I am not an expert in semigroup theory, and I am studying a compact linear operator $A$ on a Banach space $X$ for which one can define the uniformly continuous semigroup $(e^{tA})_{t \geq 0}$.
Under the hypotheses above, one can prove the existence of a constant $M \geq 1$ such that
$$ \| e^{tA} \| \leq M e^{t \beta}, \quad t \geq 0, \quad \beta = \sup_{\lambda \in \sigma(A)} Re \lambda $$
I now wish to consider a sequence of compact linear operators $(A_n)_{n \in \mathbb{N}}$ on $X$ hence, for all $n \in \mathbb{N}$ there is a constant $M_n \geq 1$ such that
$$ \| e^{tA_n} \| \leq M_n e^{t \beta_n}, \quad t \geq 0, \quad \beta_n = \sup_{\lambda \in \sigma(A_n)} Re \lambda $$
If $A_n \to A$ in the operator norm, can anything be said about $M_n$ and $\beta_n$? Are the sequences $(M_n)_n$ and $(\beta_n)_n$ bounded? Do they converge and, if yes, do they converge to $M$ and $\beta$, respectively?