Measures of weak non-compactness in 𝐿₁(𝜇)-spaces

Chen, Dongyang

doi:10.1090/proc/16414

Measures of weak non-compactness in $L_{1}(\mu )$-spaces
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by Dongyang Chen

Proc. Amer. Math. Soc. 152 (2024), 617-629

DOI: https://doi.org/10.1090/proc/16414

Published electronically: November 17, 2023

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Abstract:

Disjoint sequence methods from the theory of Riesz spaces are used to study measures of weak non-compactness in $L_{1}(\mu )$-spaces. A principal new result of the present paper is the following: Let $E$ be an abstract $M$-space. Then \begin{align*} \omega (B)&=\sup \{\limsup \limits _{n\rightarrow \infty }\rho _{B}(x_{n}):(x_{n})_{n}\subseteq B_{E} \operatorname {disjoint} \}\\ &=\inf \{\varepsilon >0:\exists x^{*}\in E^{*}_{+} \operatorname {so}\operatorname {that} B\subseteq [-x^{*},x^{*}]+\varepsilon B_{E^{*}}\}\\ &=\sup \{\limsup \limits _{n\rightarrow \infty }\rho _{B}(x_{n}):(x_{n})_{n}\subseteq B_{E} \operatorname {weakly}\operatorname {null} \}\\ &=\sup \{\operatorname {ca}_{\rho _{B}}((x_{n})_{n}):(x_{n})_{n}\subseteq (B_{E})_{+} \operatorname {increasing} \}\\ &=\sup \{\limsup \limits _{n\rightarrow \infty }\|x^{*}_{n}\|:(x^{*}_{n})_{n}\subseteq \operatorname {Sol}(B)\operatorname {disjoint}\}\\ &=\sup \{\limsup \limits _{n\rightarrow \infty }\sup \limits _{x^{*}\in B}|\langle x^{*},x_{n}\rangle |:(x_{n})_{n}\subseteq B_{E}\operatorname {disjoint} \}\\ \end{align*} for every norm bounded subset $B$ of $E^{*}$.

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