A new approach to measure  of non-compactness of Banach spaces

CHENG Li-Xin; CHENG Qing-Jin; SHEN Qin-Rui; TU Kun; ZHANG Wen

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A new approach to measure of non-compactness of Banach spaces

CHENG Li-Xin 1,CHENG Qing-Jin 1 * #,SHEN Qin-Rui 2,TU Kun 3,ZHANG Wen 1

1.School of Mathematical Sciences, Xiamen University, Xiamen 361005

2.School of mathematics and statistics, Minnan Normal University, Zhangzhou 363000

3.School of Mathematical Sciences, Yangzhou University, Yanghzou 225000

*Correspondence author

#Submitted by

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Funding: Specialized Research Fund for the Doctoral Program of Higher Education （No.20130121110032）

Opened online: 3 May 2017

Accepted by: none

Citation: CHENG Li-Xin,CHENG Qing-Jin,SHEN Qin-Rui.A new approach to measure of non-compactness of Banach spaces[OL]. [ 3 May 2017] http://en.paper.edu.cn/en_releasepaper/content/4729878

This paper aims to deal with measures of noncompactness of a Banach space $X$ in a new way: Assume that $mathfrak C$ is the collection of all nonempty bounded closed convex sets of $X$, $mathfrak Ksubsetmathfrak C$ consisting of all compact convex sets and $Omega$ is the closed unit ball of the dual $X^*$. Then (1); $mathfrak C$ is a normed semigroup endowed with the set addition $Aoplus B=overline{A+B}$, the usual scaler multiplication of sets and endowed with the norm $||cdot||$ defined for $Cinmathfrak C$ by $||C||=sup_{cin C}|c|$; (2); $J: mathfrak C ightarrow C_b(Omega)$ defined by $JC=sup_{cin C}langlecdot,c angle$ is a positively linear order isometry; further (3); both $E_mathfrak C=overline{Jmathfrak C-Jmathfrak C}$ and $E_mathfrak K=overline{Jmathfrak K-Jmathfrak K}$ are Banach sublattices and $E_mathfrak K$ is a lattice ideal of $E_mathfrak C$;(4) the quotient space $Q(E_mathfrak C)equiv E_mathfrak C/E_mathfrak K$ is an abstract $M$ space; consequently, it is order isometric to a sublattice $T(E_mathfrak C/E_mathfrak K)$ of a $C(K)$ space for some compact Hausdorff space $K$.

Keywords:Measure of non-compactness; normed semigroup; Banach lattice; abstract $M$ space; Banach space

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