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 operators of Kislyakov. On one hand, we discuss Pietsch-type domination results for $(\ell^s_p,\ell_p)$-summing operators. In this direction, we provide a negative answer to a question raised by Blasco and Signes, and we also give new insight on a result by Botelho and Santos. On the other hand, we extend to this setting the classical theorem of Kwapień characterizing those operators which factor as $S_1\circ S_2$, where $S_2$ is absolutely $p$-summing and $S_1^*$ is absolutely $q$-summing ($1<p,q<\infty$ and $1/p+1/q \leq 1$).
Let $1\le p\le q<\infty$ and let $X$ be a $p$-convex Banach function space over a $σ$-finite measure $μ$. We combine the structure of the spaces $L^p(μ)$ and $L^q(ξ)$ for constructing the new space $S_{X_p}^{\,q}(ξ)$, where $ξ$ is a probability Radon measure on a certain compact set associated to $X$. We show some of its properties, and the relevant fact that every $q$-summing operator $T$ defined on $X$ can be continuously (strongly) extended to $S_{X_p}^{\,q}(ξ)$. This result turns out to be a mixture of the Pietsch and Maurey-Rosenthal factorization theorems, which provide (strong) factoriz…
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 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.
We give conditions that ensure that an operator satisfying a Piestch domination in a given setting also satisfies a Piestch domination in a different setting. From this we derive that a bounded mutlilinear operator $T$ is Lipschitz $p$-summing if and only if the mapping $ f_T(x_1\otimes\cdots \otimes x_n):=T(x_1,\ldots, x_n)$ is Lipschitz $p$-summing. The results are based on the projective tensor norm. An example with the Hilbert tensor norm is provided to show that the statement may not hold when a reasonable cross-norm other than the projective tensor norm is considered.
In this paper we provide an abstract aproach to the study of classes of multiple summing multilinear operators between Banach spaces. The main purpose is unify the study of several known classes and results, for example multiple $(p, q_1,\dots, q_n)$-summing operators, multiple mixing $(s, q, p)$-summing operators and multiple strong $(s, q, p)$-mixing summing operators. We also define new class of multiple summing multilinear operator that are particular cases of our construction and, therefore, satisfy the results proved in the paper.
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 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…
We give some new characterizations of strictly Lipschitz p-summing operators. These operators have been introduced in order to improve the Lipschitz p-summing operators. Therefore, we adapt this definition for constructing other classes of Lipschitz mappings which are called strictly Lipschitz p-nuclear and strictly Lipschitz (p,r,s)-summing operators. Some interesting properties and factorization results are obtained for these new classes.
We present an abstract result that characterizes the coincidence of certain classes of linear operators with the class of Cohen strongly summing linear operators. Our argument is extended to multilinear operators and, as a consequence, we establish a few alternative characterizations for the class of Cohen strongly summing multilinear operators.
We prove a new result on multiple summing operators and among other applications, we provide a new extension of Littlewood's $4/3$ inequality to $m$-linear forms.
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.
We consider the space of molecules endowed with the transpose version of the Chevet-Saphar norm and we identify its dual space with the space of Lipschitz strongly p-summing operators. We also extend some old results to the category of Lipschitz mappings and we give a factorization result of Lipschitz (p,r,s)-summing 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.
We construct a general framework that generates classes of multilinear operators between Banach spaces which encompasses, as particular cases, the several classes of summing type multilinear operators that have been studied individually in the literature. Summing operators by blocks in the isotropic and anisotropic cases are taken into account. The classes we create are shown to be Banach ideals of multilinear operators and applications to coherence and coincidence theorems are provided.
We prove a general factorization theorem for Lipschitz summing operators in the context of metric spaces which recovers several linear and nonlinear factorization theorems that have been proved recently in different environments. New applications are also given.
In this article, we study the ideals of mid $p$-summing operators. We obtain representation of these operator ideals by tensor norms. These tensor norms are defined by using a particular kind of sequential dual of the class of mid $p$-summable sequences. As a consequence, we prove a characterization of the adjoints of weakly and absolutely mid $p$-summing operators in terms of the operators that are defined by the transformation of dual spaces of certain vector-valued sequence spaces.
We prove that the multiple summing norm of multilinear operators defined on some $n$-dimensional real or complex vector spaces with the $p$-norm may be written as an integral with respect to stables measures. As an application we show inclusion and coincidence results for multiple summing mappings. We also present some contraction properties and compute or estimate the limit orders of this class of operators.
Considering the successful theory of multiple summing multilinear operators as a prototype, we introduce the classes of multiple Cohen strongly p-summing multilinear operators and polynomials. The adequacy of these classes under the viewpoint of the theory of multilinear and polynomial ideals and holomorphy types is discussed in detail. Some abstract results are also proved in the abstract setting of the full general Pietsch Domination Theorem due to Pellegrino, Santos and Seoane-Sepúlveda.
