Operators with finite ascent

A continuous linear operator T on a Banach space X is said to have finite ascent if each operator T, C, has stabilizing kernel, i.e. for some n ( n may depend on ) ker() n = ker() n+ . Easy examples are provided by normal operators in Hubert 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 Hubert 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 X(F) denotes the analytic spectral subspace with respect to the closed subset FCC then X(F) is closed (Proposition 4.14). If we restrict ourselves to Hubert 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.