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 Lo,o, a local vector valued Bollobás type theorem. We say that (X, Y) has this property whenever given ε>0 and an operador T:X→Y, there is η=η(ε,T) such that if x satisfies ∥T(x)∥>1−η, then there exists x0∈SX such that x0≈x and T itself attains its norm at x0. This can be seen as a strong (although local) Bollobás theorem for operators. We prove that the pair (X, Y) has the Lo,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⊗ˆπY,K) satisfies the Lo,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 (Lp(μ)×Lq(ν);K) cannot satisfy the Lo,o for bilinear forms.