On Sequence Spaces Due to lth Order q-Difference Operator and its Spectrum

We present a quantum analog \(\nabla ^{l}_q\) of the lth order backward difference operator \(\nabla ^{l}\) and analyze its basic properties. We study the sequence spaces \(c(\nabla ^{l}_q)\) and \(c_0(\nabla ^{l}_q)\) defined as the domain of \(\nabla ^{l}_q\) in the spaces c and \(c_0\), respectively. Some basic properties, inclusion relations, Schauder basis, and \(\alpha\)-,\(\beta\)-, and \(\gamma\)-duals of the spaces \(c_0(\nabla ^{l}_q)\) and \(c(\nabla ^{l}_q)\) are obtained. Some theorems characterizing matrix transformations related to these spaces are stated and proved. Finally, we analyze the spectral divisions of \(\nabla ^{l}_q\) over the space \(c_0.\)