  In geometry group theory, one of the milestones is M. Gromov's polynomial
growth theorem: Finitely generated groups have polynomial growth if and only if
they are virtually nilpotent. Inspired by M. Gromov's work, we introduce the
growth types of weighted Hardy spaces. In this paper, we focus on the weighted
Hardy spaces of polynomial growth, which cover the classical Hardy space,
weighted Bergman spaces, weighted Dirichlet spaces and much broader. Our main
results are as follows. $(1)$ We obtain the boundedness of the composition
operators with symbols of analytic automorphisms of unit open disk acting on
weighted Hardy spaces of polynomial growth, which implies the multiplication
operator $M_z$ is similar to $M_{\varphi}$ for any analytic automorphism
$\varphi$ on the unit open disk. Moreover, we obtain the boundedness of
composition operators induced by analytic functions on the unit closed disk on
weighted Hardy spaces of polynomial growth. $(2)$ For any Blaschke product $B$
of order $m$, $M_B$ is similar to $\bigoplus_{1}^m M_z$, which is an
affirmative answer to a generalized version of a question proposed by R.
Douglas in 2007. $(3)$ We also give counterexamples to show that the
composition operators with symbols of analytic automorphisms of unit open disk
acting on a weighted Hardy space of intermediate growth could be unbounded,
which indicates the necessity of the setting of polynomial growth condition.
Then, the collection of weighted Hardy spaces of polynomial growth is almost
the largest class such that Douglas's question has an affirmative answer. $(4)$
Finally, we give the Jordan representation theorem and similarity
classification for the analytic functions on the unit closed disk as
multiplication operators on a weighted Hardy space of polynomial growth.
