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Non-complemented Spaces of Operators, Vector Measures, and co

Published online by Cambridge University Press:  20 November 2018

Paul Lewis  and
Polly Schulle
Paul Lewis
Affiliation:
Department of Mathematics, University of North Texas, Denton, TX 76203-1430 USAe-mail: lewis@unt.edu
Polly Schulle
Affiliation:
Department of Mathematics, Richland College, Dallas, TX 75243-2199 USAe-mail: PSchulle@dcccd.edu
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Abstract

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The Banach spaces $L(X,Y),K(X,Y),{{L}_{{{w}^{*}}}}({{X}^{*}},Y)$, and ${{K}_{{{w}^{*}}}}({{X}^{*}},Y)$ are studied to determine when they contain the classical Banach spaces ${{c}_{o}}$ or ${{l}_{\infty }}$. The complementation of the Banach space $K(X,Y)$ in $L(X,Y)$ is discussed as well as what impact this complementation has on the embedding of ${{c}_{o}}$ or ${{l}_{\infty }}$ in $K(X,Y)$ or $L(X,Y)$. Results of Kalton, Feder, and Emmanuele concerning the complementation of $K(X,Y)$ in $L(X,Y)$ are generalized. Results concerning the complementation of the Banach space ${{K}_{{{w}^{*}}}}({{X}^{*}},Y)$ in ${{L}_{{{w}^{*}}}}({{X}^{*}},Y)$ are also explored as well as how that complementation affects the embedding of ${{c}_{o}}$ or ${{l}_{\infty }}$ in ${{K}_{{{w}^{*}}}}({{X}^{*}},Y)$ or ${{L}_{{{w}^{*}}}}({{X}^{*}},Y)$. The ${{l}_{p}}$ spaces for $1\,=\,p\,<\,\infty $ are studied to determine when the space of compact operators from one ${{l}_{p}}$ space to another contains ${{c}_{o}}$. The paper contains a new result which classifies these spaces of operators. A new result using vector measures is given to provide more efficient proofs of theorems by Kalton, Feder, Emmanuele, Emmanuele and John, and Bator and Lewis.

Keywords

46B20spaces of operatorscompact operatorscomplemented subspacesw* – w-compact operators
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 3 , 01 September 2012 , pp. 548 - 554
Copyright
Copyright © Canadian Mathematical Society 2012

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