Spectral subspaces of subscalar and related operators
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- by T. L. Miller, V. G. Miller and M. M. Neumann PDF
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Abstract:
For a bounded linear operator $T\in L(X)$ on a complex Banach space $X$ and a closed subset $F$ 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 $T$ 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 $p,$ where $E_{T}(F)$ denotes the largest linear subspace $Y$ of $X$ for which $\left ( \lambda -T\right ) Y=Y$ for all $\lambda \in \mathbb {C} \setminus F.$ Moreover, for a wide class of operators $T$ that satisfy growth conditions of polynomial or Beurling type, it is shown that $X_{T}(F)$ is closed and equal to $E_{T}(F).$References
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Additional Information
- 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
- M. M. Neumann
- Affiliation: Department of Mathematics and Statistics, Mississippi State University, PO Drawer MA, Mississippi State, Mississippi 39762
- Email: neumann@math.msstate.edu
- Received by editor(s): August 22, 2002
- Received by editor(s) in revised form: January 14, 2003
- Published electronically: October 3, 2003
- Communicated by: Joseph A. Ball
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 1483-1493
- MSC (2000): Primary 47A11; Secondary 47B37, 47B40
- DOI: https://doi.org/10.1090/S0002-9939-03-07217-4
- MathSciNet review: 2053356