**Abstract:** We establish the convergence of pseudospectra for closed operators acting in
different Hilbert spaces and converging in the generalised norm resolvent
sense. As an assumption, we exclude the case that the limiting operator has
constant resolvent norm on an open set. We extend the class of operators for
which it is known that the latter cannot happen by showing that if the
resolvent norm is constant on an open set, then this constant is the global
minimum. We present examples exhibiting various resolvent norm behaviours
and illustrating the applicability of our results.

The talk is based on:
S. Bögli and P. Siegl: Remarks on the convergence of pseudospectra. Integral
Equation Operator Theory 80 (2014), 303-321.