Spectrum, Resolvent Set and the Krein-Rutman Theorem

Abstract

In the last two chapters we shall discuss some spectral properties of a (linear) norm bounded operator T in a complex Banach space V; in particular we deal with the case that V is a Banach lattice and the operator T is positive. In the present section we restrict ourselves, however, to the case that V is a Banach space. Not everything will be proved. The reader is assumed to be familiar with (or at least to accept without full proof) some elementary results about compact sets and compact operators in a Banach space. We have to explain first what is meant by spectral properties of an operator T. We assume, therefore, that T is a norm bounded operator mapping V into itself, i.e., T ∈ ß (V) in the notation introduced in section 18. Recall that ß (V) is a Banach space (see Theorem 18.2). Choose a complex number λ and, for brevity, denote the operator T —λI(I the identity operator) by T_λ. Assume now that T_λ maps V onto the whole of V in a one-one way, so that therefore the inverse operator (T_λ)^-1 exists with domain V, and assume also that (T_λ)^-1 is norm bounded, so (T_λ)^-1 ∈ B(V). The set of complex numbers λ for which this situation occurs is called the resolvent set of T. We shall denote this set by ρ(T). The complementary set (that is to say, the set of all complex λ for which T — λI does not have a norm bounded inverse with domain V) is called the spectrum of T and denoted by σ(T). Operator spectra have been extensively investigated for many different classes of operators.