Resolvent and Spectrum

Abstract

Let T be a linear operator whose domain D(T) and range R(T) both lie in the same complex linear topological space X. We consider the linear operator

$${T_\lambda } = \lambda I - T,$$

, where λ is a complex number and I the identity operator. The distribution of the values of λ for which T _λ has an inverse and the properties of the inverse when it exists, are called the spectral theory for the operator T. We shall thus discuss the general theory of the inverse of T _λ .