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We study the Bishop-Phelps-Bollobás property for numerical radius (in short, BPBp-nu) and find sufficient conditions for Banach spaces to ensure the BPBp-nu. Among other results, we show that

Let

In this paper we will discuss the density of numerical radius attaining operators, actually on a stronger property called Bishop-Phelps-Bollobás property for numerical radius. Let us present first a short account on the known results about numerical radius attaining operators. Motivated by the study of norm attaining operators initiated by J. Lindenstrauss in the 1960s, Sims [

Motivated by the work [

A Banach space

Notice that if a Banach space

It is shown in [

The content of this paper is the following. First, we introduce in Section

Let us introduce some notations for later use. The

Analogously to what is done in [

Let

It is immediate that a Banach space

An immediate consequence of the compactness of the unit ball of a finite-dimensional space is the following result. It was previously known to A. Guirao (private communication).

Let

Let

Suppose that

By compactness again, we may assume that

We may also give the following easy result concerning duality.

Let

We will use that

By reflexivity, it is enough to show that

We do not know whether the result above is valid in the nonreflexive case.

For a Banach space which is both uniformly convex and uniformly smooth, we get a property which is weaker than BPBp-nu. This result was known to A. Guirao (private communication).

Let

Notice that the uniform smoothness of

Let us discuss a little bit about the equivalence between the property in the result above and the BPBp-nu. For convenience, let us introduce the following definition.

A Banach space

Notice that the only difference between this concept and the BPBp-nu is the normalization of the operator

The following result is immediate. We include a proof for the sake of completeness.

Let

The necessity is clear. For the converse, assume that we have

We do not know whether the hypothesis of

Putting together Propositions

Let

Let us comment that every complex Banach space

(a) Complex Banach spaces which are uniformly smooth and uniformly convex satisfy the BPBp-nu.

(b) In particular, for every measure

(c) For every measure

In this section, we will show that

Let

As a first step, we have to start dealing with finite regular positive Borel measures, for which a representation theorem for operators exists.

Let

To prove this proposition, we need some background on representation of operators on Lebesgue spaces on finite regular positive Borel measures and several preliminary lemmas.

Let

Let

We will also use that, given an arbitrary measure

Let

From now on,

Suppose that there exist a nonnegative simple function

Let

Suppose that

Suppose that

Then there exist a nonnegative simple function

Since

We claim that

So we have

We also claim that, for each

Now, we define

Finally, we define the measure

We are now ready to present the proof of the main result in the case of finite regular positive Borel measures.

Let

Finally, we may give the proof of the main result in full generality.

Notice that the Kakutani representation theorem (see [

Fix

Now, write

Our goal here is to prove that the density of numerical radius attaining operators does not imply the BPBp-nu. Actually, we will show that, among separable spaces, there is no isomorphic property implying the BPBp-nu other than finite-dimensionality.

We need to relate the BPBp-nu to the Bishop-Phelps-Bollobás property for operators which, as mentioned in the introduction, was introduced in [

The next result relates the BPBp-nu to the BPBp for operators in a particular case. We will deduce our example from it.

If

Before proving this proposition, we will use it to get the main examples of this section. The first example shows that the density of numerical radius attaining operators does not imply the BPBp-nu.

The example above can be extended to get the result that every infinite-dimensional separable Banach space can be renormed to fail the BPBp-nu. This follows from the fact that every infinite-dimensional separable Banach space can be renormed to be strictly convex but not uniformly convex (this result can be proved “by hand”; an alternative categorical argument for it can be found in [

Every infinite-dimensional separable Banach space can be renormed to fail the weak-BPBp-nu (and so, in particular, to fail the BPBp-nu).

We need the following result which is surely well known. As we have not found a reference, we include a nice and easy proof kindly given to us by Vladimir Kadets. We recall that, given a Banach space

Let

Fix

Let

To finish the section with the promised proof of Theorem

Let

Let

Assume that an operator

We first show the case of

We next show the case of

Since

Note that

Suppose that

Let

Otherwise,

We define the operator

Indeed, write