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Zbl 0783.47028
Laursen, K.B.
Operators with finite ascent. (English)
[J] Pac. J. Math. 152, No.2, 323-336 (1992). ISSN 0030-8730

The author studies operators with finite ascent. A continuous linear operator on a Banach space is said to have finite ascent if each operator $T-\lambda$, $\lambda\in \bbfC$, has stabilizing kernel, i.e. for some $n$ ($n$ may depend on $\lambda$) $\ker(T-\lambda)\sp n= \ker(T- \lambda)\sp{n+1}$. 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 {\it 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 $X\sb T(F)$ is closed, where $X\sb T(F)$ denotes the analytic spectral subspace with respect to the closed subset $F\subset \bbfC$. 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.
[V.Stukopin (Rostov-na-Donu)]

MSC 2000:: *47A75 Eigenvalue problems (linear operators)
47A53 (Semi-)Fredholm operators; index theories
46H40 Automatic continuity

Keywords: operators with finite ascent; stabilizing kernel; polynomial growth condition; dominant operators; totally paranormal operators; analytic spectral subspace; automatic continuity

Citations: Zbl 0693.47001

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