Operator Theory and Free Harmonic Analysis

点击翻译按钮获取中文摘要

Abstract Bercovici Hari Bercovici plans to work in two basic directions. The first is related with Voiculescu's free probability theory and harmonic analysis. An important direction here is the sudy of limit laws for the multiplicative free convolution. Unlike classical harmonic analysis, the study of multiplicative convolution cannot simply be reduced to the additive case by taking logarithms. It is already known that there are some unexpected `Poisson like' probability distributions related with multiplicative free convolution. An important obstacle is the correct formulation of the classical limit theorems, even the central limit theorem, in the free multiplicative case. The second line of Bercovici's research is in operator theory and its interactions with function theory, control theory, and other areas. Among the specific problems in this area are the model theories of restrictions and compressions of operators, and the study of dual algebras generated by commuting families of operators. He also plans to return to the study of skew Toeplitz operators beyond the fairly well understood scalar case. In order to give some motivation for the research in this proposal I would like to recall that in the current description of subatomic processes one has to deal with probabilistic processes. Thus, for instance, the location of a particle is not known with precision at any given moment. One has instead the probability that that particle appears in a given region of space. Some versions of noncommutative probability theory deal with some of the aspects of quantum chemistry for instance. Free probability theory is likely to occur at some point in the quantum description of nature. Besides, it is an interesting object of study because it provides, in some sense, the `only' alternative probability theory. The operator theoretical part of this project stems to a great extent from problems raised by the control theory of large structures. Some earlier results in this theory were proposed a s the main control mechanism of the NASP. While Bercovici's expertise in the practical side of this theory is limited, control theorists can formulate their problems in theoretical terms, thus making the application of operator theoretical methods possible.