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.