit ion, then u E C(n) and ul an = g.) Connections t o the celebrate d Fejer-Riesz proof of Riemann 's conformal mapping theorem (published as part of the Introduction in a pap er by Rad 6' (Acta Sci. Math. (S zeged), 1 (1922/23) , 240-251. (reprinte d as pp . 841- 843 of the Appendix to Fejer 's 'Gesammelte Arb eiten ')) are transpar ent . The basic characterist ics of subharmonic functi ons and t heir relati on to potent ial t heory was clarified by Frigyes Riesz in two consecutive pap ers {69}, {70}. A function f E C 2 (n ) is subharmonic, if and only if Dof ~ 0 on n. The main result is t hat any subharmonic function can be represented as a pot ential plus a harmonic function. If f : n ---t [-00, (0 ) is a subharmonic function and, as befor e,

n' is

a bounded dom ain with

n' c n,

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Elbert and B. M . Garay

then

(5)

f (x)

=

r I' (z , y) dp,y + h(x ),

In!

xE D'

wher e r is th e fund amental solution of Laplace's equat ion .6.u = 0 on IR n , p,y is a (uniquely defined) Borel measure and h : D' - t IR is a harmonic function. It is worth compa ring (5) with Green 's classical representation formula

w(x) =

r r (x, y).6.w(y) dy + Jan! r ( w(y) 8r~x, y) - r (x, y) 8:(y) ) as; ou

I n!

Vu

x E D' for C 2 (D) functi ons. (Note also th at th e function defined by the boundary int egral is harmonic .) If f = w E C 2(D) then dp,y = .6.w(y ) dy. In t he book writ t en by T ibor Rado [143] several variations of formula (5) can be found. Some of t hem originate from Frigyes Riesz. (Ra do reviewed the development of the theory of subharmonic functions till 1937 but he was not concerned wit h its applications to par tial differential equations.) T he late period of Frigyes Riesz had lit tle relevance to different ial equatio ns. Techniques applied in his 1938- 48 series of papers on ergodic theory belong to genera l functi onal ana lysis and the theory of measur e and integ ration . However , he considers ergodic t heory as par t of a dynamical system t heory for meas ure-preserving transformations and interprets, occasiona lly, his own resul ts from t his view-point . It is amaz ing how much t he old Riesz felt th e imp ortan ce of function al itera tions! He asked (himself and) each reader of Matematikai Lap ok (3 (1952), Problem No. 54) to "determine the set of t hose comp lex init ial points Zl for which, given a comp lex parameter a, t he iteration Zn+l = (z; + a)/2 is converging." T he Ed itor of the Problem Session had waited two years for t he solut ion in vain when he stated (5 (1954), p. 283) that Problem No. 54 was ent irely open and promised t o publish the soluti on any time it arrives. (Actually, P roblem No. 54 concerns t he st ructure (of t he complement) of Julia set s for the quadratic family, a basic pr oblem in fractal geometry . .. Unfortunately, Mandelbrot was apparent ly unaware of t he existence of Mat ematikai Lap ok and publi shed his results somewhere else . . . ) The works of F. Riesz and Haar on linear integral equations. It is imp ortant to mention that int egral equat ions played a fund ament al role in the formulation of the pap ers on compa ct linear operat ors by Frigyes

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Riesz, as well as in the emergence of the whole functional analysis . Alfred Haar, a colleague of Frigyes Riesz at Kolozsvar and, later, at Szeged was also concerned with integral equations. As a student in Gottingen, while applying Hilbert's method elaborated for the problem flu = 0, ulan = 9 in his first paper, Haar {30} discusses the reducibility of the boundary value problem

to integral equations and proves Fredholm type alternative theorems. Haar's inequality for partial equations of the first order. The most important results in differential equations of Hungarian mathematics in the first half of the twentieth century are attached to the name of Alfred Haar. He made essential contributions to both the theory of general equations of the first order and of quasilinear elliptic equations of the second order. His results in the latter topic are related to his pioneering work in the field of the calculus of variations, to put it more precisely, in the study of the minimal surface problem and of the minimal surface equation

(1 + u;)u xx - 2uxu yuxy + (1 + u;)uyy = O. Here we sum up Alfred Haar's papers on partial differential equations of the first order based on his lecture delivered in Bologna in 1928 {39}. Consider a finite interval [Xl, X2] the triangle

T = { (x, y)

E

c

JR. Given a parameter a

> 0, define

JR2 I 0 ~ Y ~ (2a)-1(x2 - xd and Xl + ay ~ X ~ X2 - ay}.

