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

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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/}
}
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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/