arXiv subjects

Pedro Tradacete

Publications and source records attributed to Pedro Tradacete.

At least 55 records · Page 3Linked to original sources

arXiv2019

Daugavet property in tensor product spaces

We study the Daugavet property in tensor products of Banach spaces. We show that $L_1(μ)\widehat{\otimes}_\varepsilon L_1(ν)$ has the Daugavet property when $μ$ and $ν$ are purely non-atomic measures. Also, we show that $X\widehat{\otimes}_πY$ has the Daugavet property provided $X$ and $Y$ are $L_1$-preduals with the Daugavet property, in particular spaces of continuous functions with this property. With the same tecniques, we also obtain consequences about roughness in projective tensor products as well as the Daugavet property of projective symmetric tensor products.

math.FA↗

arXiv2019

The least doubling constant of a metric measure space

We study the least doubling constant $C_{(X,d)}$, among all doubling measures $μ$ supported on a metric space $(X,d)$. In particular, we prove that for every metric space with more than one point, $C_{(X,d)}\ge 2$. We also describe some further properties of $C_{(X,d)}$ and compute its value for several important examples.

math.CA↗

arXiv2018

Random unconditional convergence of vector-valued Dirichlet series

We study random unconditionality of Dirichlet series in vector-valued Hardy spaces $\mathcal H_p(X)$. It is shown that a Banach space $X$ has type 2 (respectively, cotype 2) if and only if for every choice $(x_n)_n\subset X$ it follows that $(x_n n^{-s})_n$ is Random unconditionally convergent (respectively, divergent) in $\mathcal H_2(X)$. The analogous question on $\mathcal H_p(X)$ spaces for $p\neq2$ is also explored. We also provide explicit examples exhibiting the differences between the unconditionality of $(x_n n^{-s})_n$ in $\mathcal H_p(X)$ and that of $(x_n z^n)_n$ in $H_p(X)$.

math.FA↗

arXiv2018

The free Banach lattices generated by $\ell_p$ and $c_0$

We prove that, when $2<p<\infty$, in the free Banach lattice generated by $\ell_p$ (respectively by $c_0$), the absolute values of the canonical basis form an $\ell_r$-sequence, where $\frac{1}{r} = \frac{1}{2} + \frac{1}{p}$ (respectively an $\ell_2$-sequence). In particular, in any Banach lattice, the absolute values of any $\ell_p$ sequence always have an upper $\ell_r$-estimate. Quite surprisingly, this implies that the free Banach lattices generated by the nonseparable $\ell_p(Γ)$ for $2<p<\infty$, as well as $c_0(Γ)$, are weakly compactly generated whereas this is not the case for $1\leq p\leq 2$.

math.FA↗

arXiv2018

$p$-regularity and weights for operators between $L^p$-spaces

We explore the connection between $p$-regular operators on Banach function spaces and weighted $p$-estimates. In particular, our results focus on the following problem. Given finite measure spaces $μ$ and $ν$, let $T$ be an operator defined from a Banach function space $X(ν)$ and taking values on $L^p (v d μ)$ for $v$ in certain family of weights $V\subset L^1(μ)_+$: we analyze the existence of a bounded family of weights $W\subset L^1(ν)_+$ such that for every $v\in V$ there is $w \in W$ in such a way that $T:L^p(w d ν) \to L^p(v d μ)$ is continuous uniformly on $V$. A condition for the existence of such a family is given in terms of $p$-regularity of the integration map associated to a certain vector measure induced by the operator $T$.

math.FA↗

arXiv2017

Valuations on Banach lattices

We provide a general framework for the study of valuations on Banach lattices. This complements and expands several recent works about valuations on function spaces, including $L_p(μ)$, Orlicz spaces and spaces $C(K)$ of continuous functions on a compact Hausdorff space. In particular, we study decomposition properties, boundedness and integral representation of continuous valuations.

math.FA↗

arXiv2017

Continuity and representation of valuations on star bodies

It is shown that every continuous valuation defined on the $n$-dimensional star bodies has an integral representation in terms of the radial function. Our argument is based on the non-trivial fact that continuous valuations are uniformly continuous on bounded sets. We also characterize the continuous valuations on the $n$-dimensional star bodies that arise as the restriction of a measure on $\mathbb R^n$.

math.MG↗

arXiv2017

$(p,q)$-regular operators between Banach lattices

We study the class of $(p,q)$-regular operators between quasi-Banach lattices. In particular, a representation of this class as the dual of a certain tensor norm for Banach lattices is given. We also provide some factorization results for $(p,q)$-regular operators yielding new Marcinkiewicz-Zygmund type inequalities for Banach function spaces. An extension theorem for $(q, \infty)$-regular operators defined on a subspace of $L_q$ is also given.

