  Convergence of operators acting on a given Hilbert space is an old and well
studied topic in operator theory. The idea of introducing a related notion for
operators acting on arying spaces is natural. However, it seems that the first
results in this direction have been obtained only recently, to the best of our
knowledge. Here we consider sectorial operators on scales of Hilbert spaces. We
define a notion of convergence that generalises convergence of the resolvents
in operator norm to the case when the operators act on different spaces and
show that this kind of convergence is compatible with the functional calculus
of the operator and moreover implies convergence of the spectrum. Finally, we
present examples for which this convergence can be checked, including
convergence of coefficients of parabolic problems. Convergence of a manifold
(roughly speaking consisting of thin tubes) towards the manifold's skeleton
graph plays a prominent role, being our main application.
