Expansive operators which are power bounded or algebraic

Given Hilbert space operators $P,T\in B(\H), P\geq 0$ invertible, $T$ is $(m,P)-$ expansive (resp., $(m,P)-$ isometric) for some positive integer $m$ if $\triangle_{T^*,T}^m(P)=\sum_{j=0}^m(-1)^j\left(\begin{array}{clcr}m\\j\end{array}\right){T^*}^jPT^j\leq 0$ (resp., $\triangle_{T^*,T}^m(P)=0$). An $(m,P)-$ expansive operator $T$ is power bounded if and only if it is a $C_{1\cdot}-$ operator which is similar to an isometry and satisfies $\triangle_{T^*,T}^n(Q)=0$ for some positive invertible operator $Q\in B(\H)$ and all integers $n\geq 1$. If, instead, $T$ is an algebraic $(m,I)-$ expansive operator, then either the spectral radius $r(T)$ of $T$ is greater than one or $T$ is the perturbation of a unitary by a nilpotent such that $T$ is $(2n-1, I)-$ isometric for some positive integers $m_0 \leq m$, $m_0$ odd, and $n \geq \frac{m_0 +1}{2}$.