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Pilar Rueda

Publications and source records attributed to Pilar Rueda.

17 recordsLinked to original sources
arXiv2023

Banach lattices of homogeneous polynomials not containing $c_0$

First we develop a technique to construct Banach lattices of homogeneous polynomials. We obtain, in particular, conditions for the linear spans of all positive compact and weakly compact $n$-homogeneous polynomials between the Banach lattices $E$ and $F$, denoted by ${\cal P}_{\cal K}^r(^n E; F)$ and $\mathcal{P}_{\mathcal{W}}^r(^n E; F)$, to be Banach lattices with the polynomial regular norm. Next we study when the following are equivalent for ${\cal I} = {\cal K}$ or ${\cal I} = {\cal W}$: (1) The space $\mathcal{P}^r(^n E; F)$ of regular polynomials contains no copy of $c_0$. (2) ${\cal P}_{\mathcal{I}}^r(^n E; F)$ contains no copy of $c_0$. (3) ${\cal P}_{\mathcal{I}}^r(^n E; F)$ is a projection band in $\mathcal{P}^r(^n E; F)$. (4) Every positive polynomial in $\mathcal{P}^r(^n E; F)$ belongs to ${\cal P}_{\cal I}^r(^nE;F)$. The result we obtain in the compact case can be regarded as a lattice polynomial Kalton theorem. Most of our results and examples are new even in the linear case $n = 1$.

math.FA↗
arXiv2016

Some applications of the Hölder inequality for mixed sums

We use the Hölder inequality for mixed exponents to prove some optimal variants of the generalized Hardy--Littlewood inequality for $m$-linear forms on $\ell _{p}$ spaces with mixed exponents. Our results extend recent results of Araujo et al.

math.FA↗
arXiv2016

The approximation property and Lipschitz mappings on Banach spaces

We present an overview to the approximation property, paying especial attention to the recent results relating the approximation property to ideals of linear operators and Lipschitz ideals. We complete the paper with some new results on approximation of Lipschitz mappings and their relation to linear operator ideals.

math.FA↗
arXiv2016

Some inclusion results for interpolated summing operator ideals and integrability improvement of vector valued functions

Consider a Banach space valued measurable function $f$ and an operator $u$ from the space where {$f$} takes values. If $f $ is Pettis integrable, a classical result due to J. Diestel shows that composing it with $u$ gives a Bochner integrable function $u \circ f$ whenever $u$ is absolutely summing. In a previous work we have shown that a well-known interpolation technique for operator ideals allows to prove under some requirements that a composition of a $p$-Pettis integrable function with a $q$-summing operator provides an $r$-Bochner integrable function. In this paper a new abstract inclusion theorem for classes of {abstract} summing operators is shown and applied to the class of interpolated operator ideals. Together with the results of the {aforementioned} paper, it provides more results on the relation about the integrability of the function $u \circ f$ and the summability properties of $u$.

math.FA↗
arXiv2016

Summability of multilinear forms on classical sequence spaces

We present an extension of the Hardy--Littlewood inequality for multilinear forms. More precisely, let $\mathbb{K}$ be the real or complex scalar field and $m,k$ be positive integers with $m\geq k\,$ and $n_{1},\dots ,n_{k}$ be positive integers such that $n_{1}+\cdots +n_{k}=m$. ($a$) If $(r,p)\in (0,\infty )\times \lbrack 2m,\infty ]$ then there is a constant $D_{m,r,p,k}^{\mathbb{K}}\geq 1$ (not depending on $n$) such that $$ \left( \sum_{i_{1},\dots ,i_{k}=1}^{n}\left| T\left( e_{i_{1}}^{n_{1}},\dots ,e_{i_{k}}^{n_{k}}\right) \right| ^{r}\right) ^{% \frac{1}{r}}\leq D_{m,r,p,k}^{\mathbb{K}} \cdot n^{max\left\{ \frac{% 2kp-kpr-pr+2rm}{2pr},0\right\} }\left| T\right| $$ for all $m$-linear forms $T:\ell_{p}^{n}\times \cdots \times \ell_{p}^{n}\rightarrow \mathbb{K}$ and all positive integers $n$. Moreover, the exponent $max\left\{ \frac{2kp-kpr-pr+2rm}{2pr},0\right\} $ is optimal. ($b$) If $(r, p) \in (0, \infty) \times (m, 2m]$ then there is a constant $% D_{m,r,p, k}^{\mathbb{K}}\geq 1$ (not depending on $n$) such that $$ \left( \sum_{i_{1},\dots ,i_{k}=1}^{n }\left| T\left( e_{i_{1}}^{n_{1}},\dots ,e_{i_{k}}^{n_{k}}\right) \right| ^{r }\right) ^{% \frac{1}{r }}\leq D_{m,r,p, k}^{\mathbb{K}} \cdot n^{ max \left\{\frac{% p-rp+rm}{pr}, 0\right\}}\left| T\right| $$ for all $m$-linear forms $T:\ell_{p}^{n}\times \cdots \times \ell_{p}^{n}\rightarrow \mathbb{K}$ and all positive integers $n$. Moreover, the exponent $max \left\{\frac{p-rp+rm}{pr}, 0\right\}$ is optimal. The case $k=m$ recovers a recent result due to G. Araujo and D. Pellegrino.

