Paul W. Lewis

Some mapping properties of representing measures

SummaryThis paper studies relationships between operators on continuous function spaces and properties of associated vector measures given by Riesz Representation Theorems.

Dunford-Pettis sets

Bibasic sequences are used to study relative weak compactness and relative norm compactness of Dunford-Pettis sets. A Banach space X has the Dunford-Pettis property provided that every weakly compact operator with domain X and range an arbitrary Banach space Y maps weakly compact sets in X into norm compact sets in Y. Localizing this notion, a bounded subset A of X is said to be a Dunford-Pettis subset of X if T(A) is relatively norm compact in Y whenever T: X -Y is a weakly compact operator. Consequently, a Banach space X has the Dunford-Pettis property if and only if each of its weakly compact sets is a Dunford-Pettis set. The survey article by Diestel [5] is an excellent source of information about classical results in Banach spaces which relate to the Dunford-Pettis property. Kevin Andrews utilized Dunford-Pettis sets in a study of the Bochner integral in [1]. In Theorem 1 of [1], Andrews showed that a subset K of X is a Dunford-Pettis subset of X if and only if limn (sup{ jx* (x) I x C K}) = 0 whenever (x*) is a weakly null sequence in X* ( = the continuous linear dual of X). In Corollary 4 of [1], Andrews used this characterization to show that if X has the Dunford-Pettis property and ?1 does not embed in X, then the space Ll(pLi,X) has the Dunford-Pettis property. E. Bator showed in [2] that a dual space has the weak Radon-Nikodym property if and only if each Dunford-Pettis subset of X is relatively compact. In addition to reproducing Bators result, Emmanuele [8] established several other structure properties for Banach spaces in which all Dunford-Pettis sets are relatively compact. Since every bounded subset of a Banach space X whose dual space X* has the Schur property is a Dunford-Pettis subset of X, it is clear that there are Dunford-Pettis sets which are not relatively weakly compact. However, we note that Odell [13, p. 377] showed that every sequence in a Dunford-Pettis set has a weakly Cauchy subsequence. In this paper we study Dunford-Pettis sets which fail to be relatively norm or weakly compact. The following definitions and notation will be helpful. A sequence (Xnv, f2) in XxX* is called bibasic [14, p. 85], [4] if (xn) is a basic sequence in X, (fn) is a basic sequence in X*, and fi*(xj) = ?ij. If (Xn, fn) is a bibasic sequence, Xo = [Xn], Received by the editors April 14, 1998 and, in revised form, March 15, 2000. 2000 Mathematics Subject Classification. Primary 46B20; Secondary 46B15, 46B45.

Uniform exhaustivity and Banach lattices

SummaryThe measure theoretic notion of strong boundedness is extended to the setting of a real σ-complete Banach lattice X; the lattice theoretic concept which corresponds to strong boundedness is termed exhaustivity. Uniformly exhaustive subsets of X are studied, and various characterizations are presented. One of these characterizations involves the order continuity of the norm on X. The exhaustive elements of certain classical lattices are cataloged. Applications are given to spaces of measures.

STRONGLY BOUNDED REPRESENTING MEASURES AND CONVERGENCE THEOREMS

Let K be a compact Hausdorff space, X a Banach space and C ( K, X ) the Banach space of all continuous functions f : K → X endowed with the supremum norm. In this paper we study weakly precompact operators defined on C ( K, X ).

Dunford-Pettis Properties and Spaces of Operators

J. Elton used an application of Ramsey theory to show that if X is an infinite dimensional Banach space, then c0 embeds in X, l1 embeds in X, or there is a subspace of X that fails to have the Dunford–Pettis property. Bessaga and Pelczynski showed that if c0 embeds in X ∗, then l∞ embeds in X∗. Emmanuele and John showed that if c0 embeds in K(X,Y ), then K(X,Y ) is not complemented in L(X,Y ). Classical results from Schauder basis theory are used in a study of Dunford–Pettis sets and strong Dunford–Pettis sets to extend each of the preceding theorems. The space Lw∗ (X ∗,Y ) of w∗−w continuous operators is also studied. Mathematics Department, University of Wisconsin-River Falls, River Falls, WI 54022-5001, USA e-mail: ioana.ghenciu@uwrf.edu Department of Mathematics, University of North Texas, Denton, TX 76203-1430, USA e-mail: lewis@unt.edu Received by the editors August 19, 2004. AMS subject classification: Primary: 46B20; secondary: 46B28.

