Nuclearity and multipliers between Banach spaces

Abstract

Let be a Banach sequence space with a monotone norm ‖ · ‖ , in which the canonical system (ei) is a normalized symmetric basis. Let Λ be the class of such spaces and D( , ) be the space of all diagonal operators (that is multipliers) acting between ,  ∈ Λ. In [2], Djakov and Ramanujan considered the special case of multipliers on the class of Orlicz sequence spaces and proved that for Orlicz functions, if = lM and  = lN , then D(lM , lN) = lM∗ N , where lM∗ N := sup{N (st) − M(t) : t ∈ (0, 1)}. We consider the general form of multipliers on the class Λ and evaluate for some well known Banach sequence spaces. In Theorem 2.7, it is observed that quasidiagonal isomorphisms of different -Kothe spaces implies nuclearity which coincide with the common multipliers (Δ( , ) := D( , )∩D(, )) of the corresponding spaces ∈ Λ and some results of [1] and [7] become the consequence of this theorem. Mathematics Subject Classification: 46A45, 47A99