Alexandra Bellow

Alexandra Bellow (formerly Alexandra Ionescu Tulcea), is a mathematician from Bucharest, Romania, who has made substantial contributions to the fields of ergodic theory, probability and analysis.

Contents 1 Biography 2 Mathematical work 3 Academic honours, awards, recognition 4 Professional editorial activities 5 Publications 6 Mathematics-related and other publications 7 See also 8 References

Biography

She was born in Bucharest, Romania, August 30, 1935, as Alexandra Bagdasar. Her parents were both physicians. Her mother, Florica Bagdasar, was a child psychiatrist. Her father, Dumitru Bagdasar, was a neurosurgeon (in fact, he founded the Romanian school of neurosurgery, after having obtained his training in Boston, at the clinic of the world pioneer of neurosurgery, Dr. Harvey Cushing). For more details about her parents and her early life, see article ^[1] She received her M.S. in Mathematics from the University of Bucharest in 1957, where she met and married her first husband, Cassius Ionescu Tulcea. She came to the United States in 1957, and received her Ph.D from Yale in 1959 under the direction of Shizuo Kakutani. After receiving her degree, she worked as a research associate at Yale from 1959 until 1961, and as an Assistant professor at the University of Pennsylvania from 1962 to 1964. From 1964 until 1967 she was an Associate professor at the University of Illinois. In 1967 she moved to Northwestern University as a professor of mathematics. She was at Northwestern until her retirement in 1996, when she became Professor Emeritus.

During her marriage to Ionescu Tulcea (1956–1969) she and her husband wrote a number of papers together, as well as the research monograph [3] on Lifting Theory.

Alexandra’s second husband was the acclaimed writer Saul Bellow who reached his worldwide zenith, the Nobel Prize (1976), during this marriage (1974–1985). Alexandra features in Bellow's book Jerusalem and Back (1976), and more critically, in his late roman à clef, Ravelstein - which was written fifteen years after their divorce.^[2] The decade of the nineties was for Alexandra a period of personal and professional fulfillment, brought about by her marriage in 1989 to the outstanding mathematician, Alberto P. Calderón. For more details about her personal and professional life see her autobiographical article.^[3]

Mathematical work

Some of her early work involved properties and consequences of lifting. A ‘lifting’ is a linear and multiplicative mapping which selects one function from each equivalence class of bounded measurable functions; there are also natural generalizations of this notion to abstract-valued functions. Lifting theory, which had started with the pioneering paper's of John von Neumann and later Dorothy Maharam, came into its own in the 1960s and 70’s with the work of the Ionescu Tulceas and provided the definitive treatment for the representation theory of linear operators arising in probability, the process of disintegration of measures. The Ergebnisse monograph ^[4] became a standard reference in this area. The following are some other striking applications obtained by the Ionescu Tulceas and Bellow respectively:

By applying a lifting to a stochastic process, one obtains a ‘separable’ process; this gives a rapid proof of Doob's theorem concerning the existence of a separable modification of a stochastic process (also a ‘canonical’ way of obtaining the separable modification),^[5]

By applying a lifting to a ‘weakly’ measurable function with values in a weakly compact set of a Banach space, one obtains a strongly measurable function; this gives a one line proof of Phillips’s classical theorem ( also a ‘canonical’ way of obtaining the strongly measurable version ).^[6]

We say that a set H of measurable functions satisfies the "separation property" if any two distinct functions in H belong to distinct equivalence classes. The range of a lifting is always a set of measurable functions with the "separation property". The following ‘metrization criterion’ gives some idea why the functions in the range of a lifting are so much better behaved:

Let H be a set of measurable functions with the following properties : (I) H is compact ( for the topology of pointwise convergence ); (II) H is convex; (III) H satisfies the "separation property". Then H is metrizable.^[7]

The proof of the existence of a lifting commuting with the left translations of an arbitrary locally compact group is highly non-trivial. It makes use of approximation by Lie groups, and martingale-type arguments taylored to the group structure.^[8]

In the early 1960s she worked with C Ionescu Tulcea on martingales taking values in a Banach space. ^[9] In a certain sense paper this work launched the study of vector-valued martingales, with the first proof of the ‘strong’ almost everywhere convergence for martingales taking values in a Banach space with (what later became known as) the Radon–Nikodym property; this, by the way, opened the doors to a new area of analysis, the “geometry of Banach spaces”. These ideas were later extended by Bellow to the theory of ‘uniform amarts’,( in the context of Banach spaces, uniform amarts are the natural generalization of martingales, quasi-martingales and possess remarkable stability properties, such as optional sampling ), now an important chapter in probability theory.

