The norm of matrix operators on Cesàro weighted sequence space. (English) Zbl 07143555
Summary: Let \(A=(a_{n,k})_{n,k\geq 1}\) be a matrix operator, \((v_n)^\infty_{n=1}\) and \((w_n)^\infty_{n=1}\) be two non-negative sequences and \(p\geq 1\). This paper is focused on the problem of finding the infimum of those \(U\), satisfying the following inequality:
\[
\left( \sum^\infty_{n=1} w_n\left| \frac{1}{n} \sum^\infty_{k=1} \sum^n_{i=1} a_{i,k} x_k \right|^p\right)^{\frac{1}{p}} \leq U \left( \sum^\infty_{n=1}v_n |x_n|^p\right)^{\frac{1}{p}},
\]
for all sequences \(x\in l_p\). We apply our results to well-known matrix operators such as backward difference, quasi-summability, weighted mean, Hilbert and Nörlund.
MSC:
47B37 | Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) |
15A45 | Miscellaneous inequalities involving matrices |
15A60 | Norms of matrices, numerical range, applications of functional analysis to matrix theory |
26D15 | Inequalities for sums, series and integrals |
40C05 | Matrix methods for summability |
40G05 | Cesàro, Euler, Nörlund and Hausdorff methods |
47A30 | Norms (inequalities, more than one norm, etc.) of linear operators |
Keywords:
matrix operator; norm; quasi-summability matrix; weighted mean matrix; Hilbert matrix; weighted sequence spaceReferences:
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