  In [U] (among other results), M. Uchiyama gave the necessary and sufficient
conditions for contractions to be similar to the unilateral shift $S$ of
multiplicity $1$ in terms of norm-estimates of complete analytic families of
eigenvectors of their adjoints. In [G2], it was shown that this result for
contractions can't be extended to power bounded operators. Namely, a cyclic
power bounded operator was constructed which has the requested norm-estimates,
is a quasiaffine transform of $S$, but is not quasisimilar to $S$. In this
paper, it is shown that the additional assumption on a power bounded operator
to be quasisimilar to $S$ (with the requested norm-estimates) does not imply
similarity to $S$. A question whether the criterion for contractions to be
similar to $S$ can be generalized to polynomially bounded operators remains
open.
  Also, for every cardinal number $2\leq N\leq \infty$ a power bounded operator
$T$ is constructed such that $T$ is a quasiaffine transform of $S$ and
$\dim\ker T^*=N$. This is impossible for polynomially bounded operators.
Moreover, the constructed operators $T$ have the requested norm-estimates of
complete analytic families of eigenvectors of $T^*$.
