bibkey: audenaert2008normcompression authors: K. M. R. Audenaert year: 2008 title: “On a norm compression inequality for 2×N partitioned block matrices” doi: 10.1016/j.laa.2007.08.007 url: https://arxiv.org/abs/math/0702186v2 claim: “Conjecture 1, upper branch: Schatten norm compression for arbitrary 2×N compatible block matrices at p ≥ 2.” strata_touched:

• D5/S3/Quantum/NormCompression/UpperBranchReduction license: citation-only triage: anchor

Norm compression for two block rows

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DOI: 10.1016/j.laa.2007.08.007

Source: https://arxiv.org/abs/math/0702186v2

Linear Algebra and its Applications 428 (2008), 781–795. The arXiv preprint’s page 2 contains the Schatten norm, the compression matrix and Conjecture 1; Section 4 is entitled “Proofs in Special Cases”. The publisher’s DOI metadata matches the author, title, volume and pages.

On page 1, the source defines compression in words:

The norm compression of a block-partitioned matrix T = [T₍ᵢⱼ₎] w.r.t. a given matrix norm ||.|| is a matrix obtained from T by replacing each of its blocks by their norm: [||T₍ᵢⱼ₎||].

On page 2, the Schatten norm is

$$\Vert A{\Vert }_p=(\mathrm{Tr}(|A{|}^p){)}^{1/p}.$$

The compression definition, on the same page, is introduced by:

I denote by 𝒞ₚ(T) its Schatten p-norm compression,

$${\mathcal{C}}_p(T)=\left(\begin{array}{ccc}\Vert A_1{\Vert }_p& \cdots & \Vert A_N{\Vert }_p\\ \Vert B_1{\Vert }_p& \cdots & \Vert B_N{\Vert }_p\end{array}\right).$$

Conjecture 1, page 2, verbatim:

Let T be a general matrix partitioned in 2 × N blocks, and let 𝒞ₚ(T) be its norm compression using the Schatten p-norm, then the following norm compression inequalities hold:

$$\Vert T{\Vert }_p\ge \Vert {\mathcal{C}}_p(T){\Vert }_p,1\le p\le 2,$$ $$\Vert T{\Vert }_p\le \Vert {\mathcal{C}}_p(T){\Vert }_p,p\ge 2.$$

The Lean encoding uses Matrix.toEuclideanLin and LinearMap.singularValues: the trace power is the finite sum of the p-th powers of singular values, and the Schatten norm is its real 1/p-th power. The scalar compression entries are nonnegative real block norms cast to ℂ. The two row spaces are arbitrary finite types; the finite column dimension may depend on the column index.

The upper-branch three-column reduction preserves the compression Gram matrix and increases the original Schatten norm after nonnegative square-root rescaling. This is a reduction of the upper branch, not a proof of Conjecture 1. AUD-UP on 2 < p < 4 remains open. The lower-branch reduction requires trace-power concavity and is not established by this module.