n-th roots of a non-negative operator. Conditions for uniqueness

In the present paper we study which restrictions must be imposed on the n-th roots of certain non-negative closed operators A on a Banach space so that these roots are unique.Counterexamples are given to show that the two results on this subject in the previous literature are incorrect.Finally, we obtain an explicit formula relating the canonical root A^1/n to another given non-negative n-th root B, and this allows us to establish the conditions for a given element to yield the same value by both roots. This point of view, which had not been considered up to now, provides simple conditions for global uniqueness that need only to be checked on the subspaces D(A)=_n1D(Aⁿ) and R(A)=_n1R(Aⁿ).