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.
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.
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.
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.
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…
We show that for any operator $T:l_\infty^N\to Y$, where $Y$ is a Banach space, that its cotype 2 constant, $K_2(T)$, is related to its $(2,1)$-summing norm, $π_{2,1}(T)$, by $K_2(T) \le c \log\log N π_{2,1}(T) $. Thus, we can show that there is an operator $T:C(K)\to Y$ that has cotype 2, but is not 2-summing.
It is well known in Banach space theory that for a finite dimensional space $E$ there exists a constant $c_E$, such that for all sequences $(x_k)_k \subset E$ one has \[ \summ_k \noo x_k \rrm \kl c_E \pl \sup_{\eps_k \pm 1} \noo \summ_k \eps_k x_k \rrm \pl .\] Moreover, if $E$ is of dimension $n$ the constant $c_E$ ranges between $\sqrt{n}$ and $n$. This implies that absolute convergence and unconditional convergence only coincide in finite dimensional spaces. We will characterize Banach spaces $X$, where the constant $c_E \sim \sqrt{n}$ for all finite dimensional subspaces. More generally, we…
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.
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.
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.
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.
In this note we show that the well known coincidence results for scalar-valued homogeneous polynomials can not be generalized in some natural directions.
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.
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 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.
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.
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.
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.
Let $E$ be a Banach space that does not contain any copy of $\ell^1$ and $\A$ be a non commutative $C^*$-algebra. We prove that every absolutely summing operator from $\A$ into $E^*$ is compact, thus answering a question of Pełczynski. As application, we show that if $G$ is a compact metrizable abelian group and $Λ$ is a Riesz subset of its dual then every countably additive $\A^*$-valued measure with bounded variation and whose Fourier transform is supported by $Λ$ has relatively compact range. Extensions of the same result to symmetric spaces of measurable operators are also presented.
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
We show that given a positive integer $m$, a real number $p\in\left[ 2,\infty\right)$ and $1\leq s<p^{\ast}$ the set of non--multiple $\left( r;s\right)$--summing $m$--linear forms on $\ell_{p}\times\cdots\times \ell_{p}$ contains, except for the null vector, a closed subspace of maximal dimension whenever $r<\frac{2ms}{s+2m-ms}$. This result is optimal since for $r\geq\frac{2ms}{s+2m-ms}$ all $m$--linear forms on $\ell_{p}\times \cdots\times\ell_{p}$ are multiple $\left( r;s\right)$--summing. In particular, among other results, we generalize a result related to cotype (from 2010) due to Botel…
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…
In the article is introduced a new class of Banach spaces that are called sub B-convex. Namely, a Banach space X is said to be B -convex if it may be represented as a direct sum l_1+ W, where W is B-convex. It will be shown that any separable sub B-convex Banach space X may be almost isometrically embedded in a separable Banach space G(X) of the same cotype as X, which has a series of properties. Namely, G(X) is an approximate envelope, i.e. any separable Banach space which is finitely representable in G(X) may be almost isometrically embedded into G(X); G(X)is almost isotropic; G(X) is existe…
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
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…
If $ϕ$ is a submeasure satisfying an appropriate lower estimate we give a quantitative result on the total mass of a measure $μ$ satisfying $0\leμ\leϕ.$ We give a dual result for supermeasures and then use these results to investigate convexity on non-locally convex quasi-Banach lattices. We then show how to use these results to extend some factorization theorems due to Pisier to the setting of quasi-Banach spaces. We conclude by showing that if $X$ is a quasi-Banach space of cotype two then any operator $T:C(Ω)\to X$ is 2-absolutely summing and factors through a Hilbert space and discussing g…
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.
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.
