A non-linear Bishop–Phelps–BollobÁs type theorem

Abstract

The main aim of this paper is to prove a Bishop-Phelps-Bollobás type theorem on the unital uniform algebra A w * u (B X * ) consisting of all w * -uniformly continuous functions on the closed unit ball B X * which are holomorphic on the interior of B X * . We show that this result holds for A w * u (B X * ) if X * is uniformly convex or X * is the uniformly complex convex dual space of an order continuous absolute normed space. The vector-valued case is also studied.In 1961, Bishop and Phelps proved that the set of norm attaining functionals is dense in the dual space [5]. They questioned whether the same result holds for bounded linear operators and the answer was given two years later by Lindenstrauss [17] who gave a counterexample proving that in general the answer is false. However, he also presented conditions on a Banach space to get positive results. On the other hand, Bollobás proved a stronger version for linear functionals which is nowdays known as the Bishop-Phelps-Bollobás theorem [6]. This says that functionals and points which they almost attain their norms can be simultaneously approximated by norm attaining functionals and points which they attain their norms. We highlight this theorem because it is the main motivation for the present work. If X is a Banach space, ε ∈ (0, 2) and (x, x * ) ∈ B X × B X * satisfy the following inequalitythen there is (y, y * ) ∈ S X × S X * such that |y * (y)| = 1, y − x < ε and y * − x * < ε (this version can be found in [10, Corollary 2.4]).Since 2008, with the seminal paper by Acosta, Aron, García and Maestre [2], a lot of attention had been paid in the attempt to get Bishop-Phelps-Bollobás type theorems for bounded linear operators, homogeneous polynomials and multilinear mappings by putting conditions on the Banach spaces as Lindenstrauss did. Our aim here is to get a Bishop-Phelps-Bollobás type theorem for holomorphic functions. In [1] the authors showed that if X is a complex Banach space with Radon-Nikodým property and if A u (B X ) stands for the space of all uniformly continuous functions on the closed unit ball B X which are holomorphic on the interior endowed 2010 Mathematics Subject Classification. Primary: 46B04; Secondary: 46G20.

Generalized Numerical Index of Function Algebras

“…A version of the Bishop-Phelps-Bollobás type theorem for holomorphic functions has been shown [5,13]. In the following theorem, we present a similar result.…”