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: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))=||(F QJ)T(BX)||C(K), ∀T ∈ B(X).
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
.
DOI: 10.1007/s10473-020-0301-8
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