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On the Structure of Hermitian‐Symmetric Inequalities
Author(s) -
Fitzgerald Carl H.,
Horn Roger A.
Publication year - 1977
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-15.3.419
Subject(s) - hermitian matrix , mathematics , pure mathematics , inequality , hermitian symmetric space , mathematical analysis , hermitian manifold , geometry , curvature , ricci curvature
Quadratic inequalities between Hermitian and symmetric forms are considered. Such inequalities are shown to be equivalent to the positive definiteness of a related Hermitian matrix. Several applications of this observation are made. A representation of a positive definite Hermitian form as the sum of squares of linear forms is extended to an analogous representation for matrices which satisfy Hermitian‐symmetric inequalities. A generalization of the Schur Product Theorem is proved. A new characterization is given of infinitely divisible Hermitian‐symmetric inequalities. These results are extended to the more general case of bilinear Hermitian‐symmetric inequalities. Several applications in analytic function theory and an application to the Carathéodory moment problem are discussed.