  This paper extends two recent improvements in the Hilbert space setting of
the well-known Katznelson-Tzafriri theorem by establishing both a version of
the result valid for bounded representations of a large class of abelian
semigroups and a quantified version for contractive representations. The paper
concludes with an outline of an improved version of the Katznelson-Tzafriri
theorem for individual orbits, whose validity extends even to certain unbounded
representations.
