  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.
