Does strong convergence of generators imply the strong convergence of the corresponding semigroups?

Question

Let

• $E$ be a $\mathbb R$-Banach space
• $(T_n(t))_{t\ge0}$ and $(T(t))_{t\ge0}$ be $C^0$-semigroups on $E$ with generator $(\mathcal D(A_n),A_n)$ and $(\mathcal D(A),A)$, respectively, for $n\in\mathbb N$

Suppose $\mathcal D(A_n)\subseteq\mathcal D(A)$ for all $n\in\mathbb N$ and $$\left\|A_nx-Ax\right\|_E\xrightarrow{n\to\infty}0\;\;\;\text{for all }x\in\mathcal D(A)\tag1.$$ Are we able to conclude $$\left\|T_n(t)x-T(t)x\right\|_E\xrightarrow{n\to\infty}0\;\;\;\text{for all }x\in E\tag2$$ for all $t>0$? If not, are there mild conditions which allow the conclusion?

Answer 1

As mentioned in the comments, you have to be careful with the domains of the generators, but morally it is true that strong convergence of the generators implies strong convergence of the semigroups. Here are sufficient conditions to make it rigorous (you can probably weaken them):

Assume that

• there are constants $M,\beta>0$ such that $\lVert T_n(t)\rVert\leq Me^{\beta t}$ for $t\geq 0$, $n\in\mathbb{N}$,
• there is a core $D_0$ of $A$ contained in $D(A_n)$ for all $n\in\mathbb{N}$ such that $A_n x\to A x$ for $x\in D_0$.

Then $T_n(t)\to T(t)$ strongly for all $t\geq 0$ (in fact uniformly on compact intervals).

For reference see Kato, Perturbation Theory, Theorems VIII.1.5 and IX.2.16.