Drazin invertible $(m,P)$-expansive operators

Abstract

A Hilbert space operator $T\in B$ is $(m,P)$-expansive, for some positive integer $m$ and operator $P\in B$, if $\sum_{j=0}^m{(-1)^j\left(\begin{array}{clcr}m\\j\end{array}\right)T^{*j}PT^j}\leq 0$. No Drazin invertible operator $T$ can be $(m,I)$-expansive, and if $T$ is $(m,P)$-expansive for some positive operator $P$, then necessarily $P$ has a decomposition $P=P_{11}\oplus 0$. If $T$ is $(m,|T^n|^2)$-expansive for some positive integer $n$, then $T^n$ has a decomposition $T^n=\left(\begin{array}{clcr}U_1P_1 & X\\0 & 0\end{array}\right)$; if also $\left(\begin{array}{clcr}I_1 & X\\X^* & X^*X\end{array}\right)\geq I$, then $\left(\begin{array}{clcr}P_1U_1 & P_1X\\0 & 0\end{array}\right)$ is $(m,I)$-expansive and $\left(\begin{array}{clcr}P^{\frac{1}{2}}_1U_1P^{\frac{1}{2}}_1 & P_1^{\frac{1}{2}}X\\0 & 0\end{array}\right)$ is $(m,I)$-expansive in an equivalent norm on $H$.