Gines Lopez | Searcharxiv

arXiv2019

Norm attaining operators of finite rank

We provide sufficient conditions on a Banach space $X$ in order that there exist norm attaining operators of rank at least two from $X$ into any Banach space of dimension at least two. For example, a rather weak such condition is the existence of a non-trivial cone consisting of norm attaining functionals on $X$. We go on to discuss density of norm attaining operators of finite rank among all operators of finite rank, which holds for instance when there is a dense linear subspace consisting of norm attaining functionals on $X$. In particular, we consider the case of Hilbert space valued operators where we obtain a complete characterization of these properties. In the final section we offer a candidate for a counterexample to the complex Bishop-Phelps theorem on $c_0$, the first such counterexample on a certain complex Banach space being due to V. Lomonosov.

math.FA↗

arXiv2017

Some geometric properties of Read's space

We study geometric properties of the Banach space $\mathcal{R}$ constructed recently by C.~Read (arXiv 1307.7958) which does not contain proximinal subspaces of finite codimension greater than or equal to two. Concretely, we show that the bidual of $\mathcal{R}$ is strictly convex, that the norm of the dual of $\mathcal{R}$ is rough, and that $\mathcal{R}$ is weakly locally uniformly rotund (but it is not locally uniformly rotund). Apart of the own interest of the results, they provide a simplification of the proof by M.~Rmoutil (J.\ Funct.\ Anal.\ 272 (2017), 918--928) that the set of norm-attaining functionals over $\mathcal{R}$ does not contain any linear subspace of dimension greater than or equal to two. Note that if a Banach space $X$ contains proximinal subspaces of finite codimension at least two, then the set of norm-attaining functionals over $X$ contain two-dimensional linear subspaces of $X^*$. Our results also provides positive answer to the questions of whether the dual of $\mathcal{R}$ is smooth and of whether $\mathcal{R}$ is weakly locally uniformly rotund (J.\ Funct.\ Anal.\ 272 (2017), 918--928). Finally, we present a renorming of Read's space which is smooth, whose dual is smooth, and such that its set of norm-attaining functionals does not contain any linear subspace of dimension greater than or equal to two, so the renormed space does not contain proximinal subspaces of finite codimension greater than or equal to two.

math.FA↗