Expansive operators which are power bounded or algebraic

Given Hilbert space operators $P,T\in B(\stackrel{˝}{)},P\ge 0$ 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\mathrm{.},$\triangle_{T^*,T}^m(P)=0$)\mathrm{.}An$(m,P)-$expansiveoperator$T$ is power bounded if and only if it is a $C_{1\cdot }-$ operator which is similar to an isometry and satisfies ${\mathrm{△}}_{T^{\ast },T}^n(Q)=0$ for some positive invertible operator $Q\in B(\stackrel{˝}{)}$ and all integers $n\ge 1$. 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 $m_0\le m$, $m_0$ odd, and $n\ge \frac{m_0+1}{2}$.