Minimal Factorization, Linear Systems and Integral Operators

Abstract

In three lectures a survey is given of recent work on factorization and invertibility problems concerning systems and integral operators. In the first lecture the emphasis is on minimal factorization of linear time invariant systems. Wiener-Hopf integral equations with matrix-valued kernels form the main topic of the second lecture. The classical approach to such equations is combined with the method of factorization discussed in the first lecture to obtain explicit formulas for the resolvent kernel and the Fredholm characteristics. Special attention will be paid to equations with analytic symbols and to Wiener-Hopf equations appearing in transport theory. In the third lecture integral operators are viewed as transfer operators of linear time varying systems with well-posed boundary conditions. This approach allows one to study invertibility and factorization of integral operators in terms of inversion and decoupling of systems. The main part of the work reported on is joint work (in different groupings) with H. Bart, I. Gohberg, F. van Schagen and P. Van Dooren. It is a pleasure to thank I. Gohberg for several discussions about the plan for these lectures.