Irreducible operator semigroups such that AB and BA are proportional Original Research Article

Let image be an irreducible semigroup of compact linear operators on a Banach space of dimension at least 2 with the property that the products AB and BA are proportional for each pair of elements A and B in image. We show that image is necessarily a nilpotent matrix group of nilpotency class 2 with possible addition of the zero matrix. We also study generalizations of the property.