Operators generated by Morse-Smale mappings

TL;DR

This paper investigates weighted shift operators induced by Morse-Smale mappings in $L^2$ spaces, providing conditions for invertibility and closedness of the image of $B-\lambda I$, using a novel graph decomposition approach.

Contribution

It introduces a new graph-based framework to analyze spectral properties of operators generated by Morse-Smale mappings, offering necessary and sufficient conditions for invertibility.

Findings

Characterization of invertibility of $B-\lambda I$

Conditions for the non-closedness of $Im(B-\lambda I)$

Introduction of oriented graph decomposition for spectral analysis

Abstract

Weighted shift operators $B$ in space $L^2(X,\mu)$ that are induced by Morse-Smale type of mappings are considered. A description of the properties of $B-\lambda I$ for $\lambda$ belonging to spectrum $\Sigma(B)$ is given. In particular, there is the necessary and sufficient condition that $B-\lambda I$ be a one-sided invertible and the condition that set $Im(B-\lambda I)$ be non-closed. These conditions use a new notation: an oriented decomposition of oriented graph $G(X,\alpha)$ generated by mapping $\alpha$