Operators with finite ascent (Q1188087)

The author studies operators with finite ascent. A continuous linear operator on a Banach space is said to have finite ascent if each operator \(T-\lambda\), \(\lambda\in \mathbb{C}\), has stabilizing kernel, i.e. for some \(n\) (\(n\) may depend on \(\lambda\)) \(\ker(T-\lambda)^ n= \ker(T- \lambda)^{n+1}\). The author considers the following three examples of operators with finite ascent: the class of linear operator which satisfy a polynomial growth condition, which was studied by \textit{B. A. Barnes} [Pac. J. Math. 138, No. 2, 209-219 (1989; Zbl 0693.47001)]; the dominant operators; and the totally paranormal operators (TPN). A main result is that if \(T\) be a TPN operator, then \(X_ T(F)\) is closed, where \(X_ T(F)\) denotes the analytic spectral subspace with respect to the closed subset \(F\subset \mathbb{C}\). The author also proves that if \(T\) is TPN in Hilbert space without eigenvalues then algebraic and analytic spectral subspaces coincide. At the end the author applies this result to automatic continuity theory.