On a Norm Compression Inequality for 2XN Partitioned Block Matrices

Details On a Norm Compression Inequality for 2XN Partitioned Block Matrices On a Norm Compression Inequality for 2XN Partitioned Block Matrices

2007-02-07

arxiv.org/abs/math/0702186v2

Mathematics

Functional Analysis

22 pages; mistake in section on duality corrected, additional results added

Scientific paper

We conjecture the following so-called norm compression inequality for $2\times N$ partitioned block matrices and the Schatten $p$-norms: for $p\ge 2$, $$ ||(\begin{array}{cccc} A_1 & A_2 & ... & A_N B_1 & B_2 & >... & B_N \end{array})||_p \le ||(\begin{array}{cccc} ||A_1||_p & ||A_2||_p & ... & ||A_N||_p \ ||B_1||_p & ||B_2||_p & ... & ||B_N||_p \end{array})||_p $$ while for $1\le p\le 2$ the ordering of the inequality is reversed. This inequality includes Hanner's inequality for matrices as a special case. We prove several special cases of this inequality and give examples for $3\times 3$ and larger partitionings where it does not hold.

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