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Javier Meri

Publications and source records attributed to Javier Meri.

At least 19 recordsLinked to original sources
arXiv2023

A numerical range approach to Birkhoff-James orthogonality with applications

The main aim of this paper is to provide characterizations of Birkhoff-James orthogonality (BJ-orthogonality in short) in a number of families of Banach spaces in terms of the elements of significant subsets of the unit ball of their dual spaces, which makes the characterizations more applicable. The tool to do so is a fine study of the abstract numerical range and its relation with the BJ-orthogonality. Among other results, we provide a characterization of BJ-orthogonality for spaces of vector-valued bounded functions in terms of the domain set and the dual of the target space, which is applied to get results for spaces of vector-valued continuous functions, uniform algebras, Lipschitz maps, injective tensor products, bounded linear operators with respect to the operator norm and to the numerical radius, multilinear maps, and polynomials. Next, we study possible extensions of the well-known Bhatia-\v{S}emrl theorem on BJ-orthogonality of matrices, showing results in spaces of vector-valued continuous functions, compact linear operators on reflexive spaces, and finite Blaschke products. Finally, we find applications of our results to the study of spear vectors and spear operators. We show that no smooth point of a Banach space can be BJ-orthogonal to a spear vector of $Z$. As a consequence, if $X$ is a Banach space containing strongly exposed points and $Y$ is a smooth Banach space with dimension at least two, then there are no spear operators from $X$ to $Y$. Particularizing this result to the identity operator, we show that a smooth Banach space containing strongly exposed points has numerical index strictly smaller than one. These latter results partially solve some open problems.

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arXiv2023

Generating operators between Banach spaces

We introduce and study the notion of generating operators as those norm-one operators $G\colon X\longrightarrow Y$ such that for every $0<\delta<1$, the set $\{x\in X\colon \|x\|\leq 1,\ \|Gx\|>1-\delta\}$ generates the unit ball of $X$ by closed convex hull. This class of operators includes isometric embeddings, spear operators (actually, operators with the alternative Daugavet property), and other examples like the natural inclusions of $\ell_1$ into $c_0$ and of $L_\infty[0,1]$ into $L_1[0,1]$. We first present a characterization in terms of the adjoint operator, make a discussion on the behaviour of diagonal generating operators on $c_0$-, $\ell_1$-, and $\ell_\infty$-sums, and present examples in some classical Banach spaces. Even though rank-one generating operators always attain their norm, there are generating operators, even of rank-two, which do not attain their norm. We discuss when a Banach space can be the domain of a generating operator which does not attain its norm in terms of the behaviour of some spear sets of the dual space. Finally, we study when the set of all generating operators between two Banach spaces $X$ and $Y$ generates all non-expansive operators by closed convex hull. We show that this is the case when $X=L_1(\mu)$ and $Y$ has the Radon-Nikod\'ym property with respect to $\mu$. Therefore, when $X=\ell_1(\Gamma)$, this is the case for every target space $Y$. Conversely, we also show that a real finite-dimensional space $X$ satisfies that generating operators from $X$ to $Y$ generate all non-expansive operators by closed convex hull only in the case that $X$ is an $\ell_1$-space.

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arXiv2019

On the numerical index with respect to an operator

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 numerical indices with respect to all norm-one operators on $\mathcal{L}(X,Y)$. We show that $\mathcal{N}(\mathcal{L}(X,Y))=\{0\}$ when $X$ or $Y$ is a real Hilbert space of dimension greater than one and also when $X$ or $Y$ is the space of bounded or compact operators on an infinite-dimensional real Hilbert space. For complex Hilbert spaces $H_1$, $H_2$ of dimension greater than one, we show that $\mathcal{N}(\mathcal{L}(H_1,H_2))\subseteq \{0,1/2\}$ and the value $1/2$ is taken if and only if $H_1$ and $H_2$ are isometrically isomorphic. Besides, $\mathcal{N}(\mathcal{L}(X,H))\subseteq [0,1/2]$ and $\mathcal{N}(\mathcal{L}(H,Y))\subseteq [0,1/2]$ when $H$ is a complex infinite-dimensional Hilbert space and $X$ and $Y$ are arbitrary complex Banach spaces. We also show that $\mathcal{N}(\mathcal{L}(L_1(\mu_1),L_1(\mu_2)))\subseteq \{0,1\}$ and $\mathcal{N}(\mathcal{L}(L_\infty(\mu_1),L_\infty(\mu_2)))\subseteq \{0,1\}$ for arbitrary $\sigma$-finite measures $\mu_1$ and $\mu_2$, in both the real and the complex cases. Also, we show that the Lipschitz numerical range of Lipschitz maps can be viewed as the numerical range of convenient bounded linear operators with respect to a bounded linear operator. Further, we provide some results which show the behaviour of the value of the numerical index when we apply some Banach space operations, as constructing diagonal operators between $c_0$-, $\ell_1$-, or $\ell_\infty$-sums of Banach spaces, composition operators on some vector-valued function spaces, and taking the adjoint to an operator.

