Published online by Cambridge University Press: 18 December 2014
Let A be an invertible operator on a complex Banach space X. For a given α ≥ 0, we define the class $\mathcal{D}$Aα(ℤ) (resp.
$\mathcal{D}$Aα (ℤ+)) of all bounded linear operators T on X for which there exists a constant CT>0, such that
$\begin{equation*}\Vert A^{n}TA^{-n}\Vert \leq C_{T}\left( 1+\left\vertn\right\vert \right) ^{\alpha },\end{equation*}$
$\mathcal{D}$Aα (ℤ) in the case when the spectrum of A is real or is a singleton. If T ∈
$\mathcal{D}$A(ℤ) (=
$\mathcal{D}$A0(ℤ)), some estimates for the norm of AT-TA are obtained. Some results for the class
$\mathcal{D}$Aα (ℤ+) are also given.