Sectoriality and essential spectrum of non symmetric graph Laplacians

TL;DR

This paper investigates the spectral properties of a non self-adjoint Laplacian on directed graphs, establishing conditions for sectoriality and comparing its essential spectrum with a related self-adjoint operator.

Contribution

It introduces necessary conditions for sectoriality of non symmetric graph Laplacians and compares their essential spectrum with a constructed self-adjoint operator.

Findings

Necessary conditions for sectoriality are established.

A comparison between the essential spectra of non self-adjoint and self-adjoint operators is provided.

The paper advances understanding of spectral properties of directed graph Laplacians.

Abstract

We consider a non self-adjoint Laplacian on a directed graph with non symmetric edge weights. We give necessary conditions for this Laplacian to be sectorial. We introduce a special self-adjoint operator and compare its essential spectrum with that of the non self-adjoint Laplacian considered.