On Schwarz type inequalities

Abstract

We show Schwarz type inequalities and consider their converses. A continuous function f : [0, oo) → [0, oo) is said to be semi-operator monotone on (a, b) if {f(t1/2)} 2 is operator monotone on (a 2 ,b 2 ). Let T be a bounded linear operator on a complex Hilbert space H and T = U|T| be the polar decomposition of T. Let 0 ≤ A, B E B(H) and ∥Tx∥ ≤ ∥Ax∥, ∥T*y∥ ≤ ∥By∥ for x,y E H. (1) If a non-zero function f is semi-operator monotone on (0,∞), then |(Tx,y)| ≤ ∥f(A)x∥ ∥g(B)y∥ for x,y E H, where g(t) = t/f(t). (2) If f,g are semi-operator monotone on (0, ∞), then |(Uf(|T|)g(|T|)x, y)| ≤ ∥f(A)x∥ ∥g(B)y∥ for x, y ∈ H. Also, we show converses of these inequalities, which imply that semi-operator monotonicity is necessary.