Suppose A is a bounded
operator on the Banach space ℬ such that A or A∗ is one-to-one. In this note, we
point out a relation between the commutant of A, the commutants of its powers, and
operators which intertwine A and λA, where λ is a root of unity. A consequence of
this relation is that the commutants of A and An are different if and only if there is
an operator Y , not zero, that satisfies AY = λY A, where λn= 1, λ≠1. Combining
this with Rosenblum’s theorem, we see that if the spectra of A and XA are disjoint,
the commutant of A is the same as that of A2. We also use the theorem to give a
counterexample to a conjecture of Deddens concerning intertwining analytic Toeplitz
operators.
