Title: The Essential Spectrum of a Sequence of Linear Operators in Banach Spaces
Author(s): Aymen Ammar, Noui Djaidja, Aref Jeribi
Pages: 1-7
Cite as:
Aymen Ammar, Noui Djaidja, Aref Jeribi, The Essential Spectrum of a Sequence of Linear Operators in Banach Spaces, Int. J. Anal. Appl., 15 (1) (2017), 1-7.


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.

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