# Article

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Keywords:
strongly continuous semigroups; Riesz operators; polynomially Riesz operators
Summary:
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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