## About

140

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Introduction

Miguel Martin obtained his Ph.D. degree in 2000 at the University of Granada, Granada, Spain, where he is currently Full Professor. His main research interests are in functional analysis, especially in geometry of Banach spaces, with particular interest in numerical ranges of operators, norm attaining operators, isometries of Banach spaces, Daugavet property, and the geometry of Lipschitz-free Banach spaces, among others. He has advised six PhD students and is advising one more.

Additional affiliations

March 2012 - August 2012

August 2003 - November 2016

December 2000 - July 2003

Education

September 1996 - September 2000

September 1991 - June 1996

## Publications

Publications (140)

We introduce extensions of ∆-points and Daugavet points in which slices are replaced by relative weakly open subsets (super ∆-points and super Daugavet points) or by convex combinations of slices (ccs ∆-points and ccs Daugavet points). These notions represent the extreme opposite to denting points, points of continuity, and strongly regular points....

For Banach spaces X and Y, a bounded linear operator T:X⟶Y⁎ is said to weak-star quasi attain its norm if the σ(Y⁎,Y)-closure of the image by T of the unit ball of X intersects the sphere of radius ‖T‖ centred at the origin in Y⁎. This notion is inspired by the quasi-norm attainment of operators introduced and studied in [5]. As a main result, we p...

We study the relationship between the residuality of the set of norm attaining functionals on a Banach space and the residuality and the denseness of the set of norm attaining operators between Banach spaces. Our first main result says that if C is a bounded subset of a Banach space X which admit an LUR renorming satisfying that, for every Banach s...

For Banach spaces X and Y , a bounded linear operator T : X -> Y* is said to weak-star quasi attain its norm if the σ(Y*,Y)-closure of the image by T of the unit ball of X intersects the sphere of radius ||T|| centred at the origin in Y*. This notion is inspired by the quasi-norm attainment of operators introduced and studied in [5]. As a main resu...

We provide a characterization of the Radon–Nikodým property for a Banach space Y in terms of the denseness of bounded linear operators into Y which attain their norm in a weak sense, which complement the one given by Bourgain and Huff in the 1970s for domain spaces. To this end, we introduce the following notion: an operator $$T:X \longrightarrow Y...

We prove that every JB$^*$-triple $E$ (in particular, every $C^*$-algebra) satisfying the Daugavet property also satisfies the stronger polynomial Daugavet property, that is, every weakly compact polynomial $P\colon E \longrightarrow E$ satisfies the Daugavet equation $\|\hbox{id}_{E} + P\| = 1+\|P\|$. The analogous conclusion also holds for the al...

We show that all the symmetric projective tensor products of a Banach space X have the Daugavet property provided X has the Daugavet property and either X is an $$L_1$$ L 1 -predual (i.e., $$X^{*}$$ X ∗ is isometric to an $$L_1$$ L 1 -space) or X is a vector-valued $$L_1$$ L 1 -space. In the process of proving it, we get a number of results of inde...

We study the relationship between the residuality of the set of norm attaining functionals on a Banach space and the residuality and the denseness of the set of norm attaining operators between Banach spaces. Our first main result says that if $C$ is a bounded subset of a Banach space $X$ which admit an LUR renorming satisfying that, for every Bana...

We study the Bishop–Phelps–Bollobás property for numerical radius restricted to the case of compact operators (BPBp-nu for compact operators in short). We show that \(C_0(L)\) spaces have the BPBp-nu for compact operators for every Hausdorff topological locally compact space L. To this end, on the one hand, we provide some techniques allowing to pa...

We show that the numerical index of any operator ideal is less than or equal to the minimum of the numerical indices of the domain space and the range space. Further, we show that the numerical index of the ideal of compact operators or the ideal of weakly compact operators is less than or equal to the numerical index of the dual of the domain spac...

We study the Bishop-Phelps-Bollob\'as property for numerical radius restricted to the case of compact operators (BPBp-nu for compact operators in short). We show that $C_0(L)$ spaces have the BPBp-nu for compact operators for every Hausdorff topological locally compact space $L$. To this end, on the one hand, we provide some techniques allowing to...

We show that all the symmetric projective tensor products of a Banach space $X$ have the Daugavet property provided $X$ has the Daugavet property and either $X$ is an $L_1$-predual (i.e.\ $X^*$ is isometric to an $L_1$-space) or $X$ is a vector-valued $L_1$-space. In the process of proving it, we get a number of results of independent interest. For...

