Bilinear forms, Schur multipliers, complete boundedness and duality

TL;DR

This paper explores the relationships between bilinear forms, Schur multipliers, and operator space theory, providing norm optimal factorization results and establishing duality in the context of complete boundedness.

Contribution

It introduces norm optimal versions of Grothendieck-type inequalities and demonstrates duality between bilinear forms and Schur multipliers using operator space theory.

Findings

Norm optimal factorization results for matrices related to Grothendieck's inequalities.

Duality between spaces of bilinear forms and Schur multipliers under complete boundedness.

Extension of classical inequalities to the operator space setting.

Abstract

Grothendieck's inequalities for operators and bilinear forms imply some factorization results for complex m x n matrices. Based on the theory of operator spaces and completely bounded mappings we present norm optimal versions of these results and two norm optimal factorization results related to the Schur product. We show that the spaces of respectively bilinear forms and Schur multipliers are conjugate duals to each other with respect to their completely bounded norms.