The below boundedness of matrix operators on the special Lorentz sequence spaces

Abstract

Let A=(an,k)n,k⩾0 be a non-negative matrix. Denote by Lℓp(w),weakℓq(A) the supremum of those L, satisfying the following inequality:supB{1(#B)1−1q∑n∈B(∑k=0∞an,kxk)}⩾L{∑n=0∞wnxnp}1p, where x={xk}k=0∞∈ℓp(w), x⩾0, w={wn}k=0∞ is a non-negative decreasing sequence and the supremum is taken over all nonempty subset B of non-negative integers with finite cardinal. In this paper we focus on the evaluation of Lℓp(w),weakℓq(At) for a lower triangular matrix A, where 0<q⩽p<1. A lower estimate is obtained. Moreover, in this paper a Hardy type formula is obtained for Lℓp,weakℓq(Hμα) where Hμα is the generalized Hausdorff matrix, 0<q⩽p<1 and α>0. A similar result is also established for the case in which (Hμα)t is replaced by Hμα. In particular, we apply our results to summability matrices, weighted mean matrices, Nörlund matrices and some special generalized Hausdorff matrices such as generalized Cesàro, generalized Hölder, generalized Gamma and generalized Euler matrices. Our results provide some analogue to those given by R. Lashkaripour, G. Talebi, Lower bound for matrix operators on the Euler weighted sequence space ew,pθ (0<p<1), Rend. Circ. Mat. Palermo 61 (1) (2012) 1–12; and C.-P. Chen, K.-Z. Wang, Operator norms and lower bounds of generalized Hausdorff matrices, Linear Multilinear Algebra 59 (3) (2011) 321–337.