## Abstract and Applied Analysis

### Fine Spectra of Upper Triangular Triple-Band Matrices over the Sequence Space ${\ell }_{p}$ ($\mathrm{0})

#### Abstract

The fine spectra of lower triangular triple-band matrices have been examined by several authors (e.g., Akhmedov (2006), Başar (2007), and Furken et al. (2010)). Here we determine the fine spectra of upper triangular triple-band matrices over the sequence space ${\ell }_{p}$. The operator $A\left(r,s,t\right)$ on sequence space on ${\ell }_{p}$ is defined by $A\left(r,s,t\right)x=\left(r{x}_{k}+s{x}_{k+1}+t{x}_{k+2}{\right)}_{k=0}^{\infty }$, where $x=\left({x}_{k}\right)\in {\ell }_{p}$, with $0. In this paper we have obtained the results on the spectrum and point spectrum for the operator $A\left(r,s,t\right)$ on the sequence space ${\ell }_{p}$. Further, the results on continuous spectrum, residual spectrum, and fine spectrum of the operator $A\left(r,s,t\right)$ on the sequence space ${\ell }_{p}$ are also derived. Additionally, we give the approximate point spectrum, defect spectrum, and compression spectrum of the matrix operator $A\left(r,s,t\right)$ over the space ${\ell }_{p}$ and we give some applications.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 342682, 10 pages.

Dates
First available in Project Euclid: 27 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393511782

Digital Object Identifier
doi:10.1155/2013/342682

Mathematical Reviews number (MathSciNet)
MR3035358

Zentralblatt MATH identifier
06209258

#### Citation

Karaisa, Ali; Başar, Feyzi. Fine Spectra of Upper Triangular Triple-Band Matrices over the Sequence Space ${\ell }_{p}$ ( $\mathrm{0}&lt;p&lt;\infty$ ). Abstr. Appl. Anal. 2013 (2013), Article ID 342682, 10 pages. doi:10.1155/2013/342682. https://projecteuclid.org/euclid.aaa/1393511782

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