数学物理学报(英文版) ›› 2020, Vol. 40 ›› Issue (3): 603-613.doi: 10.1007/s10473-020-0301-8

• 论文 •    下一篇

A VIEWPOINT TO MEASURE OF NON-COMPACTNESS OF OPERATORS IN BANACH SPACES

沈钦锐   

  1. School of Mathematics and Statistics, Minnan Normal University, Zhangzhou 363000, China
  • 收稿日期:2018-04-05 修回日期:2019-09-28 出版日期:2020-06-25 发布日期:2020-07-17
  • 作者简介:Qinrui SHEN,E-mail:qinrui327@163.com
  • 基金资助:
    The project supported in part by the National Natural Science Foundation of China (11801255)

A VIEWPOINT TO MEASURE OF NON-COMPACTNESS OF OPERATORS IN BANACH SPACES

Qinrui SHEN   

  1. School of Mathematics and Statistics, Minnan Normal University, Zhangzhou 363000, China
  • Received:2018-04-05 Revised:2019-09-28 Online:2020-06-25 Published:2020-07-17
  • Supported by:
    The project supported in part by the National Natural Science Foundation of China (11801255)

摘要: This article is committed to deal with measure of non-compactness of operators in Banach spaces. Firstly, the collection C(X) (consisting of all nonempty closed bounded convex sets of a Banach space X endowed with the uaual set addition and scaler multiplication) is a normed semigroup, and the mapping J from C(X) onto F(Ω) is a fully order-preserving positively linear surjective isometry, where Ω is the closed unit ball of X* and F(Ω) the collection of all continuous and w*-lower semicontinuous sublinear functions on X* but restricted to Ω. Furthermore, both EC=JC-JC and EK=JK-JK are Banach lattices and EK is a lattice ideal of EC. The quotient space EC/EK is an abstract M space, hence, order isometric to a sublattice of C(K) for some compact Haudorspace K, and (FQJ)C which is a closed cone is contained in the positive cone of C(K), where Q:ECEC/EK is the quotient mapping and F:EC/EKC(K) is a corresponding order isometry. Finally, the representation of the measure of non-compactness of operators is given:Let BX be the closed unit ball of a Banach space X, then
μ(T)=μ(T(BX))=||(F QJ)T(BX)||C(K), ∀TB(X).

关键词: Measure of non-compactness, measure of non-compactness of operators, Banach lattice, Banach space

Abstract: This article is committed to deal with measure of non-compactness of operators in Banach spaces. Firstly, the collection C(X) (consisting of all nonempty closed bounded convex sets of a Banach space X endowed with the uaual set addition and scaler multiplication) is a normed semigroup, and the mapping J from C(X) onto F(Ω) is a fully order-preserving positively linear surjective isometry, where Ω is the closed unit ball of X* and F(Ω) the collection of all continuous and w*-lower semicontinuous sublinear functions on X* but restricted to Ω. Furthermore, both EC=JC-JC and EK=JK-JK are Banach lattices and EK is a lattice ideal of EC. The quotient space EC/EK is an abstract M space, hence, order isometric to a sublattice of C(K) for some compact Haudorspace K, and (FQJ)C which is a closed cone is contained in the positive cone of C(K), where Q:ECEC/EK is the quotient mapping and F:EC/EKC(K) is a corresponding order isometry. Finally, the representation of the measure of non-compactness of operators is given:Let BX be the closed unit ball of a Banach space X, then
μ(T)=μ(T(BX))=||(F QJ)T(BX)||C(K), ∀TB(X).

Key words: Measure of non-compactness, measure of non-compactness of operators, Banach lattice, Banach space

中图分类号: 

  • 47H08