The ascent and descent of weighted conditional expectation operators

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