On the geometry around an operator between Banach spaces

Abstract

The aim of this thesis is to study and analyse different notions related to the geometry of the space of all bounded linear operators between Banach spaces around a fixed operator. The dissertation follows a compendium form, and its content is organized in six chapters, each of them corresponding to an independent paper. Chapter I [1] consists of a thorough study of the numerical index with respect to an operator between Banach spaces. This notion is the best constant of equivalence between the usual operator norm and the numerical radius with respect to an operator. We first provide some tools to study this concept and present some results dealing with the numerical index with respect to adjoint operators and rank-one operators. Then, we focus on the set of values of the numerical indices with respect to all norm-one operators between two Banach spaces. For instance, we show that this set is trivial when the domain or the codomain is a real Hilbert space or the space of compact operators on a real Hilbert space. We also study this set for some classical Banach spaces such as the real Lp space or the space of continuous functions on a compact Hausdorff topological space. Additionally, we show that the concept of Lipschitz numerical range for Lipschitz self-maps of a Banach space is a particular case of numerical range with respect to a convenient linear operator between two different Banach spaces. To finish this chapter, we provide some results showing the behaviour of this concept when we apply some Banach space operations, such as constructing diagonal operators between some sums of Banach spaces, considering composition operators between vector-valued function spaces, taking the adjoint of an operator, and composing two operators. In Chapter II [3], we deal with the Banach numerical index (i.e., the numerical index with respect to the identity operator) of operator ideals and tensor products. We show that the numerical index of any operator ideal endowed with the operator norm is less than or equal to the minimum of the numerical indices of the domain and of the codomain. We present stronger inequalities for the numerical indices of the spaces of compact and weakly compact operators, which allow to give interesting examples. We also prove that the numerical index of a projective or injective tensor product of Banach spaces is less than or equal to the numerical index of any of the factors. Furthermore, we show that if a projective tensor product of two Banach spaces has the Daugavet property and one of the factor satisfies that its unit ball is slicely countably determined or that its dual contains a point of Fréchet differentiability of the norm, then the other factor inherits the Daugavet property. On the other hand, if an injective tensor product of two Banach spaces has the Daugavet property and one of the factors has a point of Fréchet differentiability of the norm, then the other factor has the Daugavet property. Chapter III [5] and IV [6] are devoted to the problem of calculating the numerical index of the real Lp space of dimension two. To do so, we first deal with two-dimensional real spaces endowed with an absolute and symmetric norm and give a lower bound for the numerical index of such spaces. Moreover, we show that in many instances the numerical index coincides with the given bound and, as a consequence, we compute the numerical index of the real Lp space of dimension two for values of p between 3/2 and 3. In our second approach, we directly work in the real Lp space of dimension two and calculate its numerical index for values of p between 6/5 and 3/2, as well as between 2 and 6. In Chapter V [2], we introduce and study the concept of generating operator. These operators are those which generate the unit ball of the domain by closed convex hull of the points where the operator almost attains its norm. We provide a useful characterization of generating operators in terms of the geometry of some subsets of the dual spaces. Additionally, we study the relationship between generating operators and norm-attainment. While generating operators having rank one and those whose domain has the Radon-Nikodým property attain their norm, there are generating operators, even of rank two, which do not attain their norm. We also discuss the possibility for a Banach space X to be the domain of a generating operator which does not attain its norm in terms of the behaviour of some sets of the dual of X. Furthermore, we study the properties of the set of all generating operators between two Banach spaces X and Y. In this line, we show that this set generates the unit ball of the space of all bounded linear operators from X to Y by closed convex hull when X is the space of absolutely summable sequences and that this is the only possibility for real finite-dimensional spaces. In chapter VI [4], we present a widely applicable approach to address Birkhoff-James orthogonality by using its connection with abstract numerical range. More precisely, we characterize Birkhoff-James orthogonality and smooth points in a Banach space Z in terms of the actions of functionals in a subset of its dual which is one-norming for Z. This general approach can be applied in several cases to obtain known results, such as the characterization of Birkhoff-James orthogonality in the space of operators between Banach spaces endowed with the operator norm or with the numerical radius, as well as new results on Birkhoff-James orthogonality in spaces of vector-valued bounded functions and in their subspaces. Furthermore, we provide applications to the study of spear vectors and spear operators. Specifically, we prove that no smooth point of a Banach space Z can be Birkhoff-James orthogonal to a spear vector of Z. In the case when Z is the space of all bounded linear operators between Banach spaces, this leads to obstructive results for the existence of spear operators and for a Banach space to have numerical index one.