On the fine spectrum of the operator Q(r,s,t,u) over ℓp and bvp, 1<p<∞

Abstract

We specify the fine spectrum of the quadruple band matrix operator $Q(r,s,t,u)$ over the sequence spaces ${\ell }_p$ and $b{v}_p$, where $1<p<\infty$. The quadruple band matrix $Q(r,s,t,u)$ is the general state of ${\mathrm{\Delta }}^3$, $D(r,0,0,s)$, $B(r,s,t)$, ${\mathrm{\Delta }}^2$, $B(r,s)$, $\mathrm{\Delta }$, right shift and Zweier matrices, where ${\mathrm{\Delta }}^3$, $B(r,s,t)$, ${\mathrm{\Delta }}^2$, $B(r,s)$ and $\mathrm{\Delta }$ are called third-order difference, triple band, second-order difference, double band (generalized difference) and difference matrix, respectively.