A MULTIPLIER GLIDING HUMP PROPERTY FOR SEQUENCE SPACES

Charles Swartz

doi:10.4067/s0716-09172001000100002

Abstract

We consider the Banach-Mackey property for pairs of vector spaces E and E0 which are in duality. Let A be an algebra of sets and assume that P is an additive map from A into the projection operators on E. We define a continuous gliding hump property for the map P and show that pairs with this gliding hump property and another measure theoretic property are Banach-Mackey pairs,i.e., weakly bounded subsets of E are strongly bounded. Examples of vector valued function spaces, such as the space of Pettis integrable functions, which satisfy these conditions are given.

Highlights

Absract gliding hump assumptions have been used to treat a number of topics in sequence spaces;for example, Noll used a ”strong gliding hump” property to establish the weak sequential completeness of the beta dual of a sequence space ([N] ; see [BF] for a list of various gliding hump properties for sequence spaces)
In this paper we introduce a gliding hump assumption involving multipliers from a scalar sequence space which is useful in establishing uniform boundedness results for a vector-valued sequence space and its beta dual; in particular, our results establish Banach-Mackey properties for sequence spaces
E has the weak λ gliding hump property if whenever {Ik} is an increasing sequence of intervals and {xk} is a bounded sequence in E, there is a subsequence {nk} such that the coordinate sum tkχInk xk belongs to E for every t ∈ λ

Summary

Introduction

E has the strong λ gliding hump property (strong λ-GHP) if whenever {Ik} is an increasing sequence of intervals and {xk} is a bounded sequence in E, for every t = {tk} ∈ λ the coordinate sum of the series tkχIkxk belongs to E. E has the weak λ gliding hump property (weak λ-GHP) if whenever {Ik} is an increasing sequence of intervals and {xk} is a bounded sequence in E, there is a subsequence {nk} such that the coordinate sum tkχInk xk belongs to E for every t ∈ λ. Proof: Let {Ik} be an increasing sequence of intervals and {xk} ⊂ E be bounded. We give examples of non-complete scalar sequence spaces with weak lp − GHP .

Results

Conclusion