  We study two abstract scenarios, where an operator family has a certain
minimality property. In both scenarios, it is shown that norm, spectrum and
resolvent are the same for all family members. Both abstract settings are
illustrated by practically relevant examples, including discrete Schr\"odinger
operators with periodic, quasiperiodic, almost-periodic, Sturmian and
pseudo-ergodic potential. The main tool is the method of limit operators, known
from studies of Fredholm operators and convergence of projection methods. We
close by connecting this tool to the study of subwords of the operator
potential.
