Proper improvements of well-known perturbation estimations for the parallel sum

Abstract

Let A:B be the parallel sum of the positive semidefinite matrices A and B. In this paper, we first establish an operator inequality involving A:B and then apply it to obtain some norm upper bounds for the perturbations of A:B. Our results lead to considerable improvements of the well-known inequalities due to Anderson and Duffin (1969) [2]. Examples are also presented to illustrate the sharpness of the obtained norm upper bounds.