Alexandra Bellow - Mathematical Work

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 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),

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 ).

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.

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.

In the early 1960s she worked with C Ionescu Tulcea on martingales taking values in a Banach space. 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 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, 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 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 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 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

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 in her vita. Paper 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 to give a negative answer to a long standing conjecture of E. Hopf. Later, Bellow and Krengel 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,. 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, 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.

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

Mathematicians, who in their 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 “strong sweeping out property” – a notion formalized by A. Bellow;– plays a central role in this area of research.