




Volume 36 • Number 2 • 2013 

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Weyl's Type Theorem and a Local Growth Condition
M. H. M. Rashid
Abstract.
A bounded linear operator $T\in\mathbf{L(\mathbb{X})}$ acting on a Banach space satisfies a local growth condition of order $m$ for some positive integer $m,$ $T\in \text{loc}(G_m)$, if for every closed subset $F$ of the set of complex numbers and every $x$ in the glocal spectral subspace $\mathbb{X}_T (F)$ there exists an analytic function $f:\mathbb{C}\setminus F\rightarrow \mathbb{X}$ such that $(T\lambda I)f(\lambda)\equiv x$ and $\left\Vert{f(\lambda)}\right\Vert\leq M\left[\text{dist}(\lambda,F)\right]^{m}\left\Vert{x}\right\Vert$ for some $M>0$ (independent of $F$ and $x$). In this paper, we study the stability of generalized BrowderWeyl theorems under perturbations by finite rank operators, by nilpotent operators and, more generally, by algebraic and Riesz operators commuting with $T$.
2010 Mathematics Subject Classification: Primary 47A53, 47A55; Secondary 47A10, 47A11, 47A20
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