bibkey: schur1911boundedforms authors: Issai Schur year: 1911 title: “Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen” doi: 10.1515/crll.1911.140.1 url: https://doi.org/10.1515/crll.1911.140.1 claim: “Schur’s test bounds a nonnegative symmetric kernel by its weighted row sums; the finite complex quadratic-form estimate is the version used here.” strata_touched:

• D5/S3/Quantum/FermionicCorrelationalBound/OccupationHilbertContractions license: citation-only triage: anchor

Weighted quadratic-form Schur estimate

Verified locator

DOI: 10.1515/crll.1911.140.1

Source: https://doi.org/10.1515/crll.1911.140.1

Journal für die reine und angewandte Mathematik 140, pages 1–28. Crossref identifies the original article by its title and DOI.

Application

For a finite index set, a symmetric nonnegative real kernel K and strictly positive weights f, the weighted edge inequality gives

$$\mathrm{Re}\sum _{i,j}{K}_{ij}\overline{z_i}{z}_j\le \sum _i|z_i{|}^2\frac{{\sum }_j{K}_{ij}{f}_j}{f_i}.$$

This is the finite quadratic-form version of the classical test. The configuration-dependent fermionic comparison weights and the fourth-moment row estimate are separate results of the present settlement.