Aspects of (p.q) summing multipliers

Abstract

A sequence (uj )j∈N of operators in L(X, Y ) is a (p, q)-summing multiplier (or (p, q)-summing sequence of operators), in short (uj ) ∈ `πp,q (X, Y ), if there exists a constant C > 0 such that, for any finite collection of vectors x1, x2, . . . xn in X, it holds that ³Xn j=1 kujxjk p ´1/p ≤ C sup n³Xn j=1 |x ∗ xj | q ´1/q ; x ∗ ∈ BX∗ o . Some examples of these operators, inclusions between the spaces and connections with spaces of multipliers are presented.∗ Mathematics Subject Classification (2000): 47B10. Key words: Summing operators, vector-valued multipliers. 1. Introduction. Let X and Y be two real or complex Banach spaces and let E(X) and F(Y ) be two Banach spaces whose elements are defined by sequences of vectors in X and Y (containing any eventually null sequence in X or Y ). A sequence of operators (un) ∈ L(X, Y ) is called a multiplier sequence from E(X) to F(Y ) if there exists a constant C > 0 such that ° °(ujxj ) n j=1 ° ° F (Y ) ≤ C ° °(xj ) n j=1 ° ° E(X) for all finite families x1, . . . , xn in X. The set of all of multiplier sequences is denoted by (E(X), F(Y )). For the study of such multipliers for the cases of E(X) and F(Y ) corresponding to vector-valued Hardy spaces, vector-valued Bergman spaces, vector-valued BMOA or spaces of vector valued Bloch functions the reader is referred to [AB1, Bl1, Bl2, Bl3, Bl4].