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 L(X) for which sup(n greater than or equal to 0) parallel to A(n)TA(-n) parallel to < infinity. Suppose that Sp(A) = {1} and T is in B-A boolean AND BA-1. A bound is given on parallel to AT A(-1) - T parallel to in terms of the spectral radius of the commutator. Replacing the condition T in BA-1 by the weaker condition parallel to A(-n)TA(n)parallel to = o(e(epsilon root n)), as n --> infinity for every epsilon > 0, an extension of the Deddens-Stampfli-Williams results on the commutant of A is given.