Lower bounds of Copson type for Nörlund matrices

AbstractLet A=(an,k)n,k⩾0 be a non-negative matrix. Denote by Lp,q(A) the supremum of those L satisfying ‖AX‖q⩾L‖X‖p(X∈ℓp,X⩾0), and define L(p),q(A)=Lp,q(A)(p>0). We derive a range for the value of Lp,q(AWNM), where 0<q⩽p<1 and AWNM denotes the Nörlund matrix associated with the weight function W. By the continuity of L(·),q(AWNM), we show that this range is best possible. It is also proved that there exists a unique ξ∈(q,1] such that L(·),q(AWNM) maps [q,ξ] onto [1,‖W‖q/‖W‖1] and this mapping is continuous and strictly increasing. The case Lp,q((AWNM)t) with -∞<p,q