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.
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 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 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.
在本文中，我们将展示如何\emph{the metric theory of tensor products}开发的Grothendieck完全适合在信道容量，在\emph{Shannon's information theory}的中心议题的研究。此外，在过去的几年里，香农的理论已经被推广到量子环境，让\emph{quantum information theory}介入。本文研究了限制辅助纠缠下量子信道的经典容量。特别地，这些容量包括量子信道的经典容量和无限纠缠辅助的经典容量。为了处理量子情形，我们将使用$p$求和映射的非对易形式。更准确地说，我们证明了（产品状态）的经典容量的量子信道与限制辅助纠缠可以表示为一个完全$p$ -求和范数的导数。
In this work we show that, given a linear map from a general operator space into the dual of a C$^*$-algebra, its completely bounded norm is upper bounded by a universal constant times its $(1,cb)$-summing norm. This problem is motivated by the study of quantum XOR games in the field of quantum information theory. In particular, our results imply that for such games entangled strategies cannot be arbitrarily better than those strategies using one-way classical communication.
In the present paper, we introduce and investigate a new class of positively $p$-nuclear operators that are positive analogues of right $p$-nuclear operators. One of our main results establishes an identification of the dual space of positively $p$-nuclear operators with the class of positive $p$-majorizing operators that is a dual notion of positive $p$-summing operators. As applications, we prove the duality relationships between latticially $p$-nuclear operators introduced by O. I. Zhukova and positively $p$-nuclear operators. We also introduce a new concept of positively $p$-integral opera…
We prove the existence of free objects in certain subcategories of Banach lattices, including $p$-convex Banach lattices, Banach lattices with upper $p$-estimates, and AM-spaces. From this we immediately deduce that projectively universal objects exist in each of these subcategories, extending results of Leung, Li, Oikhberg and Tursi (\emph{Israel J.\ Math.}~2019). In the $p$-con\-vex and AM-space cases, we are able to explicitly identify the norms of the free Banach lattices, and we conclude by investigating the structure of these norms in connection with nonlinear $p$-summing maps.
Fractional spline wavelet systems are considered in the work. Molecular structure of elements of such systems admits estimates connecting norms of fractional integrals' images and pre--images in Besov spaces.
Integral representations of projective norm-attaining tensors
Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València（瓦伦西亚理工大学纯数学与应用数学研究所）；Czech Technical University in Prague（布拉格捷克技术大学）；Universidad de Zaragoza（萨拉戈萨大学）；Hanyang University（汉阳大学）；Universidad de Valencia（瓦伦西亚大学）
In this paper, the sharp maximal theorem is generalized to mixed-norm ball Banach function spaces, which is defined as Definition 2.7. As an application, we give a characterization of BMO via the boundedness of commutators of fractional integral operators on mixed-norm Lebesgue spaces. Moreover, the characterization of homogeneous Lipschitz space is also given by the boundedness of commutators of fractional integral operators on mixed-norm Lebesgue spaces. Finally, two applications of Corollary 6.4 are given.
A characterization of t-normed integrals was obtained in \cite{CLM} for finite compacta and in \cite{Rad} for the general case. Such characterization establishes a correspondence between the space of capacities and homogeneous respect t-norm monotone normed functionals preserving the maximum operation of comonotone functions. In fact these theorems we can consider as non-additive and non-linear analogues of well-known Riesz Theorem about a correspondence between the set of $σ$-additive regular Borel measures and the set of linear positively defined functionals. We discuss optimality of such ch…
Monte-Carlo method for multiple parametric integrals calculation and solving of linear integral Fredholm equations of a second kind, with confidence regions in uniform norm
