Lower bound for matrix operators on the Euler weighted sequence space $e_{w,p}^{\theta}$ (0<p<1)

Abstract

Let A=(a _n,k ) _n,k≥0 be a non-negative matrix. Denote by $L_{l_{p} (w),~e_{w,q}^{\theta}}(A)$ the supremum of those L, satisfying the following inequality:

where x≥0, x∈l _p (w) and w=(w _n ) is a decreasing, non-negative sequence of real numbers. In this paper, first we introduce the Euler weighted sequence space, $e_{w,p}^{\theta}~(0< p < 1)$, of non-absolute type which is the p-normed space included in the space l _p (w). 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≤p<1. Also in this paper 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≤p<1. In particular, we apply our results to summability matrices, weighted mean matrices, Nörlund matrices, Cesàro matrices, Hölder matrices and Gamma matrices which were recently considered in (Bennett in Linear Algebra Appl. 82:81–98, 1986; Bennett in Can. J. Math. 44:54–74, 1991; Chen and Wang in Linear Algebra Appl. 420:208–217, 2007; Lashkaripour and Foroutannia in Proc. Indian Acad. Sci. Math. Sci. 116:325–336, 2006; Lashkaripour and Foroutannia in J. Sci., Islam. Repub. Iran 18(1):49–56, 2007; Lashkaripour and Foroutannia in Czechoslov. Math. J. 59(134):81–94, 2009). Our results generalize some work of Altay, Başar, Mursaleen and the present authors in (Altay and Başar in Ukr. Math. J. 56(12), 2004; Altay et al. in Inf. Sci. 176:1450–1462, 2006; Lashkaripour and Talebi in Bull. Iran. Math. Soc., 2011).