  In this study, we present some differences that arise in the spectral
analysis of pseudodifferential operators with constant coefficients, when we
use the Fr\'echet structure instead of the Banach structure.
  Here we show that this change in the topology implies in significant changes
on the behavior of the Laplace's spectrum, for instance, the resolvent set
vanishes, even with a bounded domain.
  The notion of exponential dichotomy for Fr\'echet Spaces introduced by the
author and its connection with the separation of spectrum and dichotomy
inspired us to make this work.
