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BIRKHOFF ORTHOGONALITY IN CLASSICAL $M$-IDEALS

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

Published online by Cambridge University Press:  08 November 2016

PAWEŁ WÓJCIK
PAWEŁ WÓJCIK*
Affiliation:
Institute of Mathematics, Pedagogical University of Cracow, Podchora̧żych 2, 30-084 Kraków, Poland email pwojcik@up.krakow.pl
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Abstract

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The Birkhoff orthogonality has been recently intensively studied in connection with the geometry of Banach spaces and operator theory. The main aim of this paper is to characterize the Birkhoff orthogonality in ${\mathcal{L}}(X;Y)$ under the assumption that ${\mathcal{K}}(X;Y)$ is an $M$-ideal in ${\mathcal{L}}(X;Y)$. Moreover, we survey the known results, as well as giving some new and more general ones. Furthermore, we characterize an approximate Birkhoff orthogonality in ${\mathcal{K}}(X;Y)$.

Keywords

Birkhoff–James orthogonalityM-idealsmoothnesscompact operatorextreme point

MSC classification

Primary: 46B20: Geometry and structure of normed linear spaces
Secondary: 46C50: Generalizations of inner products (semi-inner products, partial inner products, etc.) 47L05: Linear spaces of operators
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 103 , Issue 2 , October 2017 , pp. 279 - 288
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

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