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Vector-valued numerical radius and $\sigma$-porosity

Mohammed Bachir
Dec 2022
摘要
It is well known that under certain conditions on a Banach space $X$, the setof bounded linear operators attaining their numerical radius is a dense subset.We prove in this paper that if $X$ is assumed to be uniformly convex anduniformly smooth then the set of bounded linear operators attaining theirnumerical radius is not only a dense subset but also the complement of a$\sigma$-porous subset. In fact, we generalize the notion of numerical radiusto a large class $\mathcal{Z}$ of vector-valued operators defined from $X\timesX^*$ into a Banach space $W$ and we prove that the set of all elements of$\mathcal{Z}$ strongly (up to a symmetry) attaining their {\it numericalradius} is the complement of a $\sigma$-porous subset of $\mathcal{Z}$ andmoreover the {\it "numerical radius"} {\it Bishop-Phelps-Bollob\'as property}is also satisfied for this class. Our results extend (up to the assumption on$X$) some known results in several directions: $(1)$ the density is replaced bybeing the complement of a $\sigma$-porous subset, $(2)$ the operators attainingtheir {\it numerical radius} are replaced by operators strongly (up to asymmetry) attaining their {\it numerical radius} and $(3)$ the results areobtained in the vector-valued framework for general linear and non-linearvector-valued operators (including bilinear mappings and the classical space ofbounded linear operators).
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