Ascent and Descent of Weighted Composition Operators on Lorentz spaces

TL;DR

This paper investigates the conditions under which weighted composition operators on Lorentz spaces have finite or infinite ascent and descent, providing theoretical insights and supporting examples.

Contribution

It characterizes the ascent and descent of weighted composition operators on Lorentz spaces based on properties of the transformation and weight functions.

Findings

Conditions for finite and infinite ascent of operators identified

Criteria for finite and infinite descent established

Examples illustrating the theoretical results provided

Abstract

The aim of this article is to detect the ascent and descent of weighted composition operators on Lorentz spaces. We investigate the conditions on the measurable transformation $T$ and the complex-valued measurable function $u$ defined on measure space $(X, \mathcal{A}, \mu)$ that cause the weighted composition operators on Lorentz space $L(p, q)$, $1 < p \leq \infty, 1 \leq q \leq \infty$ to have finite or infinite ascent (descent). We also give a number of examples in support of our findings.