On the complete boundedness of the Schur block product
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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*}References
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Additional Information
- Erik Christensen
- Affiliation: Institute for Mathematical Sciences, Universitetsparken 5, 2100 Copenhagen, Denmark
- MR Author ID: 189224
- Email: echris@math.ku.dk
- Received by editor(s): January 2, 2018
- Received by editor(s) in revised form: April 16, 2018, and April 21, 2018
- Published electronically: October 31, 2018
- Communicated by: Stephan Ramon Garcia
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 523-532
- MSC (2010): Primary 15A69, 46L07, 81P68; Secondary 46N50, 47L25, 81T05
- DOI: https://doi.org/10.1090/proc/14202
- MathSciNet review: 3894892