Ces\`{a}ro Operators on Rooted Directed Trees

In this paper, we introduce and investigate the notion of the Ces\'aro operator $C_{\mathscr T}$ on a rooted directed tree $\mathscr T.$ When $\mathscr T$ 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 $\ell^2(\mathbb N).$ We prove that for every narrow rooted directed tree $\mathscr T$, $C_{\mathscr T}$ is bounded, with norm bounded above by twice the width of $\mathscr T.$ When the tree is not narrow, this boundedness result no longer holds. Beyond several spectral properties, assuming $\mathscr T$ is leafless and narrow, we show that $C_{\mathscr T}$ is subnormal if and only if $\mathscr T$ is isomorphic to the rooted directed tree without any branching vertex. In particular, this demonstrates that the verbatim analogue of Kriete-Trutt theorem fails in the context of Ces\'aro operators on rooted directed trees. Nonetheless, under the same hypotheses, $C_{\mathscr T}$ is always a compact perturbation of a subnormal operator.