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 ${\sum }_{j=0}^m(-1{)}^j\left(\begin{array}{c}m\\ j\end{array}\right)T^{\ast j}P{T}^j\le 0$. 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 $T^n=\left(\begin{array}{cl}{U}_1{P}_1& X\\ 0& 0\end{array}\right)$; if also $\left(\begin{array}{cl}{I}_1& X\\ X^{\ast }& X^{\ast }X\end{array}\right)\ge I$, then $\left(\begin{array}{cl}{P}_1{U}_1& P_{1}X\\ 0& 0\end{array}\right)$ is $(m,I)$-expansive and $\left(\begin{array}{cl}{P}_1^{\frac{1}{2}}{U}_1{P}_1^{\frac{1}{2}}& P_1^{\frac{1}{2}}X\\ 0& 0\end{array}\right)$ is $(m,I)$-expansive in an equivalent norm on $H$.