The semigroup generated by a similarity orbit or a unitary orbit of an operator

Abstract

Abstract. Let T be an invertible operator that is not a scalar modulo the ideal of compact operators. We show that the multiplicative semigroup generated by the similarity orbit of T is the group of all invertible operators. If, in addition, T is a unitary operator, then the multiplicative semigroup generated by the unitary orbit of T is the group of all unitary operators.