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OPTIMAL EXTENSIONS FOR POSITIVE ORDER CONTINUOUS OPERATORS ON BANACH FUNCTION SPACES

Published online by Cambridge University Press:  13 August 2013

O. DELGADO
O. DELGADO*
Affiliation:
Departamento de Matemática Aplicada I, E. T. S. de Ingeniería de Edificación, Universidad de Sevilla, Avenida Reina Mercedes, 4 A, 41012 Sevilla, Spain e-mail: olvido@us.es
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Abstract

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In this paper we give conditions under which a positive order continuous operator T defined on a Banach function space can be extended, preserving the order continuity in a certain optimal way. The optimal domain for T turns out to be a space of weakly integrable functions with respect to a vector measure (defined on a δ-ring) canonically associated to T. A similar result is obtained when T is σ-order continuous and we want to preserve the σ-order continuity. We apply these results to kernel operators.

Keywords

Primary 46G10Secondary 46E3046B42
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 56 , Issue 3 , September 2014 , pp. 481 - 501
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013 

References

REFERENCES

1.Aliprantis, C. D. and Burkinshaw, O., Positive operators (Academic Press, New York, NY, 1985).Google Scholar
2.Brooks, J. K. and Dinculeanu, N., Strong additivity, absolute continuity and compactness in spaces of measures, J. Math. Anal. Appl. 45 (1974), 156–175.Google Scholar
3.Calabuig, J. M., Delgado, O., Juan, M. A. and Sánchez Pérez, E. A., On the Banach lattice structure of L 1w of a vector measure on a δ-ring, Collect. Math. (accepted) (to appear), doi: 10.1007/s13348-013-0081-8Google Scholar
4.Calabuig, J. M., Delgado, O. and Sánchez Pérez, E. A., Factorizing operators on Banach function spaces through spaces of multiplication operators, J. Math. Anal. Appl. 364 (2010), 88–103.Google Scholar
5.Curbera, G. P., Volterra convolution operators with values in rearrangement invariant spaces, J. London Math. Soc. 60 (1999), 258–268.Google Scholar
6.Curbera, G. P. and Ricker, W. J., Optimal domains for kernel operators via interpolation, Math. Nachr. 244 (2002), 47–63.Google Scholar
7.Curbera, G. P. and Ricker, W. J., Optimal domains for the kernel operator associated with Sobolev's inequality, Studia Math. 158 (2003), 131–152; Corregenda to Optimal domains for the kernel operator associated with Sobolev's inequality, Studia Math. 170 (2005), 217–218.Google Scholar
8.Delgado, O., L 1-spaces of vector measures defined on δ-rings, Arch. Math. 84 (2005), 432–443.Google Scholar
9.Delgado, O. and Soria, J., Optimal domain for the Hardy operator, J. Funct. Anal. 244 (2007), 119–133.Google Scholar
10.Diestel, J. and Uhl, J. J. Jr., Vector measures, American Mathematical Society Surveys, 15 (American Mathematical Society, Providence, RI, 1997).Google Scholar
11.Lewis, D. R., On integrability and summability in vector spaces, Illinois J. Math. 16 (1972), 294–307.CrossRefGoogle Scholar
12.Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces, vol. II (Springer-Verlag, Berlin, Germany, 1979).CrossRefGoogle Scholar
13.Luxemburg, W. A. J. and Zaanen, A. C., Riesz spaces I (North-Holland, Amsterdam, Netherlands, 1971).Google Scholar
14.Masani, P. R. and Niemi, H., The integration theory of Banach space valued measures and the Tonelli–Fubini theorems. I. Scalar-valued measures on δ-rings, Adv. Math. 73 (1989), 204–241.CrossRefGoogle Scholar
15.Masani, P. R. and Niemi, H., The integration theory of Banach space valued measures and the Tonelli–Fubini theorems. II. Pettis integration, Adv. Math. 75 (1989), 121–167.CrossRefGoogle Scholar
16.Mockenhaupt, G. and Ricker, W. J., Optimal extension of the Hausdorff–Young inequality, J. Reine Angew. Math. 620 (2008), 195–211.Google Scholar
17.Okada, S. and Ricker, W. J., Optimal domains and integral representations of convolution operators in Lp(G), Integr. Equ. Oper. Theory 48 (2004), 525–546.Google Scholar
18.Zaanen, A. C., Riesz spaces II (North-Holland, Amsterdam, Netherlands, 1983).Google Scholar