We prove that the class of positive operators from L∞(μ) to L1(ν) has the Bishop-Phelps-Bollobás property for any positive measures μ and ν. The same result also holds for the pair (c0, ℓ1). We also provide an example showing that not every pair of Banach lattices satisfies the Bishop-Phelps-Bollobás property for positive operators.

We provide sufficient conditions on a Banach space X in order that there exist norm attaining operators of rank at least two from X into any Banach space of dimension at least two. For example, a rather weak such condition is the existence of a nontrivial cone consisting of norm attaining functionals on X. We go on to discuss density of norm attain...

We show that the numerical index of any operator ideal is less than or equal to the minimum of the numerical indices of the domain and the range. Further, we show that the numerical index of the ideal of compact operators or the ideal of weakly compact operators is less than or equal to the numerical index of the dual of the domain, and this result...

We introduce a weakened notion of norm attainment for bounded linear operators between Banach spaces which we call \emph{quasi norm attaining operators}. An operator $T\colon X \longrightarrow Y$ between the Banach spaces $X$ and $Y$ is quasi norm attaining if there is a sequence $(x_n)$ of norm one elements in $X$ such that $(Tx_n)$ converges to s...

We study the stability behavior of the Bishop-Phelps-Bollob\'as property for Lipschitz maps (Lip-BPB property). This property is a Lipschitz version of the classical Bishop-Phelps-Bollob\'as property and deals with the possibility of approximating a Lipschitz map that almost attains its (Lipschitz) norm at a pair of distinct points by a Lipschitz m...

We study Banach spaces whose group of isometries acts micro-transitively on the unit sphere. We introduce a weaker property, inherited by one-complemented subspaces, that we call uniform micro-semitransitivity. We prove a number of results about both micro-transitive and uniformly micro-semitransitive spaces. In particular, they are uniformly conve...

https://www.cambridge.org/core/journals/journal-of-the-institute-of-mathematics-of-jussieu/article/equivalent-norms-with-an-extremely-nonlineable-set-of-norm-attaining-functionals/F0A9770CFA90401940E45E5F2355870E/share/0cf3e969446dc025f12e219148f6389976c7f958
We present a construction that enables one to find Banach spaces $X$ whose sets $NA(X)$ o...

In this paper, we introduce and study a Lipschitz version of the Bishop–Phelps–Bollobás property (Lip-BPB property). This property deals with the possibility of making a uniformly simultaneous approximation of a Lipschitz map F and a pair of points at which F almost attains its norm by a Lipschitz map G and a pair of points such that G strongly att...

We show that for every 1<n<∞, there exists a Banach space Xn containing proximinal subspaces of codimension n but no proximinal finite codimensional subspaces of higher codimension. Moreover, the set of norm-attaining functionals of Xn contains n-dimensional subspaces, but no subspace of higher dimension. This gives an n-by-n version of the solutio...

We classify several notions of norm attaining Lipschitz maps which were introduced previously, and present the relations among them in order to verify proper inclusions. We also analyze some results for the sets of Lipschitz maps satisfying each of these properties to be dense or not in Lip0(X,Y). For instance, we characterize a Banach space Y with...

We characterise the class of those Banach spaces in which every convex combination of slices of the unit ball intersects the unit sphere as the class of those spaces in which every convex combination of slices of the unit ball contains two points at distance exactly two. Also, we study when the convex combinations of slices of the unit ball are rel...

We study the density of the set SNA(M, Y) of those Lipschitz maps from a (complete pointed) metric space M to a Banach space Y which strongly attain their norm (i.e. the supremum defining the Lipschitz norm is actually a maximum). We present new and somehow counterintuitive examples, and we give some applications. First, we show that SNA(T, Y) is n...

We classify several notions of norm attaining Lipschitz maps which were introduced previously, and present the relations among them in order to verify proper inclusions. We also analyze some results for the sets of Lipschitz maps satisfying each of these properties to be dense or not in $\Lip(X,Y)$. For instance, we characterize a Banach space $Y$...

We study Banach spaces whose group of isometries acts micro-transitively on the unit sphere. We introduce a weaker property, which one-complemented subspaces inherit, that we call uniform micro-semitransitivity. We prove a number of results about both micro-transitive and uniformly micro-semitransitive spaces, including that they are uniformly conv...

