Subordinated discrete semigroups of operators

Given a power-bounded linear operator T in a Banach space and a probability F on the non-negative integers, one can form a 'subordinated' operator S = ∑k≥0 F(k)Tk. We obtain asymptotic properties of the subordinated discrete semigroup (Sn : n = 1, 2, ⋯) under certain conditions on F. In particular, we study probabilities F with the property that S satisfies the Ritt resolvent condition whenever T is power-bounded. Examples and counterexamples of this property are discussed. The hypothesis of power-boundedness of T can sometimes be replaced by the weaker Kreiss resolvent condition.