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 sigmaTk in L(X, Y ). First, if lambda is a scalar sequence space, we say that the series sigmaTk is lambda multiplier convergent for a locally convex topology tau on L(X, Y ) if the series sigmat kTk is tau convergent for every t = {t k} <FONT FACE=Symbol>Î</FONT>lambda . We establish conditions on &#955; which guarantee that a &#955; multiplier convergent series in the weak or strong operator topology is lambda 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 sigmaTk is E multiplier convergent in a locally convex topology eta on Y if the series sigmaTk x k is &#951; convergent for every x = {xk} <FONT FACE=Symbol>Î</FONT> E. We consider a gliding hump property on E which guarantees that a series sigmaTk which is E multiplier convergent for the weak topology of Y is E multiplier convergent for the strong topology of Y