math.FA↗

arXiv2017

Calderón-Lozanovskii interpolation on quasi-Banach lattices

We consider the Calderón-Lozanovskii construction $φ(X_0,X_1)$ in the context of quasi-Banach lattices and provide an extension of a result by V. I. Ovchinnikov concerning the associated interpolation methods $φ^c$ and $φ^0$. Our approach is based on the interpolation properties of $(\infty,1)$-regular operators between quasi-Banach lattices.

math.FA↗

arXiv2017

Interpolation and extrapolation of strictly singular operators between $L_p$ spaces

We study the interpolation and extrapolation properties of strictly singular operators between different $L_p$ spaces. To this end, the structure of strictly singular non-compact operators between $L_p-L_q$ spaces is analyzed. Among other things, we clarify the relation between strict singularity and the $L$-characteristic set of an operator. In particular, Krasnoselskii's interpolation theorem for compact operators is extended to the class of strictly singular operators.

math.FA↗

arXiv2017

The free Banach lattice generated by a Banach space

The free Banach lattice over a Banach space is introduced and analyzed. This generalizes the concept of free Banach lattice over a set of generators, and allows us to study the Nakano property and the density character of non-degenerate intervals on these spaces, answering some recent questions of B. de Pagter and A.W. Wickstead. Moreover, an example of a Banach lattice which is weakly compactly generated as a lattice but not as a Banach space is exhibited, thus answering a question of J. Diestel.

math.FA↗

arXiv2016

A Jordan-like decomposition theorem for valuations on star bodies

We show that every radial continuous valuation $V:\mathcal S_0^n\rightarrow \mathbb R$ defined on the $n$-dimensional star bodies $\mathcal S_0^n$, and verifying $V(\{0\})=0$, can be decomposed as a sum $V=V^+-V^-$, where both $V^+$ and $V^-$ are positive radial continuous valuations on $\mathcal S_0^n$ with $V^+(\{0\})=V^-(\{0\})=0$. As an application, we show that radial continuous rotationally invariant valuations $V$ on $\mathcal S_0^n$ can be characterized as the applications on star bodies which can be written as $$V(K)=\int_{S^{n-1}}θ(ρ_K)dm,$$ where $θ:[0,\infty)\rightarrow \mathbb R$ is a continuous function, $ρ_K$ is the radial function associated to $K$ and $m$ is the Lebesgue measure on $S^{n-1}$. This completes recent work of the second named author, where an analogous result is proved for the case of {\em positive} radial continuous rotationally invariant valuations.

math.MG↗

arXiv2016

Radial continuous valuations on star bodies and star sets

We show that a radial continuous valuation defined on the $n$-dimensional star bodies extends uniquely to a continuous valuation on the $n$-dimensional bounded star sets. Moreover, we provide an integral representation of every such valuation, in terms of the radial function, which is valid on the dense subset of the simple Borel star sets. We also show that every radial continuous valuation defined on the $n$-dimensional star bodies can be decomposed as a sum $V=V^+-V^-$, where both $V^+$ and $V^-$ are positive radial continuous valuations.

math.MG↗

arXiv2016

Almost band preservers

We study the stability of band preserving operators on Banach lattices. To this end the notion of $\varepsilon$-band preserving mapping is introduced. It is shown that, under quite general assumptions, a $\varepsilon$-band preserving operator is in fact a small perturbation of a band preserving one. However, a counterexample can be produced in some circumstances. Some results on automatic continuity of $\varepsilon$-band preserving maps are also obtained.

math.FA↗

arXiv2016

Geometric properties of infinite graphs and the Hardy-Littlewood maximal operator

We study different geometric properties on infinite graphs, related to the weak-type boundedness of the Hardy-Littlewood maximal averaging operator. In particular, we analyze the connections between the doubling condition, having finite dilation and overlapping indices, uniformly bounded degree, the equidistant comparison property and the weak-type boundedness of the centered Hardy-Littlewood maximal operator. Several non-trivial examples of infinite graphs are given to illustrate the differences among these properties.

math.CA↗

arXiv2015

Weakly compactly generated Banach lattices

We study the different ways in which a weakly compact set can generate a Banach lattice. Among other things, it is shown that in an order continuous Banach lattice $X$, the existence of a weakly compact set $K \subset X$ such that $X$ coincides with the band generated by $K$, implies that $X$ is WCG.

math.FA↗

arXiv2015

Disjointly homogeneous Banach lattices and applications

This is a survey on disjointly homogeneous Banach lattices and their applicactions. Several structural properties of this class are analyzed. In addition we show how these spaces provide a natural framework for studying the compactness of powers of operators allowing for a unified treatment of well-known results.

math.FA↗

arXiv2015

Almost disjointness preservers

We study the stability of disjointness preservers on Banach lattices. In many cases, we prove that an "almost disjointness preserving" operator is well approximable by a disjointess preserving one. However, this approximation is not always possible, as our examples show.

math.FA↗