Sequence spaces derived by the triple band generalized Fibonacci difference operator

Abstract

AbstractIn this article we introduce the generalized Fibonacci difference operator$\mathsf{F}(\mathsf{B})$F(B)by the composition of a Fibonacci band matrix and a triple band matrix$\mathsf{B}(x,y,z)$B(x,y,z)and study the spaces$\ell _{k}( \mathsf{F}(\mathsf{B}))$ℓk(F(B))and$\ell _{\infty }(\mathsf{F}(\mathsf{B}))$ℓ∞(F(B)). 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$\ell _{k}(\mathsf{F}(\mathsf{B}))$ℓk(F(B))and$\ell _{\infty }(\mathsf{F}(\mathsf{B}))$ℓ∞(F(B))to space$\mathsf{Y}\in \{\ell _{\infty },c_{0},c,\ell _{1},cs_{0},cs,bs\}$Y∈{ℓ∞,c0,c,ℓ1,cs0,cs,bs}and obtain the necessary and sufficient condition for a matrix operator to be compact from the spaces$\ell _{k}(\mathsf{F}(\mathsf{B}))$ℓk(F(B))and$\ell _{\infty }(\mathsf{F}(\mathsf{B}))$ℓ∞(F(B))to$\mathsf{Y}\in \{ \ell _{\infty }, c, c_{0}, \ell _{1},cs_{0},cs,bs\} $Y∈{ℓ∞,c,c0,ℓ1,cs0,cs,bs}using the Hausdorff measure of non-compactness.