Quasi-Compact Operators of Diagonal Type And Their Perturbations

Abstract

In this chapter, we gather the elements of Functional Analysis which will allow us to develop the method briefly sketched in Section 1.1. First, we state some important properties of quasi-compact operators whose powers behave asymptotically like those of finite rank diagonalizable operators. Then, we study the perturbations of certain of these operators. Except for Corollary 111.5 which concerns the study of a positive kernel, ß is a general Banach space and Q is a bounded linear operator on ß. The topological dual space of ß is denoted by ß'. If φ ∈ ß' and f ∈ ß, we set < φ,f >= φ(f). The adjoint of Q is denoted by Q*. An eigenvalue of Q with modulus r(Q) is called a peripheral eigenvalue.