Orlicz - Pettis Theorems for Multiplier Convergent Operator Valued Series

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 multiplier convergent theorems for a series P Tk in L(X,Y ). First, if λ is a scalar sequence space, we say that the series P Tk is λ multiplier convergent for a locally convex topology τ on L(X,Y ) if the series P tkTk is τ convergent for every t = {tk} ∈ 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 P Tk is E multiplier convergent in a locally convex topology η on Y if the series P Tkxk is η convergent for every x = {xk} ∈ E. We consider a gliding hump property on E which guarantees that a series P Tk which is E multiplier convergent for the weak topology of Y is E multiplier convergent for the strong topology of Y .