A Viewpoint to Measure of Non-Compactness of Operators in Banach Spaces

Abstract

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 $$\mu(T)=\mu(T(B_X))=\parallel(FQJ)\overline{T(B_X)}\parallel_{\mathcal{C}(\mathcal{K})}, \forall{T}\in{B(X)}.$$