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Norms and lower bounds of some matrix operators on Fibonacci weighted difference sequence space. (English) Zbl 1427.47013

Summary: Norm of an operator \(T:X \rightarrow Y\) is the best possible value of \(U\) satisfying the inequality \[\|Tx\|_Y \le U\|x\|_X,\] and lower bound for \(T\) is the value of \(L\) satisfying the inequality \[\|Tx\|_Y \ge L\|x\|_X,\] where \(\Vert.\Vert_X\) and \(\Vert.\Vert_Y\) are the norms on the spaces \(X\) and \(Y\), respectively. The main goal of this paper is to compute norms and lower bounds for some matrix operators from the weighted sequence space \(\ell_p(w)\) into a new space called as Fibonacci weighted difference sequence space. For this purpose, we firstly introduce the Fibonacci difference matrix \(\tilde{F}\) and the space consisting of sequences whose \(\tilde{F}\)-transforms are in \(\ell_p(\tilde{w})\).

MSC:

47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
26D15 Inequalities for sums, series and integrals
46A45 Sequence spaces (including Köthe sequence spaces)
40G05 Cesàro, Euler, Nörlund and Hausdorff methods
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