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The main idea of the present paper is to compute the spectrum and the fine spectrum of the generalized difference operator  over the sequence spaces . The operator  denotes a triangular sequential band matrix  defined by  with  for , where  or , ; the set nonnegative integers and  is either a 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  over the sequence spaces  and . These results are more general and comprehensive than the spectrum of the difference operators , , ,  , and   and include some other special cases such as the spectrum of the operators , , and  over the sequence spaces  or

Some Spectral Aspects of the Operator over the Sequence Spaces and