198,785 research outputs found
On the Dunford-Pettis property
No full text"A Banach space E has the Dunford-Pettis property (DPP) if every weakly compact operator on E is a Dunford-Pettis operator, that is, takes weakly convergent sequences into norm convergent sequences. For many years it remained an open question whether the Banach space of all continuous E -valued functions on a compact Hausdorff space K has the DPP if E has. This question was answered in the negative in 1983 by M. Talagrand [Israel J. Math. 44 (1983), no. 4, 317–321;] who constructed a Banach space E with the DPP and a weakly compact operator from C([0,1],E) into c 0 that is not a Dunford-Pettis operator.
The author and B. Rodríguez-Salinas introduced [Arch. Math. (Basel) 47 (1986), no. 1, 55–65;] a more general class of operators that they called almost Dunford-Pettis. An operator T from C(K,E) into X whose representing measure has a semivariation continuous at ∅ said to be almost Dunford-Pettis if, for every weakly null sequence (x n ) in E and every bounded sequence (φ n ) in C(K) , we have lim n→∞ T(φ n x n )=0 . In that same paper they posed the problem of characterizing those Banach spaces E such that, for all compact Hausdorff spaces K , every weakly compact operator on C(K,E) is almost Dunford-Pettis. In the paper under review the author shows that such spaces are precisely those with the Dunford-Pettis property. In particular, the main result of the paper is that the following conditions are equivalent for a Banach space E : (a) For any compact Hausdorff space K , every weakly compact operator on C(K,E) is almost Dunford-Pettis; (b) every weakly compact operator on C([0,1],E) is almost Dunford-Pettis; (c) every weakly compact operator from C([0,1],E) into c 0 is almost Dunford-Pettis; (d) E has the Dunford-Pettis property."CAICYTDepto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasTRUEpu
On the class of order Dunford-Pettis operators
No full textsummary:We characterize Banach lattices and on which the adjoint of each operator from into which is order Dunford-Pettis and weak Dunford-Pettis, is Dunford-Pettis. More precisely, we show that if and are two Banach lattices then each order Dunford-Pettis and weak Dunford-Pettis operator from into has an adjoint Dunford-Pettis operator from into if, and only if, the norm of is order continuous or has the Schur property. As a consequence we show that, if and are two Banach lattices such that or has the Dunford-Pettis property, then each order Dunford-Pettis operator from into has an adjoint which is Dunford-Pettis if, and only if, the norm of is order continuous or has the Schur property
A note on L-Dunford-Pettis sets in a topological dual Banach space
No full textsummary:The present paper is devoted to some applications of the notion of L-Dunford-Pettis sets to several classes of operators on Banach lattices. More precisely, we establish some characterizations of weak Dunford-Pettis, Dunford-Pettis completely continuous, and weak almost Dunford-Pettis operators. Next, we study the relationships between L-Dunford-Pettis, and Dunford-Pettis (relatively compact) sets in topological dual Banach spaces
Dunford-Pettis Sets and Operators
No full textSets in Banach spaces that are mapped into norm compact sets by weakly compact operators (called Dunford-Pettis sets) are studied in general and in the spaces L(,1) ((mu),X), C((OMEGA),X), and P(,1)((mu),X). It is shown that if X is a Banach space with the Dunford-Pettis property and X contains no copy (,1), then L(,1)((mu),X) has the Dunford-Pettis property. Furthermore, if X has the Dunford-Pettis property and M is a subset of L(,1)((mu),X) that satisfies any of the extant criteria for weak compactness in L(,1)((mu),X), then it is shown that M is a Dunford-Pettis set. Various classes of Dunford-Pettis operators on L(,1) ((mu),X) are examined from the point of view of measurability properties of representing kernels. The relationship between structural properties of operators on C((OMEGA),X) and L(,(INFIN))((mu),X*) and properties of their representing measures is explored.Made available in DSpace on 2014-12-14T13:09:45Z (GMT). No. of bitstreams: 1
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Previous issue date: 1980Embargo set by: Seth Robbins for item 68347
Lift date: Forever
Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDsRestricted to the U of I community idenfinitely during batch ingest of legacy ETDsU of I Only113 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1980
Unbounded absolutely weak Dunford-Pettis operators
No full textIn the present article, we expose various properties of unbounded absolutely weak Dunford-Pettis and unbounded absolutely weak compact operators on a Banach lattice E. In addition to their topological and lattice properties, we investigate relationships between M-weakly compact operators, L-weakly compact operators, and order weakly compact operators with unbounded absolutely weak Dunford-Pettis operators. We show that the square of any positive uaw-Dunford-Pettis (M-weakly compact) operator on an order continuous Banach lattice is compact. Many examples are given to illustrate the essential conditions
A complement of positive weak almost Dunford-Pettis operators on Banach lattices
No full textsummary:In this paper, we give some necessary and sufficient conditions such that each positive operator between two Banach lattices is weak almost Dunford-Pettis, and we derive some interesting results about the weak Dunford-Pettis property in Banach lattices
About positive weak Dunford–Pettis operators on Banach lattices
No full textAbstractWe characterize Banach lattices for which each positive weak Dunford–Pettis operator from a Banach lattice into another dual Banach lattice is almost Dunford–Pettis. Also, we give some sufficient and necessary conditions for which the class of positive weak Dunford–Pettis operators coincides with that of positive Dunford–Pettis operators, and we derive some consequences
Geographies of Growth, Decline and Restructuring: the Rise and Fall (privatization) of the State-owned Steel Sector and the Trajectories of Steel Localities in the Italian Mezzogiorno
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