Absolute Schur algebras and unbounded matrices

Let p, q, r be real numbers such that p, q, r greater than or equal to 1, a nd let B be a Banach algebra. Let B(l(p), l(q)) denote the set of all matri ces which define bounded linear transformations from l(p) into l(q). The se t S-r(B) = {A = [a(jk)] : a(jk) is an element of B and A([r]) = [\\a(jk)\\(r) ] is an element of B(l(p), l(q))} of infinite matrices over B; is shown to be a Banach algebra under the Schu r product operation, and the norm \\\A\\\(p,q,r) = \\A([r])\\(1/r). For r g reater than or equal to 2 and B = C, the complex field, S-p = S-p(C) contai ns the set B(l(p), l(q)). For r = 2, S-2 contains the bounded matrices B(l( p), l(q)) as an ideal.