A conjecture on the complete boundedness of Schur multipliers

Abstract

Schur multipliers are a concept from functional analysis that have various uses in mathematics. In this thesis we provide an introduction of the aforementioned Schur multipliers and the associated Schatten p-classes. We prove a number of results and introduce some concepts of functional analysis in order to get to the central topic: a conjecture by Pisier regarding Schur multipliers. For p equal to 1, 2 or infinity all Schur bounded multipliers are completely bounded, and the completely bounded norm of a Schur multiplier is in fact equal to its operator norm. On the other hand, for p unequal to 1, 2 or infinity Pisier conjectures that there exist bounded, but not completely bounded Schur multipliers. Whereas the first part of the thesis is spent on studying the theoretical nature of the problem, in the second part we perform a number of numerical computations yielding insight into the problem. For a number of random finite-dimensional Schur multipliers and various p we approximated the operator norm using the BFGS minimization algorithm. This resulted in us posing a new conjecture that the completely bounded norm is equal to the norm for any Schur multiplier for any p, i.e. we suggest Pisier’s conjecture is false. Finally, we suggest further studies that can be done.