A characterization of a local vector valued Bollob\'as theorem

TL;DR

This paper characterizes a local vector-valued Bollobás property for operators, providing conditions under which pairs of Banach spaces exhibit this property, and extends known results in the geometry of Banach spaces and tensor products.

Contribution

It introduces two characterizations for the property {f L}$_{o,o}$, generalizes previous results, and offers a complete characterization for tensor products under certain geometric conditions.

Findings

The pair $(X, Y)$ has {f L}$_{o,o}$} for compact operators iff $(X, ext{K})$ does for linear functionals.

Characterization of when $(X imes Y, ext{K})$ satisfies {f L}$_{o,o}$ for linear functionals under strict convexity or Kadec-Klee.

$(L_p( u) imes L_q( u); ext{K})$ does not satisfy {f L}$_{o,o}$} for bilinear forms.

Abstract

In this paper, we are interested in giving two characterizations for the so-called property {\bf L}$_{o,o}$, a local vector valued Bollob\'as type theorem. We say that $(X, Y)$ has this property whenever given $\eps > 0$ and an operador $T: X \rightarrow Y$, there is $\eta = \eta(\eps, T)$ such that if $x$ satisfies $\|T(x)\| > 1 - \eta$, then there exists $x_0 \in S_X$ such that $x_0 \approx x$ and $T$ itself attains its norm at $x_0$. This can be seen as a strong (although local) Bollob\'as theorem for operators. We prove that the pair $(X, Y)$ has the {\bf L}$_{o,o}$ for compact operators if and only if so does $(X, \K)$ for linear functionals. This generalizes at once some results due to D. Sain and J. Talponen. Moreover, we present a complete characterization for when $(X \pten Y, \K)$ satisfies the {\bf L}$_{o,o}$ for linear functionals under strict convexity or Kadec-Klee property assumptions in one of the spaces. As a consequence, we generalize some results in the literature related to the strongly subdifferentiability of the projective tensor product and show that $(L_p(\mu) \times L_q(\nu); \K)$ cannot satisfy the {\bf L}$_{o,o}$ for bilinear forms.