Proyecciones
Vol. 23, No 1, pp. 31-49, May 2004.
Universidad Católica del Norte
Antofagasta - Chile

ORLICZ - PETTIS THEOREMS FOR
MULTIPLIER CONVERGENT OPERATOR
VALUED SERIES

CHARLES SWARTZ
New Mexico State University, USA

Received November 2003. Accepted March 2004.

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 ST_k in L(X, Y ). First, if l is a scalar sequence space, we say that the series ST_k is l multiplier convergent for a locally convex topology t on L(X, Y ) if the series St_kT_k is t convergent for every t = {t_k} Îl . We establish conditions on λ which guarantee that a λ multiplier convergent series in the weak or strong operator topology is l 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 ST_k is E multiplier convergent in a locally convex topology h on Y if the series ST_kx_k is η convergent for every x = {x_k} Î E. We consider a gliding hump property on E which guarantees that a series ST_k 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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Charles Swartz
Mathematics Department
New Mexico state University
Las Cruces, NM 88003
USA
e-mail : cswartz@nmsu.edu