Riesz Projections and Isolated Points of the Spectrum

Abstract

In this and the following chapter, we will discuss the spectrum of a closed operator A. We know that we can form a disjoint decomposition of. The discrete spectrum of A, σ_d(A), consists of isolated eigenvalues of finite algebraic multiplicity, and σ_ess(A), the essential spectrum of A, is the remaining part of the spectrum. We will study σ_d(A) in this chapter through the projection operators that can be obtained from the resolvent of A for each distinct eigenvalue in σ_d(A). We develop the basic theory of these projections, called Riesz projections. These operators provide a powerful tool for the study of the discrete spectrum of closed operators.