  This paper considers the observability of nonlinear systems from a Koopman
operator theoretic perspective--and in particular--the effect of symmetry on
observability. We first examine an infinite-dimensional linear system
(constructed using independent Koopman eigenfunctions) such that its
observability is equivalent to the observability of the original nonlinear
system. Next, we derive an analytic relation between symmetry and nonlinear
observability; it is shown that symmetry in the nonlinear dynamics is reflected
in the symmetry of the corresponding Koopman eigenfunctions, as well as
presence of repeated Koopman eigenvalues. We then proceed to show that the loss
of observability in symmetric nonlinear systems can be traced back to the
presence of these repeated eigenvalues. In the case where we have a sufficient
number of measurements, the nonlinear system remains unobservable when these
functions have symmetries that mirror those of the dynamics. The proposed
observability framework provides insights into the minimum number of the
measurements needed to make an unobservable nonlinear system, observable. The
proposed results are then applied to a network of nano-electromechanical
oscillators coupled via a symmetric interaction topology.
