Jensen's operator and applications to mean inequalities for operators in Hilbert space (Q656085)

Let \(\mathcal{F}([a,b],\mathbb{R})\) denote the set of all continuous convex functions on an interval \([a,b]\). Then Jensen's operator \(\mathcal{J} : \mathcal{F}([a,b];\mathbb{R}) \times \mathcal{B}_h(H) \times [a,b] \times \mathbb{R}_+^2 \to \mathcal{B}_+(H)\) is defined by \[ \mathcal{J}(f,D,\delta,p)=p_1f(D) + p_2f(\delta)I - (p_1 + p_2)f \left(\frac{p_1D+p_2\delta I}{p_1+p_2}\right), \] where \(p = (p_1,p_2)\), \(aI \leq D \leq bI\), and \(I\) denotes the identity operator on the Hilbert space \(H\). In this paper, the authors investigate some properties of Jensen's operator, find lower and upper bounds for it, and establish some bounds for the spectra of Jensen's operator by means of the discrete Jensen's functional (see [\textit{S. S. Dragomir}, \textit{J. E. Pečarić} and \textit{L.-E. Persson}, Acta Math. Hung. 70, No. 1--2, 129--143 (1996; Zbl 0847.26013)]) and finally get refinements of previously known mean inequalities for operators acting on Hilbert spaces.