Matrix norm inequalities and the relative Dixmier property
1988
If x is a selfadjoint matrix with zero diagonal and non-negative entries, then there exists a decomposition of the identity into k diagonal orthogonal projections {pm} for which
$$\parallel \sum p_m xp_m \parallel \leqslant (1/k)\parallel x\parallel $$
From this follows that all bounded matrices with non-negative entries satisfy the relative Dixmier property or, equivalently, the Kadison Singer extension property. This inequality fails for large Hadamard matrices. However a similar inequality holds for all matrices with respect to the Hilbert-Schmidt norm with constant k−1/2 and for Hadamard matrices with respect to the Schatten 4-norm with constant 21/4k−1/2.
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