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Numerical radius inequalities for \(2\times 2\) operator matrices. (English) Zbl 1267.47008
The authors present some upper bounds for the numerical radius of a general \(2\times 2 \) operator matrix \(\left[\begin{matrix} A & B \\ C & D \end{matrix} \right]\) and give a generalization of the known double inequality \(\frac{1}{2}\|A\| \leq w(A)\leq \|A\|\,\,(A\in B(H))\). They also establish some lower bounds for \(\left[\begin{matrix} A & B \\ 0 & 0 \end{matrix} \right]\) and obtain some numerical radius inequalities for sums and products of operators and thus extend some inequalities of S. S. Dragomir [in: C. Bandle (ed.) et al., Inequalities and applications. Proceedings of the conference on inequalities and applications, Noszvaj, Hungary, September 9–15, 2007. Basel: Birkhäuser. International Series of Numerical Mathematics 157, 135–146 (2009; Zbl 1266.26036)].

MSC:
47A12 Numerical range, numerical radius
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
47A63 Linear operator inequalities
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
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