Abstract
This chapter covers fundamental results on the spectral theory, including Fredholm alternative theorem and Nussbaum’s theorem on the radius of essential spectrum for bounded linear operators; growth bound and essential growth bound of linear operators; the relationship between the spectrum of semigroups and the spectrum of their infinitesimal generators; spectral decomposition of the state space; and asynchronous exponential growth of linear operators. The estimates of growth bound and essential growth bound of linear operators will be used in proving the center manifold theorem in Chapter 10.1007/978-3-030-01506-0_6