On a Spectral Subdivision of the Operator $\Delta_i^2$ over the Sequence Spaces $c_0$ and $\ell_1$

P. Baliarsingh, S. Dutta

Abstract

The main objective ofthis paper is to determine the spectrum and the fine spectrum ofthe difference operator $\Delta_i^2$ over the sequence spaces $c_0$ and $\ell_1$. For any sequence $x=(x_k)$ in $c_0$ or $\ell_1$, the generalized difference operator $\Delta_i^2$ over $c_0$ or $\ell_1$ is defined by $(\Delta_i^2(x))_k =\sum_{i=0}^2\frac{(-1)^i}{i+1}\binom{2}{i}x_{k-i}=x_k-x_{k-1}+\frac{1}{3}x_{k-2}$, with $x_k = 0$ for $k < 0$. Moreover, we compute the spectrum, the point spectrum, the residual spectrum and the continuous spectrumof the difference operator $\Delta_i^2$ over the basic sequencespaces $c_0$ and $\ell_1$.