The ascent and descent of weighted conditional expectation operators

TL;DR

This paper investigates the properties of weighted conditional expectation operators on $L^p$ spaces, establishing that their ascent and descent are finite and equal to 2, and explores conditions for power and Cesaro boundedness.

Contribution

It provides new results on the ascent, descent, and boundedness conditions of weighted conditional expectation operators, extending operator theory in this context.

Findings

Ascent of $M_wEM_u$ is always finite and equals 2.

Under weak conditions, descent of $M_wEM_u$ is finite and equals 2.

Characterization of Cesaro boundedness for $M_wEM_u$ and its relation to $ ilde{T}$.

Abstract

In this paper we prove that the ascent of a weighted conditional expectation operator of the form of $M_wEM_u$ on $L^p$-spaces is always finite and is equal to 2. Also we get that under a weak condition the descent of $M_wEM_u$ is finite and is equal to 2 too. Moreover, we give some necessary and sufficient conditions for $M_wE M_u$ to be power bounded. In the sequel we apply some results in operator theory on ascent and descent to $M_wEM_u$. Finally we find that $T=M_wEM_u$ is Cesaro bounded if and only if $\widehat{T}$ is Cesaro bounded.