Decompositions of block Schur product

Given two m×n matrices A=(aij) and B=(bij) with entries in B(H) for some Hilbert space H, the Schur block product is the m×n matrix A□B:=(aijbij). There exists an m×n matrix S=(sij) with entries from B(H) such that S is a contraction operator and A□B=(diag(AA∗))1/2S(diag(B∗B))1/2. The analogus result for the block Schur tensor product ⊠ defined by Horn and Mathias in \cite{HM} holds too. This kind of decomposition of the Schur product seems to be unknown, even for scalar matrices. Based on the theory of random matrices we show that the set of contractions S, which may appear in such a decomposition, is a \textit{thin} set in the ball of all contractions.