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

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### 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.

### Most cited references4

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### A Spectral Theory for Locally Convex Algebras

(1965)
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### Semigroups of operators in locally convex spaces

(1968)
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### Strongly continuous semigroups on some Fréchet spaces

(2014)
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### Author and article information

###### Journal
2014-08-21
###### Article
10.1017/S0013091515000310
1408.5037

Primary 47D06, Secondary 46A04, 34G10
Proc. Edinb. Math. Soc. (2016) 59, no. 3, 801-810
6 pages
math.FA

Functional analysis