On generalized Fibonacci difference sequence spaces and compact operators

Abstract

In this paper, we introduce Fibonacci backward difference operator F Delta(Bl) of fractional order l by the composition of Fibonacci band matrix F and difference operator Delta(Bl) of fractional order l, defined by (Delta(Bl)x)(s) = Sigma(infinity)(i=0)(-1)(i) Gamma(l + 1)/i!Gamma(l - i + 1) x(s-i) and introduce sequence spaces l(k)(F Delta(Bl)) and l(infinity)(F Delta(Bl)). We present some topological properties, obtain Schauder basis and determine alpha-, beta- and gamma-duals of the spaces l(k)(F Delta(Bl)) and l(infinity)(F Delta Bl). We characterize certain classes of matrix mappings from the spaces l(k)(F Delta(Bl)) and l(infinity)(F Delta(Bl)) to any of the space l(infinity), c, c(0) or l(1). Finally we compute necessary and sufficient conditions for a matrix operator to be compact on the spaces l(k)(F Delta(Bl)) and l(infinity)(F Delta(Bl)).