It is well known that every compact operator acting from a symmetric function space with an absolutely continuous norm is narrow. On the other hand, there exists a non-narrow linear continuous functional on the space $L_∞$. We find some sufficient conditions on a functional from $L_∞^*$ to be narrow.
[1] Aliprantis C. D., Burkinshaw O. Positive operators. - Dordrecht: Springer, 2006. - XVI, 367 p.
[2] Eds. Johnson W. B., Lindenstrauss J. Handbook of the geometry of Banach spaces. Vol.I. - Amsterdam: Elsevier, 2001. - X, 1005 p.
[3] O. V. Maslyuchenko, V. V. Mykhaylyuk, M. M. Popov. A lattice approach to narrow operators // Positivity. - Online First. - 2008. - DOI 10.1007/s 11117-008-2193-z. - 37 p.
[4] Plichko A. M., Popov M. M. Symmetric function spaces on atomless probability spaces // Diss Math. (Rozpr. mat.) - 1990. - 306. - P. 1-85.
[5] Rodin V. A., Semenov E. M. Rademacher series in symmetric spaces // Anal. Math. - 1975. - 1. - p. 207-222.
- ACS Style
- Krasikova, I.V. Sufficient conditions for narrowness of functionals in the space $L_∞$. Bukovinian Mathematical Journal. 2018, 1
- AMA Style
- Krasikova IV. Sufficient conditions for narrowness of functionals in the space $L_∞$. Bukovinian Mathematical Journal. 2018; 1(454).
- Chicago/Turabian Style
- Iryna Volodymyrivna Krasikova. 2018. "Sufficient conditions for narrowness of functionals in the space $L_∞$". Bukovinian Mathematical Journal. 1 no. 454.