Let $T(t),t\geq 0$, be a $C\_0$-semigroup on a Banach space $X$. If $A$ is the infinitesimal generator of $T(t),t\geq 0$, then
$$T(t)x=\lim\_{n\infty}(I-\frac{t}{n}A)^{-n}x$$
for every $x \in X, t\geq 0$, and the limit is uniform on any bounded interval $[a,b]\subset [0,\infty)$. This is the exponential formula.


I wonder if one could affirm that the sequence $(I-\frac{t}{n}A)^{-n}$ converges in norm to $T(t)$ and the convergence is uniform on any bounded interval $[a,b]\subset [0,\infty)$?



On exponential formula for $C_0$ semigroups
If you have an analytic semigroup (generated by a so-called [sectorial operator](https://en.wikipedia.org/wiki/Analytic_semigroup)), then the answer is well-known and yes, even convergence rates are possible. This and generalizations were proved by many, see for example the summary and references in


[M. Crouzeix, S. Larsson, S. Piskarev, V. Thomée: The stability of rational approximations of analytic semigroups, BIT 33 (1993), 74-84](https://doi.org/10.1007/BF01990345), Theorem 3.


In general it is possible to obtain convergence rates for "nice" initial values. A nice summary of the existing results and vast generalizations are available in this paper by [Alexander Gomilko and Yuri Tomilov](https://arxiv.org/abs/1301.4406).


ADDED: After discussion with Delio, let me mention that the result (in Hilbert spaces) is implicitely contained in


[H. FUJITA & A. MIZUTANI,](https://doi.org/10.2969/jmsj/02840749) "On the finite element method for parabolic equations. I;
Approximation of holomorphic semi-groups," J. Math. Soc. Japan, v. 28, 1976, pp. 749-771.


