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

## TL;DR

This paper extends classical determinant inequalities like Hadamard's and Fischer's to operators on Hilbert spaces, providing new proofs, generalizations, and a conceptual framework based on Jensen's inequality and operator theory.

## Contribution

It introduces new generalizations of determinant inequalities for operators in von Neumann algebras and offers a unified framework using Jensen's inequality and operator convex functions.

## Key findings

- Generalized Hadamard and Fischer inequalities for positive operators.
- Provided new proofs for inequalities in von Neumann algebras.
- Established conditions for equality in these inequalities.

## 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.