Expansive operators which are power bounded or algebraic

Abstract

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 ${△}_{T^{\ast },T}^m(P)={\sum }_{j=0}^m(-1{)}^j\left(\begin{array}{c}m\\ j\end{array}\right){T^{\ast }}^{j}P{T}^j\le 0$ (resp., ${△}_{T^{\ast },T}^m(P)=0$). 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 ${△}_{T^{\ast },T}^n(Q)=0$ 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 $n\ge \frac{m_0+1}{2}$.

14 pages