Chapter 6 - Complements; other Work and further Results on Product Integration

Summary

In the present chapter, we shall discuss additional results and generalizations of the theory that we have presented so far. A large part of our discussion will focus on the work of other authors. Some of this work, e.g., product integration of nonlinear operator-valued functions, has not been developed in the previous chapters and will only be discussed briefly here. This is partly due to the fact that for these results, a really systematic and complete presentation is simply not available at the present time. On the other hand, what we have presented above has been somewhat determined by our own predispositions; a treatise of reasonable length covering all aspects of product integration in detail is hardly feasible, and we have therefore selected material which is of the most interest to us and which at the same time is not readily accessible in the modern literature. Nevertheless, the results we shall discuss in this chapter are of considerable significance and importance and certainly deserve some attention in any general discussion of product integration.

If we consider the theory of product integration of matrix-valued functions presented in Chapter 1 as a starting point, then there are several possible directions for generalization. For example, a first type of generalization might focus on reducing to a minimum the regularity assumptions on the function A(x) being product integrated with regard to its dependence on x, while not changing essentially the nature of the values of this function.