María D. Acosta

There is no bilinear Bishop-Phelps theorem

We answer a question posed by R. Aron, C. Finet and E. Werner, on the bilinear version of the Bishop-Phelps theorem, by exhibiting an example of a Banach spaceX such that the set of norm-attaining bilinear forms onX×X is not dense in the space of all continuous bilinear forms.

The Bishop-Phelps-Bollobás Theorem for bilinear forms

In this paper we provide versions of the Bishop-Phelps-Bollobás Theorem for bilinear forms. Indeed we prove the first positive result of this kind by assuming uniform convexity on the Banach spaces. A characterization of the Banach space Y satisfying a version of the Bishop-Phelps-Bollobás Theorem for bilinear forms on 1 × Y is also obtained. As a consequence of this characterization, we obtain positive results for finite-dimensional normed spaces, uniformly smooth spaces, the space C(K) of continuous functions on a compact Hausdorff topological space K and the space K(H) of compact operators on a Hilbert space H. On the other hand, the Bishop-Phelps-Bollobás Theorem for bilinear forms on 1 × L1(μ) fails for any infinite-dimensional L1(μ), a result that was known only when L1(μ) = 1.

Every real Banach space can be renormed to satisfy the denseness of numerical radius attaining operators

First we show that every real Banach space satisfying a certain property, calledβ (used by Lindenstrauss and Partington) verifies the denseness of the numerical radius attaining operators. Using this result and a renorming theorem by Partington we conclude that every Banach space is isomorphic to a new one satisfying the denseness of the numerical radius attaining operators.

Norm Attaining Operators on Some Classical Banach Spaces

We show that for the Kothe space X = c0 + 1(w), equipped with the Luxemburg norm, the set of norm attaining operators from X into any infinite-dimensional strictly convex Banach space Y is not dense in the space of all bounded operators. The same assertion holds for any infinite-dimensional L1(μ). This gives the first example of a classical space X satisfying the previous property. We also prove that all the spaces c0 + 1(w) are isomorphic for a large class of weights w.

