Spectral Theory of Linear Operators

Abstract

Let A be a linear operator which maps a usually complex linear vector space U into itself. Thus both its domain D(A) and its range R(A) are contained in U. We consider a pencil of operators A _λ = λI − A generated by scalar numbers λ ∈ ℂ where I is the identity operator on U. By carefully examining this pencil a lot of information can be extracted about the structure of the operator A. The principal task of the spectral theory of linear operators is to determine the set of allowable complex numbers λ, which is called the resolvent set, assuring the existence of a bounded inverse of the operator A _λ and to illuminate the relations between this set and its complement in the complex plane called the spectrum and those between the operators A and A _λ ⁻¹, if the latter exists. It is quite natural to expect that these relations will generally be affected by the topology chosen on the vector space. Therefore, we must anticipate that the spectral theory will have somewhat different characteristics on two fundamentally important topological vector spaces, namely, normed spaces and inner product spaces. In order to be precise, we could say that we would observe additional remarkable features in inner product spaces. We shall see that we shall only be capable to develop an interesting and conceptually appealing spectral theory mostly for closed, continuous or compact operators.