On the structure of weakly compact subsets of Hilbert spaces and applications to the geometry of Banach spaces

In this paper, we introduce weakly compact version of the weakly countably determined (WCD) property, the strongWCD(SWCD) property. A Banach spaceXis said to beSWCDif there s a sequence (An) of weak∗compact subsets ofX∗∗such that ifK⊂Xis weakly compact, there is an(nm)⊂Nsuch thatK⊂⋂m=1∞Anm⊂X. In this case, (An) is called a strongly determining sequence forX. We show thatSWCG⇒SWCDand that the converse does not hold in general. In fact,Xis a separableSWCDspace if and only if (X, weak) is anℵ0-space. Usingc0for an example, we show how weakly compact structure theorems may be used to construct strongly determining sequences.

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WEAKLY COMPACT SUBSETS OF BANACH SPACES

Surveys in General Topology ◽

10.1016/b978-0-12-584960-9.50021-3 ◽

1980 ◽

pp. 479-494 ◽

Cited By ~ 6

Author(s):

Michael L. Wage

Keyword(s):

Banach Spaces ◽

Weakly Compact ◽

Compact Subsets

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Weakly compact operators and the strong* topology for a Banach space

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210509001486 ◽

2010 ◽

Vol 140 (6) ◽

pp. 1249-1267 ◽

Cited By ~ 3

Author(s):

Antonio M. Peralta ◽

Ignacio Villanueva ◽

J. D. Maitland Wright ◽

Kari Ylinen

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Hilbert Spaces ◽

Strong Topology ◽

Weakly Compact ◽

Bounded Linear ◽

Convex Topology ◽

Linear Maps ◽

Locally Convex Topology ◽

Locally Convex

The strong* topology s*(X) of a Banach space X is defined as the locally convex topology generated by the seminorms x ↦ ‖Sx‖ for bounded linear maps S from X into Hilbert spaces. The w-right topology for X, ρ(X), is a stronger locally convex topology, which may be analogously characterized by taking reflexive Banach spaces in place of Hilbert spaces. For any Banach space Y , a linear map T : X → Y is known to be weakly compact precisely when T is continuous from the w-right topology to the norm topology of Y. The main results deal with conditions for, and consequences of, the coincidence of these two topologies on norm bounded sets. A large class of Banach spaces, including all C*-algebras and, more generally, all JB*-triples, exhibit this behaviour.

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Weakly compactly generated locally convex spaces

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100053706 ◽

1977 ◽

Vol 82 (1) ◽

pp. 85-98 ◽

Cited By ~ 8

Author(s):

Richard J. Hunter ◽

John Lloyd

Keyword(s):

Banach Spaces ◽

Compact Subset ◽

Locally Convex Spaces ◽

Weakly Compact ◽

Permanence Properties ◽

Locally Convex ◽

Compactly Generated ◽

Convex Spaces ◽

Compact Subsets

AbstractLocally convex spaces which are generated by a weakly compact subset (or a sequence of weakly compact subsets) and their subspaces are studied. Various characterizations and the permanence properties of these spaces are obtained. Certain results valid for weakly compactly generated Banach spaces are extended. These spaces are shown to have sequential properties which extend well-known properties of separable locally convex spaces.

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Sets of periods for chaotic linear operators

Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas ◽

10.1007/s13398-020-00996-z ◽

2021 ◽

Vol 115 (2) ◽

Author(s):

J. A. Conejero ◽

F. Martínez-Giménez ◽

A. Peris ◽

F. Rodenas

Keyword(s):

Banach Spaces ◽

Hilbert Spaces ◽

Complete Characterization ◽

Linear Operators

AbstractWe provide a complete characterization of the possible sets of periods for Devaney chaotic linear operators on Hilbert spaces. As a consequence, we also derive this characterization for linearizable maps on Banach spaces.

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