A characterization of a local vector valued Bollobás Theorem

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

Norm-attaining operators which satisfy a Bollobás type theorem

This is a pre-print of an article published in Banach Journal of Mathematical Analysis. The final authenticated version is available online at: https://doi.org/10.1007/s43037-020-00113-7In this paper, we are interested in studying the set A(parallel to center dot parallel to) (X, Y) of all norm-attaining operators T from X into Y satisfying the following: given epsilon > 0, there exists eta such that if parallel to Tx parallel to > 1 - eta, then there is x(0) such that parallel to x(0) - x parallel to < epsilon and T itself attains its norm at x(0). We show that every norm one functional on c(0) which attains its norm belongs to A(parallel to center dot parallel to) (c(0), K). Also, we prove that the analogous result holds neither for A(parallel to center dot parallel to) (l(1), K) nor A(parallel to center dot parallel to) (l(infinity), K). Under some assumptions, we show that the sphere of the compact operators belongs to A(parallel to center dot parallel to) (X, Y) and that this is no longer true when some of these hypotheses are dropped. The analogous set A(nu)(X) for numerical radius of an operator instead of its norm is also defined and studied. We present a complete characterization for the diagonal operators which belong to the sets A(parallel to center dot parallel to) (X, X) and A(nu)(X) when X = c(0) or l(p). As a consequence, we get that the canonical projections P-N on these spaces belong to our sets. We give examples of operators on infinite dimensional Banach spaces which belong to A(parallel to center dot parallel to) (X, X) but not to A(nu)(X) and vice-versa. Finally, we establish some techniques which allow us to connect both sets by using direct sums