Spectral subspaces of subscalar and related operators

Miller, T.; Miller, V.; Neumann, M.

doi:10.1090/S0002-9939-03-07217-4

Spectral subspaces of subscalar and related operators
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by T. L. Miller, V. G. Miller and M. M. Neumann PDF

Proc. Amer. Math. Soc. 132 (2004), 1483-1493 Request permission

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 $E_{T}(F).$

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