A continuous linear
operator T on a Banach space X is said to have finite ascent if each operator
T − λ, λ ∈C, has stabilizing kernel, i.e. for some n (n may depend on λ)
ker(T − λ)n=ker(T − λ)n+1. Easy examples are provided by normal operators in
Hilbert space. For operators with finite ascent a description of the maximal algebraic
subspaces (defined below) is readily obtained, thus opening up the possibility of
automatic continuity theory involving these operators and their intertwiners. This
paper makes a beginning. We see that operators with finite ascent have the
single valued extension property. We also characterize some of the analytic
spectral subspaces, but the main results are obtained for certain subclasses of
operators with finite ascent. Three such classes are considered: a class enjoying
a polynomial growth condition, very recently introduced by Barnes, the
dominant operators in Hilbert space, studied by many authors, and, mostly, the
totally paranormal operators, introduced in §4. A main result is that if T
belongs to this class and if XT(F) denotes the analytic spectral subspace with
respect to the closed subset F ⊆ C then XT(F) is closed (Proposition 4.14). If
we restrict ourselves to Hilbert space and to such a T without eigenvalues
then the algebraic and the analytic spectral subspaces coincide (Proposition
4.15). This result allows us to tap into the already existing fund of automatic
continuity theory; the paper includes a sample, intended to illustrate the
possibilities.
