Operators Whose Conjugation Orbits Satisfy Polynomial Growth Conditions

Let $A$ be a bounded linear operator on a complex Banach space $X.$ For a given $\alpha \geq 0,$ we consider the class $\mathcal{D}_{A}^{\alpha }\left( \mathbb{R} \right)$ of all bounded linear operators $T$ 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 $\mathcal{D}_{A}^{\alpha }\left( \mathbb{R} \right)$ in the case when the spectrum of $A$ consists of one point. These results are linked to the decomposability of $A.$ Some estimates for the norm of the commutator $AT-TA$ are obtained in the case $0\leq \alpha <1.$