The Essential Spectrum of a Sequence of Linear Operators in Banach Spaces

• Main Authors: Aymen Ammar, Noui Djaidja, Aref Jeribi
• Format: Article
• Language: English
• Published: Etamaths Publishing 2017-09-01
• Series: International Journal of Analysis and Applications
• Online Access: http://etamaths.com/index.php/ijaa/article/view/1311

<p>In this work we introduce some essential spectra $(\sigma_{ei}, i=1,...,5)$ of a sequence of closed linear operators $(T_{n})_{n\in\mathbb{N}}$ on Banach space, we prove that if $(T_{n})_{n\in\mathbb{N}}$ converges in the generalized sense to a closed linear operator $T$, then there exists $n_{0}\in \mathbb{N}$ such that, for every $n\geq n_{0}$, we have $\sigma_{ei}(\lambda _{0}-(T_{n}+B))\subseteq \sigma _{ei}(\lambda_{0}-(T+B)), i=1,...,5$, where $B$ is a bounded linear operator, and $\lambda _{0}\in \mathbb{C}$. The same treatment is made when $(T_{n}-T)$ converges to zero compactly.</p>