Finally, let z : T

---+ JR

be a C l function and set M = max{lz(x,o)11 Xl ~

X ~ X2}' Assume that, for some constants b > 0 and c > 0, IZy(x,y)1 ~alzx(x ,y)1 +blz(x,y)1 +c whenever

(x ,y)ET.

Then (6)

Iz(x, y) I ~ M eby + cb- l (eby -

1)

whenever

(x, y) E T.

Inequality (6) - just like its one-variable counterpart, the famous Gronwall lemma in the theory of ordinary differential equations - has farreaching consequences. As a preliminary, we may state that the problem

256

A.

F(uy,ux , u, x, y) = 0, uls

nc

Elbert and B. M . Garay

= h (h

S - t ]R is a cont inuous function , SC an , ]R2) can be transformed to the form (7) under very genera l condit ions.

Let 9 : [Xl,X2J - t ]R be a continuous function and let F : ]R2 X T be a cont inuous function with th e prop erty th at

IF(p , z,

X,

y) - F(p, z, X , y)! ~ alp -

-t

]R

pi + bi z - zl

whenever (p, z, X, y), (p, z, X, y) E ]R2 X T. Applying inequality (6) to th e difference of two possible solutions, it is immediate th at th e first ord er problem

(7) has at most one C l solution on T {36}. Formerly, uniqu eness results were known only for C 2 solutions and proved within the framework of th e theory of characterist ics. Inequalit y (6) has some consequences to cha racterist ics in return {37}. As has been mentioned by Hadamard in his addit ional comment publi shed jointly with Haar's paper {36}, Haar 's inequality leads not only to a result on uniqu eness but also to the assert ion that the C l solutions of equation (7) on the tri angle T depend continuously on function g. Haar finished his lecture delivered at Bologna with t he extension of inequality (6) to systems of partial differential equations with a simple st ructure. His promi se to devote anot her pap er to t he topic could not be fulfilled due to his early death. Incident ally, Alfred Haar was already concerned with syst ems of first order partial differential equat ions in one of his early ar ticles {41} the coauthor of which was Tod or Karman, one of his fellow students in Cottingen. Haar's existence and uniqueness theorem in the calculus of variations. We are going to sum up Alfred Haar 's work on calculus of vari ati ons based on his own lecture held in Hamburg in 1930 {40} as well as on t he monograph of Tibor Rado [142J who himself achieved fund ament al results in thi s field. The focus of th e discussion will be placed on Alfred Haar 's exist ence and uniqueness theorem {35}. In the proof a lemma originatin g from Tibor Rado {63} plays an important part which, by its external form , is a geometric statement on saddle surfaces but in essence is an a priori est imat e for the gradient of solutions to quasilinear elliptic equat ions which will be reviewed separate ly here. Naturally, the abst rac t existe nce and uniqueness theorem has important consequences for the classical minimal surface

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257

problem {34} which motivates it . Finally, we are going to discuss these consequences. Emphasis will be laid on the regularity properties of the solution, more exactly, on the analytic feature of the minimal surface. Consider a bounded domain 0 C JR2 with a convex Jordan curve r = ao as its boundary, a 0 2 function F : JR2 - t JR, and a continuous function r.p : r - t JR. The variational problem

(8)

I( u) =

J'r~ F(p, q) dx dy

----+

min

~r=~

is called regular if the Hessian matrix of F is positive definite for all p, q E JR2 . Of course, p = U x and q = u y . The Euler-Lagrange equation of problem (8) is

(9)

Fppu x x

+ 2Fpqux y + Fqqu yy =

O.

In the most important special case, i.e. in the minimal surface problem we have F(p , q) = (1

+ p2 + q2)1/2,

and thus equation (9) simplifies to

(10) Using geometrical terms equation (10) expresses that the mean curvature of a minimal surface equals zero. As a preliminary, observe that the functional I can be defined for all elements of the function class I:

= {u :

0

-t

JR I u is a Lipschitz function}

and the double integral in (8) is understood in the sense of Lebesgue. The validity of the formula Area(u,O)

=

Jin

(1 + u;

+ u;)1 /2 dxdy , u E I:

was proven first by Zoard Geocze {25} where Area (u, 0) stands for the area of surface S = {( x ,y,u(x,y)) E JR3 I (x,y) E O} as defined by Lebesgue for continuous surfaces in his famous doctoral thesis. Also, observe that both (9) and (10) are quasilinear elliptic equations. The eigenvalues of the coefficient matrix of equation (10) are )'1 (p, q) = 1 + p2 + q2 and A2(p, q) = 1. Since the ratio AdA2 is unbounded, (10) is nonuniformly elliptic. It is well-known that the solvability of a Dirichlet boundary value

A.

