CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCMB
Publications        
Abstract view

Non-complemented Spaces of Operators, Vector Measures, and $c_o$

  Published:2011-05-06
 Printed: Sep 2012
  • Paul Lewis,
    Department of Mathematics, University of North Texas, Denton, TX 76203-1430 USA
  • Polly Schulle,
    Department of Mathematics, Richland College, Dallas, TX 75243-2199 USA
Format:   HTML   LaTeX   MathJax   PDF  

Abstract

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 $\ell_\infty$. 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 $\ell_\infty$ 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 $\ell_\infty$ in $K_{w^*}(X^*, Y)$ or $L_{w^*}(X^*, Y)$. The $\ell_p$ spaces for $1 = p < \infty$ are studied to determine when the space of compact operators from one $\ell_p$ 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: spaces of operators, compact operators, complemented subspaces, $w^*-w$-compact operators spaces of operators, compact operators, complemented subspaces, $w^*-w$-compact operators
MSC Classifications: 46B20 show english descriptions Geometry and structure of normed linear spaces 46B20 - Geometry and structure of normed linear spaces
 

© Canadian Mathematical Society, 2014 : https://cms.math.ca/