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 B_X be the closed unit ball of a Banach space X, then

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