




Volume 37 • Number 4 • 2014 

•
An Operator Karamata Inequality
M. S. Moslehian, M. Niezgoda and R. Rajić
Abstract.
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
Full text: PDF








