Local spectral properties and \(\nu \)-convergence

In this paper we prove that if \(\left\{ T_{n}\right\} \) $\left\{T_n\right\}$ is a sequence of bounded linear operators on a complex Banach space X which converges to two different bounded linear operators T and U in \(\nu \) $\nu$ -convergence, then T and U share many local spectral properties. Furthermore, we investigate the spectral \(\nu \) $\nu$ -continuity for the local spectrum.