Index theorems for relatively strictly singular perturbations of infinitesimal generators and their application to stabilizability problems

A number of invariance principles for closed, densely defined linear operators are discussed and proved. The first invariance principle establishes an index theorem for closed, densely defined operators under relatively strictly singular perturbation. The second principle, established as a by-product of results on asymptotics, establishes the invariance of the characteristic of property of generating a C₀-semigroup under a negative spectrum shift by a relatively compact perturbation. The third invariance principle establishes the invariance of the characteristic number of a C₀-semigroup under a negative spectrum perturbation