Eigenvalue inequalities for positive block matrices with the inradius of the numerical range
Résumé
We prove the operator norm inequality, for a positive matrix partitioned into four blocks in M n , A X X * B ∞ ≤ ∥A + B∥ ∞ + δ(X), where δ(X) is the diameter of the largest possible disc in the numerical range of X. This shows that the inradius ε(X) := δ(X)/2 satisfies ε(X) ≥ ∥X∥ ∞ − ∥(|X * | + |X|)/2∥ ∞. 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, λ 3 A X X * B ≤ λ 2 (A + B) + r. We think that r is optimal and we propose a conjecture related to a norm inequality of Hayashi.
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