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 $c$

Abstract

The upper triangular double band matrix $U(a_{0},a_{1},a_{2};b_{0},b_{1},b_{2})$ is defined on a Banach sequence space by $U(a_{0},a_{1},a_{2};b_{0},b_{1},b_{2})(x_{n})=(a_{n}x_{n}+b_{n}x_{n+1})_{n=0}^{\infty}$where $a_{x}=a_{y},~b_{x}=b_{y}$ for $x\equiv y~(mod3)$. The class of the operator$U(a_{0},a_{1},a_{2};b_{0},b_{1},b_{2})$includes, in particular, the operator $U(r,s)$ 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 $U(a_{0},a_{1},a_{2};b_{0},b_{1},b_{2})$ over the sequence space $c$.