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

The growth bound for strongly continuous semigroups on Fr\'echet spaces