Inner spectral radius of positive operator matrices (Q927614)

Let \(H\) be a complex Hilbert space and \(B(H)\) be the algebra of all bounded linear operators on \(H\). For \(a \in B(H)\), let \(m(a)\), \(\sigma(a)\), \(W(a)\), \(r(a)\), \(w(a)\), \(i(a)\), 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 \(A= (a_{ij}I)_{n\times n} \in M_n(B(H))\) and \(\tilde A = (a_{ij})_{n\times n} \in M_n( \mathbb R)\) are nonnegative, then \(w_i(A) \leq w_i(\tilde A)\), \(m(A)\leq m(\tilde A)\) and \(i(A) \leq i(\tilde A)\). Moreover, if \(A= (a_{ij})_{n\times n}\) is a positive operator matrix, then \(m(A) \leq \min_{1\leq i\leq n} \{m(a_{ii})\}\).