Absolutely Summing Mappings, Nuclear Mappings and Convolution Equations

The material of this book was taught in the discipline Topics on Functional Analysis of the Graduate Program of IMECC-UNICAMP during the first semester of 2005.The results of Chapter 9 are new and extend the Existence and Approximations Theorems for convolution equations presented by C.P. Gupta in his PHD dissertation at University of Rochester in 1968 (see [5]). Of course, the theorems of this chapter, as well as those in Gupta's dissertation, are the infinite dimensional versions of well known results proved by B. Malgrange (see [9]).In order to get the above results we wrote Chapter 8, where we introduced and proved theorems on quasi-nuclear holomorphic mappings between Banach spaces.Chapter 8, with new results and extensions of the nuclear mappings considered before by Gupta (see [5]) and Matos (see [12]), is essential for the construction of the quasi-nuclear mappings.In Chapter 5 we considered $(p;m(s; q))$-summing mappings, first studied in Matos [14]. The new features in this chapter are the introduction of the exponential type $(p;m(s; q))$-summing mappings and the division results for them. These division theorems play an important role in Chapter 9.In Chapters 4 and 6 we consider linear and non-linear $(m(s; p); q)$-summing mappings. Soares in [20] considered holomorphic, multilinear and polynomial mixing summing mappings, special cases of the mappings considered in Chapter 6. The results of this Chapter 6 are new. It would be nice if someone could find for these mappings similar results to those proved in Chapters 9,8 and 5.The results of Chapters 1,2 and 3 are all known and they are there in order to motivate and prove results used in the others chapters.Since the length of the new material proved here forbids the publication of it in some journal, we opted to publish this book, in a limited edition, in order to make it accessible to the interested researchers of the area.I want to thank Vinicius Vieira Favaro for the careful reading of the first version of this book and also for pointing out several mistakes and misprints of that version.