A Characterization of a Local Vector Valued Bollobás Theorem

In this paper, we are interested in giving two characterizations for the so-called property L$$_{o,o}$$ o , o , a local vector valued Bollobás type theorem. We say that (X, Y) has this property whenever given $$\varepsilon > 0$$ ε > 0 and an operador $$T: X \rightarrow Y$$ T : X → Y , there is $$\eta = \eta (\varepsilon , T)$$ η = η ( ε , T ) such that if x satisfies $$\Vert T(x)\Vert > 1 - \eta $$ ‖ T ( x ) ‖ > 1 - η , then there exists $$x_0 \in S_X$$ x 0 ∈ S X such that $$x_0 \approx x$$ x 0 ≈ x and T itself attains its norm at $$x_0$$ x 0 . This can be seen as a strong (although local) Bollobás theorem for operators. We prove that the pair (X, Y) has the L$$_{o,o}$$ o , o for compact operators if and only if so does $$(X, \mathbb {K})$$ ( 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 $$(X \widehat{\otimes }_\pi Y, \mathbb {K})$$ ( X ⊗ ^ π Y , K ) satisfies the L$$_{o,o}$$ 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 $$(L_p(\mu ) \times L_q(\nu ); \mathbb {K})$$ ( L p ( μ ) × L q ( ν ) ; K ) cannot satisfy the L$$_{o,o}$$ o , o for bilinear forms.