On the complete boundedness of the Schur block product

TL;DR

This paper provides a Stinespring representation of the Schur block product on matrices over a C*-algebra, establishing its complete boundedness and deriving new inequalities related to matrix diagonals and norms.

Contribution

It introduces a new Stinespring representation of the Schur block product, demonstrating its complete boundedness and deriving novel inequalities for matrix diagonals and norms.

Findings

Established a Stinespring representation for the Schur block product.

Proved the complete boundedness of the Schur block product.

Derived new inequalities involving matrix diagonals and operator norms.

Abstract

We give a Stinespring representation of the Schur block product, say (*), on pairs of square matrices with entries in a C*-algebra as a completely bounded bilinear operator of the form: A:=(a_{ij}), B:= (b_{ij}): A (*) B := (a_{ij}b_{ij}) = V* pi(A) F pi(B) V, such that V is an isometry, pi is a *-representation and F is a self-adjoint unitary. This implies an inequality due to Livshits and two apparently new ones on diagonals of matrices. ||A (*) B|| \leq ||A||_r ||B||_c operator, row and column norm; - diag(A*A) \leq A* (*) A \leq diag(A*A), and for all vectors f, g: |<A(*)B f,g> |^2 \leq < diag(AA*) g, g> <diag(B*B) f,f> .