On the factorable spaces of absolutely \(p\)-summable, null, convergent, and bounded sequences. (English) Zbl 1491.46007
Summary: Let \(F\) denote the factorable matrix and \(X\in\{ \ell_p,c_0,c, \ell_\infty\}\). In this study, we introduce the domains \(X(F)\) of the factorable matrix in the spaces \(X\). Also, we give the bases and determine the alpha-, beta- and gamma-duals of the spaces \(X(F)\). We obtain the necessary and sufficient conditions on an infinite matrix belonging to the classes \((\ell_p(F),\ell_\infty)\), \((\ell_p(F),f)\) and \((X,Y(F))\) of matrix transformations, where \(Y\) denotes any given sequence space. Furthermore, we give the necessary and sufficient conditions for factorizing an operator based on the matrix \(F\) and derive two factorizations for the Cesàro and Hilbert matrices based on the Gamma matrix. Additionally, we investigate the norm of operators on the domain of the matrix \(F\). Finally, we find the norm of Hilbert operators on some sequence spaces and deal with the lower bound of operators on the domain of the factorable matrix.
MSC:
| 46A45 | Sequence spaces (including Köthe sequence spaces) |
| 46B45 | Banach sequence spaces |
| 40C05 | Matrix methods for summability |
| 40G05 | Cesàro, Euler, Nörlund and Hausdorff methods |
| 47B37 | Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) |
| 47B39 | Linear difference operators |
| 47A30 | Norms (inequalities, more than one norm, etc.) of linear operators |
| 26D15 | Inequalities for sums, series and integrals |
Keywords:
domain of factorable matrix; almost convergence; weighted mean matrix; Hilbert matrix; gamma matrix; Cesàro matrixReferences:
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