A characterization of a local vector valued Bollobás theorem

Abstract

In this paper, we are interested in giving two characterizations for the so-called property {\bf L}${}_{o,o}$, a local vector valued Bollobás type theorem. We say that $(X,Y)$ has this property whenever given $\eps > 0$ and an operador $T:X\to Y$, there is $η= η(\eps, T)$ such that if $x$ satisfies $\Vert T(x)\Vert >1-\eta$, 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) Bollobás 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 $(L_p(μ) \times L_q(ν); \K)$ cannot satisfy the {\bf L}${}_{o,o}$ for bilinear forms.

10 pages