Cesà ro Operators on Rooted Directed Trees

Abstract

In this paper, we introduce and investigate the notion of the Cesáro operator $C_{\mathcal{T}}$ on a rooted directed tree $\mathcal{T}.$ When $\mathcal{T}$ is the rooted tree with no branching vertex, then $C_{\mathcal{T}}$ is unitarily equivalent to the classical Cesáro operator $C_0$ on the sequence space ${\ell }^2(\mathbb{N}).$ We prove that for every narrow rooted directed tree $\mathcal{T}$, $C_{\mathcal{T}}$ is bounded, with norm bounded above by twice the width of $\mathcal{T}.$ When the tree is not narrow, this boundedness result no longer holds. Beyond several spectral properties, assuming $\mathcal{T}$ is leafless and narrow, we show that $C_{\mathcal{T}}$ is subnormal if and only if $\mathcal{T}$ 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, $C_{\mathcal{T}}$ is always a compact perturbation of a subnormal operator.

Revised Version. Substantial revision, 14 pages