On the fine spectrum of the generalized difference operator B(r, s) over the sequence spaces c0 and c

By B(X), we also denote the set of all bounded linear operators on X into itself. If X is any Banach space and T ∈ B(X), then the adjoint T∗ of T is a bounded linear operator on the dual X∗ of X defined by (T∗ f )(x)= f (Tx) for all f ∈ X∗ and x ∈ X . LetX = {θ} be a nontrivial complex normed space and T : (T)→ X a linear operator defined on a subspace (T)⊆ X . We do not assume thatD(T) is dense inX , or that T has closed graph {(x,Tx) : x ∈ D(T)} ⊆ X ×X . We mean by the expression “T is invertible” that there exists a bounded linear operator S : R(T)→ X for which ST = I on D(T) and R(T)= X ; such that S= T−1 is necessarily uniquely determined, and linear; the boundedness of Smeans that T must be bounded below, in the sense that there is k > 0 for which ‖Tx‖ ≥ k‖x‖ for all x ∈D(T). Associated with each complex number, α is the perturbed operator