UNCONDITIONAL CONVERGENCE IN THE STRONG OPERATOR TOPOLOGY AND ℓ

In this paper we study non-complemented spaces of operators and the embeddability of l ∞ in the spaces of operators L ( X , Y ), K ( X , Y ) and K w * ( X *, Y ). Results of Bator and Lewis [ 2 , 3 ] ( Bull. Pol. Acad. Sci. Math. 50 (4) (2002), 413–416; Bull. Pol. Acad. Sci. Math. 549 (1) (2006), 63–73), Emmanuele [ 8–10 ] ( J. Funct. Anal. 99 (1991), 125–130; Math. Proc. Camb. Phil. Soc. 111 (1992), 331–335; Atti. Sem. Mat. Fis. Univ. Modena 42 (1) (1994), 123–133), Feder [ 11 ] ( Canad. Math. Bull. 25 (1982), 78–81) and Kalton [ 16 ] ( Math. Ann. 208 (1974), 267–278), are generalised. A vector measure result is used to study the complementation of the spaces W ( X , Y ) and K ( X , Y ) in the space L ( X , Y ), as well as the complementation of K ( X , Y ) in W ( X , Y ). A fundamental result of Drewnowski [ 7 ] ( Math. Proc. Camb. Phil. Soc. 108 (1990), 523–526) is used to establish a result for operator-valued measures, from which we obtain as corollaries the Vitali–Hahn–Saks–Nikodym theorem, the Nikodym Boundedness theorem and a Banach space version of the Phillips Lemma.

Properties ( V) and ( w V) on C(Ω X)

A formal series Σ x n in a Banach space X is said to be weakly unconditionally converging, or alternatively weakly unconditionally Cauchy ( wuc ) if Σ| x *( x n )| x * ∈ X *. A subset K of X * is called a V -subset of X * if for each wuc series Σ x n in X . Further, the Banach space X is said to have property ( V ) if the V -subsets of X * coincide with the relatively weakly compact subsets of X *. In a fundamental paper in 1962, Pelczynski [ 10 ] showed that the Banach space X has property ( V ) if and only if every unconditionally converging operator with domain X is weakly compact. In this same paper, Pelczynski also showed that all C (Ω) spaces have property ( V ), and asked if the abstract continuous function space C (Ω, X ) has property ( F ) whenever X has property ( F ).

PERMANENCE PROPERTIES OF ABSOLUTE CONTINUITY CONDITIONS

Publisher Summary This chapter discusses the permanence of operator-theoretic properties of vector measures given by Riesz theorems under three separate notions of absolute continuity. It discusses whether compactness and weak compactness of operators are preserved under various absolute continuity conditions. A representing measure is countably additive if and only if it is regular. If each of n and m is a representing measure, then n is absolutely continuous with respect to m(n 0, there is a δ > 0. The reflexivity of E immediately implies that n(A) is a weakly compact operator for each A ∈ Σ.

Vector measures and the strong operator topology

A fundamental result of Nigel Kalton is used to establish a result for operator valued measures which has improved versions of the Vitali-Hahn-Saks Theorem, Phillipss Lemma, the Orlicz-Pettis Theorem and other classical results as straightforward corollaries.

Errata and addendum: tensor products and Dunford¿Pettis sets

Ghenciu and Lewis introduced the notion of a strong Dunford?Pettis set and used this notion to study the presence or absence of isomorphic copies of c0 in Banach spaces. The authors asserted that they could obtain a fundamental result of J. Elton without resorting to Ramsey theory. While the stated theorems are correct, unfortunately there is a flaw in the proof of the first theorem in the paper which also affects subsequent corollaries and theorems. The difficulty is discussed, and Eltons results are employed to establish a Schauder basis proposition which leads to a quick proof of the theorem in question. Additional results where questions arise are discussed on an individual basis