The growth bound for strongly continuous semigroups on Fréchet spaces

Abstract: We introduce the concepts of growth and spectral bound for strongly continuous semigroups acting on Fr\'echet spaces and show that the Banach space inequality $s(A)\leqslant\omega_0(T)$ extends to the new setting. Via a concrete example of an even uniformly continuous semigroup we illustrate that for Fr\'echet spaces effects with respect to these bounds may happen that cannot occur on a Banach space.