On Lattice Summing Operators

Jerzy Szulga
1983 Proceedings of the American Mathematical Society  
Given a Banach space E and a Banach lattice L, necessary and sufficient conditions on E and L are given such that every lattice summing operator T: E -L (cf. Introduction) is absolutely summing. 1. Introduction. The concept of absolutely summing operators has a certain natural analogue when the range space is a Banach lattice. Namely, an operator T: E -L is called lattice summing, if for every sequence (x") in E such that 1xn converges unconditionally, the series 2 | Txn | converges in L. Of
more » ... rse, if e.g. L is an L,-space or E is an L,-space and L is a Hubert space, then both notions coincide. The aim of this paper is to characterize all pairs E, L for which this happens. In 1979 Yanovskii [10] investigated problems related to lattice summability. In particular, he formulated a conjecture that, if all lattice summing operators acting on the same Banach lattice L are absolutely summing, then L is isomorphic to Lx. However, it follows from our characterization that spaces of cotype 2 have this property. Another by-product of this paper is a still different formulation of the Dubinsky-Pekzynski-Rosenthal property n^C^" E) = B(tx, E).
doi:10.2307/2043699 fatcat:v3w62quuo5ftrajblklf4f4hxe