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In this study, we give another generalization of second order backward difference operator ∇2 by introducing its quantum analog ∇q2. The operator ∇q2 represents the third band infinite matrix. We construct its domains c0(∇q2) and c(∇q2) in the spaces c0 and c of null and convergent sequences, respectively, and establish that the domains c0(∇q2) and c(∇q2) are Banach spaces linearly isomorphic to c0 and c, respectively, and obtain their Schauder bases and α-, β- and γ-duals. We devote the last section to determine the spectrum, the point spectrum, the continuous spectrum and the residual spectrum of the operator ∇q2 over the Banach space c0 of null sequences.

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AbstractIn this article we introduce Tribonacci sequence spaces ℓp(T) (1 ≤ p ≤ ∞) derived by the domain of a newly defined regular Tribonacci matrix. We give some topological properties, inclusion relation, obtain the Schauder basis and determine the α-, β- and γ-duals of the new spaces. We characterize the matrix classes on ℓp(T). Finally, we give some geometric properties of the space ℓp(T).

In this article we introduce binomial difference sequence spaces of fractional order α, $\begin{array}{} b_p^{r,s} \end{array}$ (Δ(α)) (1 ≤ p ≤ ∞) by the composition of binomial matrix, Br,s and fractional difference operator Δ(α), defined by (Δ(α)x)k = $\begin{array}{} \displaystyle \sum\limits_{i=0}^{\infty}(-1)^i\frac{\Gamma(\alpha+1)}{i!\Gamma(\alpha-i+1)}x_{k-i} \end{array}$. We give some topological properties, obtain the Schauder basis and determine the α, β and γ-duals of the spaces. We characterize the matrix classes ( $\begin{array}{} b_p^{r,s} \end{array}$(Δ(α)), Y), where Y ∈ {ℓ∞, c, c0, ℓ1} and certain classes of compact operators on the space $\begin{array}{} b_p^{r,s} \end{array}$(Δ(α)) using Hausdorff measure of non-compactness. Finally, we give some geometric properties of the space $\begin{array}{} b_p^{r,s} \end{array}$(Δ(α)) (1 < p < ∞).