  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.
