Refinements of norm and numerical radius inequalities

Abstract

Several refinements of norm and numerical radius inequalities of bounded linear operators on a complex Hilbert space are given. In particular, we show that if A is a bounded linear operator on a complex Hilbert space, then 14A*A+AA*≤18A+A*2+A−A*2+c2(A+A*)+c2(A−A*)≤w2(A) and 12A∗A+AA∗−14(A+A∗)2(A−A∗)212≤w2(A)≤12A∗A+AA∗, where ∥⋅∥, w(⋅) and c(⋅) are the operator norm, the numerical radius and the Crawford number, respectively. Further, we prove that if A, D are bounded linear operators on a complex Hilbert space, then AD∗≤∫01(1−t)(A2+D2)/2+tAD∗I2dt12≤12A2+D2, where |A|2=A∗A and |D|2=D∗D. This is a refinement of a well-known inequality obtained by Bhatia and Kittaneh.