In 1960 D. S. Ornstein constructed an example of a non-singular transformation on the Lebesgue space of the unit interval, which does not admit a σ – finite invariant measure equivalent to Lebesgue measure, thus solving a long-standing problem in ergodic theory. A few years later, R. V. Chacon gave an example of a positive (linear) isometry of L1 for which the individual ergodic theorem fails in L1. Her work ^[10] unifies and extends these two remarkable results. It shows, by methods of Baire Category, that the seemingly isolated examples of non-singular transformations first discovered by Ornstein and later by Chacon, were in fact the typical case.

Beginning in the early 1980s Bellow began a series of papers that has brought about a revival of that important area of ergodic theory dealing with limit theorems and the delicate question of pointwise a.e. convergence. This was accomplished by exploiting the interplay with probability and harmonic analysis, in the modern context ( the Central Limit Theorem, transference principles, square functions and other singular integral techniques are now part of the daily arsenal of people working in this area of ergodic theory ) and by attracting a number of talented mathematicians who have been very active in this area.

One of the Two problems that she raised at the Oberwolfach meeting on “Measure Theory” in 1981,^[11] was the question of the validity, for ƒ in L₁, of the pointwise ergodic theorem along the ‘sequence of squares’, and along the ‘sequence of primes’ (A similar question was raised independently, a year later, by H. Furstenberg),.^[12] This problem was solved several years later by J. Bourgain, for f in L_p, p > 1 in the case of the ‘squares’ and for p > (1 + √3)/2 in the case of the ‘primes’ (the argument was pushed through to p > 1 by M. Wierdl; the case of L₁ however had remained open). Bourgain was awarded the Fields Medal in 1994, in part for this work in ergodic theory.

It was U. Krengel who first gave, in 1971, an ingenious construction of an increasing sequence of positive integers along which the pointwise ergodic theorem fails in L1 for every ergodic transformation. The existence of such a “bad universal sequence” came as a surprise. Bellow showed ^[13] that every lacunary sequence of integers is in fact a “bad universal sequence” in L1. Thus lacunary sequences are ‘canonical’ examples of “bad universal sequences”.

Later she was able to show ^[14] that from the point of view of the pointwise ergodic theorem, a sequence of positive integers may be “good universal” in Lp, but “bad universal” in Lq, for all 1≤q<p . This was rather startling and answered a question raised by R. Jones.

A central place in this area of research is occupied by the “strong sweeping out property” (that a sequence of linear operators may exhibit). This describes the situation when almost everywhere convergence breaks down even in L∞ and in the worst possible way. Instances of this appear in several of her papers, see for example [59, 61, 63, 65, 66] in her vita. Paper [65] was an extensive and systematic study of the “strong sweeping out” property (s.s.o.), giving various criteria and numerous examples of (s.s.o.). This project involved many authors and a long period of time to complete.

Working with U. Krengel, she was able ^[15] to give a negative answer to a long standing conjecture of E. Hopf. Later, Bellow and Krengel ^[16] working with A. P. Calderón were able to show that in fact the Hopf operators have the “strong sweeping out” property.

In the study of aperiodic flows, sampling at nearly periodic times, as for example, t_n = n + ε(n), where ε is positive and tends to zero, does not lead to a.e. convergence; in fact strong sweeping out occurs,.^[17] This shows the possibility of serious errors when using the ergodic theorem for the study of physical systems. Such results can be of practical value for statisticians and other scientists.

In the study of discrete ergodic systems, which can be observed only over certain blocks of time [a,b], one has the following dichotomy of behavior of the corresponding averages: either the averages converge a.e. for all functions in L1, or the strong sweeping out property holds. This depends on the geometric properties of the blocks, see.^[18]

The following are some examples of the impact of her work on other mathematicians.

