Daniel Pellegrino
In this note we show that the well known coincidence results for scalar-valued homogeneous polynomials can not be generalized in some natural directions.
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In this paper we introduce an abstract approach to the notion of absolutely summing multilinear operators. We show that several previous results on different contexts (absolutely summing, almost summing, Cohen summing) are particular cases of our general results.
In this paper, we develop the theory of absolutely summing multipolynomials. Among other results, we generalize and unify previous works of G. Botelho and D. Pellegrino concerning absolutely summing polynomials/multilinear mappings in Banach spaces with unconditional Schauder basis.
In this short note we present some new results concerning cotype and absolutely summing multilinear operators.
We study the ideals of linear operators between Banach spaces determined by the transformation of vector-valued sequences involving the new sequence space introduced by Karn and Sinha \cite{karnsinha} and the classical spaces of absolutely, weakly and Cohen strongly summable sequences. As applications, we prove a new factorization theorem for absolutely summing operators and a contribution to the existence of infinite dimensional spaces formed by non-absolutely summing operators is given.
In this paper we investigate the connections between the several different extensions of the concept of absolutely summing operators.
We prove new summability properties for multilinear operators on $\ell_p$ spaces. An important tool for this task is a better understanding of the interplay between almost summing and absolutely summing multilinear operators.
In this note we obtain new coincidence theorems for absolutely summing multilinear mappings between Banach spaces. We also prove that our results, in general, can not be improved.
This paper has a twofold purpose: to present an overview of the theory of absolutely summing operators and its different generalizations for the multilinear setting, and to sketch the beginning of a research project related to an objective search of \textquotedblleft perfect\textquotedblright \ multilinear extensions of the ideal of absolutely summing operators. The final section contains some open problems that may indicate lines for future investigation.
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.
A famous result due to Grothendieck asserts that every continuous linear operator from $\ell_{1}$ to $\ell_{2}$ is absolutely $(1,1)$-summing. If $n\geq2,$ however, it is very simple to prove that every continuous $n$-linear operator from $\ell_{1}\times...\times\ell_{1}$ to $\ell_{2}$ is absolutely $(1;1,...,1) $-summing, and even absolutely $(\frac{2}% {n};1,...,1) $-summing$.$ In this note we deal with the following problem: Given a positive integer $n\geq2$, what is the best constant $g_{n}>0$ so that every $n$-linear operator from $\ell_{1}\times...\times\ell_{1}$ to $\ell_{2}$ is absolut…
In this note we solve, except for extremely pathological cases, a question posed by Puglisi and Seoane-Sepulveda on the lineability of the set of bounded non-absolutely summing linear operators. We also show how the idea of the proof can be adapted to several related situations.
In this note we prove the optimality of a family of known coincidence theorems for absolutely summing multilinear operators. We connect our results with the theory of multiple summing multilinear operators and prove the sharpness of similar results obtained via the complex interpolation method.
In this paper we study absolutely summing mappings on Banach spaces by exploring the cotype of their domains and ranges. It is proved that every $n$% -linear mapping from $\mathcal{L}_{\infty}$-spaces into $\mathbb{K}$ is $% (2;2,...,2,\infty)$-summing and also shown that every $n$-linear mapping from $\mathcal{L}_{\infty}$-spaces into $F$ is $(q;2,...,2)$-summing whenever $F$ has cotype $q.$ We also give new examples of analytic summing mappings and polynomial and multilinear versions of a linear Extrapolation Theorem.
In the short note we prove that for every $0<p<1$, there exists an infinite dimensional closed linear subspace of $\mathcal{L}\left( \ell_{p};\ell_{p}\right) $ every nonzero element of which is non $(r,s)$-absolutely summing operator for the real numbers $r,s$ with $1\leq s\leq r<\infty$. This improve a result obtained in \cite{DanielT}.
