• $E$ 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.


2 Answers 2


You may check Section 4 (especially, Theorem 4.2 and Proposition 4.3) in the book:

Kazufumi Ito, Franz Kappel: Evolution Equations and Approximations, World Scientific (2002).


There is a freely accessible wersion of the results presented by Huji Kolp:

Ito, Kazufumi; Kappel, Franz, The Trotter-Kato theorem and approximation of PDEs, Math. Comput. 67, No. 221, 21-44 (1998). ZBL0893.47025.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.