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…
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.
We investigate lineability/spaceability problems within the setting of multilinear summing operators on quasi-Banach sequence spaces. Furthermore, we deal with the contemporary geometric notions of pointwise-lineability and $(α,β)$-lineability. Among other results, we prove that the class of multilinear operators taking values on $\ell_q \, (0<q \leq \infty)$ that are absolutely but not multiple summing is pointwise $\mathfrak{c}$-spaceable when non-empty. Spaceability in other classes, such as Dunford-Pettis operators and multilinear operators on general summable scalar families, is also stud…
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.
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 explore the concept of multilinear operators that are multiple almost summing and present a new concept of type and cotype of multilinear operators and investigate the conditions for this new concept to recover the original concept of type and type for multilinear operators. We also show that these classes are Banach multi-ideals and establish the coherence and compatibility of these classes with the generated homogeneous polynomials.
Let $X,Y$ and $Z$ be Banach spaces, and let $\prod_p(Y,Z) (1\leq p<\infty)$ denote the space of $p$-summing operators from $Y$ to $Z$. We show that, if $X$ is a {\it \$}$_\infty$-space, then a bounded linear operator $T: X\hat \otimes_εY\longrightarrow Z$ is 1-summing if and only if a naturally associated operator $T^#: X\longrightarrow \prod_1(Y,Z)$ is 1-summing. This result need not be true if $X$ is not a {\it \$}$_\infty$-space. For $p>1$, several examples are given with $X=C[0,1]$ to show that $T^#$ can be $p$-summing without $T$ being $p$-summing. Indeed, there is an operator $T$ on $C[0…
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…
Unifying several directions of the development of the study of summing multilinear operators between Banach spaces, we construct a general framework that studies, under one single definition, multilinear operators that are summing with respect to sums taken over any number of indices, iterated and non-iterated sums (the isotropic and the anisotropic cases), sums over arbitrary blocks and over several different sequence norms. A large number of special classes of multilinear operators and of methods of generating classes of multilinear operators are recovered as particular instances. Ideal prop…
We constructively determine the Pietsch measure of the 2-summing multiplication operator \[\mathcal{M}_u:\ell^{\infty} \rightarrow H^p, \quad (φ_I) \mapsto \sum φ_Ix_Ih_I. \] Our construction of the Pietsch measure for the multiplication operator $\mathcal{M}_u$ involves the Haar coefficients of $u$ and its atomic decomposition.
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}.
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.
We apply the geometric approach provided by $Σ$-operators to develop a theory of $p$-summability for multilinear operators. In this way, we introduce the notion of Lipschitz $p$-summing multilinear operators and show that it is consistent with a general panorama of generalization: Namely, they satisfy Pietsch-type domination and factorization theorems and generalizations of the inclusion Theorem, Grothendieck's coincidence Theorems, the weak Dvoretsky-Rogers Theorem and a Lindenstrauss-Pelczyńsky Theorem. We also characterize this new class in tensorial terms by means of a Chevet-Saphar-type t…
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 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.
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…
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.
We prove when a Banach ideal of linear operators defined, or characterized, by the transformation of vector-valued sequences is maximal. Known results are recovered as particular cases and new information is obtained. To accomplish this task we study a tensor quasi-norm determined by the underlying sequence classes. The duality theory for these tensor quasi-norms is also developed.
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.
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.
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.
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.
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…
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…
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…
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…
The main purpose of this paper is the study of a~new class of summing multilinear operators acting from the product of Banach lattices with some nontrivial lattice convexity. A~mixed Pietsch-Maurey-Rosenthal type factorization theorem for these operators is proved under weaker convexity requirements than the ones that are needed in the Maurey-Rosenthal factorization through products of $L^q$-spaces. A~by-product of our factorization is an extension of multilinear operators defined by a~$q$-concavity type property to a~product of special Banach function lattices which inherit some lattice-geome…
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 $H^2(δ^2…
We consider the Carleson embeddings of the classical Hardy spaces (on the disk) into a L p ($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
A Banach space $X$ has the $2$-summing property if the norm of every linear operator from $X$ to a Hilbert space is equal to the $2$-summing norm of the operator. Up to a point, the theory of spaces which have this property is independent of the scalar field: the property is self-dual and any space with the property is a finite dimensional space of maximal distance to the Hilbert space of the same dimension. In the case of real scalars only the real line and real $\ell_\infty^2$ have the $2$-summing property. In the complex case there are more examples; e.g., all subspaces of complex $\ell_\in…
It is shown that the p-summing norm of any operator with n-dimensional domain can be well-aproximated using only ``few" vectors in the definition of the p-summing norm. Except for constants independent of n and log n factors, ``few" means n if 1<p<2 and n^{p/2} if 2<p<infinity.
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.
