When does strong convergence imply convergence in operator norm?

Question

I have a $C_0$-semigroup $(T_t)_{t\ge0}$ and I want to show

$\lim_{t \to \infty} T_t =0 $ with respect to the operator norm.

After some effort, I was able to prove

$\lim_{t \to \infty} T_t =0 $ with respect to the strong operator topology.

Now I know that in general strong convergence doesn't imply convergence in operator norm. Therefore my question is the following:

What additional conditions on $(T_t)_{t\geq 0}$ or its generator $A$ are sufficient for strong convergence to imply convergence in operator norm?

I would be happy to be provided with some references too.

Answer 1

Since your semigroup is $C_0$, if it satisfies some condition according to which "strong convergence implies convergence in operator norm", then it is uniformly continuous (which I suspect is not the case). Therefore, you probably should try another strategy. For this, the following result can be useful.

For a $C_0$-semigroup $(T(t))_{t\geq 0}$ on a Banach space $X$ with generator $A$, the following assertions are equivalent.

1. $\|T(t)\|\overset{t\to\infty}{\longrightarrow}0$ (uniform stability).
2. $e^{\varepsilon t}\|T(t)\|\overset{t\to\infty}{\longrightarrow}0$ for some $\varepsilon>0$ (uniform exponential stability).
3. $e^{\varepsilon t}\|T(t)x\|\overset{t\to\infty}{\longrightarrow}0$ for all $x\in X$ and some $\varepsilon>0$.
4. $\|T(t_0)\|< 1$ for some $t_0>0$.
5. $r(T(t_0))<1$ for some $t_0>0$, where $r$ is the spectral radius.
6. $\displaystyle \int_0^\infty\|T(t)x\|^p\, dt<\infty$ for some/all $p\in[1,\infty)$ and all $x\in X$ (Datko-Pazy Theorem).
7. If $(T(t))_{t\geq 0}$ is eventually norm continuous:
   $s(A)<0$, where $s$ is the spectral bound.
8. If $X$ is Hilbert:
   $\{\lambda\mid \operatorname{Re}\lambda>0\}\subset\rho(A)$ and $\displaystyle \sup_{\operatorname{Re}\lambda>0}\|(\lambda I-A)^{-1}\|<\infty$ (Gearhart-Prüss Theorem).
9. If $X$ is Hilbert and $(T(t)_{t\geq 0}$ is of contractions:
   $\mathbf{i}\mathbb{R}\subset\rho(A)$ and $\displaystyle \limsup_{|\lambda|\to\infty}\|(\mathbf{i}\lambda I-A)^{-1}\|<\infty$ (Gearhart-Prüss Theorem).

Source: Engel & Nagel and Liu & Zheng. Other equivalences can be found in Neerven.