Geodesic
Singular symmetric functionals and Banach limits with additional invariance properties
Izvestiya. Mathematics , Tome 67 (2003) no. 6, pp. 1187-1212.

Voir la notice de l'article provenant de la source Math-Net.Ru

For symmetric spaces of measurable functions on the real half-line, we study the problem of existence of positive linear functionals monotone with respect to the Hardy–Littlewood semi-ordering, the so-called symmetric functionals. Two new wide classes of symmetric spaces are constructed which are distinct from Marcinkiewicz spaces and for which the set of symmetric functionals is non-empty. We consider a new construction of singular symmetric functionals based on the translation-invariance of Banach limits defined on the space of bounded sequences. We prove the existence of Banach limits invariant under the action of the Hardy operator and all dilation operators. This result is used to establish the stability of the new construction of singular symmetric functionals for an important class of generating sequences. 
@article{IM2_2003_67_6_a3,
     author = {P. G. Dodds and B. De Pagter and A. A. Sedaev and E. M. Semenov and F. A. Sukochev},
     title = {Singular symmetric functionals and {Banach} limits with additional invariance properties},
     journal = {Izvestiya. Mathematics },
     pages = {1187--1212},
     publisher = {mathdoc},
     volume = {67},
     number = {6},
     year = {2003},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2003_67_6_a3/}
}
TY  - JOUR
AU  - P. G. Dodds
AU  - B. De Pagter
AU  - A. A. Sedaev
AU  - E. M. Semenov
AU  - F. A. Sukochev
TI  - Singular symmetric functionals and Banach limits with additional invariance properties
JO  - Izvestiya. Mathematics 
PY  - 2003
SP  - 1187
EP  - 1212
VL  - 67
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2003_67_6_a3/
LA  - en
ID  - IM2_2003_67_6_a3
ER  - 
%0 Journal Article
%A P. G. Dodds
%A B. De Pagter
%A A. A. Sedaev
%A E. M. Semenov
%A F. A. Sukochev
%T Singular symmetric functionals and Banach limits with additional invariance properties
%J Izvestiya. Mathematics 
%D 2003
%P 1187-1212
%V 67
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2003_67_6_a3/
%G en
%F IM2_2003_67_6_a3
P. G. Dodds; B. De Pagter; A. A. Sedaev; E. M. Semenov; F. A. Sukochev. Singular symmetric functionals and Banach limits with additional invariance properties. Izvestiya. Mathematics , Tome 67 (2003) no. 6, pp. 1187-1212. http://geodesic.mathdoc.fr/item/IM2_2003_67_6_a3/

[1] Dixmier J., “Existence de traces non normales”, C. R. Acad. Sci. Paris, 262 (1966), A1107–A1108 | MR

[2] Connes A., Noncommutative Geometry, Academic Press, San Diego, 1994 | MR | Zbl

[3] Dodds P. G., de Pagter B., Semenov E. M., Sukochev F. A., “Symmetric functionals and singular traces”, Positivity, 2 (1998), 47–75 | DOI | MR | Zbl

[4] Dodds P. G., de Pagter B., Sedaev A. A., Semenov E. M., Sukochev F. A., “Singulyarnye simmetrichnye funktsionaly”, Zapiski nauch. sem. POMI, 290, POMI, SPb., 2002, 42–71 | Zbl

[5] Lorentz G. G., “A contribution to the theory of divergent sequences”, Acta Math., 80 (1948), 167–190 | DOI | MR

[6] Sucheston L., “Banach limits”, Amer. Math. Monthly, 74 (1967), 308–311 | DOI | MR | Zbl

[7] Krein S. G., Petunin Yu. I., Semenov E. M., Interpolyatsiya lineinykh operatorov, Nauka, M., 1978 | MR

[8] Bennett C., Sharpley R., Interpolation of Operators, Academic Press, London, 1988 | MR

[9] Brudnyi Yu. A., Kruglyak N. Ya., Interpolation Functors and Interpolation Spaces, V. 1, North Holland, Amsterdam, 1991 | MR | Zbl

[10] Berg I., Lëfstrëm I., Interpolyatsionnye prostranstva. Vvedenie, Mir, M., 1980 | MR

[11] Rid M., Saimon B., Metody sovremennoi matematicheskoi fiziki. I. Funktsionalnyi analiz, Mir, M., 1977 | MR

[12] Lozanovskii G. Ya., “O lokalizovannykh funktsionalakh v vektornykh strukturakh”, Teor. funktsii, funkts. analiz i ikh prilozh., 19 (1974), 66–80 | MR | Zbl