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I know that every analytic $C_0$-semigroup is differentiable and then every differentiable semigroup is norm continuous.

I want to know where uniform continuity fits in the above picture.

My intuition is that since the generator of a uniformly continuous semigroup is bounded, it is of the form $(e^{tA})_{t\geq 0}$ for some bounded operator $A$, and being the generalization of "exponential" it should certainly be analytic (or at least differentiable). On the other hand, the implication "uniform continuity implies analyticity" seems questionable.

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    $\begingroup$ The answer is yes and you find it in any book on semigroups. In few words we may define $e^{zA} $ by a power series. $\endgroup$
    – Giorgio Metafune
    Commented Apr 2, 2022 at 8:08
  • $\begingroup$ Something is wrong in your premisses, since $C^0$-semigroup are not necessarily norm-continuous. $\endgroup$
    – Denis Serre
    Commented Apr 2, 2022 at 9:59
  • $\begingroup$ @DenisSerre But I never said that they are? $\endgroup$
    – Guest
    Commented Apr 2, 2022 at 19:13
  • $\begingroup$ @GiorgioMetafune Thanks! $\endgroup$
    – Guest
    Commented Apr 2, 2022 at 19:13
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    $\begingroup$ @Guest You are not quoting correctly, in fact, uniform continuous isn't the same as immediately uniform continuous, the same for diffierentiable. $\endgroup$
    – ahdahmani
    Commented Mar 29, 2023 at 10:52

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