### Fine Spectrum of the Generalized Difference Operator $\Delta_{uv}$ on the Sequence Space $c_0$

#### Abstract

The purpose of this paper is to determine spectrum and fine spectrum of the operator $\Delta_{uv}$ on the sequence space $c_0$. The operator $\Delta_{uv}$ on sequence space $c_0$ is defined as $\displaystyle \Delta_{uv} x=(u_n x_n + v_{n-1} x_{n-1})_{n=0}^{\infty}$ satisfying certain conditions, where $x_{-1}=0$ and $x=(x_n) \in c_0$. In this paper we have obtained the results on the spectrum and point spectrum for the operator $\Delta_{uv}$ on the sequence space $c_0$. Further, the results on continuous spectrum, residual spectrum and fine spectrum of the operator $\Delta_{uv}$ on sequence space $c_0$ are also derived.

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