Singular value and arithmetic-geometric mean inequalities for operators (Q714426)

The classical arithmetic-geometric mean inequality for positive numbers \(a\) and \(b\) is important in functional analysis, matrix theory, electrical networks, etc. Several unitarily invariant norm and singular value inequalities of the arithmetic-geometric mean type for matrices and Hilbert space operators have been obtained by many authors (K. M. R. Audenaert, X.-Z. Zhan and R. Mathias). In this article, a singular value inequality for sums and products of Hilbert space operators is given. This inequality generalizes several recent singular value inequalities, and includes the case when \(A, B\) and \(X \) are positive operators on a complex Hilbert space \(H\). The author's analysis is based on the majorization of singular values and the matrix arithmetic-geometric mean inequality. Other singular value inequalities for sums and products of operators are presented. Related arithmetic-geometric mean inequalities for operators are also discussed.