  By means of a new technique, we develop further a discrete subordination
approach to the functional calculus of power bounded and Ritt operators
initiated by N. Dungey in [19]. This allows us to show, in particular, that
(infinite) convex combinations of powers of Ritt operators are Ritt. Moreover,
we provide a unified framework for several main results on discrete
subordination from [19] and answer a question left open in [19]. The paper can
be considered as a complement to [26] for the discrete setting.