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arXiv2017

Spear operators between Banach spaces

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 weaker called the alternative Daugavet property (if rank-one operators $T$ satisfy the requirements), and one stronger called lushness, and we develop a complete theory about the relations between these three properties. To do this, the concepts of spear vector and spear set play an important role. Further, we provide with many examples among classical spaces, being one of them the lushness of the Fourier transform on $L_1$. We also study the relation of these properties with the Radon-Nikod\'ym property, with Asplund spaces, with the duality, and we provide some stability results. Further, we present some isometric and isomorphic consequences of these properties as, for instance, that $\ell_1$ is contained in the dual of the domain of every real operator with infinite rank and the alternative Daugavet property, and that these three concepts behave badly with smoothness and rotundity. Finally, we study Lipschitz spear operators (that is, those Lipschitz operators satisfying the Lipschitz version of the equation above) and prove that (linear) lush operators are Lipschitz spear operators.

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arXiv2016

On a second numerical index for Banach spaces

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, in many cases, the behaviour of this second numerical index differs from the one of the classical numerical index. As main results, we prove that Hilbert spaces have second numerical index one and that they are the only spaces with this property among the class of Banach spaces with one-unconditional basis and non-trivial Lie algebra. Besides, an application to the Bishop-Phelps-Bollob\'as property for numerical radius is given.

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arXiv2014

Two refinements of the Bishop-Phelps-Bollob\'as modulus

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, Bollob\'as proved the nowadays known as the Bishop-Phelps-Bollob\'as theorem, which allows to approximate at the same time a functional and a vector in which it almost attains the norm. Very recently, two Bishop-Phelps-Bollob\'as moduli of a Banach space have been introduced [J. Math. Anal. Appl. 412 (2014), 697--719] to measure, for a given Banach space, what is the best possible Bishop-Phelps-Bollob\'as theorem in this space. In this paper we present two refinements of the results of that paper. On the one hand, we get a sharp general estimation of the Bishop-Phelps-Bollob\'as modulus as a function of the norms of the point and the functional, and we also calculate it in some examples, including Hilbert spaces. On the other hand, we relate the modulus of uniform non-squareness with the Bishop-Phelps-Bollob\'as modulus obtaining, in particular, a simpler and quantitative proof of the fact that a uniformly non-square Banach space cannot have the maximum value of the Bishop-Phelps-Bollob\'as modulus.

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arXiv2011

Lushness, numerical index 1 and the Daugavet property in rearrangement invariant spaces

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 $[0,1]$ is $L_1[0,1]$; the same space is the only r.i.\ separable function space on $[0,1]$ with the Daugavet property over the reals.

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arXiv2010

On the numerical radius of operators in Lebesgue spaces

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 when the dimension of $L_p(\mu)$ is greater than one. We also give lower bounds for the best constant of equivalence between the numerical radius and the operator norm in $L_p(\mu)$ for atomless $\mu$ when restricting to rank-one operators or narrow operators.

