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Complemented Subspaces of Linear Bounded Operators

  Published:2011-05-20
 Printed: Sep 2012
  • Manijeh Bahreini,
    University of Isfahan, Department of Mathematics, Isfahan 81745-163, Iran
  • Elizabeth Bator,
    University of North Texas, Department of Mathematics, Denton, Texas 76203-1430
  • Ioana Ghenciu,
    University of Wisconsin-River Falls, Department of Mathematics, River Falls, WI 54022-5001
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Abstract

We study the complementation of the space $W(X,Y)$ of weakly compact operators, the space $K(X,Y)$ of compact operators, the space $U(X,Y)$ of unconditionally converging operators, and the space $CC(X,Y)$ of completely continuous operators in the space $L(X,Y)$ of bounded linear operators from $X$ to $Y$. Feder proved that if $X$ is infinite-dimensional and $c_0 \hookrightarrow Y$, then $K(X,Y)$ is uncomplemented in $L(X,Y)$. Emmanuele and John showed that if $c_0 \hookrightarrow K(X,Y)$, then $K(X,Y)$ is uncomplemented in $L(X,Y)$. Bator and Lewis showed that if $X$ is not a Grothendieck space and $c_0 \hookrightarrow Y$, then $W(X,Y)$ is uncomplemented in $L(X,Y)$. In this paper, classical results of Kalton and separably determined operator ideals with property $(*)$ are used to obtain complementation results that yield these theorems as corollaries.
Keywords: spaces of operators, complemented subspaces, compact operators, weakly compact operators, completely continuous operators spaces of operators, complemented subspaces, compact operators, weakly compact operators, completely continuous operators
MSC Classifications: 46B20, 46B28 show english descriptions Geometry and structure of normed linear spaces
Spaces of operators; tensor products; approximation properties [See also 46A32, 46M05, 47L05, 47L20]
46B20 - Geometry and structure of normed linear spaces
46B28 - Spaces of operators; tensor products; approximation properties [See also 46A32, 46M05, 47L05, 47L20]
 

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