Expansive operators and Drazin invertibility

Abstract

A Hilbert space operator $T\in B({{{\mathcal {H}}}})$ is (m, P)-expansive, for some integer $m\ge 1$ and positive operator $P\in B({{{\mathcal {H}}}})$, if $\sum _{j=0}^m{(-1)^j\left( \begin{array}{clcr}m\\ j\end{array}\right) T^{*j}PT^j}\le 0$. 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 $P=P_{11}\oplus 0$ and the Drazin inverse $T_d$ of T is (m, P)-contractive. In particular, a Drazin invertible (2, P)-expansive operator is P isometric. If T is $(m,|T^n|^2)$-expansive for some positive integer n, then $T^n$ has a decomposition $T^n=\left( \begin{array}{clcr}U_1P_1 & X\\ 0 & 0\end{array}\right)$, the operator $P_1U_1$ is m-expansive, the operator $P^{\frac{1}{2}}_1U_1P^{\frac{1}{2}}_1$ is m-expansive in an equivalent norm, the operator $\left( \begin{array}{clcr}P_1U_1 & P_1X\\ 0 & 0\end{array}\right)$ is (m, C)-expansive and the operator $\left( \begin{array}{clcr}P^{\frac{1}{2}}_1U_1P^{\frac{1}{2}}_1 & P_1^{\frac{1}{2}}X\\ 0 & 0\end{array}\right)$ is (m, D)-expansive for some operators $C, D\ge 0$.