Polar decomposition of order bounded disjointness preserving operators
HTML articles powered by AMS MathViewer
- by Karim Boulabiar and Gerard Buskes PDF
- Proc. Amer. Math. Soc. 132 (2004), 799-806 Request permission
Abstract:
We constructively prove (i.e., in ZF set theory) a decomposition theorem for certain order bounded disjointness preserving operators between any two Riesz spaces, real or complex, in terms of the absolute value of another order bounded disjointness preserving operator. In this way, we constructively generalize results by Abramovich, Arensen and Kitover (1992), Grobler and Huijsmans (1997), Hart (1985), Kutateladze, and Meyer-Nieberg (1991).References
- Y. A. Abramovich and A. K. Kitover, Inverses of disjointness preserving operators, Mem. Amer. Math. Soc. 143 (2000), no. 679, viii+162. MR 1639940, DOI 10.1090/memo/0679
- Y. A. Abramovich, E. L. Arenson, and A. K. Kitover, Banach $C(K)$-modules and operators preserving disjointness, Pitman Research Notes in Mathematics Series, vol. 277, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992. MR 1202880
- Charalambos D. Aliprantis and Owen Burkinshaw, Positive operators, Pure and Applied Mathematics, vol. 119, Academic Press, Inc., Orlando, FL, 1985. MR 809372
- S. J. Bernau, Orthomorphisms of Archimedean vector lattices, Math. Proc. Cambridge Philos. Soc. 89 (1981), no. 1, 119–128. MR 591978, DOI 10.1017/S030500410005800X
- G. Buskes and A. van Rooij, Small Riesz spaces, Math. Proc. Cambridge Philos. Soc. 105 (1989), no. 3, 523–536. MR 985689, DOI 10.1017/S0305004100077902
- B. de Pagter, A note on disjointness preserving operators, Proc. Amer. Math. Soc. 90 (1984), no. 4, 543–549. MR 733403, DOI 10.1090/S0002-9939-1984-0733403-7
- B. de Pagter, $f$-Algebras and Orthomorphisms, Ph.D. Dissertation, Leiden, 1981.
- J. J. Grobler and C. B. Huijsmans, Disjointness preserving operators on complex Riesz spaces, Positivity 1 (1997), no. 2, 155–164. MR 1658332, DOI 10.1023/A:1009746711470
- D. R. Hart, Some properties of disjointness preserving operators, Nederl. Akad. Wetensch. Indag. Math. 47 (1985), no. 2, 183–197. MR 799079, DOI 10.1016/1385-7258(85)90006-X
- Graham Jameson, Ordered linear spaces, Lecture Notes in Mathematics, Vol. 141, Springer-Verlag, Berlin-New York, 1970. MR 0438077, DOI 10.1007/BFb0059130
- Mathieu Meyer, Les homomorphismes d’espaces vectoriels réticulés complexes, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 17, 793–796 (French, with English summary). MR 622421
- Mathieu Meyer, Le stabilisateur d’un espace vectoriel réticulé, C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 5, Aii, A249–A250. MR 433191
- Peter Meyer-Nieberg, Banach lattices, Universitext, Springer-Verlag, Berlin, 1991. MR 1128093, DOI 10.1007/978-3-642-76724-1
- A. W. Wickstead, Extensions of orthomorphisms, J. Austral. Math. Soc. Ser. A 29 (1980), no. 1, 87–98. MR 566279, DOI 10.1017/S1446788700020966
- A. C. Zaanen, Riesz spaces. II, North-Holland Mathematical Library, vol. 30, North-Holland Publishing Co., Amsterdam, 1983. MR 704021, DOI 10.1016/S0924-6509(08)70234-4
Additional Information
- Karim Boulabiar
- Affiliation: Département de Mathématiques, Faculté des Sciences de Bizerte, Université de 7 Novembre à Carthage, 7021-Zarzouna, Tunisia
- Email: karim.boulabiar@ipest.rnu.tn
- Gerard Buskes
- Affiliation: Department of Mathematics, University of Mississippi, University, Mississippi 38677
- Email: mmbuskes@olemiss.edu
- Received by editor(s): February 20, 2002
- Received by editor(s) in revised form: July 29, 2002, and October 30, 2002
- Published electronically: August 21, 2003
- Additional Notes: The second named author gratefully acknowledges support from the Department of the Navy, Office of Naval Research Grant, no. N00014-01-1-0322
- Communicated by: N. Tomczak-Jaegermann
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 799-806
- MSC (2000): Primary 46A40, 47B65
- DOI: https://doi.org/10.1090/S0002-9939-03-07007-2
- MathSciNet review: 2019958