General upper bounds for the numerical radius on complex Hilbert space

In this work, we show that if {A(i)}(i-1)(m) i and{X-i}(i-1)(m) are two sets of bounded linear operators on the complex Hilbert space H, then for every n is an element of N and m> 2, we have w(A(1)(n-1) (Sigma(m-1)(i=0) A(m-i) Xm-i A(i+1)(*))(A(1)(*))(n-1)) <= parallel to A(1)parallel to(2n-2) (2 parallel to A(1) parallel to parallel to A(m) parallel to + Sigma(m-1)(j=2)parallel to A(j) parallel to(2)) w(E) and w(A(1)(n-1) A(2) X-2 (A(1)(*))(n) +/- A(1)(n)X(1)A(2)(*) (A(1)*)(n-1)) = 2 parallel to A(1)parallel to(2n-1) parallel to A(2)parallel to w(left perpendicular (X2) (0) (X1) (0) right perpendicular), where w(.) is the numerical radius and E= [(Xm) (0) ... (0) (X1)]. 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].