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Inner spectral radius of positive operator matrices. (English) Zbl 1173.47012
Let H be a complex Hilbert space and B(H) be the algebra of all bounded linear operators on H. For aB(H), let m(a), σ(a), W(a), r(a), w(a), i(a), and W i (a) denote the minimum moduli, spectrum, numerical range, spectral radius, numerical radius, inner spectral radius and inner numerical radius of a, respectively. The authors give some inequalities for these. For example, if A=(a ij I) n×n M n (B(H)) and A ˜=(a ij ) n×n M n () are nonnegative, then w i (A)w i (A ˜), m(A)m(A ˜) and i(A)i(A ˜). Moreover, if A=(a ij ) n×n is a positive operator matrix, then m(A)min 1in {m(a ii )}.
MSC:
47A63Operator inequalities
47A50Equations and inequalities involving linear operators, with vector unknowns
47A30Operator norms and inequalities
47A10Spectrum and resolvent of linear operators