Operators with bounded conjugation orbits

For a bounded invertible operator $A$ on a complex Banach space $X,$ let $B_A$ be the set of operators $T$ in $\mathcal{L} (X)$ for which $\sup_{n \geq 0} \Vert A^n T A^{-n}\Vert < \infty.$ Suppose that $Sp(A) = \{1\}$ and $T$ is in $B_A \cap B_{A^{-1}}.$ A bound is given on $\Vert ATA^{-1} - T\Vert$ in terms of the spectral radius of the commutator. Replacing the condition $T$ in $B_{A^{-1}}$ by the weaker condition $\Vert A^{-n} T A^n\Vert = o(e^{\epsilon\sqrt{n}}),$ as $n \to \infty$ for every $\epsilon>0$, an extension of the Deddens-Stampfli-Williams results on the commutant of $A$ is given.