My Way in Mathematics: From Ergodic Theory Through Scattering to J-inner Matrix Functions and Passive Linear Systems Theory

Abstract

Some of the main mathematical themes that I have worked on, and how one theme led to another, are reviewed. Over the years I moved from the subject of my Master’s thesis on entropy in ergodic theory to scattering theory and the Nehari problem (in work with V.M. Adamjan and M.G. Krein) and then (in my second thesis) to passive linear stationary systems (including the Darlington method), to generalized bitangential interpolation and extension problems in special classes of matrix-valued functions, and then (in work with H. Dym) to the theory of de Branges reproducing kernel Hilbert spaces and their applications to direct and inverse problems for integral and differential systems of equations and to prediction problems for second-order vector-valued stochastic processes and (in work with O. Staffans) to new developments in the theory of passive linear stationary systems in the direction of state/signal systems theory. The role of my teachers (A.A. Bobrov, V.P. Potapov and M.G. Krein) and my former graduate students will also be discussed.