On the fine spectra of the generalized rth difference operator Δvr on the sequence space l1

Abstract The main purpose of this paper is to determine the spectrum and the fine spectrum of the r th difference operator Δ ν r on sequence space l 1 . The operator Δ ν r over l 1 is defined by Δ ν r ( x ) = ( Δ ν r x k ) k = 0 ∞ = ( Δ ν r - 1 x k - Δ ν r - 1 x k - 1 ) k = 0 ∞ with x k = ν k = 0 for k 0 , where x = ( x k ) ∈ l 1 , r ∈ N and ν = ( ν k ) is either constant or strictly decreasing sequence of positive real numbers satisfying certain conditions.

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