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.