The Hadamard determinant inequality -- extensions to operators on a Hilbert space (Q1707945)

This is an interesting paper. First, for a version of the classical determinant inequalities such as the Hadamard's and Fischer's inequalities, the author presents an alternative proof. These were originally proved by \textit{W. B. Arveson} [Am. J. Math. 89, 578--642 (1967; Zbl 0183.42501)] for positive operators in von Neumann algebras with a tracial state. Then, some Fischer-type inequalities for determinants of perturbed positive-definite matrices are improved and extended to the framework of finite von Neumann algebras. A conceptual setting is presented for viewing these inequalities as manifestations of Jensen's inequality. This is done in conjunction with the theory of operator monotone and operator convex functions on the nonnegative real numbers. Emphasis is put on recording necessary and sufficient conditions for equality to hold.