Cesàro operators on rooted directed trees

In this paper, we introduce and investigate the notion of the Cesáro operator C T on a rooted directed tree T . When T is the rooted tree with no branching vertex, then C T is unitarily equivalent to the classical Cesáro operator C 0 on the sequence space ℓ 2 ( N ) . We prove that for every narrow rooted directed tree T , C T is bounded, with norm bounded above by twice the square root of the width of T . When the tree is not narrow, this boundedness result no longer holds. Beyond several spectral properties, assuming T is leafless and narrow, we show that C T is subnormal if and only if 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 Cesáro operators on rooted directed trees. Nonetheless, under the same hypotheses, C T is always a compact perturbation of a subnormal operator.