Mathematicians who in their seminal papers, answered questions raised by A. Bellow:

1. • J. Bourgain, in the paper “On the maximal ergodic theorem for certain subsets of the integers”, Israel Journal Math., vol. 61, No. 1 (1988), pp. 39–72.
   • M. A. Akcoglu, A. del Junco and W. M. F. Lee, in the paper “A solution to a problem of A. Bellow”, Almost everywhere convergence II ( ed. A. Bellow and R. Jones ), Academic Press, 1991, pp. 1–7.
   • Vitaly Bergelson, J. Bourgain and M. Boshernitzan, in the paper “Some results on non-linear recurrence”, Journal d’Analyse Math., vol. 62 (1994), pp. 29–46; see 72.

The following younger mathematicians, collaborators of A. Bellow, have continued to work on pointwise a.e. convergence in ergodic theory, and in particular on “subsequence ergodic theory”:

R. Jones, V. Losert, K Reinhold-Larsson, J. Rosenblatt, M. Wierdl ( and their own collaborators ).

The “strong sweeping out property” – a notion formalized by A. Bellow – plays a central role in this area of research.^[19]

Academic honours, awards, recognition

• 1977–80 Member, Visiting Committee, Harvard Mathematics Department
• 1980 Fairchild Distinguished Scholar Award, Caltech, Winter Term
• 1987 Humboldt Prize, Alexander von Humboldt Foundation, Bonn, Germany
• 1991 Emmy Noether Lecture, San Francisco, Ca., U.S.A
• 1997 International Conference in Honor of Alexandra Bellow, on the occasion of her retirement, held at Northwestern University, October 23–26, 1997. A Proceedings of this Conference appeared as a special issue of the Illinois Journal of Mathematics, Fall 1999, Vol. 43, No. 3.

Professional editorial activities

• 1974–77 Editor, Transactions of the American Mathematical Society
• 1980–82 Associate Editor, Annals of Probability
• 1979– Associate Editor, Advances in Mathematics

