Approximation of upper bound for matrix operators on the Fibonacci weighted sequence spaces

Let A = (a(n, k))(n, k >= 0) be a non-negative matrix. Denote by parallel to A parallel to(lp)(w),F-w, (p), the infimum of those U satisfying the following inequality: {Sigma(infinity)(n=0) wn (Sigma(n)(k=0) f(k)(2)/f(n)f(n+1) Sigma(infinity)(i=0) a(k),(i)x(i))(p)}(1/p) <= U(Sigma(infinity)(n=0) w(n)x(n)(p))(1/p) , where x >= 0, x is an element of l(p)(w) and {f(n)}(n=0)(infinity) is the Fibonacci numbers sequence and {w(n)}(n=0)(infinity) is a decreasing, non-negative sequence of real numbers. In this paper, first, the Fibonacci weighted sequence space F-w,F- p(1 <= p < infinity) is introduced. Then, we focus on the evaluation of parallel to A parallel to l(p)(w), F-w,F- p, where A is the Hausdorff matrix operator or the Norlund matrix operator or the transpose of the Norlund matrix operator. For the case of Hausdorff matrices, a Hardy-type formula is established as an upper estimate. Also, a general upper estimate is established for the case of Norlund matrices and their transpose. In particular, we apply our results to Cesaro matrices, Holder matrices, Euler matrices and Gamma matrices.