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A finiteness theorem for holomorphic Banach bundles

Jürgen Leiterer (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Let E be a holomorphic Banach bundle over a compact complex manifold, which can be defined by a cocycle of holomorphic transition functions with values of the form id + K where K is compact. Assume that the characteristic fiber of E has the compact approximation property. Let n be the complex dimension of X and 0 q n . Then: If V X is a holomorphic vector bundle (of finite rank) with H q ( X , V ) = 0 , then dim H q ( X , V E ) < . In particular, if dim H q ( X , 𝒪 ) = 0 , then dim H q ( X , E ) < .

A new characterization of Eberlein compacta

Luis Oncina (2001)

Studia Mathematica

We give a sufficient and necessary condition for a Radon-Nikodým compact space to be Eberlein compact in terms of a separable fibre connecting weak-* and norm approximation.

A new proof of James' sup theorem.

Marianne Morillon (2005)

Extracta Mathematicae

We provide a new proof of James' sup theorem for (non necessarily separable) Banach spaces. One of the ingredients is the following generalization of a theorem of Hagler and Johnson: "If a normed space E does not contain any asymptotically isometric copy of l1, then every bounded sequence of E' has a normalized l1-block sequence pointwise converging to 0".

A quantitative version of Krein's theorem.

M. Fabian, P. Hájek, V. Montesinos, V. Zizler (2005)

Revista Matemática Iberoamericana

A quantitative version of Krein's Theorem on convex hulls of weak compact sets is proved. Some applications to weakly compactly generated Banach spaces are given.

An approximation property with respect to an operator ideal

Juan Manuel Delgado, Cándido Piñeiro (2013)

Studia Mathematica

Given an operator ideal , we say that a Banach space X has the approximation property with respect to if T belongs to S T : S ( X ) ¯ τ c for every Banach space Y and every T ∈ (Y,X), τ c being the topology of uniform convergence on compact sets. We present several characterizations of this type of approximation property. It is shown that some of the existing approximation properties in the literature may be included in this setting.

An Example Concerning Valdivia Compact Spaces

Kalenda, Ondrej (1999)

Serdica Mathematical Journal

∗ Supported by Research grants GAUK 190/96 and GAUK 1/1998We prove that the dual unit ball of the space C0 [0, ω1 ) endowed with the weak* topology is not a Valdivia compact. This answers a question posed to the author by V. Zizler and has several consequences. Namely, it yields an example of an affine continuous image of a convex Valdivia compact (in the weak* topology of a dual Banach space) which is not Valdivia, and shows that the property of the dual unit ball being Valdivia is not an isomorphic...

Application of ( L ) sets to some classes of operators

Kamal El Fahri, Nabil Machrafi, Jawad H'michane, Aziz Elbour (2016)

Mathematica Bohemica

The paper contains some applications of the notion of Ł sets to several classes of operators on Banach lattices. In particular, we introduce and study the class of order ( L ) -Dunford-Pettis operators, that is, operators from a Banach space into a Banach lattice whose adjoint maps order bounded subsets to an ( L ) sets. As a sequence characterization of such operators, we see that an operator T : X E from a Banach space into a Banach lattice is order Ł -Dunford-Pettis, if and only if | T ( x n ) | 0 for σ ( E , E ' ) for every weakly null...

Characterizations of weakly compact sets and new fixed point free maps in c₀

P. N. Dowling, C. J. Lennard, B. Turett (2003)

Studia Mathematica

We give a basic sequence characterization of relative weak compactness in c₀ and we construct new examples of closed, bounded, convex subsets of c₀ failing the fixed point property for nonexpansive self-maps. Combining these results, we derive the following characterization of weak compactness for closed, bounded, convex subsets C of c₀: such a C is weakly compact if and only if all of its closed, convex, nonempty subsets have the fixed point property for nonexpansive mappings.

Compact embeddings of Brézis-Wainger type.

Fernando Cobos, Thomas Kühn, Tomas Schonbek (2006)

Revista Matemática Iberoamericana

Let Ω be a bounded domain in Rn and denote by idΩ the restriction operator from the Besov space Bpq1+n/p(Rn) into the generalized Lipschitz space Lip(1,-α)(Ω). We study the sequence of entropy numbers of this operator and prove that, up to logarithmic factors, it behaves asymptotically like ek(idΩ) ~ k-1/p if α > max (1 + 2/p + 1/q, 1/p). Our estimates improve previous results by Edmunds and Haroske.

Compact operators whose adjoints factor through subspaces of l p

Deba P. Sinha, Anil K. Karn (2002)

Studia Mathematica

For p ≥ 1, a subset K of a Banach space X is said to be relatively p-compact if K n = 1 α x : α B a l l ( l p ' ) , where p’ = p/(p-1) and x l p s ( X ) . An operator T ∈ B(X,Y) is said to be p-compact if T(Ball(X)) is relatively p-compact in Y. Similarly, weak p-compactness may be defined by considering x l p w ( X ) . It is proved that T is (weakly) p-compact if and only if T* factors through a subspace of l p in a particular manner. The normed operator ideals ( K p , κ p ) of p-compact operators and ( W p , ω p ) of weakly p-compact operators, arising from these factorizations,...

Compact spaces that do not map onto finite products

Antonio Avilés (2009)

Fundamenta Mathematicae

We provide examples of nonseparable compact spaces with the property that any continuous image which is homeomorphic to a finite product of spaces has a maximal prescribed number of nonseparable factors.

Compactness criteria in function spaces

Monika Dörfler, Hans G. Feichtinger, Karlheinz Gröchenig (2002)

Colloquium Mathematicae

The classical criterion for compactness in Banach spaces of functions can be reformulated into a simple tightness condition in the time-frequency domain. This description preserves more explicitly the symmetry between time and frequency than the classical conditions. The result is first stated and proved for L ² ( d ) , and then generalized to coorbit spaces. As special cases, we obtain new characterizations of compactness in Besov-Triebel-Lizorkin, modulation and Bargmann-Fock spaces.

Compactness in L¹ of a vector measure

J. M. Calabuig, S. Lajara, J. Rodríguez, E. A. Sánchez-Pérez (2014)

Studia Mathematica

We study compactness and related topological properties in the space L¹(m) of a Banach space valued measure m when the natural topologies associated to convergence of vector valued integrals are considered. The resulting topological spaces are shown to be angelic and the relationship of compactness and equi-integrability is explored. A natural norming subset of the dual unit ball of L¹(m) appears in our discussion and we study when it is a boundary. The (almost) complete continuity of the integration...

Corrections to the paper “The boundedness of certain sublinear operator in the weighted variable Lebesgue spaces“

Rovshan A. Bandaliev (2013)

Czechoslovak Mathematical Journal

In this paper the author proved the boundedness of the multidimensional Hardy type operator in weighted Lebesgue spaces with variable exponent. As an application he proved the boundedness of certain sublinear operators on the weighted variable Lebesgue space. The proof of the boundedness of the multidimensional Hardy type operator in weighted Lebesgue spaces with a variable exponent does not contain any mistakes. But in the proof of the boundedness of certain sublinear operators on the weighted...

Countable products of spaces of finite sets

Antonio Avilés (2005)

Fundamenta Mathematicae

We consider the compact spaces σₙ(Γ) of subsets of Γ of cardinality at most n and their countable products. We give a complete classification of their Banach spaces of continuous functions and a partial topological classification.

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