Hari Bercovici will work on several aspects of operator theory and function theory, which arise in the study of free random variables and other areas. His tools are intrinsic to these areas, but also involve input from other areas, such as combinatorics. One important theme is the use of the combinatorial Littlewood-Richardson rule in the study of eigenvalue problems for compact operators on a Hilbert space, or for selfadjoint elements in a finite von Neumann algebra. This rule, and its continuous extensions, also plays a role in a different direction concerning the classification of invariant subspaces of certain operators. Another important theme is the study of weak and strong limit laws in free probability, as well as in monotonic probability theory. There are many problems here where methods of classical function theory yield interesting and sometimes unexpected results. Other problems of operator theory to be considered concern the spectral Nevanlinna-Pick problem and its analogues, hyperinvariant subspaces, dual algebras, and p-entropies.
This project will broaden mathematical knowledge in specific theoretical areas, as well as seek applications in related areas of mathematics, control theory, and computer science. Hari Bercovici will seek the participation of graduate and, when possible, undergraduate students. This will contribute to the training of future scientists.
This project involves theoretical research in the general area of operator theory and operator algebras. These areas have intrinsic interest as well as connections to other areas of pure and applied mathematics, for instance, control theory. We begin with the intellectual contributions supported by this award. The results in operator theory include a reduction of the long-standing invariant subspace problem to a very special class of Hilbert space operators. In another direction, a characterization of the possible spectral properties of a sum of self-adjoint operators, each of which has known spectral properties. Other results pertain to the spectral properties of contractive operators in a variety of classes. A new direction of reasearch was initiated in the study of systems of commuting isometric operators on a Hilbert space. The study of a single isometry, started by von Neumann, led to many developments in operator theory and in the study of time series. It is expected that commuting systems of isometries would become equally important once they are better understood. This study involves both direct geoemtric methods and analytic tools concerning the boundary behavior of functions in the unit disk. The results in operator algebras are centered on the embedding problem for finite factors (in the sense of von Neumann) and questions about the analytic aspects of free probability theory. It was shown that, as far as simple spectral data is concerned, any sum of selfadjoint elements ina finite factor can be realized inside a special factor (technically, the ultrapower of the hyperfinite II-1 factor). This can be viewed as a first step towards answering an old question of Connes, and it involves the development of fairly complicated combinatorial tools. (Some of the combinatorial problems raised in this research were solved with the collaboration of undergraduate students.) The work in free probability pertains to the existence of absolutely continuous parts in free convolutions of measures, as well as the description of the possible weak limits for sums or products of infinitesimal and free random variables. The results briefly described above are contained in approximately twenty works published in refereed journals and conference proceedings. Other significant technical publications include a new edition of the book "Harmonic analysis of operators on Hilbert space". This work was out of print, and the new edition contains much new material. Among the broader impacts of the activities generated by this award, several graduate students have been trained, including three graduated Ph. D.s, and another three who will gaduate soon. Funds from this award were also used to support visitors and thus foster scientific collaboration. Another significant activity is the research activity of two undergraduate students who participated in REU programs, and whose results will soon be published. Both of these students are now in graduate school (computer science and mathematics, respectively). Another activity with broader impacts is the publication of the English translation of Hadamard's classical text on plane geometry. It is hoped that this book will contribute to the education of talented high school students, as well as to the training of future teachers of mathematics.
