Abstract
This chapter discusses the asymptotic behavior of ( C 0 ) semigroups on Hilbert space. The semigroups of interest discussed in this chapter are the uniformly bounded and the contraction semigroups. The chapter discusses the application of the mean ergodic theorem in “big” Hilbert spaces. The first such space is the set of Hilbert–Schmidt operators on H . The second is the space of pseudo-periodic locally square integrable H -valued function on R . In the chapter, bound states and decaying and scattering states of a contraction semigroup are described. The existence and characterization of periodic and pseudo-periodic solutions of time dependent evolution equations are discussed in the chapter. An asymptotic equipartition of energy, scattering theory for abstract wave equations, and other equations of higher order in time are described in the chapter. Wiener's Theory is discussed and a result of this theorem is also described in the chapter.
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