Orlicz-Pettis theorems for multiplier convergent operator valued series

• Swartz, Charles ^[1]
  1. [1] New Mexico State University

     New Mexico State University

     Estados Unidos

     
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 23, Nº. 1, 2004, págs. 61-72
• Idioma: inglés
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• Resumen

  • 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 ∑ Tₕ in L(X, Y ). First, if λ is a scalar sequence space, we say that the series ∑ Tₕ is λ multiplier P convergent for a locally convex topology τ on L(X, Y ) if the series ∑ tₕTₕ is τ convergent for every t = {tₕ} ∈ 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 ∑ Tₕ is E multiplier convergent in a locally convex topology η on Y if the series ∑ Tₕxₕ is η convergent for every x = {xₕ} ∈ E. We consider a gliding hump property on E which guarantees that a series ∑ Tₕ which is E multiplier convergent for the weak topology of Y is E multiplier convergent for the strong topology of Y .