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Refinements of Choi-Davis-Jensen’s inequality. (English) Zbl 1314.47022

Summary: Let \(\Phi_1, \dots, \Phi_n\) be strictly positive linear maps from a unital \(C^\ast\)-algebra \(\mathcal A\) into a \(C^\ast\)-algebra \(\mathcal B\) and let \(\Phi =\sum^n_{i=1}\Phi_i\) be unital. If \(f\) is an operator convex function on an interval \(J\), then for every self-adjoint operator \(A \in \mathcal A\) with spectrum contained in \(J\), the following refinement of the Choi-Davis-Jensen inequality holds: \[ f (\Phi (A)) \leq \sum^n_{i=1} \Phi_i(I)^{\frac {1}{2}}f \left (\Phi_i(I)^{-\frac {1}{2}} \Phi_i(A) \Phi_i(I)^{-\frac {1}{2}} \right) \Phi_i(I)^{\frac {1}{2}} \leq \Phi (f(A)). \]

MSC:

47A63 Linear operator inequalities
47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
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