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Complemented Subspaces of Linear Bounded Operators

Published online by Cambridge University Press:  20 November 2018

Manijeh Bahreini ,
Elizabeth Bator  and
Ioana Ghenciu
Manijeh Bahreini
Affiliation:
University of Isfahan, Department of Mathematics, Isfahan 81745-163, Irane-mail: mebahreini@math.ui.ac.ir
Elizabeth Bator
Affiliation:
University of North Texas, Department of Mathematics, Denton, Texas 76203-1430e-mail: bator@unt.edu
Ioana Ghenciu
Affiliation:
University of Wisconsin–River Falls, Department of Mathematics, River Falls, WI 54022-5001e-mail: ioana.ghenciu@uwrf.edu
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Abstract

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We study the complementation of the space $W\left( X,Y \right)$ of weakly compact operators, the space $K\left( X,Y \right)$ of compact operators, the space $U\left( X,Y \right)$ of unconditionally converging operators, and the space $CC\left( X,Y \right)$ of completely continuous operators in the space $L\left( X,Y \right)$ of bounded linear operators from $X$ to $Y$. Feder proved that if $X$ is infinite-dimensional and ${{c}_{0}}\,\to \,Y$, then $K\left( X,Y \right)$ is uncomplemented in $L\left( X,Y \right)$. Emmanuele and John showed that if ${{c}_{0}}\,\to \,K(X,\,Y)$, then $K\left( X,Y \right)$ is uncomplemented in $L\left( X,Y \right)$. Bator and Lewis showed that if $X$ is not a Grothendieck space and ${{c}_{0}}\,\to \,Y$, then $W\left( X,Y \right)$ is uncomplemented in $L\left( X,Y \right)$. In this paper, classical results of Kalton and separably determined operator ideals with property $\left( * \right)$ are used to obtain complementation results that yield these theorems as corollaries.

Keywords

46B2046B28spaces of operatorscomplemented subspacescompact operatorsweakly compact operatorscompletely continuous operators
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 3 , 01 September 2012 , pp. 449 - 461
Copyright
Copyright © Canadian Mathematical Society 2012

References

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