Spectral analysis using ascent, descent, nullity and defect

Abstract

In this paper the fine structure of the spectrum of a closed linear operator T is studied in terms of the ascent, descent, nullity and defect of the operators )~- T. Several characterizations of poles of the resolvent operator Ra(T) are obtained:and these are used to characterize certain classes of operators, e.g., the class of meromorphic operators. Much of the underlying algebraic theory was developed by A. E. Taylor [17] and M. A. Kaashoek [8]. Their notation will be used throughout this paper. 1. Algebraic Properties of Ascent, Descent, Nullity and Defect Let X be a linear space over the field of complex numbers and let T be a linear operator with domain ~(T) and range ~(T) in X. The ascent of T, ~(T), is the smallest nonnegative integer p such that the null manifolds Jf(T p) and A/,(Tp+ i) are equal. If no such p exists, set ~(T) = ~. The descent of T, 6(T), is the smallest nonnegative integer q such that the range spaces ~(T q) and ~(Tq+ 1) are equal. If no such q exists, set 6(T)= ~. The nullity of T, n(T), is the dimension of A/'(T). The defect of T, d(T), is the dimension of the quotient space X/~t(T). Let r(T)= {2~CI ~(2- T )=6(2- T )=0}. When X is a Banach space and T is a closed linear operator, r(T)= 0(T), the resolvent set of T.