Proceedings of the American Mathematical Society

Author(s): T. L. Miller; V. G. Miller; M. M. Neumann
Journal: Proc. Amer. Math. Soc. 132 (2004), 1483-1493.
MSC (2000): Primary 47A11; Secondary 47B37, 47B40
Posted: October 3, 2003
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Abstract: For a bounded linear operator $T\in L(X)$ on a complex Banach space

and a closed subset

of the complex plane $\mathbb{C},$ this note deals with algebraic representations of the corresponding analytic spectral subspace $X_{T}(F)$ from local spectral theory. If

is the restriction of a generalized scalar operator to a closed invariant subspace, then it is shown that $X_{T}(F)=E_{T}(F)=\bigcap_{\hspace{0.03cm}\lambda \notin F}\left( \lambda -T\right) ^{\hspace{0.03cm}p}X$ for all sufficiently large integers

where $E_{T}(F)$ denotes the largest linear subspace

for which $\left( \lambda -T\right) Y=Y$ for all $\lambda \in \mathbb{C} \setminus F.$ Moreover, for a wide class of operators

that satisfy growth conditions of polynomial or Beurling type, it is shown that $X_{T}(F)$ is closed and equal to $E_{T}(F).$

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T. L. Miller
Affiliation: Department of Mathematics and Statistics, Mississippi State University, PO Drawer MA, Mississippi State, Mississippi 39762
Email: miller@math.msstate.edu

V. G. Miller
Affiliation: Department of Mathematics and Statistics, Mississippi State University, PO Drawer MA, Mississippi State, Mississippi 39762
Email: vivien@math.msstate.edu

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