×

Narrow operators on lattice-normed spaces. (English) Zbl 1253.47024

The paper is devoted to a generalization of the notions of narrow and order narrow operators to the setting of linear maps defined on lattice normed spaces. The main results generalize the corresponding theorems of O. Maslyuchenko, V. Mykhaylyuk and M. Popov [Positivity 13, No. 3, 459–495 (2009; Zbl 1183.47033)] proved for operators defined on vector lattices. Let us mention two of them.
Theorem 5.1. Let \(E, F\) be Dedekind complete vector lattices with \(E\) atomless and \(F\) an ideal of an order continuous Banach lattice, and let \((V,E)\) be a Banach-Kantorovich space. Then every (bo)-continuous dominated linear operator \(T: V \to F\) is (bo)-order narrow if and only if \(\mathbf{|} T \mathbf{|}: E \to F\) is order narrow.
Let \((V,E),(W,F)\) be lattice-normed spaces and let \(M(V,W)\) be the space of all dominated operators from \(V\) to \(W\). Let \(L_{pe}(E,F)\) denote the set of all pseudo-embeddings from \(E\) to \(F\) in the sense of [loc. cit.], and set \(L_{pe}(V,W) = \{ T \in M(V,W): \mathbf{|} T \mathbf{|} \in L_{pe}^+ (E,F) \}\).
Theorem 5.2. Let \(E,F\) be order complete vector lattices with \(E\) atomless, \(F\) an ideal of some order continuous Banach lattice, and let \((V,E)\) be a Banach-Kantorovich space. Then every (bo)-continuous dominated linear operator \(T: V \to F\) is uniquely represented in the form \(T = T_{pe} + T_{on}\), where \(T_{pe} \in L_{pe}(V,W)\) and \(T_{on}\) is a (bo)-continuous (bo)-order narrow operator.

MSC:

47B65 Positive linear operators and order-bounded operators
47B99 Special classes of linear operators
47B38 Linear operators on function spaces (general)

Citations:

Zbl 1183.47033
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[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; · Zbl 0433.46027
[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; · Zbl 0983.47025
[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; · Zbl 0910.46008
[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; · Zbl 0715.46011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.