Decompositions of Schur block products
arXiv:1811.03668
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)).
Nov. 2019, version accepted for publication. Some improvements based on the referee's comments