Publications

1. Ergodic theory of random series, Doctoral dissertation, Yale University, June 1959.
2. Ionescu Tulcea, Alexandra (1959). "Sur le prolongement analytique des séries aléatoires". Comptes Rendus Acad. Sci. Paris 248: 3396–3397. 
3. Ionescu Tulcea, Alexandra (1960). "Analytic continuation of random series". Journal Math. And Mech. 9: 339–410. 
4. Ionescu Tulcea, Alexandra; C. Ionescu Tulcea (1961). "On the decomposition and integral representation of continuous linear operators". Annali de Matematica pura ed applicata 4 (53): 63–88. 
5. Ionescu Tulcea, Alexandra; C. Ionescu Tulcea (1961). "On the lifting property I". Journal Math. Anal. and Applications 3 (3): 537–546. doi:10.1016/0022-247X(61)90075-0. 
6. Ionescu Tulcea, Alexandra; C. Ionescu Tulcea (1962). "On the lifting property II, Representation of linear operators on the spaces L_r(E), 1 ≤ r < ∞". Journal Math. And Mech. 11: 773–796. 
7. Contribution to information theory for abstract alphabets, Arkiv för Mathematik, 4, No.18, pp. 235–247 (1960).
8. (with C. Ionescu Tulcea) Abstract ergodic theorems, Proceedings Nat. Acad. Sci., U.S.A., pp. 204–206, February 1962.
9. Random series and spectra of measure-preserving transformations, Proceedings Internat. Symposium on Ergodic Theory held at Tulane Univ. October 1961, Academic Press, pp. 273–292 (1963) (Ergodic Theory, Editor: Fred B. Wright).
10. (with C. Ionescu Tulcea) Abstract ergodic theorems, Transactions Amer. Math. Soc., 107, pp. 107–124 (1963).
11. Un théorème de catégorie dans la théorie ergodique, Comptes Rendus Acad. Sci. Paris, 257, pp. 18–20 (1963).
12. (with C. Ionescu Tulcea) Problems and remarks concerning the lifting property, Technical Report, U.S. Army Research Office (Durham), (September 1963).
13. (with C. Ionescu Tulcea) On the lifting property III, Bulletin Amer. Math. Soc., 70, pp. 193–197 (1964).
14. (with C. Ionescu Tulcea) On the lifting property IV, Annales Inst. Fourier, 14, pp. 445–472 ( 1964).
15. Ergodic Properties of isometries in L_p spaces, 1 < p < ∞, Bulletin Amer. Math. Soc., 70 (3), pp. 366–371 (May 1964).
16. On the category of certain classes of transformations in ergodic theory, Transactions Amer. Math. Soc., 114, pp. 262–279 (1965).
17. On the lifting property V, Annals Math. Statistics, 36 (3), pp. 819–828 (1965).
18. (with C. Ionescu Tulcea) On the existence of a lifting commuting with the left translations of an arbitrary locallycompact group, Proceedings Fifth Berkeley Symposium on Math. Stat. And Probability, II, pp. 63–97, University of California Press (1967).
19. Sur le relèvement fort et la desintégration des mesures, Comptes Rendus Acad. Sci. Paris, 262, pp. 617–618 (March 1966).
20. Sur la domination des mesures et la desintégration des mesures, Comptes Rendus Acad. Sci. Paris, pp. 1442–1445 (June 1966).
21. Liftings compatible with topologies, Bulletin Soc. Math. de Grèce, 8, pp. 116–126 (1967).
22. On the lifting property (Invited address), Proceedings Symposium in Analysis, Queen’s University, Kingston, Ontario (June 1967).
23. (with C. Ionescu Tulcea) Liftings for abstract-valued functions and separable stochastic processes, Zeitschrift für Wahrscheinlichkeitstheorie, 13, (1969).
24. (with M. Moretz) Ergodic properties of semi-Markovian operators on the Z1- part, Zeitschrift für Wahr., 13, pp. 119–122 (1969).
25. (with C. Ionescu Tulcea) TOPICS IN THE THEORY OF LIFTINGS monograph, Springer-Verlag, Ergebnisse der Mathematik, Band 48, 1969.
26. On super-mean-valued functions and semi-polar sets, Journal Math. Anal. and Applications, 34, pp. 19–25 (1971).
27. On pointwise convergence, compactness and equicontinuity in the lifting topology I, Zeitschrift für Wahr., 26, pp. 197–205 (1973).
28. On pointwise convergence, compactness and equicontinuity II, Advances in Math., 12, pp. 171–177 ( Febr. 1974).
29. On measurability, pointwise convergence and compactness, (Invited address), AMS Meeting, April 1973 / Evanston, Bulletin Amer. Math. Soc., 80, pp. 231–236 (March 1974).
30. (with D. G. Austin and G. A. Edgar) Pointwise convergence in terms of convergence in expectation, Zeitschrift für Wahr., 30, pp. 17–26 (1974).
31. An L_p- inequality with application to Ergodic Theory, ( Invited paper), Houston Journal of Math., 1, pp. 153–159 (1975), (first issue).
32. A problem in L_p- spaces, Proceedings Internat. Conference on Measure Theory held at Oberwolfach June 1975, Springer-Verlag Lecture Notes Math. #541, pp. 381–388 (1976).
33. Concluding Remarks, Proceedings Internat. Conference on Measure Theory held at Oberwolfach June 1975, Springer-Verlag Lecture Notes Math. #541, pp. 429–430 (1976).
34. Stability properties of the class of asymptotic martingales, Bulletin Amer. Math,Soc., 82, No. 2, pp. 338–340 (March 1976).
35. On vector-valued asymptotic martingales, Proceedings Nat. Acad. Sci., U.S.A., 73, No. 6, pp. 1798–1799 (June 1976).
36. Several stability properties of the class of asymptotic martingales, Zeitschrift für Wahr., 37, pp. 275–290 (1977).
37. Les amarts uniformes, Comptes Rendus Acad, Sci. Paris, 284, Série A, pp. 1295–1298 (1977).
38. Uniform amarts: A class of asymptotic martingales for which strong almost sure convergence obtains, Zeitschrift für Wahr., 41, pp. 177–191 (1978).
39. Some aspects of the theory of vector-valued amarts (Keynote invited lecture), Vector Space Measures and Applications I, Proceedings of the 1977 Dublin Conference, Springer-Verlag Lecture Notes Math., #644, pp. 57–67 (1978).
40. Submartingale characterization of measurable cluster points (Invited paper), Advances in Probability, 4, pp. 69–80 (1978) ( Probability on Banach Spaces, Editor: James D. Kuelbs ).
