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.

Abstract


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.


Full Text: PDF

 

References


  1. A. Ammar and A. Jeribi, The weyl essential spectrum of a sequence of linear operators in Banach spaces, Indag. Math., New Ser. 27 (1) (2016), 282-295. Google Scholar

  2. S. Goldberg, Perturbations of semi-Fredholm operators by operators converging to zero compactly, Proc. Amer. Math. Soc. 45 (1974), 93-98. Google Scholar

  3. A. Jeribi, Spectral Theory and Applications of Linear Operators and Block Operator Matrices. Springer-Verlag,New York, 2015. Google Scholar

  4. A. Jeribi and N. Moalla, A characterisation of some subsets of Schechter’s essential spectrum and applications to singular transport equation, J. Math. Anal. Appl. 358 (2) (2009), 434-444. Google Scholar

  5. A.Jeribi, Une nouvelle caractrisation du spectre essentiel et application, Comp Rend.Acad.Paris serie I, 331 (2000),525-530. Google Scholar

  6. T. Kato, Perturbation theory for linear operators, Second edition. Grundlehren der Mathematischen Wissenschaften, Band 132. Springer-Verlag, Berlin-New York, 1966. Google Scholar

  7. K. Latrach, A. Jeribi, Some results on Fredholm operators, essential spectra, and application, J. Math. Anal. Appl. 225 (1998), 461-485. Google Scholar

  8. M. Schechter, Principles of Functional Analysis, Grad. Stud. Math. vol. 36, Amer. Math. Soc., Providence, 2002. Google Scholar

  9. M. Schechter, Riesz operators and Fredholm perturbations, Bull. Amer. Math. Soc. 74 (1968), 1139-1144. Google Scholar