Sequence Spaces and Spectrum of <i>q</i>-Difference Operator of Second Order

The sequence spaces ℓp(∇q2)(0≤p∞) and ℓ∞(∇q2) are introduced by using the q-difference operator ∇q2 of the second order. Apart from studying some basic properties of these spaces, we construct the basis and obtain the α-, β- and γ-duals of these spaces. Besides some matrix classes involving q-difference sequence spaces, ℓp(∇q2) and ℓ∞(∇q2) are characterized. The final section is devoted to classifying the spectrum of the q-difference operator ∇q2 over the space ℓ1 of absolutely summable sequences