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Volume 37 • Number 4 • 2014
• An Operator Karamata Inequality
M. S. Moslehian, M. Niezgoda and R. Rajić

We present an operator version of the Karamata inequality. More precisely, we prove that if $A$ is a selfadjoint element of a unital $C^*$-algebra $\mathscr{A}$ ,ρ is a state on $\mathscr{A}$ , the functions f ;g are continuous on the spectrum σ(A) of A such that $0$ < $m$$1$ ≤$f(s)$ ≤ $M$$1$, $0$ < $m$$2$ ≤ $g(s)$ ≤ $M$$2$ for all $s\in \sigma(A)$ and $K=\left(\sqrt{m_1m_2}+\sqrt{M_1M_2}\right)/\left(\sqrt{m_1M_2}+\sqrt{M_1m_2}\right)$ then $$K^{-2}\le \frac{\rho(f(A)g(A))}{\rho(f(A)) \rho(g(A))}\le K^2.$$ We also give some applications.

2010 Mathematics Subject Classification: Primary 47A63; Secondary 47B25, 15A60

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