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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 $a \in B(H)$, 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 $m(A) \leq \min_{1\leq i\leq n} \{m(a_{ii})\}$.
47A63Operator inequalities
47A50Equations and inequalities involving linear operators, with vector unknowns
47A30Operator norms and inequalities
47A10Spectrum and resolvent of linear operators