SUMMARY
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 PTk in L(X, Y ). First, if ? is a scalar sequence space, we say that the series PTk is ? multiplier P convergent for a locally convex topology t on L(X, Y ) if the series tkTk is t convergent for every t = {tk} ? ?. We establish conditions on ? which guarantee that a ? multiplier convergent series in the weak or strong operator topology is ? 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 PTk is E multiplier convergent in a locally convex topology ? on Y if the series PTkxk is ? convergent for every x = {xk} ? E. We consider a gliding hump property on E which guarantees that a series PTk which is E multiplier convergent for the weak topology of Y is E multiplier convergent for the strong topology of Y.