for generators of polynomially bounded $C_0$-groups on Banach spaces, but only in the case when generators are bounded. In 2001 M. Malejki [@Malejki] obtained necessary and sufficient conditions for a closed densely defined operator on a Banach space to be the generator of polynomially bounded $C_0$-group, in terms of the behaviour of the resolvent. In 2005 T. Eisner generalized results of Malejki to the case of polynomially bounded $C_0$-semigroups [@Eisner2]. The keystone of main results of works [@Malejki], [@Eisner2] are certain conditions on the integrability for the resolvent $$\left(A - (a+i\cdot)I \right)^{-1},$$ or the square of it, along lines parallel to the imaginary axis, where $a>0$.

Finally, in 2007 T. Eisner and H. Zwart [@Eisner1] obtained more simple characterizations of polynomial growth of a $C_0$-semigroup in terms of the first power of the resolvent of the generator. This was done in the class of operators, which have $p$-integrable resolvent for some $p>1.$ This class includes $C_0$-semigroups on Hilbert spaces and analytic $C_0$-semigroups on Banach spaces, see [@Eisner1] for details. For the overview and the prehistory of these results we refer to the Chapter III of the monograph [@Eisner3], where open questions, useful remarks and illustrative examples may also be found.

