Weyl's Type Theorem and a Local Growth Condition

M. H. M. Rashid

Bulletin of the Malaysian Mathematical Society, Tome 36 (2013) no. 2

Résumé

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 Browder-Weyl theorems under perturbations by finite rank operators, by nilpotent operators and, more generally, by algebraic and Riesz operators commuting with $T$.

Détail

Classification : Primary 47A53, 47A55; Secondary 47A10, 47A11, 47A20