The Hadamard Determinant Inequality - Extensions to Operators on a Hilbert Space

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 $[0,\infty )$. We place emphasis on documenting necessary and sufficient conditions for equality to hold.

21 pages, minor revision (see Corollary 4.3, Remark 4.12) based on referee comments, accepted for publication in JFA

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