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.

On Drazin invertibility

“…In the case of the Banach algebra L(X), T ∈ L(X) is said to be Drazin invertible (with a finite index) precisely when p(T ) = q(T ) < ∞ and this is equivalent to saying that T = T 0 ⊕ T 1 , where T 0 is invertible and T 1 is nilpotent; see [19 The concept of Drazin invertibility for bounded operators may be extended as follows.…”

Holling's ?hungry mantid? model for the invertebrate functional response considered as a Markov process. III. Stable satiation distribution

“…:P 0. There exists an s E ( w, c + w) and an The following result, stated in [12], enables us to give a more complete description of u(Aw) n n. Hence every boundary point of u(Aw) n l1 is isolated. As a consequence there are two possibilities:…”

“…p(L) is the resolvent set, and r(L) the spectral radius. N(L) and R(L) are the nullspace and range [12,18] …”

Weyl Type Theorems for Unbounded Hyponormal Operators

“…For T ∈ C(H), an isolated point λ ∈ σ(T) is said to be a pole of order p if p = p(T -λI) < ∞ and q(T -λI) < ∞ [12].…”