This chapter delves into perturbation theory for compact operators. The material collected here will subsequently furnish some of the tools that will be needed for establishing large sample properties associated with methods for principle components estimation in Chapter 9.
The definitive treatise on operator perturbation theory is that of Kato (1995). Our particular treatment of this topic focuses on two scenarios that parallel the developments in Chapter 4 and is partly motivated by the results in Dauxious, Pousse, and Romain (1982), Hall and Hosseini-Nasab (2005, 2009), and Riesz and Sz.-Nagy (1990).
First, in Section 5.1, we consider the more standard case of self-adjoint, compact operators on a Hilbert space. In that setting, we obtain bounds and expansions that allow us to measure the effect that perturbing an operator will have on its eigenvalues and eigenvectors. Section 5.2 then explores the more complicated case of operators that are not self-adjoint and presents similar results to those of Section 5.1 for singular values and vectors.
5.1 Perturbation of self-adjoint compact operators
Theorems 4.2.8 and 4.5.3 represent anomalies of sorts when viewed in terms of the overall theme of Chapter 4. They both dealt with the eigenvalues of two operators rather than just one and then provided bounds for the difference in the operators' eigenvalues in terms of some measure of the size of the overall difference between the two operators. Results ...