Domain of q-difference operator ?q(2) in the sequence spaces h and bv, and its spectrum over h

Abstract

We construct the sequence spaces h(Delta((2))(q)) = h(Delta q(2)) and bv(Delta((2))(q)) = bv ((2))(Delta q), where Delta((2))(q) is a second order q-difference operator defined by (Delta((2))(q)u)(d) = u(d) - (1 + q)u(d- 1) + qu(d-2) for all d is an element of Z(0)(+), where we presume that u(d) = 0 for d <= 0. We obtain some inclusion relations, Schauder bases, and important duals of the spaces h(Delta((2))(q)) and bv(Delta((2))(q)). We also determine necessary and sufficient condition for a matrix to map between the spaces h(Delta((2))(q)) /bv(Delta((2))(q)) and any arbitrary space mu. In the final section, we determine the point spectrum and the spectrum of the operator Delta((2))(q) over the space h.