权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Operator Multipliers

运算符乘数

基本信息

批准号：
EP/D050677/1
负责人：
Ivan Todorov
金额：
$ 15.53万
依托单位：
Queen's University Belfast
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2006
资助国家：
英国
起止时间：
2006 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FD050677%2F1
关键词：
Operator Multipliers

项目摘要

Vectors and matrices play a fundamental role in mathematics and its applications. If n is a positive integer, a vector of dimension n is an ordered collection of n real numbers. The scalar product of two vectors of the same size is the sum of the products of their corresponding components. If n and m are positive integers, a matrix of size n x m is a rectangular table of numbers with n rows and m columns. Thus, rows and columns of matrices can be viewed as vectors. If A is an n x m matrix and B an m x k matrix, the product AB is the n x k matrix whose (i,j) element is the scalar product of the i-th row of A with the j-th column of B. The Schur product A*B of A and B, on the other hand, is defined in the case A and B have the same size, and is the matrix whose components are the products of the corresponding components of A and B. In the same way, one may define vectors and matrices of infinite size; not all infinite sequences and tables are allowed now, but only those whose components are, in a certain sense that we will not define precisely, not too big . Such infinite matrices are called operators. Operator and Schur multiplication are defined similarly to the finite case. Operator multiplication has the important feature of being non-commutative: the low AB = BA does not hold for all operators A, B. This property plays a very important role in physics, where the study of operators originated. An infinite matrix A gives rise to a transformation of the set of all infinite matrices by sending B to A*B. If this transformation sends operators to operators, A is called a Schur multiplier. The study of Schur multipliers has attracted a lot of attention in Mathematical Analysis since the work of Schur in the early 20th century. Since every matrix can be viewed as a function on two integer variables, Schur multipliers can be identified with certain functions of this type. A characterisation of these functions was obtained by one of the greatest mathematicians of the 20th century, A. Grothendieck. He proved an important inequality, closely related to Schur multipliers, nowadays known as Grothendieck's inequality. In the last 25 years a new and powerful theory, called Quantised Functional Analysis, has been developed, penetrating a large part of Mathematical Analysis. It is based on the non-commutative structure of (collections of) operators. The objects of study in Classical Analysis are functions, and thus satisfy the commutative low. The new theory aims at finding non-commutative versions of results about classical objects. Functions are in this endeavour appropriately replaced by operators. Since Schur multipliers can be identified with functions, their quantisation is a well posed and timely problem, which was first addressed only very recently. The aim of the present project is to study non-commutative and multivariate versions of Schur multipliers, called operator multipliers. They will be defined to be operators and will depend not on two, but on any finite number of, variables. The objectives of the project are to formulate a framework for the study of these multivariate operator multipliers, to generalise the few known results on operator multipliers to the multivariate setting, to study specific examples of considerable importance and to provide new commutative multivariate versions of known facts. These will include the aforementioned Grothendieck's inequality. For the first time, multivariate Schur and operator multipliers will be considered, and non-commutativity will be brought into the project in a new and more general way. Due to the novelty of the research, a considerable impact is expected on various areas of Functional Analysis. Applications to related fields, such as Harmonic Analysis, are anticipated. The beneficiaries of the research will thus be researchers from scientific fields of a wide range.

向量和矩阵在数学及其应用中起着基础性的作用。如果n是一个正整数，则一个n维向量是n个真实的数的有序集合。两个相同大小的向量的标量积是其相应分量的乘积之和。如果n和m是正整数，则大小为n x m的矩阵是具有n行和m列的数字的矩形表。因此，矩阵的行和列可以被看作向量。如果A是一个n × m矩阵，B是一个m × k矩阵，则乘积AB是一个n × k矩阵，其（i，j）元素是A的第i行与B的第j列的标量积。另一方面，A和B的舒尔积A*B是在A和B具有相同大小的情况下定义的，并且是其分量是A和B的对应分量的乘积的矩阵。同样，我们可以定义无限大小的向量和矩阵;现在并不是所有的无限序列和表都是允许的，但只有那些其分量在某种意义上（我们将不精确定义）不太大的序列和表才是允许的。这样的无限矩阵称为算子。算子和Schur乘法的定义类似于有限情形。算子乘法具有不可交换的重要特征：低AB = BA并不适用于所有算子A，B。这个性质在物理学中起着非常重要的作用，算子的研究起源于物理学。无限矩阵A通过将B发送到A*B而产生所有无限矩阵的集合的变换。如果这种转换将运算符发送到运算符，则A称为Schur乘数。Schur乘子的研究自世纪初Schur的工作以来，一直是数学分析领域的研究热点。由于每个矩阵都可以被看作是两个整数变量的函数，舒尔乘数可以与这种类型的某些函数相识别。世纪最伟大的数学家之一A.格罗滕迪克他证明了一个重要的不平等，密切相关的舒尔乘数，今天被称为格罗滕迪克的不平等。在过去的25年里，一个新的和强大的理论，称为量化函数分析，已经发展，渗透了很大一部分的数学分析。它基于运算符（集合）的非交换结构。经典分析的研究对象是函数，因此满足交换律。新理论的目的是寻找关于经典对象的结果的非交换版本。在这一努力中，职能由运营商适当取代。由于舒尔乘数可以确定的功能，其量化是一个很好的和及时的问题，这是第一次解决，只是最近。本项目的目的是研究非交换和多元版本的舒尔乘数，称为运营商乘数。它们将被定义为算子，并且不依赖于两个变量，而是依赖于任何有限数量的变量。该项目的目标是制定一个框架，这些多元算子乘数的研究，概括了一些已知的结果算子乘数的多元设置，研究具体的例子相当重要，并提供新的交换多元版本的已知事实。其中包括前面提到的Grothendieck不等式。第一次，多元舒尔和算子乘数将被考虑，非交换性将被带到一个新的和更一般的方式进入项目。由于这项研究的新奇，预计将对功能分析的各个领域产生相当大的影响。预计将应用于相关领域，如谐波分析。因此，研究的受益者将是来自广泛科学领域的研究人员。