A Viewpoint to Measure of Non-Compactness of Operators in Banach Spaces
Autor: | Qinrui Shen |
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Rok vydání: | 2020 |
Předmět: |
Unit sphere
Physics Sublinear function Semigroup General Mathematics 010102 general mathematics Banach space General Physics and Astronomy Quotient space (linear algebra) 01 natural sciences 010101 applied mathematics Surjective function Combinatorics 0101 mathematics Quotient Measure of non-compactness |
Zdroj: | Acta Mathematica Scientia. 40:603-613 |
ISSN: | 1572-9087 0252-9602 |
DOI: | 10.1007/s10473-020-0301-8 |
Popis: | This article is committed to deal with measure of non-compactness of operators in Banach spaces. Firstly, the collection $$\mathcal{C}(X)$$ (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 $$\mathcal{C}(X)$$ onto $$\mathcal{F}(\Omega)$$ is a fully order-preserving positively linear surjective isometry, where Ω is the closed unit ball of X* and $$\mathcal{F}(\Omega)$$ the collection of all continuous and w*-lower semicontinuous sublinear functions on X* but restricted to Ω. Furthermore, both $$E_\mathcal{C}=\overline{J\mathcal{C}-J\mathcal{C}}$$ and $$E_\mathcal{K}=\overline{J\mathcal{K}-J\mathcal{K}}$$ are Banach lattices and $$E_\mathcal{K}$$ is a lattice ideal of $$E_\mathcal{C}$$ . The quotient space $$E_\mathcal{C}/E_\mathcal{K}$$ is an abstract M space, hence, order isometric to a sublattice of C(K) for some compact Haudorspace K, and $$(FQJ)\mathcal{C}$$ which is a closed cone is contained in the positive cone of C(K), where $$Q:E_\mathcal{C}\rightarrow{E_\mathcal{C}/E_\mathcal{K}}$$ is the quotient mapping and $$F:E_\mathcal{C}/E_\mathcal{K}\rightarrow{C(K)}$$ 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 $$\mu(T)=\mu(T(B_X))=\parallel(FQJ)\overline{T(B_X)}\parallel_{\mathcal{C}(\mathcal{K})}, \forall{T}\in{B(X)}.$$ |
Databáze: | OpenAIRE |
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