Orlicz-Pettis theorems for multiplier convergent operator valued series
DOI:
https://doi.org/10.4067/S0716-09172004000100005Keywords:
Locally convex spaces, Orlicz-Pettis theorems, continuous linear operators, convergence, scalar spaces, sequence spaces, series, locally convex topology, espacios localmente convexos, teoremas de Orlicz-Pettis, operadores lineales continuos, convergencia.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 multiplier convergent theorems for a series ? T? in L(X, Y ). First, if ? is a scalar sequence space, we say that the series ? T? is ? multiplier P convergent for a locally convex topology ? on L(X, Y ) if the series ? t?T? is ? convergent for every t = {t?} ? ?. 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 ? T? is E multiplier convergent in a locally convex topology ? on Y if the series ? T?x? is ? convergent for every x = {x?} ? E. We consider a gliding hump property on E which guarantees that a series ? T? which is E multiplier convergent for the weak topology of Y is E multiplier convergent for the strong topology of Y .
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