Unique Matrix Factorizations associated to Bilinear Forms and Schur Multipliers

TL;DR

This paper explores unique matrix factorizations linked to bilinear forms and Schur multipliers, leveraging Grothendieck's inequalities and operator space theory to identify conditions for their uniqueness.

Contribution

It establishes the uniqueness of three optimal matrix factorizations associated with bilinear forms and Schur multipliers, expanding understanding within operator space theory.

Findings

Three of the four optimal factorizations are uniquely determined.

The remaining factorization is unique under certain conditions.

The results deepen the connection between Grothendieck's inequalities and matrix factorizations.

Abstract

Grothendieck's inequalities for operators and bilinear forms imply some factorization results for complex m x n matrices. The theory of operator spaces provides a set up which describes 4 norm optimal factorizations of Grothendieck's sort. It is shown that 3 of the optimal factorizations are uniquely determined and the remaining one is unique in some cases.