The paper defines a lexicographic total order on complex numbers and a Spectral and Nilpotent Ordering (SNO) for arbitrary matrices, claiming to extend majorization, Schur-Ostrowski, and operator convexity results to the complex domain.
Generalized Choi-Davis-Jensen's Operator Inequalities and Their Applications
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abstract
The original Choi-Davis-Jensen's inequality, with its wide-ranging applications in diverse scientific and engineering fields, has motivated researchers to explore generalizations. In this study, we extend Davis-Choi-Jensen's inequality by considering a nonlinear map instead of a normalized linear map and generalize operator convex function to any continuous function defined in a compact region. The Stone-Weierstrass theorem and Kantorovich function are instrumental in formulating and proving generalized Choi-Davis-Jensen's inequalities. Additionally, we present an application of this generalized inequality in the context of statistical physics.
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Matrix Ordering through Spectral and Nilpotent Structures in Totally Ordered Complex Number Fields
The paper defines a lexicographic total order on complex numbers and a Spectral and Nilpotent Ordering (SNO) for arbitrary matrices, claiming to extend majorization, Schur-Ostrowski, and operator convexity results to the complex domain.