A note on narrow operators in $L_\infty$

I.V.Krasikova

A usual technique used in various investigations on narrow operators does not work in the case when the domain space is $L_\infty$. We show that the example of a narrow projection in an r.i. function space $E$ on $[0,1]$ with an absolutely continuous norm onto a subspace $E_1 \subset E$ isometric to $E$ constructed by A.~Plichko and M.~Popov extends to the case of $E = L_\infty$. We also prove that a sum of two narrow operators in $L_\infty$ need not be narrow, answering to a question of O.~Maslyuchenko, V.~Mykhaylyuk and M.~Popov.

1. Abramovich Yu.A., Aliprantis C.D. An invitation to operator theory. -- Graduate Studies in Math. -- V.50. -- Amer. Math. Soc., Providence, Rhode Island, 2002.

2. Aliprantis C.D., Burkinshaw O. Positive operators. -- Springer, Dordrecht, 2006.

3. Банах С.С. Курс функціонального аналізу. -- Київ: Радянська школа, 1948. Translation from the book Théorie des opérations linéaires (1932)

4. Diestel J., Uhl J.J. Vector Measures. -- AMS, Mathematical surveys, 1977.

5. Lacey H.E. The Isometrical Theory of Classical Banach Spaces. -- Springer--Verlag, Berlin--Heidelberg--New York, 1974.

6. Lindenstrauss J., Tzafriri L. Classical Banach spaces, V. 1, Sequence spaces. -- Springer--Verlag, Berlin--Heidelberg--New York, 1977.

7. Lindenstrauss J., Tzafriri L. Classical Banach spaces, V. 2, Function spaces. -- Springer--Verlag, Berlin--Heidelberg--New York, 1979.

8. Maslyuchenko O.V., Mykhaylyuk V.V., Popov M.M. A lattice approach to narrow operators // Positivity -- 2009. -- V.13 (Online version: 10.1007/s11117-008-2193-z).

9. Plichko A.M., Popov M.M. Symmetric function spaces on atomless probability spaces // Dissertationes Math. (Rozprawy Mat.). -- 1990. -- V.306. -- P.1--85.

	A note on narrow operators in $L_\infty$
Author	I.V.Krasikova yudp@mail.ru Department of Mathematics,Zaporizhzhya State University,str. Zhukovs'koho 2, Zaporizhzhya, Ukraine
Abstract	A usual technique used in various investigations on narrow operators does not work in the case when the domain space is $L_\infty$. We show that the example of a narrow projection in an r.i. function space $E$ on $[0,1]$ with an absolutely continuous norm onto a subspace $E_1 \subset E$ isometric to $E$ constructed by A.~Plichko and M.~Popov extends to the case of $E = L_\infty$. We also prove that a sum of two narrow operators in $L_\infty$ need not be narrow, answering to a question of O.~Maslyuchenko, V.~Mykhaylyuk and M.~Popov.
Keywords	narrow operator, function space, absolutely continuous norm
DOI	doi:10.30970/ms.31.1.102-106
Reference	1. Abramovich Yu.A., Aliprantis C.D. An invitation to operator theory. -- Graduate Studies in Math. -- V.50. -- Amer. Math. Soc., Providence, Rhode Island, 2002. 2. Aliprantis C.D., Burkinshaw O. Positive operators. -- Springer, Dordrecht, 2006. 3. Банах С.С. Курс функціонального аналізу. -- Київ: Радянська школа, 1948. Translation from the book Théorie des opérations linéaires (1932) 4. Diestel J., Uhl J.J. Vector Measures. -- AMS, Mathematical surveys, 1977. 5. Lacey H.E. The Isometrical Theory of Classical Banach Spaces. -- Springer--Verlag, Berlin--Heidelberg--New York, 1974. 6. Lindenstrauss J., Tzafriri L. Classical Banach spaces, V. 1, Sequence spaces. -- Springer--Verlag, Berlin--Heidelberg--New York, 1977. 7. Lindenstrauss J., Tzafriri L. Classical Banach spaces, V. 2, Function spaces. -- Springer--Verlag, Berlin--Heidelberg--New York, 1979. 8. Maslyuchenko O.V., Mykhaylyuk V.V., Popov M.M. A lattice approach to narrow operators // Positivity -- 2009. -- V.13 (Online version: 10.1007/s11117-008-2193-z). 9. Plichko A.M., Popov M.M. Symmetric function spaces on atomless probability spaces // Dissertationes Math. (Rozprawy Mat.). -- 1990. -- V.306. -- P.1--85.
Pages	102-106
Volume	31
Issue	1
Year	2009
Journal	Matematychni Studii
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