Abstract
The main objective of the present article is to determine the spectrum and the fine spectrum of the generalized difference operator \(\Delta _\nu ^r\) over the sequence space \(c_0\), where the operator \(\Delta _\nu ^r\) denotes the triangular matrix \((a_{nk})\) defined by \(\displaystyle \Delta _\nu ^{r}(x) = ( \Delta _\nu ^{r}x_{k})_{k=0}^\infty =\left( \sum _{i=0}^{r}(-1)^{i}\left( {\begin{array}{c}r\\ i\end{array}}\right) \nu _{k-i}x_{k-i}\right) ,\) with \( x_{k} = \nu _{k} = 0\) for \(k<0\), where \( x\in c_0\), \(0\ne r,k\in \mathbb {N}_0=\{0,1,2,3,...\},\) the set of non negative integers and \( \nu = (\nu _k)\) is either constant or strictly decreasing sequence of positive real numbers satisfying certain conditions. Finally, we obtain the spectrum, the point spectrum, the residual spectrum and the continuous spectrum of the operator \(\Delta _\nu ^r\) over the sequence space \(c_0\).
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More From: Proceedings of the National Academy of Sciences, India Section A: Physical Sciences
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