In this article we introduce the generalized Fibonacci difference operator F(B)\documentclass[12pt]{minimal}
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\begin{document}$\mathsf{F}(\mathsf{B})$\end{document} by the composition of a Fibonacci band matrix F and a triple band matrix B(x,y,z)\documentclass[12pt]{minimal}
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\begin{document}$\mathsf{B}(x,y,z)$\end{document} and study the spaces ℓk(F(B))\documentclass[12pt]{minimal}
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\begin{document}$\ell _{k}( \mathsf{F}(\mathsf{B}))$\end{document} and ℓ∞(F(B))\documentclass[12pt]{minimal}
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\begin{document}$\ell _{\infty }(\mathsf{F}(\mathsf{B}))$\end{document}. We exhibit certain topological properties, construct a Schauder basis and determine the Köthe–Toeplitz duals of the new spaces. Furthermore, we characterize certain classes of matrix mappings from the spaces ℓk(F(B))\documentclass[12pt]{minimal}
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\begin{document}$\ell _{k}(\mathsf{F}(\mathsf{B}))$\end{document} and ℓ∞(F(B))\documentclass[12pt]{minimal}
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\begin{document}$\ell _{\infty }(\mathsf{F}(\mathsf{B}))$\end{document} to space Y∈{ℓ∞,c0,c,ℓ1,cs0,cs,bs}\documentclass[12pt]{minimal}
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\begin{document}$\mathsf{Y}\in \{\ell _{\infty },c_{0},c,\ell _{1},cs_{0},cs,bs\}$\end{document} and obtain the necessary and sufficient condition for a matrix operator to be compact from the spaces ℓk(F(B))\documentclass[12pt]{minimal}
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\begin{document}$\ell _{k}(\mathsf{F}(\mathsf{B}))$\end{document} and ℓ∞(F(B))\documentclass[12pt]{minimal}
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\begin{document}$\ell _{\infty }(\mathsf{F}(\mathsf{B}))$\end{document} to Y∈{ℓ∞,c,c0,ℓ1,cs0,cs,bs}\documentclass[12pt]{minimal}
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\begin{document}$\mathsf{Y}\in \{ \ell _{\infty }, c, c_{0}, \ell _{1},cs_{0},cs,bs\} $\end{document} using the Hausdorff measure of non-compactness.
