On the complete boundedness of the Schur block product

Christensen, Erik

doi:10.1090/proc/14202

On the complete boundedness of the Schur block product
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Proc. Amer. Math. Soc. 147 (2019), 523-532 Request permission

Abstract:

We give a Stinespring representation of the Schur block product on pairs of square matrices with entries in a C$^*$-algebra as a completely bounded bilinear operator of the form \begin{equation*} A:=(a_{ij}), B:= (b_{ij}): A \square B := (a_{ij}b_{ij}) = V^*\lambda (A)F \lambda (B) V, \end{equation*} such that $V$ is an isometry, $\lambda$ is a *-representation, and $F$ is a self-adjoint unitary. This implies an inequality due to Livshits and two apparently new inequalities on the diagonals of matrices: \begin{align*} \|A \square B\| &\!\leq \! \|A\|_r \|B\|_c \text { operator, row, and column norm;} \\ - \mathrm {diag}(A^*A) &\!\leq \! A^*\square A \leq \mathrm {diag}(A^*A), \\ \forall \Xi , \Gamma \!\in \! \mathbb {C}^n\!\otimes \! H: |\langle (A \square B) \Xi , \Gamma \rangle | & \!\leq \! \|\big (\mathrm {diag}(B^*B)\big )^{1/2}\Xi \| \|\big (\mathrm {diag}(AA^*)\big )^{1/2}\Gamma \|. \end{align*}

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