In what sense is the limit in the definition of infinitesimal generator?

Question

Question. $Ax=\left.\frac{d^+}{dt}T(t)x\right|_{t=0}$ means that

\begin{align} \lim_{t\to 0^+} \left\| Ax-\frac{T(t)x-x}{t}\right\|_{X}=0? \end{align} Thanks.

Answer 1

Yes, unsurprisingly, your interpretation is correct.

One potential issue is, indeed, in what sense that limit exists or is taken. If the adjective is "uniform", then, yes, it should be in the uniform-operator-norm sense. But, in fact, there are many situations in which this is not the thing to do... and the limit should be in the (weaker...) "strong operator topology".