In this chapter, we investigate general spectral gap properties in \(L^{1}\) -spaces for a class of perturbed \(C_{0}\) -semigroups that differs significantly from the one considered in Chap.  3 . Specifically, we focus on spectral gap analysis of Desch’s semigroups introduced in Chap. 2. Unlike the setting of Chap.  3 , this class is not limited to contraction \(C_{0}\) -semigroups and offers a key technical advantage: the domain of the perturbed generator coincides with that of the unperturbed one. This class is analyzed using weak compactness arguments and finds applications in various settings discussed in the subsequent chapters. An additional advantage of the approach is that the construction simplifies considerably when the unperturbed semigroup is immediately norm-continuous, particularly when it is holomorphic. This situation arises in Chap.  6 , where the unperturbed semigroup is a multiplication semigroup, as well as in the context of Kolmogorov differential equations studied in Chap.  9 , and the neutron diffusion equations addressed in Chap.  10 . We also present two methods for extending the construction to \(L^{p}\) -spaces.