  Recently, the conception of slice regular functions was allowed to introduce
a new quaternionic functional calculus, among which the theory of semigroups of
linear operators was developed into the quaternionic setting, even in a more
general case of real alternative $*$-algebras. In this paper, we initiate to
study the discrete case and introduce the notion of power-bounded quaternionic
operators. In particular, by the spherical Yosida approximation, we establish a
discrete Hille-Yosida-Phillips theorem to give an equivalent characterization
of quaternionic linear operators being power-bounded. A sufficient condition of
the power-boundedness for quaternionic linear operators is also given. In
addition, a non-commutative version of the Katznelson-Tzafriri theorem (J.
Funct. Anal. 68: 313-328, 1986) for power-bounded quaternionic operators is
formulated in terms of the $S$-spectrum.
