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

doi:10.1007/s10473-020-0301-8

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

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

沈钦锐

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

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/Abstract

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 E_C=JC-JC and E_K=JK-JK are Banach lattices and E_K is a lattice ideal of E_C. The quotient space E_C/E_K 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:E_C → E_C/E_K is the quotient mapping and F:E_C/E_K → C(K) is a corresponding order isometry. Finally, the representation of the measure of non-compactness of operators is given:Let B_X be the closed unit ball of a Banach space X, then
μ(T)=μ(T(B_X))=||(F QJ)T(BX)||_C(K), ∀T ∈ B(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 E_C=JC-JC and E_K=JK-JK are Banach lattices and E_K is a lattice ideal of E_C. The quotient space E_C/E_K 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:E_C → E_C/E_K is the quotient mapping and F:E_C/E_K → C(K) is a corresponding order isometry. Finally, the representation of the measure of non-compactness of operators is given:Let B_X be the closed unit ball of a Banach space X, then
μ(T)=μ(T(B_X))=||(F QJ)T(BX)||_C(K), ∀T ∈ B(X).

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

中图分类号:

47H08

沈钦锐. A VIEWPOINT TO MEASURE OF NON-COMPACTNESS OF OPERATORS IN BANACH SPACES[J]. 数学物理学报(英文版), 2020, 40(3): 603-613.

Qinrui SHEN. A VIEWPOINT TO MEASURE OF NON-COMPACTNESS OF OPERATORS IN BANACH SPACES[J]. Acta mathematica scientia,Series B, 2020, 40(3): 603-613.

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

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

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