Alexandra Bellow

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Alexandra Bellow
Ionescu tulcea.jpg
Born (1935-08-30) 30 August 1935 (age 84)
NationalityRomanian American
Alma materUniversity of Bucharest
Yale University
Cassius Ionescu-Tulcea
(m. 1956; div. 1969)

Saul Bellow
(m. 1974; div. 1985)

Alberto Calderón
(m. 1989; died 1998)
Scientific career
InstitutionsUniversity of Pennsylvania
University of Illinois at Urbana–Champaign
Northwestern University
Doctoral advisorShizuo Kakutani

Alexandra Bellow (formerly Alexandra Ionescu Tulcea; born 30 August 1935) is a mathematician from Bucharest, Romania, who has made contributions to the fields of ergodic theory, probability and analysis.


Bellow was born in Bucharest, Romania, on 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. 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 accompanied her husband to the United States in 1957 and received her Ph.D from Yale University 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 at Urbana–Champaign. 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 Cassius Ionescu-Tulcea (1956–1969) she and her husband wrote a number of papers together, as well as the research monograph [25] on lifting theory.

Alexandra's second husband was the writer Saul Bellow who was awarded the Nobel Prize (1976), during this marriage (1975–1985). Alexandra features in Bellow's writings; she is portrayed lovingly in his memoir To Jerusalem and Back (1976), and, his novel The Dean's December (1982), more critically, satirically in his last novel Ravelstein (2000) - which was written many years after their divorce.[1][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 mathematician, Alberto P. Calderón. For more details about her personal and professional life see her autobiographical article.[3] See also her recent interview.[4]

Mathematical work[edit]

Some of her early work involved properties and consequences of lifting. Lifting theory, which had started with the pioneering papers 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[5] became a standard reference in this area.

By applying a lifting to a stochastic process, A. Ionescu Tulcea and C. Ionescu Tulcea obtained 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).[6]

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).[7][8]

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.[8][9]

The proof of the existence of a lifting commuting with the left translations of an arbitrary locally compact group, by A. Ionescu Tulcea and C. Ionescu Tulcea, is highly non-trivial. It makes use of approximation by Lie groups, and martingale-type arguments tailored to the group structure.[10]

In the early 1960s she worked with C Ionescu Tulcea on martingales taking values in a Banach space.[11] 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’,[12](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. Chacón gave an example of a positive (linear) isometry of L1 for which the individual ergodic theorem fails in L1. Her work[13] 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 Chacón, 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,[14] 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). This problem was solved several years later by J. Bourgain, for f in Lp, 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[15] 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[16] 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 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[17] to give a negative answer to a long-standing conjecture of E. Hopf. Later, Bellow and Krengel[18] 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, tn = n + ε(n), where ε is positive and tends to zero, does not lead to a.e. convergence; in fact strong sweeping out occurs.[19] 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.[20]

The following are some examples of the work of A. Bellow with other mathematicians.

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

  • Bourgain, J. (1988). "On the maximal ergodic theorem for certain subsets of the integers". Israel Journal of Mathematics. 61 (1): 39–72. doi:10.1007/bf02776301.
  • Akcoglu, M. A.; del Junco, A.; Lee, W. M. F. (1991). A. Bellow and R. Jones (eds.). "A solution to a problem of A. Bellow". Almost Everywhere Convergence II: 1–7.CS1 maint: uses editors parameter (link)
  • Bergelson, Vitaly; Bourgain, J.; Boshernitzan, M. (1994). "Some results on non-linear recurrence". Journal d'Analyse Math. 62 (72): 29–46. doi:10.1007/BF02835947.

The "strong sweeping out property", a notion formalized by A. Bellow, plays a role in this area of research.[21]

Academic honors, awards, recognition[edit]

Professional editorial activities[edit]

See also[edit]


