P-continuous vector-valued functions

Abstract

This thesis focuses on the study of p-continuous vector-valued functions. They are functions defined on a compact Hausdorff space with values in a Banach space whose range is p-compact (1 < p < oo). A subset K of a Banach space X is relatively p-compact if there exists a sequence (xn)£/p(X) ((Xn) e co(X) if p = oo) such that K c {Xn «nXn : £n [cxn|p' < 1}, where p' is the conjugate index of p (i.e., 1/p + 1/p' = 1, with l/oo = 0). From this point of view, the notion of compactness can be seen as the particular case of oo-compactness. The set of all these functions is denoted by CP(Q,X), where n is a compact Hausdorff space and X is a Banach space. The thesis has been organized as follows. In Chapter 1, we present the space of p-continuous X-valued functions, Cp(n,X), and we prove that it is isometricaliy isomorphic to C(Q)®dpX, where dp is the right Chevet-Saphar tensor norm. We also introduce the space of unconditionally p-continuous vector-valued functions in a natural way and we characterize it by a tensor product too. Tensor products play an important role in this chapter. Both characterizations rely, among others, on an important result about a-nuclear operators, which is proved. This chapter is based on [57]. Chapter 2 collects some topological properties of Cp(n,X), Namely, we obtain some resuits related to density of simple vector-valued functions in B(Z)®dpX (where B(X) denotes the space of all bounded Borei-measurabie scalar functions defined on O), complemented embeddings of C(Q) and X in Cp(n,X), and sequences in Cp(o,X). In this chapter, we also study the weak and weak* convergences of sequences in Cp(0,X). Chapter 3 focuses on a classical result of Analysis: integral representation of operators defined on continuous functions. In particular, we establish two integral representation theorems: one for operators S e L(C(0),L(X,Y)) (which extends the classical Bartle-Dunford-Schwartz representation theorem), and another for operators U e L(CP(Q,X),Y) (which extends the classical Dincuieanu-Singer representation theorem). We provide an alternative simpler proof of the latter result using the first one. We also build the needed integration theory. This chapter is based on [59]. Chapter 4 deals with the associated operator U# defined in Chapter 3. Every operator U e L(Z®aX,Y) has an associated operator U# e L(Z,L(X,Y)) defined in a natural way. In this chapter, we study the problem of the existence of an operator U e L(Z®aX,Y) such that U# = S for a given operator S e L(Z,L(X,Y)), solving a long-standing conjecture by Dincuieanu. This chapter is based on [58]. Chapter 5 is devoted to the study of absolutely (r,q)-summing operators from Cp(Q,X) to Y. We study the interplay between U, its associated operator U#, and its representing measure (built in Chapter 3). Since CM(fi,X) = C(Q,X), this encompasses not only the classical Swartz theorem about absolutely summing operators from C(fl,X) to Y but also its existing extensions, providing an improvement even to the Swartz theorem. Counterexamples are exhibited to indicate sharpness of our results. This chapter is based on [60].