Schur and operator multipliers

TL;DR

This paper reviews the development of Schur and operator multipliers, highlighting their historical origins, key theoretical advances, and connections to harmonic analysis and operator integrals.

Contribution

It summarizes recent progress in the theory of Schur and operator multipliers, including multidimensional generalizations and quantization advances.

Findings

Classical Schur multipliers characterized by Grothendieck

Measurable multipliers studied by Peller and Spronk

Multidimensional Schur and operator multipliers developed by Juschenko et al.

Abstract

Schur multipliers were introduced by Schur in the early 20th century and have since then found a considerable number of applications in Analysis and enjoyed an intensive development. Apart from the beauty of the subject in itself, sources of interest in them were connections with Perturbation Theory, Harmonic Analysis, the Theory of Operator Integrals and others. Advances in the quantisation of Schur multipliers were recently made by Kissin and Shulman. The aim of the present article is to summarise a part of the ideas and results in the theory of Schur and operator multipliers. We start with the classical Schur multipliers defined by Schur and their characterisation by Grothendieck, and make our way through measurable multipliers studied by Peller and Spronk, operator multipliers defined by Kissin and Shulman and, finally, multidimensional Schur and operator multipliers developed by Juschenko and the authors. We point out connections of the area with Harmonic Analysis and the Theory of Operator Integrals.