Given Banach spaces $X$ and $Y$, and a norm-one operator $G\in \mathcal{L}(X,Y)$, the numerical index with respect to $G$, $n_G(X,Y)$, is the greatest constant $k\geq 0$ such that $$\max_{|w|=1}\|G+wT\|\geq 1 + k \|T\|$$ for all $T\in \mathcal{L}(X,Y)$. We present some results on the set $\mathcal{N}(\mathcal{L}(X,Y))$ of the values of the numerica...

https://arxiv.org/pdf/1905.08272.pdf
We provide sufficient conditions on a Banach space $X$ in order that there exist norm attaining operators of rank at least two from $X$ into any Banach space of dimension at least two. For example, a rather weak such condition is the existence of a non-trivial cone consisting of norm attaining functionals on $X...

In this paper, we study conditions assuring that the Bishop–Phelps–Bollobás property (BPBp, for short) is inherited by absolute summands of the range space or of the domain space. Concretely, given a pair (X, Y) of Banach spaces having the BPBp,
(a)
if \(Y_1\) is an absolute summand of Y, then \((X,Y_1)\) has the BPBp;
(b)
if \(X_1\) is an absolute...

In this paper, we introduce and study a Lipschitz version of the Bishop-Phelps-Bollob\'as property (Lip-BPB property). This property deals with the possibility to make a uniformly simultaneous approximation of a Lipschitz map $F$ and a pair of points at which $F$ almost attains its norm by a Lipschitz map $G$ and a pair of points such that $G$ stro...

We study the set SNA(M,Y) of those Lipschitz maps from a (complete pointed) metric space M to a Banach space Y which (strongly) attain their Lipschitz norm (i.e. the supremum defining the Lipschitz norm is a maximum). Extending previous results, we prove that this set is not norm dense when M is a length space (or local) or when M is a closed subse...

We study approximation of operators between Banach spaces $X$ and $Y$ that nearly attain their norms in a given point by operators that attain their norms at the same point. When such approximations exist, we say that the pair $(X,Y)$ has the pointwise Bishop–Phelps–Bollobás property (pointwise BPB property for short). In this paper we mostly conce...

Given two real Banach spaces $X$ and $Y$ with dimensions greater than one, it is shown that there is a sequence $\{T_n\}_{n\in \mathbb{N}}$ of norm attaining norm-one operators from $X$ to $Y$ and a point $x_0\in X$ with $\|x_0\|=1$, such that $\|T_n(x_0)\|\longrightarrow 1$ but $\inf_{n \in \mathbb{N}} \{\mbox{dist} (x_0,\,\{x\in X: \|T_n(x)\|=\|x...

An error in the production process unfortunately led to online publication of the chapter abstracts prematurely, before incorporation of the final corrections. The version supplied here has been corrected and approved by the author [authors].

We study the set $\operatorname{SNA}(M, Y)$ of those Lipschitz operators from a (complete pointed) metric space $M$ to a Banach space $Y$ which (strongly) attain their Lipschitz norm (i.e. the supremum defining the Lipschitz norm is a maximum). Extending previous results, we prove that this set is not norm dense when $M$ is length (or local) or whe...

We prove that a version of the Bishop-Phelps-Bollobás property in which the operator is not changed is possible only if one of the involved Banach spaces is one-dimensional.

In this paper we study conditions assuring that the Bishop-Phelps-Bollob\'as property (BPBp, for short) is inherited by absolute summands of the range space or of the domain space. Concretely, given a pair (X, Y) of Banach spaces having the BPBp, (a) if Y1 is an absolute summand of Y, then (X, Y1) has the BPBp; (b) if X1 is an absolute summand of X...

We show that for every $1<n<\infty$, there exits a Banach space $X_n$ containing proximinal subspaces of codimension $n$ but no proximinal finite codimensional subspaces of higher codimension. Moreover, the set of norm-attaining functionals of $X_n$ contains $n$-dimensional subspaces, but no subspace of higher dimension. This gives a $n$-by-$n$ ver...

We complete the book with a collection of open problems.

Our goal here is to complement the previous chapter with some interesting results. We characterize lush operators when the domain space has the Radon-Nikodým Property or the codomain space is Asplund, and we get better results when the domain or the codomain is finite-dimensional or when the operator has rank one. Further, we study the behaviour of...

