Subordination and related inequalities on a C⁎−algebra

Abstract

Let X be a $C^{⁎}$-algebra, $x\in X$ and $\Vert x\Vert <1$. For a complex valued function f, which is analytic in $U=\{\lambda \in C:|\lambda |<1\}$ let $f\left(x\right)$ be defined by the following integral: $f\left(x\right)={\int }_{\Gamma }f\left(\lambda \right){(\lambda e-x)}^{-1}d\lambda$, where Γ is a positively oriented rectifiable simple closed contour in $U$ that surrounds $\sigma \left(x\right)$, the spectrum of x. In this paper Schwarz's lemma, Harnack's inequalities and results related to subordination are obtained for $f\left(x\right)$.