A sufficient condition for that an operator map is of the form of functional calculus

Abstract

Let ${\mathscr {H}}$ be a Hilbert space, J be an open interval and $B_J({\mathscr {H}})$ be the set of all self-adjoint operators on ${\mathscr {H}}$ with spectra in J. Suppose that $\phi :B_J({\mathscr {H}})\rightarrow B({\mathscr {H}})$ is an operator map satisfying

$$\begin{aligned} \phi (C^*AC+aD^*D)\le C^*\phi (A)C+\phi (a.1_{\mathscr {H}})D^*D, \end{aligned}$$

for every $A\in B_J({\mathscr {H}})$ with finite spectrum, every $a\in J$ and all operators C, D on ${\mathscr {H}}$ with $C^*C+D^*D=1_{\mathscr {H}}$. We prove that there exists a real convex function f on J such that $\phi (A)=f(A)$ for every $A\in B_J({\mathscr {H}})$ with finite spectrum and $\phi (A)\le f(A)$ for every $A\in B_J({\mathscr {H}})$. If, moreover, $\phi$ is monotone, then the equality is obtained for all $A\in B_J({\mathscr {H}})$. We apply these results to investigate the relation between the above inequality and an inequality of Jensen-type by Jensen–Mercer operator inequality.