Lower bound for matrix operators on the Euler weighted sequence space \(e_{w,p}^{\theta}\) (\(0<p<1\))
type: Item
instance of: scholarly article
title: Lower bound for matrix operators on the Euler weighted sequence space \(e_{w,p}^{\theta}\) (\(0<p<1\))
zbMATH Open document ID: 1279.47048
publication date: 2012-12-07T00:00:00Z
review text: From the abstract: ``Let \(A=\left( a_{n,k}\right) _{n,k\geq 0}\) be a nonnegative matrix. Denote by \(L_{l_{p}(w) ,e_{w,q}^{\theta }}(A) \) the supremum of those \(L\) satisfying the inequality \[ \left\{ \sum_{n=0}^{\infty }w_{n}\left( \sum_{k=0}^{n}\binom{n}{k}\left( 1-\Theta \right) ^{n-k_{\Theta }n}\sum_{j=0}^{\infty }a_{k,j}x_{j}\right) ^{q}\right\} ^{\frac{1}{q}}\geq L\left( \sum_{n=0}^{\infty }w_{n}x_{n}^{p}\right) ^{\frac{1}{p}}, \] where \(x\geq 0,\) \(x\in l_{p}\left( w\right) \) and \(w=( w_{n}) \) is a decreasing, non-negative sequence of real numbers. In this paper, first we introduce the Euler weighted sequence space, \(e_{w,q}^{\theta}\) (\(0<p<1\)) of non-absolute type, which is the \(p\)-normed space included in the space \(l_{p}\left( w\right)\). Then we focus on the evaluation of \( L_{l_{p}(w) ,e_{w,q}^{\theta }}( A^{t}) \) for a lower triangular matrix \(A\), where \(0<q\leq p<1\). Also, a Hardy type formula is established for \(L_{l_{p}(w)},e_{w,q}^{\theta }( H^{t}) \), where \(H\) is Hausdorff matrix and \(0<q\leq p<1\). In particular, we apply our results to summability matrices, weighted mean matrices, Nörlund matrices, Cesàro matrices, Hölder matrices and Gamma matrices.'' Furthermore, the authors prove the following theorems: Theorem 2.1. The set \(e_{w,p}^{\theta }\) becomes a linear space with the coordinatewise addition and scalar multiplication which is the \(p\)-normed space with the \(p\)-norm \[ ||| x||| :=\| x\|_{e_{w,p}^{\theta }}^{p}=\sum_{n=0}^{\infty }w_{n}\left| \sum_{k=0}^{n}\binom{n}{k}\left( 1-\Theta \right) ^{n-k}x_{k}\right| ^{p}. \] The inclusion \(e_{w,p}^{\theta }\subset l_{p}(w)\) holds for \(0<p<1\). If \(0<s\leq r<1\), then \(e_{w,p}^{s}\subset e_{w,p}^{r}\). Theorem 2.5. Let \(0<p<1\). Define the sequence \(b^{\left( j\right) }\left( \theta \right) =\left\{ b_{k}^{\left( j\right) }\left( \theta \right) \right\} _{k=0}^{\infty }\) of the elements of the space \(e_{w,p}^{\theta }\) by \[ b_{k}^{\left( j\right) }\lef…
Mathematics Subject Classification ID: 47B37
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 47A30
Mathematics Subject Classification ID: 40G05
Mathematics Subject Classification ID: 46A45
Mathematics Subject Classification ID: 54D55
zbMATH DE Number: 6113983
zbMATH Keywords: lower bound
zbMATH Keywords: lower triangular matrices
zbMATH Keywords: summability matrices
zbMATH Keywords: Nörlund matrices
zbMATH Keywords: Hausdorff matrices
zbMATH Keywords: weighted mean matrices
zbMATH Keywords: Euler weighted sequence space
author: R. Lashkaripour
author: Gholamreza Talebi
reviewed by: Ekrem Savaş
MaRDI profile type: Publication
full work available at URL: s12215-011-0067-x
OpenAlex ID: W1982704157
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DOI: 10.1007/S12215-011-0067-X
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 6113983