Orlicz-Pettis theorems for multiplier convergent operator valued series

Charles Swartz


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 .

Palabras clave

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.


[B] G. Bennett, Some inclusion theorems for sequence spaces, Pacific J. Math., 46, pp. 17-30, (1973).

[BL] J. Boos and T. Leiger, Some distinguished subspaces of domains of operator valued matrices, Results Math., 16, pp. 199-211, (1989).

[D] J. Diestel, Sequences and Series in Banach Spaces, Springer-Verlag, N.Y., (1984).

[DF] J. Diestel and B. Faires, Vector Measures, Trans. Amer. Math. Soc., 198, pp. 253-271, (1974).

[DS] N. Dunford and J. Schwartz, Linear Operators I, Interscience, N. Y., (1958).

[FP] M. Florencio and P. Paul, A note on λmultiplier convergent series, Casopis Pro Pest. Mat., 113, pp. 421-428, (1988).

[G] D. J. H. Garling, The β- and γ-duality of sequence spaces, Proc. Cambridge Phil. Soc., 63, pp. 963-981, (1967).

[KG] P. K. Kamthan and M. Gupta, Sequence Spaces and Series, Marcel Dekker, N. Y., (1981).

[LCC] Li Ronglu, Cui Changri and Min Hyung Cho, An invariant with respect to all admissible (X,X’)-polar topologies, Chinese Ann. Math.,3, pp. 289-294, (1998).

[S1] C. Stuart, Weak Sequential Completeness in Sequence Spaces, Ph.D. Dissertation, New Mexico State University, (1993).

[S2] C. Stuart, Weak Sequential Completeness of β-Duals, Rocky Mountain Math. J., 26, pp. 1559-1568, (1996).

[SS] C. Stuart and C. Swartz, Orlicz-Pettis Theorems for Multiplier Convergent Series, Journal for Analysis and Appl.,17, pp. 805-811, (1998).

[Sw1] C. Swartz, An Introduction to Functional Analysis,Marcel Dekker, N.Y., (1992).

[Sw2] C. Swartz, Infinite Matrices and the Gliding Hump, World Sci.Publ., Singapore, (1996).

[Sw3] C. Swartz, A multiplier gliding hump property for sequence spaces, Proy. Revista Mat., 20, pp. 19-31, (2001).

[W] A. Wilansky, Modern Methods in Topological Vector Spaces, McGraw-Hill, N. Y., (1978).

[WL] Wu Junde and Li Ronglu, Basic properties of locally convex Aspaces, Studia Sci. Math. Hungar., to appear.

DOI: http://dx.doi.org/10.4067/S0716-09172004000100005

Enlaces refback

  • No hay ningún enlace refback.