Operators determining the complete norm topology of C(K)

A. Villena

Displaying similar documents to “Operators determining the complete norm topology of C(K)”

Transitivity for linear operators on a Banach space

Bertram Yood (1999)

Studia Mathematica

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Let G be the multiplicative group of invertible elements of E(X), the algebra of all bounded linear operators on a Banach space X. In 1945 Mackey showed that if $x_{1}, \dots, x_{n}$ and $y_{1}, \dots, y_{n}$ are any two sets of linearly independent elements of X with the same number of items, then there exists T ∈ G so that $T (x_{k}) = y_{k}$ , $k = 1, \dots, n$ . We prove that some proper multiplicative subgroups of G have this property.

Simple construction of spaces without the Hahn-Banach extension property

Jerzy Kąkol (1992)

Commentationes Mathematicae Universitatis Carolinae

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An elementary construction for an abundance of vector topologies $ξ$ on a fixed infinite dimensional vector space $E$ such that $(E, ξ)$ has not the Hahn-Banach extension property but the topological dual ${(E, ξ)}^{'}$ separates points of $E$ from zero is given.

Projections from $L (X, Y)$ onto $K (X, Y)$

Kamil John (2000)

Commentationes Mathematicae Universitatis Carolinae

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Generalization of certain results in [Sap] and simplification of the proofs are given. We observe e.g.: Let $X$ and $Y$ be Banach spaces such that $X$ is weakly compactly generated Asplund space and $X^{*}$ has the approximation property (respectively $Y$ is weakly compactly generated Asplund space and $Y^{*}$ has the approximation property). Suppose that $L (X, Y) \neq K (X, Y)$ and let $1 < λ < 2$ . Then $X$ (respectively $Y$ ) can be equivalently renormed so that any projection $P$ of $L (X, Y)$ onto $K (X, Y)$ has the sup-norm greater or equal to $λ$ . ...

Uniqueness of the topology on L¹(G)

J. Extremera, J. F. Mena, A. R. Villena (2002)

Studia Mathematica

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Let G be a locally compact abelian group and let X be a translation invariant linear subspace of L¹(G). If G is noncompact, then there is at most one Banach space topology on X that makes translations on X continuous. In fact, the Banach space topology on X is determined just by a single nontrivial translation in the case where the dual group Ĝ is connected. For G compact we show that the problem of determining a Banach space topology on X by considering translation operators on X is...

The structure of Lindenstrauss-Pełczyński spaces

Jesús M. F. Castillo, Yolanda Moreno, Jesús Suárez (2009)

Studia Mathematica

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Lindenstrauss-Pełczyński (for short ℒ) spaces were introduced by these authors [Studia Math. 174 (2006)] as those Banach spaces X such that every operator from a subspace of c₀ into X can be extended to the whole c₀. Here we obtain the following structure theorem: a separable Banach space X is an ℒ-space if and only if every subspace of c₀ is placed in X in a unique position, up to automorphisms of X. This, in combination with a result of Kalton [New York J. Math. 13 (2007)], provides...

Theorems of Krein Milman type for certain convex sets of functions operators

Robert R. Phelps (1970)

Annales de l'institut Fourier

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Sufficient conditions are given in order that, for a bounded closed convex subset $B$ of a locally convex space $E$ , the set $C (X, B)$ of continuous functions from the compact space $X$ into $B$ , is the uniformly closed convex hull in $C (X, E)$ of its extreme points. Applications are made to the unit ball of bounded (or compact, or weakly compact) operators from certain Banach spaces into $C (X)$ .

Holomorphic germs on Banach spaces

Chae Soo Bong (1971)

Annales de l'institut Fourier

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Let $E$ and $F$ be two complex Banach spaces, $U$ a nonempty subset of $E$ and $K$ a compact subset of $E$ . The concept of holomorphy type $θ$ between $E$ and $F$ , and the natural locally convex topology $𝒯_{ω, θ}$ on the vector space $ℋ_{θ} (U, F)$ of all holomorphic mappings of a given holomorphy type $θ$ from $U$ to $F$ were considered first by L. Nachbin. Motived by his work, we introduce the locally convex space $ℋ_{θ} (K, F)$ of all germs of holomorphic mappings into $F$ around $K$ of a given holomorphy type $θ$ , and study its interplay with...

On decompositions of Banach spaces into a sum of operator ranges

V. Fonf, V. Shevchik (1999)

Studia Mathematica

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It is proved that a separable Banach space X admits a representation $X = X_{1} + X_{2}$ as a sum (not necessarily direct) of two infinite-codimensional closed subspaces $X_{1}$ and $X_{2}$ if and only if it admits a representation $X = A_{1} (Y_{1}) + A_{2} (Y_{2})$ as a sum (not necessarily direct) of two infinite-codimensional operator ranges. Suppose that a separable Banach space X admits a representation as above. Then it admits a representation $X = T_{1} (Z_{1}) + T_{2} (Z_{2})$ such that neither of the operator ranges $T_{1} (Z_{1})$ , $T_{2} (Z_{2})$ contains an infinite-dimensional closed subspace...