Decompositions of Schur block products

TL;DR

This paper explores decompositions of the Schur block product of matrices with entries in B(H), showing the existence of contraction matrices that facilitate such decompositions and analyzing their properties using random matrix theory.

Contribution

It introduces a decomposition of the Schur block product involving contraction matrices and demonstrates that the set of such contractions is a very thin subset of the unit ball.

Findings

Existence of a contraction matrix S for the Schur block product decomposition.

The set of possible contraction matrices S is a very thin subset of the unit ball.

Application of random matrix theory to analyze the properties of these contraction matrices.

Abstract

Given two m x n matrices A = (a_{ij}) and B=(b_{ij}) with entries in B(H), the Schur block product is the m x n matrix A \square B := (a_{ij}b_{ij}). There exists an m x n contraction matrix S = (s_{ij}), such that A \square B = diag(AA*)^(1/2) S diag(B*B)^(1/2). This decomposition is also valid for the block Schur tensor product. It is shown, via the theory of random matrices, that the set of contractions S, which may appear in such a decomposition, is a very thin subset of the unit ball of M_n(B(H)).