Factoring absolutely convergent series

Abstract

It is a well known fact that if p > l and 1 / p + l / q = l , every sequence (x.) in 11 (absolutely convergent series) can be factored in the form (x,,)=(u,,v,,) where (u.) is ha 1 p (p-absolutely convergent series) and (v.) is in t q. A related fact is the familiar exercise in advanced calculus that every sequence (x.) in 11 can be factored in the form (x.)= (u.v.) where (u.) is in Co (sequences convergent to 0) and (v.) is again in 11. Our purpose in this paper is to prove a natural generalization of this fact in the setting of K6the sequence spaces [6]. Our main result states essentially that 11 factors through every balanced Banach sequence space and its K6the dual. The proof is not easy and constitutes a nice exercise in non-linear functional analysis. Some consequences of this result concerning the structure of Banach spaces will appear in [9]. To prove the main result (the theorem in Section 2) we first establish a corresponding statement in the finite dimensional case and proceed to the infinite dimensional case via a compactness argument. Throughout this paper we assume our vector spaces to be over the field of real numbers. The complex case of the main theorem, however, follows immediately from the real case.