Cesàro Operators on Rooted Directed Trees

Abstract: In this paper, we introduce and study the notion of the Ces\'aro operator $C_{\mathscr T}$ on a rooted directed tree $\mathscr T.$ If $\mathscr T$ is the rooted tree with no branching vertex, then $C_{\mathscr T}$ is unitarily equivalent to the classical Ces\'aro operator $C_{0}$ on the sequence space $\ell^2(\mathbb N).$ Besides some basic properties related to boundedness and spectral behavior, we show that $C_{\mathscr T}$ is subnormal if and only if $\mathscr T$ is isomorphic to the rooted directed tree $\mathbb N$ provided $\mathscr T$ is locally finite of finite branching index. In particular, the verbatim analogue of Kriete-Trutt theorem fails for Ces\'aro operators on rooted directed trees. Nevertheless, for a locally finite rooted directed tree $\mathscr T$ of finite branching index, $C_{\mathscr T}$ is always a compact perturbation of a subnormal operator.