On the spectrum of the upper triangular double band matrix $U(a_{0},a_{1},a_{2}b_{0},b_{1},b_{2})$ over the sequence space $c$

Authors : Nuh DURNA, Rabia KILIÇ

Pages : 554-565

Doi:10.31801/cfsuasmas.977593

View : 6 | Download : 13

Publication Date : 2022-06-30

Article Type : Research Paper

Abstract :The upper triangular double band matrix $Uinsert ignore into journalissuearticles values(a_{0},a_{1},a_{2};b_{0},b_{1},b_{2});$ is defined on a Banach sequence space by  $Uinsert ignore into journalissuearticles values(a_{0},a_{1},a_{2};b_{0},b_{1},b_{2});insert ignore into journalissuearticles values(x_{n});=insert ignore into journalissuearticles values(a_{n}x_{n}+b_{n}x_{n+1});_{n=0}^{\infty}$ where $a_{x}=a_{y},~b_{x}=b_{y}$ for $x\equiv y~insert ignore into journalissuearticles values(mod3);$. The class of the operator $Uinsert ignore into journalissuearticles values(a_{0},a_{1},a_{2};b_{0},b_{1},b_{2});$ includes, in particular, the operator $Uinsert ignore into journalissuearticles values(r,s);$ when $a_{k}=r$ and $b_{k}=s$ for all $k\in\mathbb{N}$, with $r,s\in\mathbb{R}$ and $s\neq 0$. Also, it includes the upper difference operator; $a_{k}=1$ and $b_{k}=-1$ for all $k\in\mathbb{N}$. In this paper, we completely determine the spectrum, the fine spectrum, the approximate point spectrum, the defect spectrum, and the compression spectrum of the operator $Uinsert ignore into journalissuearticles values(a_{0},a_{1},a_{2};b_{0},b_{1},b_{2});$ over the sequence space $c$.
Keywords : Upper triangular band matrix, spectrum, fine spectrum, approximate point spectrum