Operator Multipliers

向量和矩阵在数学及其应用中起着重要的作用。如果n是正整数，则n维向量是n个实数的有序集合。两个大小相同的向量的标积是它们相应分量的乘积之和。如果n和m是正整数，则大小为n×m的矩阵是具有n行m列的矩形数表。因此，可以将矩阵的行和列视为向量。如果A是n×m矩阵，B是m×k矩阵，则乘积AB是n×k矩阵，其(i，j)元是A的第i行与B的第j列的标积。另一方面，A和B的Schur积A*B是在A和B具有相同大小的情况下定义的，是其分量是A和B的相应分量的乘积的矩阵。同样，可以定义无限大的向量和矩阵；不是所有的无限序列和表现在都被允许，但只允许那些在某种意义上我们不会精确定义的组件不太大的那些。这样的无限矩阵称为算符。算子和Schur乘法的定义类似于有限情形。算符乘法具有非对易的重要特征：低AB=BA并不适用于所有的算符A、B。这一性质在物理学中扮演着非常重要的角色，也是研究算符的发源地。无限矩阵A通过将B发送到A*B而引起所有无限矩阵的集合的变换。如果这种变换将算子发送给算子，则A称为Schur乘子。自20世纪初舒尔的工作以来，舒尔乘子的研究一直是数学分析领域的研究热点。因为每个矩阵都可以看作是关于两个整数变量的函数，所以舒尔乘子可以用这类函数来识别。20世纪最伟大的数学家之一A.Grothendieck对这些函数进行了刻画。他证明了一个重要的与Schur乘子密切相关的不等式，即今天所说的Grothendieck不等式。在过去的25年里，一个新的和强大的理论，称为量化泛函分析，已经被发展，渗透到数学分析的很大一部分。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，因而满足交换律。这一新理论旨在找到关于经典物体的结果的非对易版本。在这一努力中，职能被操作员适当地取代。由于舒尔乘子可以与函数联系在一起，所以它们的量化是一个适时且适时的问题，直到最近才首次得到解决。本项目的目的是研究舒尔乘子的非交换和多元形式，称为算符乘子。它们将被定义为运算符，并且将不依赖于两个变量，而是依赖于任意有限数量的变量。该项目的目标是制定一个研究这些多变量算子乘数的框架，将关于算子乘数的少数已知结果推广到多变量环境，研究相当重要的具体例子，并提供已知事实的新的可交换的多变量版本。这将包括前面提到的Grothendieck不等。将首次考虑多变量Schur和算子乘子，并将非交换性以一种新的更一般的方式引入到项目中。由于研究的新颖性，预计会对功能分析的各个领域产生相当大的影响。并展望了其在相关领域的应用，如谐波分析。因此，这项研究的受益者将是来自广泛科学领域的研究人员。

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.