We recall the concept of spear vector and introduce the new notion of spear set. They are both used as ``leitmotiv'' to give a unified presentation of the concepts of spear operator, lush operator, aDP, and other type of operators that will be introduced here. We collect some properties of spear sets and vectors, together with some (easy) examples...

This is the main chapter of our manuscript, as we introduce and deeply study the main definitions: the one of spear operator, the weaker of operator with the alternative Daugavet property, and the stronger of lush operator.

Our goal here is to present consequences on the Banach spaces X and Y of the fact that there is \(G\in \mathcal {L}(X,Y)\) which is a spear operator, is lush, or has the aDP.

This chapter contains an overview of the known results about Banach spaces with numerical index 1, as well as the notation and terminology we will need along the book.

We study Lipschitz spear operators. These are just the spear vectors of the space of Lipschitz operators between two Banach spaces endowed with the Lipschitz norm. The main result here is that every (linear) lush operator is a Lipschitz spear operator. We also provide with analogous results for aDP operators and for Daugavet centers.

Our aim here is to present examples of operators which are lush, spear, or have the aDP, defined in some classical Banach spaces. One of the most intriguing examples is the Fourier transform on L1, which we prove that is lush. Next, we study a number of examples of operators arriving to spaces of continuous functions. In particular, it is shown tha...

Our aim here is to provide several results on the stability of our properties for operators by several operations like absolute sums, vector-valued function spaces, and ultraproducts.

Link to the published version (free for downloading until January 31, 2018): https://authors.elsevier.com/c/1WCnj51yEMRS4
We study geometric properties of the Banach space $\mathcal{R}$ constructed recently by C.Read which does not contain proximinal subspaces of finite codimension greater than or equal to two. Concretely, we show that the bidual...

This monograph is devoted to the study of spear operators, that is, bounded linear operators $G$ between Banach spaces $X$ and $Y$ satisfying that for every other bounded linear operator $T:X\longrightarrow Y$ there exists a modulus-one scalar $\omega$ such that
$\|G + \omega\,T\|=1+ \|T\|$.
This concept extends the properties of the identity oper...

We study approximations of operators that nearly attain their norms in a given point by operators that attains their norm at the same point. When for operators between given Banach spaces $X$ and $Y$ such approximations exist, then we say that the pair $(X, Y)$ has the Bishop-Phelps-Bollob\'as point property (BPBpp for short). In this paper we most...

The aim of this manuscript is to study \emph{spear operators}: bounded linear operators $G$ between Banach spaces $X$ and $Y$ satisfying that for every other bounded linear operator $T:X\longrightarrow Y$ there exists a modulus-one scalar $\omega$ such that $$ \|G + \omega\,T\|=1+ \|T\|. $$ To this end, we introduce two related properties, one weak...

http://rdcu.be/kJfV
We continue the study of the Bishop-Phelps-Bollob\'as moduli $\Phi_X(\delta)$ and $\Phi_X^S(\delta)$ initiated in [M.~Chica, V.~Kadets, M.~Mart{í}n, S.~Moreno-Pulido, and F.~Rambla-Barreno. {J. Math. Anal. Appl.} {412} (2014), no.~2, 697--719]. In particular, for a uniformly non-square Banach space $X$ we present a simple proof...

We prove that for a given Banach space $X$, the subset of norm attaining Lipschitz functionals in $\mathrm{Lip}_0(X)$ is weakly dense but not strongly dense. Then we introduce a weaker concept of directional norm attainment and demonstrate that for a uniformly convex $X$ the set of directionally norm attaining Lipschitz functionals is strongly dens...

It is shown that the Bishop–Phelps–Bollobás theorem holds for bilinear forms on the complex C0(L1)×C0(L2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_0(L_1)\times...

We introduce a second numerical index for real Banach spaces with non-trivial Lie algebra, as the best constant of equivalence between the numerical radius and the quotient of the operator norm modulo the Lie algebra. We present a number of examples and results concerning absolute sums, duality, vector-valued function spaces\ldots which show that,...

We study the Bishop-Phelps-Bollob\'as property (BPBp for short) for compact operators. We present some abstract techniques which allows to carry the BPBp for compact operators from sequence spaces to function spaces. As main applications, we prove the following results. Let $X$, $Y$ be Banach spaces. If $(c_0,Y)$ has the BPBp for compact operators,...