  1. ^ A Bellow Novel Eulogizes a Friendship DINITIA SMITH, The New York Times, January 27, 2000
  2. ^ "România, prin ochii unui scriitor cu Nobel" (in Romanian). Evenimentul zilei. 24 March 2008. Retrieved 7 October 2014.
  3. ^ "Una vida matemática" ("A mathematical life"), this article appeared in Spanish in La Gaceta de la Real Sociedad Matematica Española, vol.5, No.1, Enero-Abril 2002, pp. 62–71.
  4. ^ "interview with Alexandra Bellow". Cite journal requires |journal= (help) (in Romanian). Adevarul. 25 October 2014
  5. ^ Ionescu Tulcea, Alexandra; Ionescu Tulcea, C. (1969). "TOPICS IN THE THEORY OF LIFTINGS". Ergebnisse der Mathematik. 48. OCLC 851370324.
  6. ^ Ionescu Tulcea, Alexandra; Ionescu Tulcea, C. (1969). "Liftings for abstract-valued functions and separable stochastic processes". Zeitschrift für Wahr. 13 (2): 114–118. doi:10.1007/BF00537015.
  7. ^ Ionescu Tulcea, Alexandra (1973). "On pointwise convergence, compactness and equicontinuity in the lifting topology I". Zeitschrift für Wahr. 26 (3): 197–205. doi:10.1007/bf00532722.
  8. ^ a b Ionescu Tulcea, Alexandra (March 1974). "On measurability, pointwise convergence and compactness". Bull. Amer. Math. Soc. 80 (2): 231–236. doi:10.1090/s0002-9904-1974-13435-x.
  9. ^ Ionescu Tulcea, Alexandra (February 1974). "On pointwise convergence, compactness and equicontinuity II". Advances in Mathematics. 12 (2): 171–177. doi:10.1016/s0001-8708(74)80002-2.
  10. ^ Ionescu Tulcea, Alexandra; Ionescu Tulcea, C. (1967). "On the existence of a lifting commuting with the left translations of an arbitrary locally compact group" (Proceedings Fifth Berkeley Symposium on Math. Stat. and Probability, II, University of California Press): 63–97. Cite journal requires |journal= (help)
  11. ^ Ionescu Tulcea, Alexandra; Ionescu Tulcea, C. (1963). "Abstract ergodic theorems" (PDF). Transactions of the American Mathematical Society. 107: 107–124. doi:10.1090/s0002-9947-1963-0150611-8.
  12. ^ Bellow, Alexandra (1978). "Uniform amarts: A class of asymptotic martingales for which strong almost sure convergence obtains". Zeitschrift für Wahr. 41 (3): 177–191. doi:10.1007/bf00534238.
  13. ^ Ionescu Tulcea, Alexandra (1965). "On the category of certain classes of transformations in ergodic theory". Transactions of the American Mathematical Society. 114 (1): 262–279. doi:10.1090/s0002-9947-1965-0179327-0. JSTOR 1994001.
  14. ^ Bellow, Alexandra (June 1982). "Two problems". Proceedings Conference on Measure Theory, Oberwolfach, June 1981, Springer-Verlag Lecture Notes Math. 945: 429–431. OCLC 8833848.
  15. ^ Bellow, Alexandra (June 1982). 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. Lecture Notes in Mathematics. 1033. pp. 74–78. doi:10.1007/BFb0099847. ISBN 978-3-540-12703-1.
  16. ^ Bellow, Alexandra (1989). "Perturbation of a sequence". Advances in Mathematics. 78 (2): 131–139. doi:10.1016/0001-8708(89)90030-3.
  17. ^ Bellow, Alexandra; Krengel, U. (1991). On Hopf's ergodic theorem for particles with different velocities. Almost Everywhere Convergence II, Proceedings Internat. Conference on Almost Everywhere Convergence in Probability and Ergodic Theory, Evanston, October 1989, Academic Press, Inc. pp. 41–47. ISBN 9781483265926.
  18. ^ Bellow, Alexandra; Calderón, A. P.; Krengel, U. (1995). "Hopf's ergodic theorem for particles with different velocities and the "strong sweeping out property"". Canadian Mathematical Bulletin. 38 (1): 11–15. doi:10.4153/cmb-1995-002-0.
  19. ^ Bellow, Alexandra; Akcoglu, M.; del Junco, A.; Jones, R. (1993). "Divergence of averages obtained by sampling a flow" (PDF). Proc. Amer. Math. Soc. 118 (2): 499–505. doi:10.1090/S0002-9939-1993-1143221-1.
  20. ^ Bellow, Alexandra; Jones, R.; Rosenblatt, J. (1990). "Convergence for moving averages". Ergodic Th. & Dynam. Syst. 10: 43–62. doi:10.1017/s0143385700005381.
  21. ^ Bellow, Alexandra; Akcoglu, M.; Jones, R.; Losert, V.; Reinhold-Larsson, K.; Wierdl, M. (1996). "The strong sweeping out property for lacunary sequences, Riemann sums, convolution powers and related matters". Ergodic Th. & Dynam. Syst. 16 (2): 207–253. doi:10.1017/S0143385700008798.
  22. ^ 2017 Class of the Fellows of the AMS, American Mathematical Society, retrieved 2016-11-06.