258

Elbert and B . M . Garay

problem for nonuniforml y elliptic quasilinear equations depends crucially on th e geomet ric assu mpt ions on the pair (f , ({J) . T he crucial assumption of Haar 's existe nce and uniqueness theorem is that t he pair (f, ({J) satisfies Hilbert 's so-called three-point condition. In other words, it is assumed that, for some constant K > 0, every set of t hree distinct points on the curve {( x , y, ({J(x , y)) E 1R3 I (x, y) E r] lies in a plane of slope ~ K . The three-point condit ion is t he assumption which implies t he property I: i= 0. It implies also the st rict convexity of f . After such preparations we are in a positi on to formulate Haar 's existence and uniqu eness t heorem {35}. Consider the variatio nal problem (8) . Assume that (8) is regular and that the t hree- point condition with const ant K is satisfied. Then (8) has a uniqu e solution in the function class I: and the Lipschitz constant of thi s solut ion is ~ K.

T. Rad6's regularity Lemma. Uniqueness is a rath er element ary consequence of the regularity assumption on (8). The st arti ng point of the existe nce proof is the observati on that the function al I : I: - 1R is lower semicont inuous (wit h respect to uniform convergence). Th e difficulty is that minimizing sequences are not precompact. A nice geomet ric property of saddle functions, which was conjectured by Haar and proved by Rado {63}, helps, instead . A conti nuous function u : n - 1R is a saddle funct ion if, given arbit rarily t hree constants a , (3" E IR and an open set (/) i=- 0,' c 0" t he maximum-minimum prin ciple

min_( u(x ,y)-(a x + (3y + , ))

=

max_(u(x ,y) - (ax +(3y +,)) (x ,y) Ef2'

min (x ,y)Ea f2'

(x ,Y)Ef2'

=

max

(u(x , y )- (a x+ (3y+ , )) , (u(x , y) - (ax+(3y +,))

(x ,y)Eaf2'

is satisfied. Rado's lemma concerns saddle functions u : n - 1R for which the pair (r; ul r ) is subject to the three-point condition with const ant K and st at es that such functions are Lipschitz cont inuous on n and t he Lipschitz constant is ~ K. Rad6's lemma was given a new and a much simpler proof by J anos Neumann {51}. Both t he original method of proving and t he one given by Neumann lead , auto matically, t o Lemma 12.6 in 'D. Gilbarg & N. S. Trudinger, Elliptic P artial Differential Equation s of the Second Order, Springer, Berlin , 1983': Let n be a bounded domain in 1R 2 , and let ({J : an - 1R be a cont inuous function satisfying t he t hree-point condit ion with constant K. Suppose

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259

u E C(n) n C 2(n) with ulan

= ip satisfies a quasilinear elliptic equation of the form auxx + 2bu xy + CU yy = 0 where a,b, c are continuous functions

in the variables (x , y, u,p, q). Then luxl , IUyl ~ K on n. (Besides the LaxMilgram lemma, which is considered 'half-Hungarian' due to Peter Lax , this is the only result in the well-known book of Gilbarg and Trudinger which originates from a Hungarian mathematician. It is worth mentioning that Lemma 12.6 and, in relation to this, the name of Tibor Rado are mentioned in the preface of this monograph. Haar's Lemma on the variation of double integrals. Applying the existence and uniqueness theorem for the variational problem (8) in the special case F(p , q) = (1 + p2 for which

+ q2)1/2

we obtain that there exists a u* E E

Area(u*,n) < Area(u,n)

for all u*

i= u

E L.

Actually, u* In is analytic. The argumentation leading to this will be outlined below. The starting point is one of Alfred Haar's earlier results {32} , namely the two-variable counterpart of Du Bois-Reymond's fundamental lemma of the calculus of variations. In order that the novelty of Haar's result should be emphasized, we would like to present Du Bois-Reymond's lemma: If f : [a, b] -r lR is a continuous function such that

l

b

f(x)
=0

for each