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M ( r , s ) -ideals of compact operators

Rainis Haller, Marje Johanson, Eve Oja (2012)

Czechoslovak Mathematical Journal

We study the position of compact operators in the space of all continuous linear operators and its subspaces in terms of ideals. One of our main results states that for Banach spaces X and Y the subspace of all compact operators 𝒦 ( X , Y ) is an M ( r 1 r 2 , s 1 s 2 ) -ideal in the space of all continuous linear operators ( X , Y ) whenever 𝒦 ( X , X ) and 𝒦 ( Y , Y ) are M ( r 1 , s 1 ) - and M ( r 2 , s 2 ) -ideals in ( X , X ) and ( Y , Y ) , respectively, with r 1 + s 1 / 2 > 1 and r 2 + s 2 / 2 > 1 . We also prove that the M ( r , s ) -ideal 𝒦 ( X , Y ) in ( X , Y ) is separably determined. Among others, our results complete and improve some well-known results...

Mapping Properties of c 0

Paul Lewis (1999)

Colloquium Mathematicae

Bessaga and Pełczyński showed that if c 0 embeds in the dual X * of a Banach space X, then 1 embeds as a complemented subspace of X. Pełczyński proved that every infinite-dimensional closed linear subspace of 1 contains a copy of 1 that is complemented in 1 . Later, Kadec and Pełczyński proved that every non-reflexive closed linear subspace of L 1 [ 0 , 1 ] contains a copy of 1 that is complemented in L 1 [ 0 , 1 ] . In this note a traditional sliding hump argument is used to establish a simple mapping property of c 0 which simultaneously...

M-complete approximate identities in operator spaces

A. Arias, H. Rosenthal (2000)

Studia Mathematica

This work introduces the concept of an M-complete approximate identity (M-cai) for a given operator subspace X of an operator space Y. M-cai’s generalize central approximate identities in ideals in C*-algebras, for it is proved that if X admits an M-cai in Y, then X is a complete M-ideal in Y. It is proved, using ’special’ M-cai’s, that if J is a nuclear ideal in a C*-algebra A, then J is completely complemented in Y for any (isomorphically) locally reflexive operator space Y with J ⊂ Y ⊂ A and...

Measures of noncompactness and normal structure in Banach spaces

J. García-Falset, A. Jiménez-Melado, E. Lloréns-Fuster (1994)

Studia Mathematica

Sufficient conditions for normal structure of a Banach space are given. One of them implies reflexivity for Banach spaces with an unconditional basis, and also for Banach lattices.

Metric ellipses in Minkowski planes.

Wu Senlin, Ji Donghai, Javier Alonso (2005)

Extracta Mathematicae

An ellipse in R2 can be defined as the locus of points for which the sum of the Euclidean distances from the two foci is constant. In this paper we will look at the sets that are obtained by considering in the above definition distances induced by arbitrary norms.

Metric spaces with the small ball property

Ehrhard Behrends, Vladimir M. Kadets (2001)

Studia Mathematica

A metric space (M,d) is said to have the small ball property (sbp) if for every ε₀ > 0 it is possible to write M as the union of a sequence (B(xₙ,rₙ)) of closed balls such that the rₙ are smaller than ε₀ and lim rₙ = 0. We study permanence properties and examples of sbp. The main results of this paper are the following: 1. Bounded convex closed sets in Banach spaces have sbp only if they are compact. 2. Precisely the finite-dimensional Banach spaces have sbp. (More generally: a complete metric...

Metrizability of the unit ball of the dual of a quasi-normed cone

L. M. García-Raffi, S. Romaguera, E. A. Sánchez-Pérez, O. Valero (2004)

Bollettino dell'Unione Matematica Italiana

We obtain theorems of metrization and quasi-metrization for several topologies of weak* type on the unit ball of the dual of any separable quasi-normed cone. This is done with the help of an appropriate version of the Alaoglu theorem which is also obtained here.

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