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In short, the question is whether norm-resolvent convergence implies operator-norm convergence of the assocoated semigroups. More specifically, assume the following:

  1. The $A_n$ generate contraction semigroups for all $n$,
  2. $A$ generates a contraction semigroup,
  3. One has $\|(A_n+1)^{-1} - (A+1)^{-1}\|\to 0$ in operator norm.

Then, can one get anything better than strong convergence for the associated semigroups (preferably operator norm convergence for some fixed t>0)?

The answer is relatively simple for analytic semigroups, since in that case one has an integral formula that expresses the semigroup in terms of the resolvent. For general $C_0$ semigroups, it's much less obvious...

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    $\begingroup$ In general, no, at least not without some uniformity with respect to z. Note that $(tA-z)^{-1}$ is continuous in the operator norm for $t>0$ and any $z$ where it is defined. But $e^{-tA}$ need not be continuous in operator norm. $\endgroup$
    – Michael Renardy
    Commented Jul 8, 2020 at 17:13

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