We show that there are compact linear operators on Banach spaces which cannot be approximated by numerical radius attaining operators.

We study the Bishop-Phelps-Bollobás property for numerical radius (for short, BPBp-nu) of operators on ℓ1-sums and ℓ∞-sums of Banach spaces. More precisely, we introduce a property of Banach spaces, which we call strongly lush. We find that if X is strongly lush and X ⊕1 Y has the weak BPBp-nu, then (X, Y ) has the Bishop-Phelps- Bollobás property...

We introduce a substitute for the concept of slice for the case of non-linear
Lipschitz functionals and transfer to the non-linear case some results about
the Daugavet and the alternative Daugavet equations previously known only for
linear operators.

We introduce a generalized approximate hyperplane series property for a pair of Banach spaces to characterize when has the Bishop–Phelps–Bollobás property. In particular, we show that has this property if X, Y are finite-dimensional, if X is a space and Y is a Hilbert space, or if X is Asplund and , where K is a compact Hausdorff space and L is a l...

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 functi...

We study the relation between the intrinsic and the spatial numerical ranges
with the recently introduced "approximated" spatial numerical range. As main
result, we show that the intrinsic numerical range always coincides with the
convex hull of the approximated spatial numerical range. Besides, we show
sufficient conditions and necessary condition...

It has been very recently discovered that there are compact linear operators
between Banach spaces which cannot be approximated by norm attaining operators.
The aim of this expository paper is to give an overview of those examples and
also of sufficient conditions ensuring that compact linear operators can be
approximated by norm attaining operator...

Extending the celebrated result by Bishop and Phelps that the set of norm
attaining functionals is always dense in the topological dual of a Banach
space, Bollobás proved the nowadays known as the Bishop-Phelps-Bollobás
theorem, which allows to approximate at the same time a functional and a vector
in which it almost attains the norm. Very recently...

It is shown that the Bishop-Phelps-Bollob\'as theorem holds for bilinear
forms on the complex $C_0(L_1)\times C_0(L_2)$ for arbitrary locally compact
topological Hausdorff spaces $L_1$ and $L_2$.

We present a sufficient condition for a Banach space to have the approximate
hyperplane series property (AHSP) which actually covers all known examples. We
use this property to get a stability result to vector-valued spaces of
integrable functions. On the other hand, the study of a possible
Bishop-Phelps-Bollob\'as version of a classical result of...

A talk presented at the IV International Hahn conference,
Chernivtsi (Ukraine), June 30 – July 5, 2014

In this paper we show that the Bishop–Phelps–Bollobás theorem holds for L(L1(μ),L1(ν))L(L1(μ),L1(ν)) for all measures μ and ν and also holds for L(L1(μ),L∞(ν))L(L1(μ),L∞(ν)) for every arbitrary measure μ and every localizable measure ν . Finally, we show that the Bishop–Phelps–Bollobás theorem holds for two classes of bounded linear operators from...

We study the Bishop-Phelps-Bollob\'as property for numerical radius (in
short, BPBp-nu) and find sufficient conditions for Banach spaces ensuring the
BPBp-nu. Among other results, we show that $L_1(\mu)$-spaces have this property
for every measure $\mu$. On the other hand, we show that every
infinite-dimensional separable Banach space can be renorm...

We introduce two Bishop-Phelps-Bollobas moduli of a Banach space which measure, for a given Banach space, what is the best possible Bishop-Phelps-Bollobas theorem in this space. We show that there is a common upper bound for these moduli for all Banach spaces and we present an example showing that this bound is sharp. We prove the continuity of the...

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 topologica...

We estimate the polynomial numerical indices of the spaces C(K) and L1(μ).

We show examples of compact linear operators between Banach spaces which
cannot be approximated by norm attaining operators. This is the negative answer
to an open question posed in the 1970's. Actually, any strictly convex Banach
space failing the approximation property serves as the range space. On the
other hand, there are examples in which the...

We study a Bishop-Phelps-Bollob\'as version of Lindenstrauss properties A and
B. For domain spaces, we study Banach spaces $X$ such that $(X,Y)$ has the
Bishop-Phelps-Bollob\'as property (BPBp) for every Banach space $Y$. We show
that in this case, there exists a universal function $\eta_X(\eps)$ such that
for every $Y$, the pair $(X,Y)$ has the BP...

