Eigenvalue inequalities for positive block matrices with the inradius of the numerical range

We prove the operator norm inequality, for a positive matrix partitioned into four blocks in Mn, parallel to [GRAPHICS] parallel to(infinity) <= parallel to A + B parallel to(infinity) + delta(X), where delta(X) is the diameter of the largest possible disc in the numerical range of X. This shows that the inradius epsilon(X) := delta(X)/2 satisfies epsilon(X) >= parallel to X parallel to(infinity) - parallel to(|X*|+ |X|)/2 parallel to(infinity). Several eigenvalue inequalities are derived. In particular, if X is a normal matrix whose spectrum lies in a disc of radius r, the third eigenvalue of the full matrix is bounded by the second eigenvalue of the sum of the diagonal block, lambda(3) ( [GRAPHICS] ) <= lambda(2) (A + B) + r. We think that r is optimal and we propose a conjecture related to a norm inequality of Hayashi.