### Spectral inclusions and stability results for strongly continuous semigroups.

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In this paper we characterize the class of polynomially Riesz strongly continuous semigroups on a Banach space $X$. Our main results assert, in particular, that the generators of such semigroups are either polynomially Riesz (then bounded) or there exist two closed infinite dimensional invariant subspaces ${X}_{0}$ and ${X}_{1}$ of $X$ with $X={X}_{0}\oplus {X}_{1}$ such that the part of the generator in ${X}_{0}$ is unbounded with resolvent of Riesz type while its part in ${X}_{1}$ is a polynomially Riesz operator.

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