Fine Spectra of Upper Triangular Triple-Band Matrices over the Sequence Space ℓ p ( 0 < p < ∞ )

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 $A(r,s,t)$ on sequence space on ${\ell }_p$ is defined by $A(r,s,t)x=(r{x}_k+s{x}_{k+1}+t{x}_{k+2}{)}_{k=0}^{\infty }$, where $x=(x_k)\in {\ell }_p$, with . In this paper we have obtained the results on the spectrum and point spectrum for the operator $A(r,s,t)$ on the sequence space ${\ell }_p$. Further, the results on continuous spectrum, residual spectrum, and fine spectrum of the operator $A(r,s,t)$ 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 $A(r,s,t)$ over the space ${\ell }_p$ and we give some applications.