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arXiv2009

On the numerical index of real $L_p(\mu)$-spaces

We give a lower bound for the numerical index of the real space $L_p(\mu)$ showing, in particular, that it is non-zero for $p\neq 2$. In other words, it is shown that for every bounded linear operator $T$ on the real space $L_p(\mu)$, one has $$ \sup{\Bigl|\int |x|^{p-1}\sign(x) T x d\mu \Bigr| : x\in L_p(\mu), \|x\|=1} \geq \frac{M_p}{12\e}\|T\| $$ where $M_p=\max_{t\in[0,1]}\frac{|t^{p-1}-t|}{1+t^p}>0$ for every $p\neq 2$. It is also shown that for every bounded linear operator $T$ on the real space $L_p(\mu)$, one has $$ \sup{\int |x|^{p-1}|Tx| d\mu : x\in L_p(\mu), \|x\|=1} \geq \frac{1}{2\e}\|T\|. $$

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arXiv2009

Slicely Countably Determined Banach spaces

We introduce the class of slicely countably determined Banach spaces which contains in particular all spaces with the RNP 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, lush spaces and Banach spaces with numerical index 1. In particular, we show that the dual of a real infinite-dimensional Banach with the alternative Daugavet property contains $\ell_1$ and that operators which do not fix copies of $\ell_1$ on a space with the alternative Daugavet property satisfy the alternative Daugavet equation.

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arXiv2009

Isometries on extremely non-complex Banach spaces

Given a separable Banach space $E$, we construct an extremely non-complex Banach space (i.e. a space satisfying that $\|Id + T^2\|=1+\|T^2\|$ for every bounded linear operator $T$ on it) whose dual contains $E^*$ as an $L$-summand. We also study surjective isometries on extremely non-complex Banach spaces and construct an example of a real Banach space whose group of surjective isometries reduces to $\pm Id$, but the group of surjective isometries of its dual contains the group of isometries of a separable infinite-dimensional Hilbert space as a subgroup.

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arXiv2008

Convexity and smoothness of Banach spaces with numerical index one

We show that a Banach space with numerical index one cannot enjoy good convexity or smoothness properties unless it is one-dimensional. For instance, it has no WLUR points in its unit ball, its norm is not Frechet smooth and its dual norm is neither smooth nor strictly convex. Actually, these results also hold if the space has the (strictly weaker) alternative Daugavet property. We construct a (non-complete) strictly convex predual of an infinite-dimensional $L_1$ space (which satisfies a property called lushness which implies numerical index~1). On the other hand, we show that a lush real Banach space is neither strictly convex nor smooth, unless it is one-dimensional. In particular, if a subspace $X$ of the real space $C[0,1]$ is smooth or strictly convex, then $C[0,1]/X$ contains a copy of $C[0,1]$. Finally, we prove that the dual of any lush infinite-dimensional real space contains a copy of $\ell_1$.

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arXiv2008

Extremely non-complex C(K) spaces

We show that there exist infinite-dimensional extremely non-complex Banach spaces, i.e. spaces $X$ such that the norm equality $\|Id + T^2\|=1 + \|T^2\|$ holds for every bounded linear operator $T:X\longrightarrow X$. This answers in the positive Question 4.11 of [Kadets, Martin, Meri, Norm equalities for operators, \emph{Indiana U. Math. J.} \textbf{56} (2007), 2385--2411]. More concretely, we show that this is the case of some $C(K)$ spaces with few operators constructed in [Koszmider, Banach spaces of continuous functions with few operators, \emph{Math. Ann.} \textbf{330} (2004), 151--183] and [Plebanek, A construction of a Banach space $C(K)$ with few operators, \emph{Topology Appl.} \textbf{143} (2004), 217--239]. We also construct compact spaces $K_1$ and $K_2$ such that $C(K_1)$ and $C(K_2)$ are extremely non-complex, $C(K_1)$ contains a complemented copy of $C(2^ω)$ and $C(K_2)$ contains a (1-complemented) isometric copy of $\ell_\infty$.

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arXiv2006

Norm equalities for operators

A Banach space $X$ has the Daugavet property if the Daugavet equation $\|\Id + T\|= 1 + \|T\|$ holds for every rank-one operator $T:X \longrightarrow X$. We show that the most natural attempts to introduce new properties by considering other norm equalities for operators (like $\|g(T)\|=f(\|T\|)$ for some functions $f$ and $g$) lead in fact to the Daugavet property of the space. On the other hand there are equations (for example $\|\Id + T\|= \|\Id - T\|$) that lead to new, strictly weaker properties of Banach spaces.

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