In Chapter 27 we established many conditions which imply SVEP at a point. The main goal of this chapter is to show that all these implications become equivalences for an important class of operators, the class of all semi-Fredholm operators. It will be also shown that, for these operators, SVEP at a point λ0 ∈ ℂ is equivalent to the finiteness of two important quantities, the ascent and the descent of the operator λ0I – T. These equivalences also provide useful information on the fine structure of the spectrum. In particular, we shall show that many spectra originating from Fredholm theory coincide whenever T or T′ have SVEP.

Ascent, descent, and semi-Fredholm operators

Let us recall the definition of some classical quantities associated with an operator. Given a linear operator T on a vector space X, it is easy to see that ker for every n ∈ ℕ.

Definition 28.1.1The ascent of T is the smallest positive integer p = p(T), whenever it exists, such that ker Tp = ker Tp+1. If such p does not exist, set p = ∞. Analogously, the descent of T is defined to be the smallest integer q = q(T), whenever it exists, such that. If such q does not exist, set q = ∞. If both p(T) and q(T) are finite, then T has finite chains.