Operators with bounded conjugation orbits

Abstract:

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 $\sup _{n \geq 0} \|A^n T A^{-n}\| < \infty .$ Suppose that $Sp(A) = \{1\}$ and $T$ is in $B_A \cap B_{A^{-1}}.$ A bound is given on $\|ATA^{-1} - T\|$ in terms of the spectral radius of the commutator. Replacing the condition $T$ in $B_{A^{-1}}$ by the weaker condition $\|A^{-n} T A^n\| = o(e^{\epsilon \sqrt {n}}),$ as $n \to \infty$ for every $\epsilon >0$, an extension of the Deddens-Stampfli-Williams results on the commutant of $A$ is given.