41. (with A. Dvoretzky) A characterization of almost sure convergence, Proceedings Second Internat. Conference on Probability in Banach Spaces, Oberwolfach June 1978, Springer-Verlag Lecture Notes Math., #709, pp. 45–65 (1979).
42. (with H. Furstenberg) An application of number theory to ergodic theory and the construction of uniquely ergodic models, Israel Journal Math. ( special volume dedicated to the memory of Shlomo Horowitz ), 33, No.3-4, pp. 231–240 (1979).
43. Mesures de Radon et espaces relèvement compacts, Comptes Rendus Acad. Sci. Paris, 289, Série A, pp. 621–624 (1979).
44. Lifting compact spaces, Proceedings Conference on Measure Theory, Oberwolfach, July 1979, Springer-Verlag Lecture Notes Math., #794, pp. 233–253 (1980).
45. (with A. Dvoretzky) On martingales in the limit, Annals of Probability, 8, pp. 602–606 (1980).
46. Martingales, amarts and related stopping time techniques ( Invited survey lecture ), Proceedings Third Internat. Conference on Probability in Banach Spaces, Tufts University, August 1980, Springer-Verlag Lecture Notes Math., # 860, pp. 9–24 (1981).
47. ( with L. Egghe ) Inégalités de Fatou généralisées, Comptes Rendus Acad. Sci. Paris, #292, Série I, pp. 847–850 ( May 1981 ).
48. ( with L Egghe ) Generalized Fatou inequalities, Annales Inst. Henri Poincaré, Section B, XVIII, No. 4, pp. 335–365 ( December 1982 ).
49. Sur la structure des suites mauvaises universelles en Théorie Ergodique, Comptes Rendus Acad. Sci. Paris, 294, pp. 55–58 ( January 1982 ).
50. Two problems, Proceedings Conference on Measure Theory, Oberwolfach, June 1981, Springer-Verlag Lecture Notes Math., #945, pp. 429–431 (1982).
51. On “bad universal” sequences in ergodic theory (II), Measure theory and its Applications, Proceedings of a Conference held at Université de Sherbrooke, Quebec, Canada, June 1982, Springer-Verlag Lecture Notes Math, #1033, pp. 74–78 (1983)
52. Editor ( together with R. Beals, A. Beck and A. Hajian ), CONFERENCE IN MODERN ANALYSIS AND PROBABILITY held in June 1982 at Yale University in honor of Professor Shizuo Kakutani, on the occasion of his retirement, Contemporary Mathematics Series, AMS., 26 (1984).
53. ( with V. Losert ) On sequences of density zero in ergodic theory, Contemporary Math. Series, AMS, 26, pp. 49–60 (1984).
54. ( with V. Losert ) The weighted pointwise ergodic theorem and the individual ergodic theorem along subsequences, Transactions Amer. Math. Soc., 288, No.1, pp. 307–345 (1985).
55. For the historical record, Proceedings Conference on Measure theory, Oberwolfach, June 1983, Springer-Verlag Lecture Notes Math., #1089, (1985).
56. ( with D. G. Austin and N. Bouzar ) The Fatou inequality revisited: variations on a theme by A. Dvoretzky, Proceedings Fifth Internat. Conference on Probability in Banach Spaces, Tufts University, July 1984, Springer-Verlag Lecture Notes Math. (1985).
57. Perturbation of a sequeence, Advances in Math., 78, No.2, pp. 131–139, (1989).
58. ( with. R. Jones and J. Rosenblatt ), Convergence for moving averages, Ergodic Th. & Dynam. Syst., 10, pp. 43–62 (1990).
59. Editor ( together with R.L. Jones ), ALMOST EVERYWHERE CONVERGENCE II, Proceedings Internat. Conference on Almost Everywhere Convergence in Probability and Ergodic Theory, Evanston, Illinois, October 1989, Academic Press, Inc. 1991.
60. ( with U. Krengel ) On Hopf’s ergodic theorem for particles with different velocities, ALMOST EVERYWHERE CONVERGENCE II everywhere convergence II, Proceedings Internat. Conference on Almost Everywhere Convergence in Probability and Ergodic Theory, Evanston, October 1989, pp. 41–47, Academic Press, Inc. (1991).
61. ( with R. Jones and J. Rosenblatt ), Almost everywhere convergence of weighted averages, Math. Annalen, 293, pp. 399–426 (1992)
62. ( with M. Akcoglu, A. del Junco and R. Jones ), Divergence of averages obtained by sampling a flow, Proceedings Amer. Math. Soc., 118, pp. 499–505 (1993).
63. ( with R. Jones and J. Rosenblatt ), Almost everywhere convergenve of convolution powers, Ergodic Th. & Dynam. Syst., 14, pp. 415–432 (1994).
64. ( with A. P. Calderón and U. Krengel ), Hopf’s ergodic theorem for particles with different velocities and the “strong sweeping out property”, Canadian Math. Bulletin, 38, No.1, pp. 11–15 (1995).
65. ( with M. Akcoglu, R. Jones, V. Losert, K. Reinhold-Larsson and M. Wierdl ), The strong sweeping out property for lacunary sequences, Riemann sums, convolution powers and related matters, Ergodic Th. & Dynam. Syst., 16, pp. 207–253 (1996).
66. ( with R. Jones ), A Banach Principle for L∞, Advances in Math., 120, No.1, pp. 155–172 (1996).
67. Transference Principles in Ergodic Theory, HARMONIC ANALYSIS AND PARTIAL DIFFERENTIAL EQUATIONS, Essays in honor of Alberto P. Calderón to celebrate his 75th birthday, Chicago Lectures in Math. Series, The University of Chicago Press, pp. 27–39, 1999 ( Editors: Michael Christ, Carlos E. Kenig and Cora Sadosky ).
68. ( with Alberto P. Calderón ), A Weak-Type Inequality for Convolution Products, in the same volume as the previous paper, pp. 41–48.
69. ( with U. Krengel ) On the L1- behavior of the maximal operator for the class of martingales adapted to a given filtration, Illinois Journal Math., 43, No.3, pp. 568–581 (1999).
70. ( with M. Akcoglu, J. Baxter and R. Jones ), On restricted weak-type (1,1); the discrete case, Israel Journal Math., 124, pp. 285–297 (2001).
71. From Riemann to Lebesgue by a.s. averaging, Invited talk given at the 6th Congress of Romanian Mathematicians, held June 28-July 4, 2007, Bucharest, Romania (to appear in the Proceedings of the Congress, 2008).
72. Editor ( together with Carlos E. Kenig and Paul Malliavin ), SELECTED PAPERS OF ALBERTO P. CALDERÓN WITH COMMENTARY, 639 pages, Collected Works, Amer. Math. Society, to appear Spring 2008.

