Inequalities involving eigenvalues for difference of operator means

Abstract

In a recent paper by Hirzallah et al. (O. Hirzallah, F. Kittaneh, M. Krnic, N. Lovriycevic, and J. Peycaric. Eigenvalue inequalities for differences of means of Hilbert space oper- ators. Linear Algebra and its Applications, 436:1516-1527, 2012.), several eigenvalue inequalities are obtained for the difference of weighted arithmetic and weighted geometric means of two positive invertible operators A and B on a separable Hilbert space under the condition that A B is com- pact. This paper aims to prove some general versions of eigenvalue inequalities for the difference of weighted arithmetic, weighted geometric and generalized Heinz means with better bounds under the same conditions.