Sequence spaces derived by the triple band generalized Fibonacci difference operator

Abstract

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

Keywords

• Fibonacci difference space
• Schauder basis
• α-, β-, γ-duals
• Matrix mappings
• Compact operators
• Hausdorff measure of non-compactness