On the stability of the spectral properties under commuting perturbations

kai Yan, Weigang Su, Xiaochun Fang


In this paper, we examine the stability of several spectral properties under commuting perturbations. In particular, we show that if $T\in L(X)$ is an isoloid operator satisfying generalized Weyl's theorem and if $F\in L(X)$ is a power finite rank operator that commutes with $T$, then generalized Weyl's theorem holds for $T+F$. In addition, we consider the permanence of Bishop's property $(\beta)$, at a point, under commuting perturbation that is an algebraic operator.

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