Grothendieck's theorem asserts that every continuous linear operator from $\ell_{1}$ to $\ell_{2}$ is absolutely $\left( 1;1\right) $-summing. In this note we prove that the optimal constant $g_{m}$ so that every continuous $m$-linear operator from $\ell_{1}\times\cdots\times\ell_{1}$ to $\ell_{2}$ is absolutely $\left( g_{m};1\right) $-summing is $\frac{2}{m+1}$. We also show that if $g_{m}<\frac{2}{m+1}$ there is $\mathfrak{c}$ dimensional linear space composed by continuous non absolutely $\left( g_{m};1\right) $-summing $m$-linear operators from $\ell_{1}\times\cdots\times\ell_{1}$ to $\el…
In this paper, we construct the abstract ideal of polynomials. We show this is an ideal of Banach and, in a second moment, we explore the question of the coherence and compatibility of the pair composed by the abstract ideals of polynomials and multilinear applications.
In this note we prove a general version of the Extrapolation Theorem, extending the classical linear extrapolation theorem due to B. Maurey. Our result shows, in particular, that the operators involved do not need to be linear.
This short note has a twofold purpose: (i) to solve the question that motivates a recent paper of D. Popa on multilinear variants of Pietsch's composition theorem for absolutely summing operators. More precisely, we remark that there is a natural perfect extension of Pietsch's composition theorem to the multilinear and polynomial settings. This fact was overlooked in the aforementioned paper; (ii) to investigate extensions of some results of the aforementioned paper for particular situations, mainly by exploring cotype properties of the spaces involved.
It is shown that the ultimate version of the nonlinear Pietsch Domination Theorem remains true, even in a stronger presentation, if one of its hypotheses is removed. More precisely, we show that no trace of (sub)-homogeneity assumption is needed. Some consequences are also presented.
Given an infinite-dimensional Banach space $X$ and a Banach space $Y$ with no finite cotype, we determine whether or not every continuous linear operator from $X$ to $Y$ is absolutely $(q;p)$-summing for almost all choices of $p$ and $q$, including the case $p=q$. If $X$ assumes its cotype, the problem is solved for all choices of $p$ and $q$. Applications to the theory of dominated multilinear mappings are also provided.
Absolutely Summing Operators on non commutative $C^*$-algebras and applications
设$E$是一个不含$\ell^1$任何拷贝的Banach空间，$\A$是一个非交换的$C^*$ -代数.我们证明了从$\A$到$E^*$的每一个绝对和算子都是紧的，从而回答了Petrizczynski的一个问题.作为应用，我们证明了：如果$G$是紧可度量化阿贝尔群，$Λ$是它的对偶的Riesz子集，则每一个可数可加的有界变差的$\A^*$值测度，其傅里叶变换由$Λ$支持，都有相对紧的值域.同样的结果也推广到对称空间的可测算子。
We consider the Carleson embeddings of the classical Hardy spaces (on the disk) into a L p ($μ$) space, where $μ$ is a Carleson measure on the unit disk. This includes the case of composition operators. We characterize such operators which are r-summing on H p , where p \textgreater{} 1 and r $\ge$ 1. This completely extends the former results on the subject and solves a problem open since the early seventies. Mathematics Subject Classification. Primary: 47B33 -- Secondary: 28A12; 30C85; 31A15; 46E20; 46E22; 47B06
Absolutely Summing Toeplitz operators on Bergman spaces in the unit ball of $\mathbb{C}^n$
本文研究了复数n维单位球上Bergman空间中Toeplitz算子的绝对可和性，并给出了其范数等价条件，同时扩展了Carleson嵌入的相关结果。
Motivated from the theory of Hilbert-Schmidt morphisms between Hilbert C*-modules over commutative C*-algebras by Stern and van Suijlekom \textit{[J. Funct. Anal., 2021]}, we introduce the notion of p-absolutely summing morphisms between Hilbert C*-modules over commutative C*-algebras. We show that an adjointable morphism between Hilbert C*-modules over monotone closed commutative C*-algebra is 2-absolutely summing if and only if it is Hilbert-Schmidt. We formulate version of Pietsch factorization problem for p-absolutely summing morphisms and solve partially
