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

In this paper we show that if \(\{T_n\}\) $\{T_n\}$ is a sequence of bounded linear operators on a complex Banach space X which \(\nu \) $\nu$ -converges to two different bounded linear operators T and U, then T and U have the same parts of the spectrum. In particular, we generalize the results of Sánchez-Perales and Djordjević (J Math Anal Appl 433:405–415, 2016) and of Ammar (Indag Math 28:424–435, 2017). We also investigate the spectral \(\nu \) $\nu$ -continuity for the surjective spectrum.