math.FA↗
arXiv2015

Surjectivity of isometries of weighted spaces of holomorphic functions and of Bloch spaces

We examine the surjectivity of isometries between weighted spaces of holomorphic functions. We show that for certain classical weights on the open unit disc all isometries of the weighted space of holomorphic functions, ${ \mathcal H}_{v_o}( Δ)$, are surjective. Criteria for surjectivity of isometries of ${ \mathcal H}_v(U)$ in terms of a separation condition on points in the image of ${ \mathcal H}_{v_o}(U)$ are also given for $U$ a bounded open set in $\mathbb{C}$. Considering the weight $v(z)= 1-|z|^2$ and the isomorphism $f\mapsto f'$ we are able to show that all isometries of the little Bloch space are surjective.

math.FA↗
arXiv2015

Improving integrability via absolute summability: a general version of Diestel's Theorem

A classical result by J. Diestel establishes that the composition of a summing operator with a (strongly measurable) Pettis integrable function gives a Bochner integrable function. In this paper we show that a much more general result is possible regarding the improvement of the integrability of vector valued functions by the summability of the operator. After proving a general result, we center our attention in the particular case given by the $(p,σ)$-absolutely continuous operators, that allows to prove a lot of special results on integration improvement for selected cases of classical Banach spaces ---including $C(K)$, $L^p$ and Hilbert spaces--- and operators ---$p$-summing, $(q,p)$-summing and $p$-approximable operators---.

math.FA↗
arXiv2015

s-Numbers sequences for homogeneous polynomials

We extend the well known theory of $s$-numbers of linear operators to homogeneous polynomials defined between Banach spaces. Approximation, Kolmogorov and Gelfand numbers of polynomials are introduced and some well-known results of the linear and multilinear settings are obtained for homogeneous polynomials.

math.FA↗
arXiv2014

Coincidence results for summing multilinear mappings

In this paper we prove coincidence results concerning spaces of absolutely summing multilinear mappings between Banach spaces. The nature of these results arises from two distinct approaches: the coincidence of two \textit{a priori} different classes of summing multilinear mappings and the summability of all multilinear mappings defined on products of Banach spaces. Optimal generalizations of known results are obtained. We also introduce and explore new techniques in the field, for example a technique to extend coincidence results for linear, bilinear and even trilinear mappings to general multilinear ones.

math.FA↗
arXiv2013

Weak compactness and strongly summing multilinear operators

Every absolutely summing linear operator is weakly compact. However, for strongly summing multilinear operators and polynomials - one of the most natural extensions of the linear case to the non linear framework - weak compactness does not hold in general. We show that a subclass of the class of strongly summing multilinear operators/polynomials, sharing its main properties such as Grothendieck's Theorem, Pietsch Domination Theorem and Dvoretzky-Rogers Theorem, has even better properties like weak compactness and a natural factorization theorem.

math.FA↗
arXiv2012

On Pietsch measures for summing operators and dominated polynomials

We relate the injectivity of the canonical map from $C(B_{E'})$ to $L_p(μ)$, where $μ$ is a regular Borel probability measure on the closed unit ball $B_{E'}$ of the dual $E'$ of a Banach space $E$ endowed with the weak* topology, to the existence of injective $p$-summing linear operators/$p$-dominated homogeneous polynomials defined on $E$ having $μ$ as a Pietsch measure. As an application we fill the gap in the proofs of some results of concerning Pietsch-type factorization of dominated polynomials.

math.FA↗
arXiv2009

Cotype and absolutely summing linear operators

Cotype is used in this paper to prove new results concerning the existence of non-absolutely summing linear operators between Banach spaces. We derive consequences that extend/generalize/ complement some classic results. We also point out that some of our results are sharp.

math.FA↗
arXiv2008

Factorization theorems for dominated polynomials

In this note we prove that the factorization theorem for dominated polynomials previously proved by the authors is equivalent to an alternative factorization scheme that uses classical linear techniques and a linearization process. However, this alternative scheme is shown not to be satisfactory until the equivalence is proved.

math.FA↗
arXiv2008

A unified Pietsch domination theorem

In this paper we prove an abstract version of Pietsch's domination theorem which unify a number of known Pietsch-type domination theorems for classes of mappings that generalize the ideal of absolutely p-summing linear operators. A final result shows that Pietsch-type dominations are totally free from algebraic conditions, such as linearity, multilinearity, etc.

math.FA↗
arXiv2008

Summability of multilinear mappings: Littlewood, Orlicz and beyond

In this paper we prove a plenty of new results concerning summabililty properties of multilinear mappings between Banach spaces, such as an extension of Littlewood's 4/3 Theorem. Among other features, it is shown that every continuous n-linear form on the disc algebra or the Hardy space is (1;2,...,2)-summing, the role of the Littlewood-Orlicz property in the theory is established and the interplay with almost summing multilinear mappings is explored.

math.FA↗