This is an announcement for the paper "A norm compression inequality for block partitioned positive semidefinite matrices" by Koenraad M.R. Audenaert.
Abstract: Let $A$ be a positive semidefinite matrix, block partitioned as $$ A=\twomat{B}{C}{C^*}{D}, $$ where $B$ and $D$ are square blocks. We prove the following inequalities for the Schatten $q$-norm $||.||_q$, which are sharp when the blocks are of size at least $2\times2$: $$ ||A||_q^q \le (2^q-2) ||C||_q^q + ||B||_q^q+||D||_q^q, \quad 1\le q\le 2, $$ and $$ ||A||_q^q \ge (2^q-2) ||C||_q^q + ||B||_q^q+||D||_q^q, \quad 2\le q. $$ These bounds can be extended to symmetric partitionings into larger numbers of blocks, at the expense of no longer being sharp: $$ ||A||_q^q \le \sum_{i} ||A_{ii}||_q^q + (2^q-2) \sum_{i<j} ||A_{ij}||_q^q, \quad 1\le q\le 2, $$ and $$ ||A||_q^q \ge \sum_{i} ||A_{ii}||_q^q + (2^q-2) \sum_{i<j} ||A_{ij}||_q^q, \quad 2\le q. $$
Archive classification: Functional Analysis
Mathematics Subject Classification: 15A60
Remarks: 24 pages
The source file(s), normcompr_v3.tex: 50189 bytes, is(are) stored in gzipped form as 0505680.gz with size 16kb. The corresponding postcript file has gzipped size 79kb.
Submitted from: kauden@imperial.ac.uk
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