Journal of Inequalities and Applications
Volume 7 (2002), Issue 6, Pages 779-785
doi:10.1155/S1025583402000401
Polar decomposition approach to Reid’s inequality
Department of Mathematics, Bishop's University, Lennoxville PQ J1M 1Z7, Canada
Received 19 March 2001; Revised 17 May 2001
Copyright © 2002 C.-S. Lin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
If S≥0 and SK is Hermitian, then |(SKx,x)≤‖K‖(Sx,x) holds for all x∈H, which is known as Reid’s inequality and was sharpened by Halmos in which ‖K‖ was replaced by r(K), the spectral radius of K. In this article we present generalizations of Reid’s and Halmos’ inequalities via polar decomposition approach. Conditions on S and SK are relaxed. Theorem regards Reid-type inequalities, and Theorem 2 contains Halmos-type inequalities.