We prove several numerical radius inequalities for certain 2×2 operator matrices. Among other inequalities, it is shown that if X, Y, Z, and W are bounded linear operators on a Hilbert space, thenAs an application of a special case of the second inequality, it is shown thatwhich is a considerable improvement of the classical inequality X 2 ≤ w(X). Here w(·) and · are the numerical radius and the usual operator norm, respectively.Mathematics Subject Classification (2010). 47A12, 47A30, 47A63, 47B15.

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It is shown that if A and B are operators on a separable complex Hilbert space and if ||| · ||| is any unitarily invariant norm, then