The Bishop–Phelps–Bollobás theorem for operators (original) (raw)

The Bishop–Phelps–Bollobás property for operators from c0 into some Banach spaces

Journal of Mathematical Analysis and Applications, 2017

We provide a version for operators of the Bishop-Phelps-Bollobás theorem when the domain space is the complex space C 0 (L). In fact, we prove that the pair (C 0 (L), Y) will satisfy the Bishop-Phelps-Bollobás property for operators for every Hausdorff locally compact space L and any C-uniformly convex space. As a consequence, this holds for Y = L p (µ) (1 ≤ p < ∞).

Bishop–Phelps–Bollobás property for certain spaces of operators

Journal of Mathematical Analysis and Applications, 2014

We characterize the Banach spaces Y for which certain subspaces of operators from L 1 (μ) into Y have the Bishop-Phelps-Bollobás property in terms of a geometric property of Y , namely AHSP. This characterization applies to the spaces of compact and weakly compact operators. New examples of Banach spaces Y with AHSP are provided. We also obtain that certain ideals of Asplund operators satisfy the Bishop-Phelps-Bollobás property.

Characterization of Banach spaces Y satisfying that the pair (ℓ∞4,Y) has the Bishop–Phelps–Bollobás property for operators

Journal of Mathematical Analysis and Applications, 2019

We study the Bishop-Phelps-Bollobás property for operators from ℓ 4 ∞ to a Banach space. For this reason we introduce an appropiate geometric property, namely the AHSp-ℓ 4 ∞. We prove that spaces Y satisfying AHSp-ℓ 4 ∞ are precisely those spaces Y such that (ℓ 4 ∞ , Y) has the Bishop-Phelps-Bollobás property. We also provide classes of Banach spaces satisfying this condition. For instance, finite-dimensional spaces, uniformly convex spaces, C 0 (L) and L 1 (µ) satisfy AHSp-ℓ 4 ∞ .

The Bishop-Phelps-Bollob 'as property for operators between spaces of continuous functions

2013

We show that the space of bounded and linear operators between spaces of continuous functions on compact Hausdorff topological spaces has the Bishop-Phelps-Bollob\'as property. A similar result is also proved for the class of compact operators from the space of continuous functions vanishing at infinity on a locally compact and Hausdorff topological space into a uniformly convex space, and for the class of compact operators from a Banach space into a predual of an L_1L_1L_1-space.

The Bishop–Phelps–Bollobás Property: a Finite-Dimensional Approach

Publications of the Research Institute for Mathematical Sciences, 2015

Our goal is to study the Bishop-Phelps-Bollobás property for operators from c0 into a Banach space. We first characterize those Banach spaces Y for which the Bishop-Phelps-Bollobás property holds for (3 ∞ , Y). Examples of spaces satisfying this condition are provided.

The Bishop-Phelps-Bollobás property for compact operators

Canadian Journal of Mathematics, 2016

We study the Bishop-Phelps-Bollobàs property (BPBp) for compact operators. We present some abstract techniques that allow us to carry the BPBp for compact operators from sequence spaces to function spaces. As main applications, we prove the following results. Let X and Y be Banach spaces. If (c 0, Y) has the BPBp for compact operators, then so do (C 0(L), Y) for every locally compactHausdorò topological space L and (X, Y) whenever X * is isometrically isomorphic to . If X * has the Radon-Nikodým property and (X), Y) has the BPBp for compact operators, then so does (L 1(μ, X), Y) for every positive measure μ; as a consequence, (L 1(μ, X), Y) has the BPBp for compact operators when X and Y are finite-dimensional or Y is a Hilbert space and X = c 0 or X = Lp (v) for any positive measure v and 1 < p < ∞. For , if (X, (Y)) has the BPBp for compact operators, then so does (X, Lp (μ, Y)) for every positive measure μ such that L 1(μ) is infinite-dimensional. If (X, Y) has the BPBp for...

The Bishop-Phelps-Bollobás version of Lindenstrauss properties A and B

Transactions of the American Mathematical Society, 2015

We study a Bishop-Phelps-Bollobás version of Lindenstrauss properties A and B. For domain spaces, we study Banach spaces X such that (X, Y ) has the Bishop-Phelps-Bollobás property (BPBp) for every Banach space Y . We show that in this case, there exists a universal function ηX (ε) such that for every Y , the pair (X, Y ) has the BPBp with this function. This allows us to prove some necessary isometric conditions for X to have the property. We also prove that if X has this property in every equivalent norm, then X is one-dimensional. For range spaces, we study Banach spaces Y such that (X, Y ) has the Bishop-Phelps-Bollobás property for every Banach space X. In this case, we show that there is a universal function ηY (ε) such that for every X, the pair (X, Y ) has the BPBp with this function. This implies that this property of Y is strictly stronger than Lindenstrauss property B. The main tool to get these results is the study of the Bishop-Phelps-Bollobás property for c0-, 1-and ∞-sums of Banach spaces.