On the Araki-Lieb-Thirring inequality in the semifinite von Neumann algebra
Let \(A\) and \(B\) be positive self-adjoint operators on a Hilbert space, and let \(f\) be any increasing continuous real-valued function on \([0,\infty[\) such that \(f(0) = 0\) and \(t\mapsto f(e^t)\) is a convex function. By the Araki-Lieb-Thirring inequality, for all \(q\geq 1\) the following inequality holds: \[ \text{Tr}f((B^{1/2} AB^{1/2})^q)\leq \text{Tr}f(B^{q/2} A^qB^{q/2}). \] In this paper, the author shows that this inequality and its extension due to \textit{J.-C. Bourin} and \textit{E.-Y. Lee} [Int. J. Math. 27, No. 9, Article ID 1650071, 19 p. (2016; Zbl 1402.15015)] hold also in the setting of von~Neumann algebras, namely, for \(\tau\)-measurable operators associated with a~semifinite von~Neumann algebra. The paper is concluded with a series of submajorization inequalities which are related to Heinz means and some of them are new even in the matrix case.
scientific article; zbMATH DE number 6667758