A Viewpoint to Measure of Non-Compactness of Operators in Banach Spaces
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
A VIEWPOINT TO MEASURE OF NON-COMPACTNESS OF OPERATORS IN BANACH SPACES∗
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Qinrui SHEN (
School of Mathematics and Statistics, Minnan Normal University, Zhangzhou 363000, China E-mail : [email protected] 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 (F QJ)C which is a closed cone is contained in the positive cone of C(K), where Q : EC → EC /EK is the quotient mapping and F : EC /EK → C(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 )) = k(F QJ)T (BX )kC(K) , ∀T ∈ B(X). Key words
Measure of non-compactness; measure of non-compactness of operators; Banach lattice; Banach space
2010 MR Subject Classification
1
47H08; 46B42; 46B50
Introduction
The measure of non-compactness was first proposed by K.Kuratowski [1] in 1930, and later called the measure of set non-compactness or the measure of Kuratowski non-compactness (denoted as α): Let X be a metric space and Q be a nonempty bounded set in X, then, n n o [ α(Q) = inf ε > 0 : Q ⊂ Si , diam(Si ) ≤ ε, Si ⊂ X, i = 1, · · · , n, n ∈ N . i=1
Let diam(Q) = sup{d(x, y)|x, y ∈ Q}, obviously, we have α(Q) ≤ diam (Q). α can be used to measure the distance between the nonempty bounded set and the compact set in a metric ∗ Received April 5, 2018; revised September 28, 2019. The project supported in part by the National Natural Science Foundation of China (11801255)
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ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
space X, and it has the following properties (where A, B represent any nonempty bounded set in X): a) α(A) = 0 ⇐⇒ A is a relatively compact set; b) α(A) = α(A); c) A ⊂ B =⇒ α(A) ≤ α(B); d) α(A ∪ B) = max{α(A), α(B)}; Furthermore, when X is a normed space, α also satisfies the following properties: e) α(kA) = |k|α(A), kA = {x|x = ka, a ∈ A}, k ∈ F ; f) α(A + B) ≤ α(A) + α(B), A + B = {x = a + b, a ∈ A, b ∈ B}; g) α(coA) = α(A); h) α(A + x0 ) = α(A), ∀x0 ∈ X. In addition, α also has a continuity, that is, for any nonempty bounded set A, B in X, for any ε > 0, there is δ > 0, |α(A) − α(B)| < ε when ρ(A, B) < δ. These properties of α are almost parallel from the properties of the diame
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