Transitivity for linear operators on a Banach space

Bertram Yood

Displaying similar documents to “Transitivity for linear operators on a Banach space”

Isometries between groups of invertible elements in Banach algebras

Osamu Hatori (2009)

Studia Mathematica

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We show that if T is an isometry (as metric spaces) from an open subgroup of the group of invertible elements in a unital semisimple commutative Banach algebra A onto a open subgroup of the group of invertible elements in a unital Banach algebra B, then $T {(1)}^{- 1} T$ is an isometrical group isomorphism. In particular, $T {(1)}^{- 1} T$ extends to an isometrical real algebra isomorphism from A onto B.

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 $λ$ . ...

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...

Banach spaces which admit a norm with the uniform Kadec-Klee property

S. Dilworth, Maria Girardi, Denka Kutzarova (1995)

Studia Mathematica

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Several results are established about Banach spaces Ӿ which can be renormed to have the uniform Kadec-Klee property. It is proved that all such spaces have the complete continuity property. We show that the renorming property can be lifted from Ӿ to the Lebesgue-Bochner space $L_{2} (Ӿ)$ if and only if Ӿ is super-reflexive. A basis characterization of the renorming property for dual Banach spaces is given.

Remarks and examples concerning distance ellipsoids

Dirk Praetorius (2002)

Colloquium Mathematicae

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We provide for every 2 ≤ k ≤ n an n-dimensional Banach space E with a unique distance ellipsoid such that there are precisely k linearly independent contact points between and $B_{E}$ . The corresponding result holds for spaces with non-unique distance ellipsoids as well. We construct n-dimensional Banach spaces E such that one distance ellipsoid has precisely k linearly independent contact points and all other distance ellipsoids have less than k-1 such points.

An indecomposable Banach space of continuous functions which has small density

Rogério Augusto dos Santos Fajardo (2009)

Fundamenta Mathematicae

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Using the method of forcing we construct a model for ZFC where CH does not hold and where there exists a connected compact topological space K of weight $ω ₁ < 2^{ω}$ such that every operator on the Banach space of continuous functions on K is multiplication by a continuous function plus a weakly compact operator. In particular, the Banach space of continuous functions on K is indecomposable.

Operators determining the complete norm topology of C(K)

A. Villena (1997)

Studia Mathematica

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For any uniformly closed subalgebra A of C(K) for a compact Hausdorff space K without isolated points and $x_{0} \in A$ , we show that every complete norm on A which makes continuous the multiplication by $x_{0}$ is equivalent to $∥ \cdot ∥_{\infty}$ provided that $x_{0}^{- 1} (λ)$ has no interior points whenever λ lies in ℂ. Actually, these assertions are equivalent if A = C(K).

The Maurey extension property for Banach spaces with the Gordon-Lewis property and related structures

P. G. Casazza, N. J. Nielsen (2003)

Studia Mathematica

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The main result of this paper states that if a Banach space X has the property that every bounded operator from an arbitrary subspace of X into an arbitrary Banach space of cotype 2 extends to a bounded operator on X, then every operator from X to an L₁-space factors through a Hilbert space, or equivalently $B (ℓ_{\infty}, X *) = Π ₂ (ℓ_{\infty}, X *)$ . If in addition X has the Gaussian average property, then it is of type 2. This implies that the same conclusion holds if X has the Gordon-Lewis property (in particular X could...

On copies of $c_{0}$ in the bounded linear operator space

Juan Carlos Ferrando, J. M. Amigó (2000)

Czechoslovak Mathematical Journal

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In this note we study some properties concerning certain copies of the classic Banach space $c_{0}$ in the Banach space $ℒ (X, Y)$ of all bounded linear operators between a normed space $X$ and a Banach space $Y$ equipped with the norm of the uniform convergence of operators.

The Dual of a Non-reflexive L-embedded Banach Space Contains $l^{\infty}$ Isometrically

Hermann Pfitzner (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

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A Banach space is said to be L-embedded if it is complemented in its bidual in such a way that the norm between the two complementary subspaces is additive. We prove that the dual of a non-reflexive L-embedded Banach space contains $l^{\infty}$ isometrically.

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...

Antiproximinal sets in the Banach space $c (X)$

S. Cobzaş (1997)

Commentationes Mathematicae Universitatis Carolinae

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If $X$ is a Banach space then the Banach space $c (X)$ of all $X$ -valued convergent sequences contains a nonvoid bounded closed convex body $V$ such that no point in $C (X) ∖ V$ has a nearest point in $V$ .

Noncommutative uniform algebras

Mati Abel, Krzysztof Jarosz (2004)

Studia Mathematica

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We show that a real Banach algebra A such that ||a²|| = ||a||² for a ∈ A is a subalgebra of the algebra $C_{ℍ} (X)$ of continuous quaternion-valued functions on a compact set X.

(Non-)amenability of ℬ(E)

Volker Runde (2010)

Banach Center Publications

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In 1972, the late B. E. Johnson introduced the notion of an amenable Banach algebra and asked whether the Banach algebra ℬ(E) of all bounded linear operators on a Banach space E could ever be amenable if dim E = ∞. Somewhat surprisingly, this question was answered positively only very recently as a by-product of the Argyros-Haydon result that solves the “scalar plus compact problem”: there is an infinite-dimensional Banach space E, the dual of which is ℓ¹, such that $ℬ (E) = (E) + ℂ i d_{E}$ . Still, ℬ(ℓ²) is...