Operators on vector sequence spaces

Abstract

In the following, (∑⊕En)p will denote the p-sum of a sequence of Banach spaces and Πm [resp. Im] are the corresponding canonical projections [resp. injections] onto [from] the coordinate spaces Em for m=1,2,⋯. The aim of the paper is the characterization of operators (continuous and linear) T on (∑⊕En)p in terms of the operators Tn=T∘In. Let F be a Banach space. It is proved that for every operator T:(∑⊕En)p→F, the sequence (Tn) is unique with respect to the following properties: (1) For every y∗∈F∗, (T∗n(y∗))∈(∑⊕E∗n)q with 1p+1q=1. (2) {(T∗n(y∗))∞n=1: y∗∈B(F∗)} is bounded in (∑⊕E∗n)q, where B(F∗) denotes the unit ball in F∗. (3) T(x)=∑∞n=1Tn(xn) for all x=(xn)∈(∑⊕En)p. (4) ∥T∥=sup{∥T∗n(y∗)∥q: y∗∈B(F∗)}. According to the above-mentioned result, if T:(∑⊕En)p→F belongs to some operator ideal I, then Tn∈I for all n=1,2,⋯. Some examples, to show that the converse statement is in general false, are discussed. However, it is then shown that if p=1, the converse holds if I=U (the family of all unconditionally converging operators), I=D (Dieudonné operators) and I=DP (Dunford-Pettis operators). In case of p>1, the converse also holds if I=ω (weakly compact operators). The c0-sum is denoted by E=(∑⊕En)0. In this case it is proved that the above-mentioned results can be sharpened to obtain the following: Let I be a closed operator ideal, contained in U. Then T∈I(E,F) if and only if Tn∈I(En,F) for all n=1,2,⋯, and limm→∞∥∑mn=1Tn∘Πn−T∥=0. The above-mentioned results are applied by the author to: (1) obtain necessary and sufficient conditions for (∑⊕En)p to have (if p=0;1) the Dunford-Pettis property; (if p=0; 1<p<∞) the reciprocal Dunford-Pettis property; (if p=0; 1<p<∞) the Dieudonné property; (if p=0; 1<p<∞) the V-property of Pełczyński; (if 1<p<∞) the Grothendieck property; (2) prove necessary and sufficient conditions for (∑⊕En)p to (if 1<p<∞) have a complemented copy of l1; (if p=1) be a Schur space; be weakly sequentially complete; contain a copy of c0; (if p=0) contain a complemented copy of lp for 1≤p<∞.