Mathematical Sciences: Hankel Operators and Their Applications

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9623231 PELLER V.V. Peller is going to continue to study superoptimal approximation by analytic operator functions. Such approximations minimize not only the supremum of the norms but also the suprema of all further singular values. In particular V.V. Peller is going to study the dependence of the Mcmillan degree of the superoptimal approximant on the Mcmillam degree of the initial function. It is not known whether the Mcmillan degree can jump. Another open problem is whether the operator of superoptimal approximation preserves the Wiener algebra. V.V. Peller is also going to study the indices in the so-called thematic factorizations in the Nehari-Takagi problem and other problem on superoptimal approximation. V.V. Peller is also going to continue his work in prediction theory. One of the most important unsolved problems is to characterize in terms of the spectral densities the completely regular vectorial stationary Gaussian processes. V.V. Peller is going to apply Hankel operators in different domains of analysis, prediction theory and control theory. It has become clear that Hankel operators play a significant role in applications. In particular they play a decisive role in H-infinity control theory. V.V. Peller used Hankel operators to study superoptimal approximations by analytic matrix functions. In his joint work with N.J. Young it was shown that under very natural assumptions such an approximation is unique and can be found constructively. Further development of the theory was given by V.V. Peller and in joint papers of V.V. Peller with S.R. Treil. Superoptimal approximations play a very important role in control theory. There are still many open problems about superoptimal approximations which are very important in applications. V.V. Peller is going to continue to work on them. Another domain of applications of Hankel operators is prediction theory. V.V. Peller obtained many strong results (partly in a joint work with Khrushchev) in this field. The re are still many open problems in this field. V.V. Peller hopes to progress using vectorial Hankel operators.