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Ideals of compact operators

Part of: Linear spaces and algebras of operators Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  09 April 2009

Åsvald Lima  and
Eve Oja
Åsvald Lima
Affiliation:
Department of Mathematics, Agder College, Gimlemoen 25J, Serviceboks 422, 4604 Kristiansand, Norway e-mail: asvald.Lima@hia.no
Eve Oja
Affiliation:
Faculty of Mathematics, Tartu University, Liivi 2-606, EE-50409 Tartu, Estonia e-mail: eveoja@math.ut.ee
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Abstract

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We give an example of a Banach space X such that K (X, X) is not an ideal in K (X, X**). We prove that if z* is a weak* denting point in the unit ball of Z* and if X is a closed subspace of a Banach space Y, then the set of norm-preserving extensions H B(x* ⊗ z*) ⊆ (Z*, Y)* of a functional x* ⊗ Z* ∈ (Z ⊗ X)* is equal to the set H B(x*) ⊗ {z*}. Using this result, we show that if X is an M-ideal in Y and Z is a reflexive Banach space, then K (Z, X) is an M-ideal in K(Z, Y) whenever K (Z, X) is an ideal in K (Z, Y). We also show that K (Z, X) is an ideal (respectively, an M-ideal) in K (Z, Y) for all Banach spaces Z whenever X is an ideal (respectively, an M-ideal) in Y and X * has the compact approximation property with conjugate operators.

Keywords

compact operatoridealM-idealapproximation property

MSC classification

Secondary: 46B20: Geometry and structure of normed linear spaces 46B28: Spaces of operators; tensor products; approximation properties 47L05: Linear spaces of operators

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 77 , Issue 1 , August 2004 , pp. 91 - 110
Copyright
Copyright © Australian Mathematical Society 2004

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