Orlicz - Pettis Theorems for Multiplier Convergent Operator Valued Series

Abstract

Let X, Y be locally convex spaces and L(X, Y) the space of continuous linear operators from X into Y. We consider 2 types of mul-tiplier convergent theorems for a series T k in L(X, Y). First, if λ is a scalar sequence space, we say that the series T k is λ multiplier convergent for a locally convex topology τ on L(X, Y) if the series t k T k is τ convergent for every t = {t k } ∈ λ. We establish conditions on λ which guarantee that a λ multiplier convergent series in the weak or strong operator topology is λ multiplier convergent in the topology of uniform convergence on the bounded subsets of X. Second , we consider vector valued multipliers. If E is a sequence space of X valued sequences, the series T k is E multiplier convergent in a locally convex topology η on Y if the series T k x k is η convergent for every x = {x k } ∈ E. We consider a gliding hump property on E which guarantees that a series T k which is E multiplier convergent for the weak topology of Y is E multiplier convergent for the strong topology of Y. 136 Charles Swartz

Uniform Convergence in B-Duals

“…In the papers [Sw6], [Sw7] gliding hump assumptions were used to establish uniform convergence results for pointwise bounded subsets of the β-dual of vector valued sequence spaces. Similar gliding hump properties were used in [Sw3] to establish Orlicz-Pettis Theorems for multiplier convergent series with respect to strong topologies on locally convex spaces and spaces of continuous linear operators. In [CL] Li and Chen used other assumptions on the space of multipliers to establish similar Orlicz-Pettis results for strong topologies.…”

An Abstract Gliding Hump Property

“…Gliding hump properties have been used to treat a number of topics in sequence spaces such as weak sequential completeness of β-duals ( [No], [St]), uniform convergence of elements in β-duals ( [StSw1]), uniform boundedness principles ( [Sw4]), Orlicz-Pettis theorems for multiplier convergent series ([StSw2], [Sw5]) and Hahn-Schur theorems for multiplier convergent series ([Sw6]). In this note we introduce an abstract gliding hump property which include the signed weak gliding hump property and the signed strong gliding hump property as well as new gliding hump properties.…”

“…We consider such theorems for scalar multipliers and operator valued series where the original topology on the space is L s (X, Y ). There do not seem to be many Orlicz-Pettis Theorems for the strong operator topology (see [StSw2] Theorem 9 and [Sw5] for examples of such results; however, there are strong results for subseries convergent series in the space of compact operators due to Kalton (see [Sw2] 10.5.6)). Let λ be a scalar sequence space which contains the space c 00 of all sequences which are eventually 0.…”