Cesàro Operators on Rooted Directed Trees
Mankunikuzhiyil Abhinand, Sameer Chavan, Sophiya S. Dharan, Thankarajan Prasad
TL;DR
The paper extends the classical Cesàro operator to the setting of rooted directed trees by defining $C_{\\mathscr T}$ on $\\ell^2(V)$ with a depth-based averaging rule. It establishes a sharp boundedness criterion for narrow trees, giving precise norm bounds and revealing rich spectral structure, including when eigenvalues occur and the nature of the spectrum. A central result is that, under leafless and narrow assumptions, subnormality characterizes the simple, non-branching tree, showing limits to a direct analogue of the Kriete–Trutt theorem in this context, while still ensuring that $C_{\\mathscr T}$ is a compact perturbation of a subnormal operator. The paper also provides a finite-branching-index decomposition, reducing $C_{\\mathscr T}$ to a finite direct sum of $C_0$ plus compact perturbations, and offers detailed examples demonstrating sharpness and the necessity of hypotheses, along with open questions on norm behavior and essential normality.
Abstract
In this paper, we introduce and investigate the notion of the Cesáro 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 Cesáro 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 Cesáro operators on rooted directed trees. Nonetheless, under the same hypotheses, $C_{\mathscr T}$ is always a compact perturbation of a subnormal operator.