General upper bounds for the numerical radius on complex Hilbert space

Abstract

In this work, we show that if {Ai}mi=1 and {Xi}mi=1  are two sets of bounded linear operators on the complex Hilbert space H, then for every n ∈ ℕ  and m > 2,  we have w(A1n–1 (Si=0 m–1 Am–i Xm–i A*i+1)(A*1)n–1) ≤ ||A1||2n–2 (2||A1|| ||Am|| + Sj=2m–1||Aj||2)w(E), and w(A1n–1A2X2(A*1)n ± A1nX1A*2(A*1)n–1) = 2||A1||2n–1 ||A2||w([0 X2 X1 0]), where w(.)  is the numerical radius and E = [0  Xm  .... X1 0].  This provides an improvement of Theorem 3 by Fong and Holbrook [3] and a generalization of Theorem 3 by Hirzallah et al. [6]. Moreover, we provide some new upper bounds for the numerical radius of off-diagonal operator matrices and provide a generalization of the main result by Abu-Omar and Kittaneh [17].