The Hadamard determinant inequality — Extensions to operators on a Hilbert space |
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Authors: | Soumyashant Nayak |
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Institution: | Smilow Center for Translational Research, University of Pennsylvania, Philadelphia, PA 19104, United States |
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Abstract: | A generalization of classical determinant inequalities like Hadamard's inequality and Fischer's inequality is studied. For a version of the inequalities originally proved by Arveson for positive operators in von Neumann algebras with a tracial state, we give a different proof. We also improve and generalize to the setting of finite von Neumann algebras, some ‘Fischer-type’ inequalities by Matic for determinants of perturbed positive-definite matrices. In the process, a conceptual framework is established for viewing these inequalities as manifestations of Jensen's inequality in conjunction with the theory of operator monotone and operator convex functions on . We place emphasis on documenting necessary and sufficient conditions for equality to hold. |
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Keywords: | Determinant inequality Hadamard–Fischer inequality Operator monotone functions Conditional expectations |
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