In the last decades many authors have become interested in the study of multilinear and polynomial generalizations of families of operator ideals (such as, for instance, the ideal of absolutely summing operators). However, these generalizations must keep the essence of the given operator ideal and there seems not to be a universal method to achieve this. The main task of this paper is to discuss, study, and introduce multilinear and polynomial extensions of the aforementioned operator ideals taking into account the already existing methods of evaluating the adequacy of such generalizations. Be…
Absolutely summing Toeplitz operators on weighted Fock spaces
本文研究加权Fock空间上复符号Toeplitz算子的绝对r-求和性，通过IDA分解和Rademacher论证等工具，给出了基于局部均值、复球平均和Berezin变换的等价刻画，并证明了准则的最优性。
We use an interpolative technique from \cite{abps} to introduce the notion of multiple $N$-separately summing operators. Our approach extends and unifies some recent results; for instance we recover the best known estimates of the multilinear Bohnenblust-Hille constants due to F. Bayart, D. Pellegrino and J. Seoane-Sepúlveda. More precisely, as a consequence of our main result, for $1\leq t<2$ and $m\in \mathbb{N}$ we prove that $$ \left(\sum_{i_{1},\dots,i_{m}=1}^{\infty}\left\vert U\left(e_{i_{1}},\dots,e_{i_{m}}\right) \right\vert^{\frac{2tm}{2+(m-1)t}}\right)^{\frac{2+(m-1)t}{2tm}} \leq \l…
For $f \in H^p(δ^2)$, $0<p\leq 2$, with Haar expansion $f=\sum f_{I \times J}h_{I\times J}$ we constructively determine the Pietsch measure of the $2$-summing multiplication operator \[\mathcal{M}_f:\ell^{\infty} \rightarrow H^p(δ^2), \quad (φ_{I\times J}) \mapsto \sum φ_{I\times J}f_{I \times J}h_{I \times J}. \] Our method yields a constructive proof of Pisier's decomposition of $f \in H^p(δ^2)$ \[|f|=|x|^{1-θ}|y|^θ\quad\quad \text{ and }\quad\quad \|x\|_{X_0}^{1-θ}\|y\|^θ_{H^2(δ^2)}\leq C\|f\|_{H^p(δ^2)}, \] where $X_0$ is Pisier's extrapolation lattice associated to $H^p(δ^2)$ and $H^2(δ^2…
It is well known that not every summability property for non linear operators leads to a factorization theorem. In this paper we undertake a detailed study of factorization schemes for summing linear and nonlinear operators. Our aim is to integrate under the same theory a wide family of classes of mappings for which a Pietsch type factorization theorem holds. We analyze the class of linear operators that are defined by a summability inequality involving a homogeneous map. Our construction includes the cases of absolutely $p$-summing linear operators, $(p,σ)$-absolutely continuous linear operat…
Generalizing classical results of the theory of absolutely summing operators, in this paper we characterize the duals of a quite large class of Banach operator ideals defined or characterized by the transformation of vector-valued sequences.
A famous theorem due to Maurey and Pisier asserts that for an infinite dimensional Banach space $E$, the infumum of the $q$ such that the identity map $id_{E}$ is absolutely $\left( q,1\right) $-summing is precisely $\cot E$. In the same direction, the Dvoretzky--Rogers Theorem asserts $id_{E}$ fails to be absolutely $\left( p,p\right) $-summing, for all $p\geq1$. In this note, among other results, we unify both theorems by charactering the parameters $q$ and $p$ for which the identity map is absolutely $\left( q,p\right)$-summing. We also provide a result that we call \textit{strings of coinc…
In this paper we obtain new inclusion and coincidence theorems for absolutely or multiple summing multilinear mappings. In particular, we derive optimal coincidence theorems of Bohnenblust-Hille type for multilinear forms on K-convex Banach spaces of cotype 2.
