Introduction

Summary

We indicate here, briefly, the content of the chapters. Chapter 1 is an elementary introduction to the product integral of a continuous matrix-valued function and its properties. (The generalization to “Lebesgue product integrals” is sketched in Section 8.) This chapter should be accessible to readers with quite minimal mathematical background. It is prerequisite for understanding of the other chapters. Chapter 2 deals with contour product integrals; the development is parallel to that of the theory of ordinary contour integrals in complex variable theory. Contour product integrals are not used in later chapters. In Chapter 3 we present a theory of product integration in a much more general setting. More mathematical background is required here, for example, familiarity with functional analysis in Banach space and some integration theory. Included are results on the equation of evolution (1) with unbounded A(x). The other chapters (except Chapter 4, Section 5) are independent of this chapter. Chapter 4 presents applications of product integration to the theory of differential equations; some new results concerning solutions of the Schrodinger equation with rather singular potentials are included. In Chapter 5 we discuss product integration of (matrix-valued) measures. Some familiarity with measure theory is assumed here, but nothing very sophisticated is required. Chapter 6 contains a discussion of work on product integration by various other authors and some remarks on generalizations of the theory. Following Chapter 6 is an appendix on matrix theory containing elementary definitions and results and a few special results which may not be familiar to all readers.