STUDIA MATHEMATICA 210 (2) (2012) Numerical radius inequalities for 2×2 operator matrices by Omar Hirzallah (Zarqa), Fuad Kittaneh (Amman) and Khalid Shebrawi (Salt) Abstract. We derive several numerical radius inequalities for 2 ×2 operator matrices. Numerical radius inequalities for sums and products of operators are given. Applications of our inequalities are also provided. 1. Introduction. Let B(H) be the space of all bounded linear operators on a complex Hilbert space H with inner product h·, ·i. The numerical range of A ∈ B(H), denoted by W (A), is the subset of the complex numbers given by W (A)= {hAx, xi : x ∈H, kxk =1}. The numerical radius of A, w(A), is defined by w(A) = sup{|λ| : λ ∈ W (A)}. It is well-known that w(·) defines a norm on B(H), which is equivalent to the usual operator norm k·k. In fact, for A ∈ B(H), we have (1.1) 1 2 kAk≤ w(A) ≤kAk. Also, it is known that w(·) is weakly unitarily invariant, that is, (1.2) w(U * AU )= w(A) for every unitary U ∈ B(H). For other properties of the numerical radius, the reader is referred to [7] and [8]. Recent numerical radius inequalities for commutators of operators and operator matrices have been given in [9] and [10]. The following numerical radius inequalities for a product of operators have been given in [6]: If A, B ∈ B(H), then (1.3) max(kA + Bk 2 , kA - Bk 2 ) -   |A * | 2 + |B * | 2   ≤ 2w(AB * ) 2010 Mathematics Subject Classification : Primary 47A12, 47A30, 47A63; Secondary 15A60, 47B15. Key words and phrases : numerical range, numerical radius, operator norm. DOI: 10.4064/sm210-2-1 [99] c  Instytut Matematyczny PAN, 2012