A new approach to measures of noncompactness of Banach spaces
Volume 240 / 2018
Abstract
We deal with measures of noncompactness on a Banach space $X$ in a new way: Assume that $\mathfrak C$ is the collection of all nonempty bounded closed convex subsets of $X$, let $\mathfrak K\subset \mathfrak C$ consist of all compact convex sets and let $\varOmega $ be the closed unit ball of the dual $X^*$. Then (1) $\mathfrak C$ is a normed semigroup endowed with the set addition $A\oplus B=\overline {A+B}$, the usual scalar multiplication of sets and the norm $|\hskip -1.5pt|\hskip -1.5pt|\cdot |\hskip -1.5pt|\hskip -1.5pt|$ defined by $|\hskip -1.5pt|\hskip -1.5pt|C|\hskip -1.5pt|\hskip -1.5pt|=\sup_{c\in C}\| c\| $ for $C\in \mathfrak C$; (2) $J: \mathfrak C\rightarrow C_b(\varOmega )$ defined by $JC=\sup_{c\in C}\langle \cdot ,c\rangle $ is a positively linear order isometry. We further show that (3) both $E_\mathfrak C=\overline {J\mathfrak C-J\mathfrak C}$ and $E_\mathfrak K=\overline {J\mathfrak K-J\mathfrak K}$ are Banach sublattices and $E_\mathfrak K$ is a lattice ideal of $E_\mathfrak C$; (4) the quotient space $Q(E_\mathfrak C)\equiv E_\mathfrak C/E_\mathfrak K$ is an abstract $M$-space; consequently, it is order isometric to a sublattice $T(E_\mathfrak C/E_\mathfrak K)$ of a $C(K)$ space for some compact Hausdorff space $K$. Moreover, (5) $TQJ\mathfrak C$ is contained in the positive cone of $C(K)$ and the ball measure of noncompactness $\beta $ of $X$ satisfies $$ \beta (B)=\| TQJ[\mathop {\overline {{\rm co}}}\nolimits B]\| _{C(K)}\hskip 1em \ \text {for all nonempty bounded} B\subset X. $$ As an application, we give an easy construction of homogeneous (respectively, regular) measures of noncompactness of $X$ via positive (respectively, evaluation) functionals on the space $C(K)$, and further we prove that every Banach space admitting a closed subspace with an infinite decomposition, in particular, containing an unconditional basic sequence has inequivalent homogeneous measures of noncompactness. Hence, this gives an affirmative answer to a question proposed by Mallet-Paret and Nussbaum.