Absolutely summing Carleson embeddings on weighted Fock spaces with $A_{\infty}$-type weights
本文通过多种技术完全刻画了加权Fock空间上Carleson嵌入的$r$-可和性，并应用于算子有界性，解决了$1<p<2$时的公开问题。
Kwapień's theorem asserts that every continuous linear operator from $\ell_{1}$ to $\ell_{p}$ is absolutely $\left( r,1\right) $-summing for $1/r=1-\left\vert 1/p-1/2\right\vert .$ When $p=2$ it recovers the famous Grothendieck's theorem. In this paper investigate multilinear variants of these theorems and related issues. Among other results we present a unified version of Kwapień's and Grothendieck's results that encompasses the cases of multiple summing and absolutely summing multilinear operators.
Absolutely summing Hankel operators on Fock spaces and the Berger-Coburn phenomenon
本文研究了Fock空间上Hankel算子的$r$-summing性质，揭示了其与符号函数$\mathrm{IDA}^{κ, p}$范数的等价关系，并探讨了Berger-Coburn现象。
Absolutely summing operators on Bergman spaces over convex domains of finite type
本文在有限型凸域的 Bergman 空间上刻画了绝对求和嵌入与Hankel算子，给出上下范数估计、显式逼近误差及Toeplitz谱渐近，并应用于复合与Volterra算子。
Noncommutative vector valued $L_p$-spaces and completely $p$-summing maps
设$E$是Blecher-Paulsen和Effros-Ruan最近发展的理论意义下的算子空间.我们引入了一个概念的$E$值的非交换$L_p$ -空间的$1 \leq p < \infty$，我们证明了由此产生的算子空间满足自然属性，例如对偶和插值。这一概念导致了“完全p-和”映射的定义，它是Pietsch-Kwapiesten意义下的$p$ -绝对和映射的算子空间模拟。这些概念扩展了Effros-Ruan先前研究的特殊情况$p=1$。
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.
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 inclusio…
Gel'fand Integration of B(E, F*)-Valued Functions With Emphasis on (q, p)-Summing Operators
本文推广了Gel'fand积分在Banach空间上的结果，探讨了(q, p)-求和算子的充分条件，并应用新方法回答了相关问题，还给出了正算子值函数的应用。
A famous result of S. Kwapień asserts that a linear operator from a Banach space to a Hilbert space is absolutely $1$-summing whenever its adjoint is absolutely $q$-summing for some $1\leq q<\infty$; this result was recently extended to Lipschitz operators by Chen and Zheng. In the present paper we show that Kwapień's and Chen--Zheng theorems hold in a very relaxed nonlinear environment, under weaker hypotheses. Even when restricted to the original linear case, our result generalizes Kwapień's theorem because it holds when the adjoint is just almost summing. In addition, a variant for $\mathca…
In this note we prove an abstract version of a result from 2002 due to Delgado and Piñero on absolutely summing operators. Several applications are presented; some of them in the multilinear framework and some in a completely nonlinear setting. In a final section we investigate the size of the set of non uniformly dominated sets of linear operators under the point of view of lineability.
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 mul…
Grothendieck's theorem asserts that every continuous linear operator from $\ell_1$ to $\ell_2$ is absolutely $(1,1)$-summing. This kind of result is commonly called coincidence result. In this paper we investigate coincidence results in the multilinear setting, showing how the cotype of the spaces involved affect such results. The special role played by $\ell_1$ spaces is also investigated with relation to interpolation of tensor products. In particular, an open problem on the interpolation of $m$ injective tensor products is solved.
In this paper we give a description of separating or disjointness preserving linear bijections on spaces of vector-valued absolutely continuous functions defined on compact subsets of the real line. We obtain that they are continuous and biseparating in the finite-dimensional case. The infinite-dimensional case is also studied.
