9401380 Bercovici The proposed work is in three directions. The first is in basic operator theory. The theory of dual algebras will be applied to study pairs of commuting operators. These techniques were most successful in the case of single operators. The second direction is related to the study of interpolation and lifting problems associated with the concept of structured singular value. These issues are interesting as problems in operator theory but they also have applications for control theory. The third direction is a development of earlier work in the framework of Voiculescu's free harmonic analysis. The convolution theory in this analysis still has many aspects which are not well understood. For instance, the domain of attraction to the semicircle distribution in the central limit theorem is not known. Problems on large deviations have not been studied in this context. This project lies at the interface of several branches of mathematics (operator theory, probability, function theory). Much of the project involves a better understanding and new results in these disciplines. Further, parts of the project have a significant interaction with control theory and systems theory, and the results might have an impact on calculations related to the control of large systems. ***
