Mathematical Sciences: Hankel Operators and Their Applications

PELLER v。v。PELLER将继续研究解析算子函数的超优逼近。这样的近似不仅使范数的上极值最小，而且使所有进一步的奇异值的上极值最小。特别是V.V. Peller将研究超优近似的麦克米伦度与初始函数的麦克米伦度的依赖关系。目前尚不清楚麦克米伦学位是否可以跳跃。另一个开放问题是超优逼近算子是否保留维纳代数。V.V. Peller也将研究所谓的主题分解中的指标在Nehari-Takagi问题和其他超优逼近问题中。V.V. Peller也将继续他的预测理论研究。一个最重要的尚未解决的问题是用谱密度来描述完全规则的矢量平稳高斯过程。V.V. Peller将把Hankel算子应用于分析、预测理论和控制理论的不同领域。很明显，汉高操作符在应用程序中发挥着重要作用。特别是它们在h∞控制理论中起着决定性的作用。Peller用Hankel算子研究了解析矩阵函数的超优逼近。在他与N.J. Young的联合研究中，证明了在非常自然的假设下，这种近似是唯一的，并且可以建设性地找到。V.V. Peller和他与S.R. Treil的联合论文进一步发展了这一理论。超优逼近在控制理论中占有非常重要的地位。超优逼近在应用中仍有许多有待解决的问题。V.V. Peller会继续研究它们。汉克尔算子的另一个应用领域是预测理论。佩勒（V.V. Peller）在这方面取得了许多强有力的成果（部分是在与赫鲁晓夫的合作中）。在这一领域仍有许多未解决的问题。V.V. Peller希望使用向量汉克尔算子取得进展。

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.