STUDIA MATHEMATICA 182 (2) (2007) Numerical radius inequalities for Hilbert space operators. II by Mohammad El-Haddad (Dammam) and Fuad Kittaneh (Amman) Abstract. We give several sharp inequalities involving powers of the numerical radii and the usual operator norms of Hilbert space operators. These inequalities, which are based on some classical convexity inequalities for nonnegative real numbers and some operator inequalities, generalize earlier numerical radius inequalities. 1. Introduction. Let B(H ) denote the C * -algebra of all bounded linear operators on a complex Hilbert space H with inner product 〈·, ·〉. For A ∈ B(H ), let w(A) and ‖A‖ denote the numerical radius and the usual operator norm of A, respectively. It is well known that w(·) defines a norm on B(H ), which is equivalent to the usual operator norm ‖·‖. In fact, for every A ∈ B(H ), (1) 1 2 ‖A‖≤ w(A) ≤‖A‖. The inequalities in (1) are sharp. The first inequality becomes an equality if A 2 = 0. The second inequality becomes an equality if A is normal. For basic properties of the numerical radius, we refer to [3] and [4]. The inequalities in (1) have been improved considerably by the second author in [8] and [9]. It has been shown in [8] and [9], respectively, that if A ∈ B(H ), then (2) w(A) ≤ 1 2 ‖|A| + |A * |‖≤ 1 2 (‖A‖ + ‖A 2 ‖ 1/2 ), where |A| =(A * A) 1/2 is the absolute value of A, and (3) 1 4 ‖A * A + AA * ‖≤ w 2 (A) ≤ 1 2 ‖A * A + AA * ‖. The inequalities in (2), which refine the second inequality in (1), have been utilized in [8] to derive an estimate for the numerical radius of the Frobenius companion matrix. Such an estimate can be employed to give new bounds for the zeros of polynomials (see, e.g., [7], [8], and references therein). If A = B + iC is the Cartesian decomposition of A, then B and C are self-adjoint, and A * A + AA * = 2(B 2 + C 2 ). Thus, the inequalities in (3) can 2000 Mathematics Subject Classification : 47A12, 47A30, 47A63, 47B15. Key words and phrases : numerical radius, operator norm, Cartesian decomposition, Jensen’s inequality, mixed Schwarz inequality, triangle inequality. [133] c  Instytut Matematyczny PAN, 2007