Title: 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.