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…
Let $X$, $Y$ and $Z$ be Banach spaces and let $U$ be a subspace of $\mathcal{L}(X^*,Y)$, the Banach space of all operators from $X^*$ to $Y$. An operator $S: U \to Z$ is said to be $(\ell^s_p,\ell_p)$-summing (where $1\leq p <\infty$) if there is a constant $K\geq 0$ such that $$ \Big( \sum_{i=1}^n \|S(T_i)\|_Z^p \Big)^{1/p} \le K \sup_{x^* \in B_{X^*}} \Big(\sum_{i=1}^n \|T_i(x^*)\|_Y^p\Big)^{1/p} $$ for every $n\in \mathbb{N}$ and every $T_1,\dots,T_n \in U$. In this paper we study this class of operators, introduced by Blasco and Signes as a natural generalization of the $(p,Y)$-summing ope…
In this paper we extend the scope of three important results of the linear theory of absolutely summing operators. The first one was proved by Bu and Kranz in \cite{BK} and it asserts that a continuous linear operator between Banach spaces takes almost unconditionally summable sequences into Cohen strongly $q$-summable sequences for any $q\geq2$, whenever its adjoint is $p$-summing for some $p\geq1$. The second of them states that $p$-summing operators with hilbertian domain are Cohen strongly $q$-summing operators ($1<p,q<\infty$), this result is due to Bu \cite{Bu}. The third one is due to K…
Let $E$ be an operator space in the sense of the theory recently developed by Blecher-Paulsen and Effros-Ruan. We introduce a notion of $E$-valued non commutative $L_p$-space for $1 \leq p < \infty$ and we prove that the resulting operator space satisfies the natural properties to be expected with respect to e.g. duality and interpolation. This notion leads to the definition of a ``completely p-summing" map which is the operator space analogue of the $p$-absolutely summing maps in the sense of Pietsch-Kwapień. These notions extend the particular case $p=1$ which was previously studied by Effro…
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…
We analyze the interplay between maximal/minimal/adjoint ideals of multilinear operators (between sequence spaces) and their associated Köthe sequence spaces. We establish relationships with spaces of multipliers and apply these results to describe diagonal multilinear operators from Lorentz sequence spaces. We also define and study some properties of the ideal of $(E;p)$-summing multilinear mappings, a natural extension of the linear ideal of absolutely $(E;p)$-summing operators.
As is well known absolute convergence and unconditional convergence for series are equivalent only in finite dimensional Banach spaces. Replacing the classical notion of absolutely summing operators by the notion of 1 summing operators \[ \summ_k || Tx_k || \leq c || \summ_k e_k \otimes x_k ||_{\ell_1\otimes_{min}E}\] in the category of operator spaces, it turns out that there are quite different interesting examples of 1 summing operator spaces. Moreover, the eigenvalues of a composition $TS$ decreases of order $n^{\frac{1}{q}}$ for all operators $S$ factorizing completely through a commutati…
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 Banac…
Within the theory of complex interpolation and theta-Hilbert spaces we extend classical results of Kwapien on absolutely (r,1)-summing operators on l_1 with values in l_p as well as their natural extensions for mixing operators invented by Maurey. Furthermore, we show that for 1<p<2 every operator T on l_1 with values in theta-type 2 spaces, theta=2/p', is Rademacher p-summing. This is another extension of Kwapien's results, and by an extrapolation procedure a natural supplement to a statement of Pisier.
Among other results we investigate $\left( α,β\right) $-lineability of the set of non-continuous $m$-linear operators defined between normed spaces as a subset of the space of all $m$-linear operators. We also give a partial answer to an open problem on the lineability of the set of non absolutely summing operators.
Let $1 \leq p <\infty$. A sequence $\lef x_n \rig$ in a Banach space $X$ is defined to be $p$-operator summable if for each $\lef f_n \rig \in l^{w^*}_p(X^*)$, we have $\lef \lef f_n(x_k)\rig_k \rig_n \in l^s_p(l_p)$. Every norm $p$-summable sequence in a Banach space is operator $p$-summable, while in its turn every operator $p$-summable sequence is weakly $p$-summable. An operator $T \in B(X, Y)$ is said to be $p$-limited if for every $\lef x_n \rig \in l_p^w(X)$, $\lef Tx_n \rig$ is operator $p$-summable. The set of all $p$-limited operators form a normed operator ideal. It is shown that ev…
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 $\ell_1$.
