Operator Multipliers

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.