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Journal of the London Mathematical Society 1977 s2-15(3):419-430; doi:10.1112/jlms/s2-15.3.419
© 1977 by London Mathematical Society
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© Oxford University Press

On the Structure of Hermitian-Symmetric Inequalities

Carl H. Fitzgerald and Roger A. Horn

Department of Mathematics, University of California San Diego, La Jolla, California 92093, U.S.A.
Department of Mathematical Sciences, The Johns Hopkins University Baltimore, Maryland 21218, U.S.A.

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.


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