Absolutely Summing Toeplitz operators on Fock spaces
本文在带一般权函数的 Fock 空间上，完整刻画了使 Toeplitz 算子为 $r$-求和（$r\ge1$）的正 Borel 测度条件。
In this note we answer a question concerning lineability of the set of non-absolutely summing operators.
In this paper we introduce a new approach to the concept of multipolynomials and generalize several results of the homogeneous polynomials and symmetric multilinear applications. We also present an abstract approach to the concept of absolutely summing multipolynomials
We study the reduced Beurling spectra $sp_{\Cal {A},V} (F)$ of functions $F \in L^1_{loc} (\jj,X)$ relative to certain function spaces $\Cal{A}\st L^{\infty}(\jj,X)$ and $V\st L^1 (\r)$, where $\jj$ is $\r_+$ or $\r$ and $X$ is a Banach space. We show that if $F$ is bounded or slowly oscillating on $\jj$ with $0\not\in sp_{\A,\f} (F)$, where $\A$ is $\{0\}$ or $C_0 (\jj,X)$ for example and $\f=\f(\r)$, then $F$ is ergodic. This result is new even for $F\in BUC(\jj,X)$ and $\A= C_0(\jj,X)$. If $F$ is ergodic and belongs to the space $ \f'_{ar}(\jj,X)$ of absolutely regular distributions and if …
We prove that every Lipschitz function from a subset of a locally compact length space to a metric tree has a unique absolutely minimal Lipschitz extension (AMLE). We relate these extensions to a stochastic game called {\bf Politics} - a generalization of a game called {\bf Tug of War} that has been used in \cite{PSSW09} to study real-valued AMLEs.
Almost sure linear independence of absolutely continuous Hilbert space-valued random vectors with respect to a special class of Hilbert space probability measures
本文研究无限维 Hilbert 空间中随机向量的线性无关性，证明在有限维子空间相对概率测度可忽略时，绝对连续分布可保证无需约束张成空间的随机维数。
Suppose $Π_1(E, F)$ is the space of all absolutely 1-summing operators between two Banach spaces $E$ and $F$. We show that if $F$ has a copy of $c_0$, then $Π_1(E, F)$ will have a copy of $c_0$, and under some conditions if $E$ has a copy of $\ell_1$ then $Π_1(E, F)$ would have a complemented copy of $\ell_1$.
In the recent paper \cite{Hos}, surjective isometries, not necessarily linear, $T: {\rm AC}(X,E) \longrightarrow {\rm AC}(Y,F)$ between vector-valued absolutely continuous functions on compact subsets $X$ and $Y$ of the real line, has been described. The target spaces $E$ and $F$ are strictly convex normed spaces. In this paper, we assume that $X$ and $Y$ are compact Hausdorff spaces and $E$ and $F$ are normed spaces, which are not assumed to be strictly convex. We describe (with a short proof) surjective isometries $T: (A,\|\cdot\|_A) \longrightarrow (B,\|\cdot\|_B)$ between certain normed su…
p-Summing Bloch mappings on the complex unit disc
本文定义复单位圆盘到 Banach 空间的 $p$-求和 Bloch 映射，证明其 Möbius 不变性与零保持子空间的内射理想性质，并建立 Bloch 型 Pietsch、Maurey 定理及分子空间对偶刻画。
This paper deals with families of matrix-valued Aleksandrov--Clark measures $\{\boldsymbolμ^α\}_{α\in\mathcal{U}(n)}$, corresponding to purely contractive $n\times n$ matrix functions $b$ on the unit disc of the complex plane. We do not make other apriori assumptions on $b$. In particular, $b$ may be non-inner and/or non-extreme. The study of such families is mainly motivated from applications to unitary finite rank perturbation theory. A description of the absolutely continuous parts of $\boldsymbolμ^α$ is a rather straightforward generalization of the well-known results for the scalar case (…
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