Mathematics-related and other publications

1. Commentaries on two papers by S. Kakutani in Ergodic Theory, SHIZUO KAKUTANI, Selected Papers Volume 2, Contemporary Mathematicians, pp. 405–408, Birkhäuser Boston Inc., 1986 ( Editor: Robert R. Kallman ).
2. “Public perception of mathematics as we enter the new millennium”, presentation at the Round Table Shaping the 21st Century ( Miguel de Guzmán – moderator - ): MATHEMATICAL GLIMPSES INTO THE 21ST CENTURY, Round Tables at the Third European Congress of Mathematics, July 2000, Barcelona, ( Editors: C. Casacuberta, J. M. Ortega, R. M. Miró-Roig and S. Xambó-Descamps ), Centro Nacional de Métodos Numéricos en Ingeniería, Barcelona, Spain, pp. 172–176 ( December 2001).
3. “Una vida matemática” ( “A mathematical life”), La Gaceta de la Real Sociedad Matemática Española, Vol. 5, No.1, Enero-Abril 2002, pp. 62–71 ( Talk given at the Royal Academy of Sciences, Madrid, Spain, in September 2001, at the festive inauguration of the project “Estimulo del Talento Matemático” for the year 2001-2002 ).
4. “Testimonio personal”, Un Homenaje a Miguel de Guzmán, (“Personal testimony”, A Homage to Miguel de Guzmán), Memorial Conference held at Universidad Complutense de Madrid, December 2004.
5. “Asclepios versus Hades în România”, (“Asclepius versus Hades in Romania”); this article appeared in Romanian, in two separate installments of Revista 22: Nr.755 [24–30 August 2004] and Nr. 756 [31 August – 6 septembrie 2004].

See also

• Saul Bellow
• Ravelstein

References

Persondata
Name	AlexandraBellow
Alternative names
Short description	Mathematician
Date of birth	August 30, 1935
Place of birth	Bucharest, Romania
Date of death
Place of death