Operators Whose Conjugation Orbits Satisfy Polynomial Growth Conditions

Let $A$ be a bounded linear operator on a complex Banach space $X\mathrm{.}$ For a given $\alpha \ge 0,$ we consider the class $$\mathcal{D}${A}^{\alpha }\left( $\mathbb{R}$ \right) $ofallboundedlinearoperators$T$on$X$ for which there exists a constant $C_T>0$, such that \begin{equation*} \left\Vert e^{tA}Te^{-tA}\right\Vert \leq C{T}\left( 1+\left\vert t\right\vert \right) ^{\alpha }, \text {} \forall t\in $\mathbb{R}$ \end{equation*} We present complete description of the class $$\mathcal{D}$_{A}^{\alpha }\left( $\mathbb{R}$ \right) $inthecasewhenthespectrumof$A$ consists of one point. These results are linked to the decomposability of $A\mathrm{.}$ Some estimates for the norm of the commutator $AT-TA$ are obtained in the case $0\le \alpha <1.$