We establish the convergence of pseudospectra in Hausdorff distance 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 a number of examples exhibiting various resolvent norm behaviours and illustrating the applicability of this characterisation compared to known results.
Boegli, S., & Siegl, P. (2014). Remarks on the Convergence of Pseudospectra. Integral Equations and Operator Theory, 80(3), 303-321. https://doi.org/10.1007/s00020-014-2178-1