Lower bounds of Copson type for Nörlund matrices

Let 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 , where 0