Proyecciones
Vol. 25, No 3, pp. 271-291, December 2006.
Universidad Católica del Norte
Antofagasta - Chile

ON OPERATOR IDEALS DEFINED BY A REFLEXIVE ORLICZ SEQUENCE SPACE

J. LÓPEZ¹, M. RIVERA², G. LOAIZA³

¹ UNIVERSIDAD POLITÉCNICA DE VALENCIA, ESPAÑA
² UNIVERSIDAD POLITÉCNICA DE VALENCIA, ESPAÑA
³ UNIVERSIDAD EAFIT, COLOMBIA

Abstract

Classical theory of tensornorms and operator ideals studies mainlythose defined by means of sequence spaces ..p. Considering Orlicz sequence spaces as natural generalization of ..p spaces, in a previous paper [12] an Orlicz sequence space was used to define a tensornorm, and characterize minimal and maximal operator ideals associated, by using local techniques. Now, in this paper we give a new characterization of the maximal operator ideal to continue our analysis of some coincidences among such operator ideals. Finally we prove some new metric properties of tensornorm mentioned above.

Key words : Maximal operator ideals. Ultraproducts of spaces, Orlicz spaces.

AMSMathematics Subject Classification : Primary 46M05, 46A32.

REFERENCES

[1] Aliprantis, C. D., Burkinshaw, O.: Positive operators. Pure and Applied Mathematics 119. Academic Press, Newe York, (1985).         [ Links ]

[2] Diestel, J. and Uhl, J. J. Jr.: Vector measures. Mathematical Surveys and Monographs. Number 15. American Mathematical Society. U. S. A. (1977).         [ Links ]

[3] De Grande-De Kimpe, N.: L-mappings between locally convex spaces, Indag. Math. 33, pp. 261-274, (1971).         [ Links ]

[4] Defant, A. and Floret, K.: Tensor norms and operator ideals. North Holland Math. Studies. Amsterdam. (1993).         [ Links ]

[5] Dubinsky, E. and Ramanujan, M. S.: On M-nuclearity. Mem. Amer. Math. Soc. 128, (1972).         [ Links ]

[6] Harksen, J.: Tensornormtopologien. Dissertation, Kiel, (1979).         [ Links ]

[7] Haydon, R., Levy, M., Raynaud, Y.: Randomly normed spaces. Hermann, (1991).         [ Links ]

[8] Heinrich, S.: Ultraproducts in Banach spaces theory. J. reine angew. Math. 313, pp. 72-104, (1980).         [ Links ]

[9] Johnson, W. B.: On finite dimensional subspaces of Banach spaces with local unconditional structure. Studia Math. 51, pp. 225-240, (1974).         [ Links ]

[10] Komura, T., Komura, Y.: Sur les espaces parfaits de suites et leurs généralisations, J. Math. Soc. Japan, 15,3, pp. 319-338, (1963).         [ Links ]

[11] Lacey, H. E.: The isometric theory of Classical Banach spaces, Springer Verlag. Berlin, Heidelberg, New York, (1974).         [ Links ]

[12] Loaiza, G., López Molina, J.A., Rivera, M.J.: Characterization of the Maximal Ideal of Operators Associated to the Tensor Norm Defined by an Orlicz Function. Zeitschrift für Analysis und ihre Anwendungen (Journal for Analysis and its Applications) 20, no.2, pp. 281.293, (2001).         [ Links ]

[13] Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces I and II, Springer Verlang. Belin, Heidelberg, New York, (1977).         [ Links ]

[14] Lindenstrauss, J., Tzafriri, L.: The uniform approximation property in Orlicz spaces, Israel J. Math. 23, 2, pp. 142-155,(1976).         [ Links ]

[15] Pelczy´nski, A., Rosenthal, H. P.: Localization techniques in L^p spaces, Studia Math. 52, pp. 263-289, (1975).         [ Links ]

[16] Pietsch, A.: Operator Ideals. North Holland Math. Library. Amsterdam, New York. (1980).         [ Links ]

[17] Rivera, M.J.: On the classes of L^l, L^l,g and quasi-L^E spaces. Preprint         [ Links ]

[18] Saphar, P.: Produits tensoriels topologiques et classes d'applications lineáires. Studia Math. 38, pp. 71-100, (1972).         [ Links ]

[19] Sims, B.: "Ultra"-techniques in Banach space theory, Queen's Papers in Pure and Applied Mathematics, 60. Ontario, (1982).         [ Links ]

[20] Tomášek, S: Projectively generated topologies on tensor products, Comentations Math. Univ. Carolinae, 11, 4 (1970).         [ Links ]

J. A. López Molina
Universidad Politécnica de Valencia
E.T.S. Ingenieros Agrónomos
Camino de Vera 46072 Valencia
Spain
e-mail : jalopez@mat.upv.es

M. J. Rivera
Universidad Politécnica de Valencia
E. T. S. Ingenieros Agrónomos
Camino de Vera 46072 Valencia
Spain
e-mail : mjrivera@mat.upv.es,

G. Loaiza
Universidad EAFIT
Departamento de Ciencias Básicas
Carrera 49 n- 7 sur-50 Medellín.
Colombia
e-mail: gloaiza@eafit.edu.co

Received : March 2006. Accepted : September 2006