We show that for spaces with 1–unconditional bases lushness, the alternative Daugavet property and numerical index 1 are equivalent. In the class of rearrangement invariant (r.i.) sequence spaces the only examples of spaces with these properties are ${{c}_{0,}}{{\ell }_{1}}$ and ${{\ell }_{\infty }}$ . The only lush r.i. separable function space on...

In this paper we show that the Bishop-Phelps-Bollob\'as theorem holds for $\mathcal{L}(L_1(\mu), L_1(\nu))$ for all measures $\mu$ and $\nu$ and also holds for $\mathcal{L}(L_1(\mu),L_\infty(\nu))$ for every arbitrary measure $\mu$ and every localizable measure $\nu$. Finally, we show that the Bishop-Phelps-Bollob\'as theorem holds for two classes...

We study the rank-1 numerical index of a Banach space, namely the infimum of the numerical radii of those rank-1 operators
on the space which have norm 1. We show that the rank-1 numerical index is always greater than or equal to 1/e. We also present properties of this index and some examples.

The editors thank all the contributors and colleagues who spared time and were prompt in refereeing work.

We prove that an onto isometry between unit spheres of finite-dimensional
polyhedral Banach spaces extends to a linear isometry of the corresponding
spaces.
Open archive in ScienceDirect: http://www.sciencedirect.com/science/article/pii/S0022247X12005203

The only infinite-dimensional complex space with 1-unconditional basis which has polynomial numerical index of order 2 equal to 1 is c0. In the real case, there is no space of this type. We also show that, in the complex case, if X is an infinite-dimensional Banach sequence space with absolute norm whose Köthe dual is norming and has polynomial num...

Let $M$ be a von Neumann algebra and let $M_\star$ be its (unique) predual.
We study when for every $\varphi\in M_\star$ there exists $\psi\in M_\star$
solving the equation $\|\varphi \pm \psi\|=\|\varphi\|=\|\psi\|$. This is the
case when $M$ does not contain type I nor type III$_1$ factors as direct
summands and it is false at least for the uniqu...

A Banach space X is said to be an extremely non-complex space if the norm equality ∥Id +T
2∥ = 1+∥T
2∥ holds for every bounded linear operator T on X. We show that every extremely non-complex Banach space has positive numerical index, it does not have an unconditional basis
and that the infimum of diameters of the slices of its unit ball is positiv...

We give a lower bound for the numerical index of the real space L
p
(µ) showing, in particular, that it is non-zero for p ≠ 2. In other words, it is shown that for every bounded linear operator T on the real space L
p
(µ), one has
$$\sup \left\{ {|\int {|x{|^{p - 1}}{\rm{sign}}(x)Tx d\mu |:x \in {L_p}\left( \mu \right), ||x|| = 1} } \right\} \g...

We study when a Banach space with absolute norm may have polynomial numerical indices equal to one. In the real case, we show that a Banach space X with absolute norm, which has the Radon–Nikodým property or is Asplund, satisfies n(2)(X)<1 unless it is one-dimensional. In the complex case, we show that the only Banach spaces X with absolute norm an...

We show that the absolute numerical index of the space $L_p(\mu)$ is $p^{-1/p} q^{-1/q}$ (where $1/p+1/q=1$). In other words, we prove that $$ \sup\{\int |x|^{p-1}|Tx|\, d\mu \, : \ x\in L_p(\mu),\,\|x\|_p=1\} \,\geq \,p^{-\frac{1}{p}} q^{-\frac{1}{q}}\,\|T\| $$ for every $T\in \mathcal{L}(L_p(\mu))$ and that this inequality is the best possible wh...

We introduce the class of slicely countably determined Banach spaces which contains in particular all spaces with the Radon–Nikodým property and all spaces without copies of $ \ell_1$. We present many examples and several properties of this class. We give some applications to Banach spaces with the Daugavet and the alternative Daugavet properties,...

For any atomless positive measureμ, the space L
1(μ) has the polynomial Daugavet property, i.e., every weakly compact continuous polynomial P:L1(m)® L1(m){P:L_1(\mu)\longrightarrow L_1(\mu)} satisfies the Daugavet equation ||Id + P||=1 + ||P||{\|{\rm Id} + P\|=1 + \|P\|}. The same is true for the vector-valued spaces L
1(μ, E),μ atomless, E arbitra...