Extensions of orthogonally additive operators

A. V. Gumenchuk; M. A. Pliev; M. M. Popov

We find natural sufficient conditions on a subset $D$ of a vector lattice $E$ under which every orthogonally additive operator $T_0\colon D \to X$, where $X$ is a vector space, can be extended to an orthogonally additive operator $T\colon E \to X$. Two theorems on the extension from lateral ideals and lateral bands, respectively, are obtained.

1. C.D. Aliprantis, O. Burkinshaw, Positive operators, Springer, Dordrecht, 2006.

2. V. Mykhaylyuk, M. Pliev, M. Popov, Laterally ordered sets and orthogonally additive operators on vector lattices, Preprint.

3. Th. Jech, Set theory, Springer, Berlin, 2003.

4. A.G. Kusraev, M.A. Pliev, Orthogonally additive operators on lattice-normed spaces, Vladikavkaz Math. J., (1999), №3, 33–43.

5. A.G. Кusraev, M.A. Pliev,Weak integral representation of the dominated orthogonally additive operators, Vladikavkaz Math. J., (1999), №4, 22–39.

6. J.M. Mazon, S. Segura de Leon, Order bounded ortogonally additive operators, Rev. Roumane Math. Pures Appl., 35 (1990), №4, 329–353.

7. J.M. Mazon, S. Segura de Leon, Uryson operators, Rev. Roumane Math. Pures Appl., 35 (1990), №5, 431–449.

8. V.V. Mykhaylyuk, M.A. Pliev, M.M. Popov, O.V. Sobchuk, Dividing measures and narrow operators, Preprint.

9. M. Pliev, Uryson operators on the spaces with mixed norm, Vladikavkaz Math. J., (2007), №3, 47–57.

10. M. Pliev, M. Popov, Dominated Uryson operators, Int. J. Math. Anal., 8 (2014), №22, 1051–1059.

11. M. Pliev, M. Popov, Narrow orthogonally additive operators, arXiv:1309.5487v1, to appear in Positivity.

12. J. Ratz, On orthogonally additive mappings, Aequationes math., 28 (1985), №1, 35–49.

	Extensions of orthogonally additive operators
Author	A. V. Gumenchuk, M. A. Pliev, M. M. Popov anna_hostyuk@ukr.net, misham.popov@gmail.com Chernivtsi Medical College, Chernivtsi National University
Abstract	We find natural sufficient conditions on a subset $D$ of a vector lattice $E$ under which every orthogonally additive operator $T_0\colon D \to X$, where $X$ is a vector space, can be extended to an orthogonally additive operator $T\colon E \to X$. Two theorems on the extension from lateral ideals and lateral bands, respectively, are obtained.
Keywords	vector lattice; orthogonally additive operator; disjointness preserving operator
Reference	1. C.D. Aliprantis, O. Burkinshaw, Positive operators, Springer, Dordrecht, 2006. 2. V. Mykhaylyuk, M. Pliev, M. Popov, Laterally ordered sets and orthogonally additive operators on vector lattices, Preprint. 3. Th. Jech, Set theory, Springer, Berlin, 2003. 4. A.G. Kusraev, M.A. Pliev, Orthogonally additive operators on lattice-normed spaces, Vladikavkaz Math. J., (1999), №3, 33–43. 5. A.G. Кusraev, M.A. Pliev,Weak integral representation of the dominated orthogonally additive operators, Vladikavkaz Math. J., (1999), №4, 22–39. 6. J.M. Mazon, S. Segura de Leon, Order bounded ortogonally additive operators, Rev. Roumane Math. Pures Appl., 35 (1990), №4, 329–353. 7. J.M. Mazon, S. Segura de Leon, Uryson operators, Rev. Roumane Math. Pures Appl., 35 (1990), №5, 431–449. 8. V.V. Mykhaylyuk, M.A. Pliev, M.M. Popov, O.V. Sobchuk, Dividing measures and narrow operators, Preprint. 9. M. Pliev, Uryson operators on the spaces with mixed norm, Vladikavkaz Math. J., (2007), №3, 47–57. 10. M. Pliev, M. Popov, Dominated Uryson operators, Int. J. Math. Anal., 8 (2014), №22, 1051–1059. 11. M. Pliev, M. Popov, Narrow orthogonally additive operators, arXiv:1309.5487v1, to appear in Positivity. 12. J. Ratz, On orthogonally additive mappings, Aequationes math., 28 (1985), №1, 35–49.
Pages	214-219
Volume	41
Issue	2
Year	2014
Journal	Matematychni Studii
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