A Version of James' Theorem for Numerical Radius

We prove that if all the rank-one bounded operators on a Banach space X attain their numerical radii, then X must be reflexive, but the converse does not hold. In fact, every reflexive space with basis can be renormed in such a way that there is a rank-one operator not attaining the numerical radius. The classical James’ Theorem [12] characterizes reflexive spaces as those Banach spaces such that every functional attains its norm. Recently, some attention has been paid to the study of denseness of numerical radius attaining operators (see, for instance, [4, 9, 3, 14, 2, 10]), a problem analogous to the well-known parallel question for the norm. The general idea gained from looking at the results is that, even if some more tricky work is required, almost any result that is true for norm does also hold for numerical radius. Since James’ Theorem can be looked at as, equivalently, a Banach space is reflexive if and only if every rank-one operator on it attains its norm, we raised the parallel version for the numerical radius. In order to pose the problem in a more precise way, let us introduce some notation. Given a Banach space X over the scalar field K (R or C), we shall denote by BX and SX , respectively, the closed unit ball and the unit sphere of X; X ∗ will be the dual space of X, and L(X) will be the Banach space of all (bounded and linear) operators from X into itself, both spaces endowed with their usual norms. Let us recall that the numerical radius of an operator T is the real number v(T ) given by v(T ) = sup{|x∗Tx| : (x, x∗) ∈ Π(X)}, where Π(X) := {(x, x∗) ∈ SX × SX∗ : x∗(x) = 1}, and such an operator T attains its numerical radius if there exists an element (x0, x ∗ 0) ∈ Π(X) such that |x∗0Tx0| = v(T ). Instead of the set Π(X), it will be useful for us to consider the set Π̂(X) given by Π̂(X) := {(x, x∗) ∈ SX × SX∗ : |x∗(x)| = 1}. It is clear that v(T ) = sup{Re x∗Tx : (x, x∗) ∈ Π̂(X)}, and T attains its numerical radius if and only if there is some element (x0, x ∗ 0) ∈ Π̂(X) with Re x∗0Tx0 = v(T ). For a complete survey about the theory of numerical range of an element in a Banach algebra, we recommend the monographs [7] and [8]. This topic has been Received 26 June 1997. 1991 Mathematics Subject Classification 47A12, 46B10. Research partially supported by DGICYT project no. PB96-1906. Bull. London Math. Soc. 31 (1999) 67–74 68 maría d. acosta and manuel ruiz galán found to be a useful tool for dealing with several questions in Banach algebras and in the geometry of Banach spaces [13]. First, we shall prove that a Banach space X on which every rank-one operator attains its numerical radius must be reflexive. Then we shall exhibit counterexamples to show that the converse does not hold. For operators T far from being compact, of course, there is no hope of finding any relationship between reflexivity and the assertion that T attains its numerical radius. For instance, if X has 1-unconditional basis, then it is easy to construct a diagonal operator not attaining its numerical radius even if the space is reflexive. To prove the announced result, we shall make use of the following maximinimax principle due to S. Simons [15, Theorem 5]. Proposition 1. Let X0 and Y0 be non-void sets, and let f : X0 × Y0 → R be a bounded function such that for every sequence {zn} of elements in co f(·, Y0) (where co denotes the convex hull ) and any sequence {tn} of positive real numbers with ∑∞ n=1 tn = 1, there exists a bounded real function z on X0 satisfying lim inf n zn 6 z 6 lim sup n zn, and the function ∑∞ n=1 tn(zn − z) attains its supremum on X0. Then inf ∅6=F⊂Y0 finite sup x∈X0 inf f(x, F) 6 sup ∅6=G⊂X0 finite inf y∈Y0 sup f(G, y). The above inequality has been used successfully to obtain new proofs of James’ Theorem (see [15]). In order to work with the above proposition, we first prove the following technical result. Lemma 1. Let X be a Banach space, and let F be a finite subset of L(X) such that ‖T‖ 6 1 for all T ∈ F . Then sup (x,x∗)∈Π̂(X) inf T∈F Re x ∗Tx = sup (x∗ ,x∗∗)∈Π̂(X∗) inf T∈F Re x ∗∗T ∗x∗. Proof. The left-hand side of the equation is clearly less than or equal to the right-hand side. In order to check the reverse inequality, let us fix ε > 0, (x∗0, x∗∗ 0 ) ∈ Π̂(X∗), and write λ := x∗∗ 0 (x∗0)−1, a scalar with |λ| = 1. Since F is finite and BX is w∗-dense in BX∗∗ , we can find z ∈ BX such that |x∗0(x∗∗ 0 − z)| < ε 2 36 and |(x∗∗ 0 − z)T ∗x∗0| < ε 3 , for all T ∈ F. (1)

The Daugavet property in rearrangement invariant spaces

We study rearrangement invariant spaces with the Daugavet property. The main result of this paper states that under mild assumptions the only nonseparable rearrangement invariant space X over an atomless finite measure space with the Daugavet property is L∞ endowed with its canonical norm. We also prove that a uniformly monotone rearrangement invariant space over an infinite atomless measure space with the Daugavet property is isometric to L1. As an application we obtain that an Orlicz space over an atomless measure space has the Daugavet property if and only if it is isometrically isomorphic to L1.

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

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 (`∞, Y ). Examples of spaces satisfying this condition are provided. 2010 Mathematics Subject Classification: Primary 46B20; Secondary 46B28, 47B99.

Norm attaining operators and James’ Theorem

Abstract There are several results relating isomorphic properties of a Banach space and the set of norm attaining functionals. Here, we show versions for operators of some of these results. For instance, a Banach space X has to be reflexive if it does not contain l1 and for some non trivial Banach space Y and positive r, the unit ball of the space of operators from X into Y is the closure (weak operator topology) of the convex hull of the norm one operators satisfying that balls centered at any of them with radius r are contained in the set of norm attaining operators. We also prove a similar result by using a very weak isometric condition on the space instead of non containing l1.

The Bishop–Phelps–Bollobás property for numerical radius of operators on L 1 ( μ )

Abstract In this paper, we introduce the notion of the Bishop–Phelps–Bollobas property for numerical radius (BPBp- ν ) for a subclass of the space of bounded linear operators. Then, we show that certain subspaces of L ( L 1 ( μ ) ) have the BPBp- ν for every finite measure μ . As a consequence we deduce that the subspaces of finite-rank operators, compact operators and weakly compact operators on L 1 ( μ ) have the BPBp- ν .