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

Abstract

We present a quantum analog del(l)(q) of the lth order backward difference operator del(l) and analyze its basic properties. We study the sequence spaces c(del(l)(q)) and c(0) (del(l)(q)) defined as the domain of del(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)(del(l)(q)) and c(del(l)(q)) are obtained. Some theorems characterizing matrix transformations related to these spaces are stated and proved. Finally, we analyze the spectral divisions of del(l)(q) over the space c(0):