Narrow operators on lattice-normed spaces

Marat Pliev

Open Mathematics, Tome 9 (2011), p. 1276-1287 / Harvested from The Polish Digital Mathematics Library

Résumé

The aim of this article is to extend results of Maslyuchenko, Mykhaylyuk and Popov about narrow operators on vector lattices. We give a new definition of a narrow operator, where a vector lattice as the domain space of a narrow operator is replaced with a lattice-normed space. We prove that every GAM-compact (bo)-norm continuous linear operator from a Banach-Kantorovich space V to a Banach lattice Y is narrow. Then we show that, under some mild conditions, a continuous dominated operator is narrow if and only if its exact dominant is so.

Détail

Publié le : 2011-01-01

Zbl 1253.47024

EUDML-ID : urn:eudml:doc:269204

Bibliographie

[1] Abramovich Y.A., Aliprantis C.D., An Invitation to Operator Theory, Grad. Stud. Math., 50, American Mathematical Society, Providence, 2002 | Zbl 1022.47001

[2] Aliprantis C.D., Burkinshaw O., Positive Operators, Springer, Dordrecht, 2006

[3] Andrews K.T., Representation of compact and weakly compact operators on the space of Bochner integrable functions, Pacific J. Math., 1981, 92(2), 257–267 | Zbl 0427.47021

[4] Bilik D., Kadets V., Shvidkoy R., Sirotkin G., Werner D., Narrow operators on vector-valued sup-normed spaces, Ilinois J. Math., 2006, 46(2), 421–441 | Zbl 1030.46014

[5] Bilik D., Kadets V., Shvidkoy R., Werner D., Narrow operators and the Daugavet property for ultraproducts, Positivity, 2005, 9(1), 45–62 http://dx.doi.org/10.1007/s11117-003-9339-9 | Zbl 1099.46009

[6] Boyko K., Kadets V., Werner D., Narrow operators on Bochner L 1-spaces, Zh. Mat. Fiz. Anal. Geom., 2002, 2(4), 358–371 | Zbl 1147.46006

[7] Enflo P., Starbird T.W., Subspaces of L 1 containing L 1, Studia Math., 1979, 65(2), 213–225

[8] Flores J., Ruiz C., Domination by positive narrow operators, Positivity, 2003, 7(4), 303–321 http://dx.doi.org/10.1023/A:1026211909760 | Zbl 1051.47032

[9] Ghoussoub N., Rosenthal H.P., Martingales, G δ-embeddings and quotients of L 1, Math. Ann., 1983, 264(3), 321–332 http://dx.doi.org/10.1007/BF01459128 | Zbl 0511.46017

[10] Johnson W.B., Maurey B., Schechtman G., Tzafriri L., Symmetric Structures in Banach Spaces, Mem. Amer. Math. Soc., 19, American Mathematical Society, Providence, 1979 | Zbl 0421.46023

[11] Kadets V.M., Kadets M.I., Rearrangements of Series in Banach Spaces, Transl. Math. Monogr., 86, American Mathematical Society, Providence, 1991 | Zbl 0743.46011

[12] Kadets V.M., Popov M.M., The Daugavet property for narrow operators in rich subspaces of the spaces C[0; 1] and L 1[0; 1], Algebra i Analiz, 1996, 8(4), 43–62 (in Russian) | Zbl 0881.47017

[13] Kadets V.M., Shvidkoy R.V., Werner D., Narrow operators and rich subspaces of Banach spaces with the Daugavet property, Studia Math., 2001, 147(3), 269–298 http://dx.doi.org/10.4064/sm147-3-5 | Zbl 0986.46010

[14] Kantorovich L.V., On a class of functional equations, Dokl. Akad. Nauk SSSR, 1936, 4(5), 211–216

[15] Kusraev A.G., Dominated Operators, Math. Appl., 519, Kluwer, Dordrecht, 2000

[16] Lindenstrauss J., Tzafriri L., Classical Banach Spaces. I, Ergeb. Math. Grenzgeb., 92, Springer, Berlin-New York, 1977 | Zbl 0362.46013

[17] Lindenstrauss J., Tzafriri L., Classical Banach Spaces. II, Ergeb. Math. Grenzgeb., 97, Springer, Berlin-New York, 1979 | Zbl 0403.46022

[18] Maslyuchenko O.V., Mykhaylyuk V.V., Popov M.M., A lattice approach to narrow operators, Positivity, 2009, 13(3), 459–495 http://dx.doi.org/10.1007/s11117-008-2193-z | Zbl 1183.47033

[19] Megginson R.E., An Introduction to Banach Space Theory, Grad. Texts in Math., 183, Springer, New York, 1998 http://dx.doi.org/10.1007/978-1-4612-0603-3

[20] Plichko A.M., Popov M.M., Symmetric Function Spaces on Atomless Probability Spaces, Dissertationes Math. (Rozprawy Mat.), 306, Polish Academy of Sciences, Warsaw, 1990