# Trotter-Kato approximation theorem for uniformly continuous approximants

Let

• $$E$$ be a $$\mathbb R$$-Banach space
• $$(T_n(t))_{t\ge0}$$ and $$(T(t))_{t\ge0}$$ be strongly continuous contraction semigroups on $$E$$ with generators $$(\mathcal D(A_n),A_n)$$ and $$(\mathcal D(A),A)$$, respectively
• $$D$$ be a core of $$(\mathcal D(A),A)$$

Consider the following assertions:

1. $$D\subseteq\mathcal D(A_n)$$ for all $$n\in\mathbb N$$ and $$\left\|A_nx-Ax\right\|_E\xrightarrow{n\to\infty}0\;\;\;\text{for all }x\in E\tag1$$
2. For each $$x\in D$$ there is a sequence $$x_n\in\mathcal D(A_n)$$, n$$\in\mathbb N$$, with $$\left\|x_n-x\right\|_E+\left\|A_nx_n-Ax\right\|_E\xrightarrow{n\to\infty}0\tag2$$
3. For each bounded interval $$I\subseteq[0,\infty)$$ and $$x\in E$$, $$\sup_{t\in I}\left\|T_n(t)x-T(t)x\right\|_E\xrightarrow{n\to\infty}0\tag3$$

By the Trotter-Kato Approximation Theorem, the following implications hold: $$\text{1.}\Rightarrow\text{2.}\Leftrightarrow\text{3.}\tag4$$

Now assume $$(T_n(t))_{t\ge0}$$ is not a semigroup on $$E$$, but on another $$\mathbb R$$-Banach space $$E_n$$. Moreover, assume there is a bounded linear operator $$\iota_n:E\to E_n$$ such that $$\sup_{n\in\mathbb N}\left\|\iota_n\right\|<\infty\tag5$$ (if necessary, assume that $$\left\|\iota_n\right\|\le1$$). Now consider the following assertions:

1. For each $$x\in D$$ there is a sequence $$x_n\in\mathcal D(A_n)$$, n$$\in\mathbb N$$, with $$\left\|x_n-\iota_nx\right\|_{E_n}+\left\|A_nx_n-\iota_nAx\right\|_{E_n}\xrightarrow{n\to\infty}0\tag5$$
2. For each bounded interval $$I\subseteq[0,\infty)$$ and $$x\in E$$, $$\sup_{t\in I}\left\|T_n(t)\iota_nx-\iota_nT(t)x\right\|_{E_n}\xrightarrow{n\to\infty}0\tag6$$

Question 1: Are we able to infer the equivalence $$\text{4.}\Leftrightarrow{5.}\tag7$$ by the known result $$(4)$$ or do we need to mimic its proof from scratch?

Question 2: Is there an easier proof of $$\text{2.}\Leftrightarrow{3.}$$ (in the former setting) available, if we assume that each $$(T_n(t))_{t\ge0}$$ is even uniformly continuous (and hence each $$A_n$$ is bounded)?

In the context of question 2 I've got the Hille-Yosida approximation theorem in mind where something similar is shown. The crucial fact therein seems that $$A_n$$ and $$T(t)$$ commute. While this should be wrong in our general setting (is it?), there might still be an easier proof available.