Polar decomposition approach to Reid's inequality. Issue 6 (2002) - Lin, C.-S.
Abstract : IfS ≥ 0 andS K is Hermitian, then| ( S K x, x ) ≤ ‖ K ‖ ( S x, x ) holds for allx ∈ H, which is known as Reid's inequality and was sharpened by Halmos in which‖ K ‖ was replaced byr ( K ), the spectral radius ofK . In this article we present generalizations of Reid's and Halmos' inequalities via polar decomposition approach. Conditions onS andS K are relaxed. Theorem regards Reid-type inequalities, and Theorem 2 contains Halmos-type inequalities.