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\vert
n\right\vert \right) ^{\alpha },
\end{equation*}
$
for all
n ∈ ℤ (resp.
n∈ ℤ
+). We present a complete description of the class
$\mathcal{D}$
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.