Daugavet type inequalities for narrow operators
in the space $L_1$

M. M. Popov

First the classical Daugavet Equation $\|I+T\| = 1 + \|T\|$ (DE) for $L_1$ was established for compact operators $T$ and then generalized for wide classes of operators in a number of papers. We prove some inequalities which generalize the DE for an arbitrary into isomorphism $J: \, L_1 \to L_1$ instead of the identity $I$.

1. Abramovich Yu. New classes of spaces on which compact operators satisfy the Daugavet equation // J. Operator Theory. – 1991. – V.25. – P.331–345.

2. Alspach D. E. Small into isomorphisms on $L_p$ spaces // Ill. J. Math. -- 1983. -- V.27, No 2. -- P.300--314.

3. Benyamini Y., Lin P.-K. An operator on $L^p$ without best compact approximation // Israel J. Math. -- 1985. -- V.51, No 4. -- P.298--304.

4. Даугавет И. К. Об одном свойстве вполне непрерывных операторов в пространстве $C$ // Успехи мат. наук. -- 1963. -- Т.18, № 5. -- С.157--158.

5. Dor L. E. On projections in $L^1$ // Ann. Math. -- 1975. -- V.102, No 3. -- P.463--474.

6. Franchetti C. Lower bounds for the norms of projections with small kernels // Bull. Austral. Math. Soc. – 1992. – V.45. – P.507–511.

7. Franchetti C., Semenov E. M. A Hilbert space characterization among function spaces // Anal. Math. – 1995. – V.21. – P.85–93.

8. Godefroy G., Kalton N. J., Li D. Operators between subspaces and quotients of $L^1$ // Indiana Univ. Math. J. -- 2000. -- V.9, No 1. -- P.245--286.

9. Holub J. R. A property of weakly compact operators on $C([0,1])$ // Proc. Amer. Math. Soc. -- 1986. -- V.97. -- P.396--398.

10. Holub J. R. Daugavet's equation and operators on $L^1(\mu)$ // Proc. Amer. Math. Soc. -- 1987. -- V.100, No~2. -- P.295--300.

11. Kadets V. M. Some remarks concerning the Daugavet equation // Quaestiones Math. – 1996. – V.19. – P.225–235.

12. Kadets V. M., Popov M. M. On the Liapunov convexity theorem with applications to sign-embeddings // Укр. мат. ж. – 1992. – V.44, No 9. – P.1192–1200.

13. Кадец В. М., Попов М. М. Свойство Даугавета для узких операторов в богатых под\-про\-странствах пространств $C[0,1]$ и $L_1[0,1]$ // Алгебра и анализ. -- 1996. -- Т.8, вып.4. -- С.43--62.

14. Kadets V.\ M., Popov M. M. Some stability theorems about narrow operators on $L_1$ and $C(K)$ // Mat. fiz., anal., geom. -- 2003. -- V.10, No 1. -- P.49--60.

15. Kadets V. M., Shvidkoy R. V. The Daugavet property for pairs of Banach spaces // Mat. fiz., anal., geom. – 1999. – V.6, No 3/4. – P.253–263.

16. Kadets V. M., Shvidkoy R. V., Sirotkin G. G., Werner D. Banach spaces with the Daugavet property // Trans. Amer. Math. Soc. – 2000. – V.352. No 2. – P.855–873.

17. Kadets V. M., Shvidkoy R. V., Werner D. Narrow operators and rich subspaces of Banach spaces with the Daugavet property // Stud. Math. – 2001. – V.147, No 3. – P.269–298.

18. Kalton N. J. The endomorphisms of $L_p \, (0 \leq p \leq 1)$ // Indiana Univ. Math. J. -- 1978. -- V.27, No 3. -- P.353--381.

19. Kalton N. J., Randrianantoanina B. Surjective isometries of rearrangement-invariant spaces // Quart. J. Math. Oxford. – 1994. – V.45. – P.301–327.

20. Kalton N. J., Peck N. T., Roberts J. W. An $F$-space sampler. -- Cambridge etc.: Cambridge Univ. Press, 1984. -- 154 pp.

21. Kwapień S. On the form of a linear operator in the space of all measurable functions // Bull. Acad. Polon. Sci. – 1973. – V.21. – P.951–954.

22. Lindenstrauss J., Tzafriri L. Classical Banach Spaces I. – Berlin etc.: Springer-Verlag, 1977. – 188 pp.

23. Lindenstrauss J., Tzafriri L. Classical Banach Spaces II. – Berlin etc.: Springer-Verlag, 1979. – 243 pp.

24. Лозановский Г. Я. О почти интегральных операторах в $KB$-пространствах // Вестн. Ленингр. ун-та. Сер. мат. мех. астроном. -- 1966. -- Т.21, № 7. -- С.35--44.

25. Oikhberg T. The Daugavet property of $C^*$-algebras and non-commutative $L_p$-spaces // Positivity. -- 2002. -- V.6. -- P.59--73.

26. Oikhberg T. Spaces of operators, the $\psi$-Daugavet property, and numerical indices // Preprint.

27. Plichko A. M., Popov M. M. Symmetric function spaces on atomless probability spaces // Dissertationes Mathematicae. – 1990. – V.306. – P.1–85.

28. Попов М. М. О нормах проекторов в $L_p(\mu)$ с ``малыми" ядрами // Функц. анализ и его прил. -- 1987. -- Т.21, вып. 2. -- С.84--85.

29. Popov M. M., Randrianantoanina B. A pseudo-Daugavet property for narrow projections in Lorentz spaces // Ill. J. Math. – 2003. (to appear).

30. Randrianantoanina B. Contractive projections in nonatomic function spaces // Proc. Amer. Math. Soc. – 1995. – V.123. – P.1747–1750.

31. Randrianantoanina B. A Daugavet type property in real rearrangement invariant function spaces // Preprint.

32. Shvidkoy R. Geometric aspects of the Daugavet property // J.Funct. Anal. – 2000. – V.176, No 2. – P.198–212.

33. Shvidkoy R. The largest linear space of operators satisfying the Daugavet equation in $L_1$ // Proc. Amer. Math. Soc. -- 2001. -- V.130, No 3. -- P.773--777.

34. Shvidkoy R. Operators and integrals in Banach spaces. – Dissertation, Univ. of Missouri-Columbia, 2001, 125 pp.

35. Wojtaszczyk P. Some remarks on the Daugavet equation // Proc. Amer. Math. Soc. – 1992. – V.115. – P.1047–1052.

	Daugavet type inequalities for narrow operators in the space $L_1$
Author	M. M. Popov popov@chv.ukrpack.net Department of Mathematics, Chernivtsi National University, Kotsiubyns'kogo str., 2,58012, Chernivtsi, Ukraine
Abstract	First the classical Daugavet Equation $\\|I+T\\| = 1 + \\|T\\|$ (DE) for $L_1$ was established for compact operators $T$ and then generalized for wide classes of operators in a number of papers. We prove some inequalities which generalize the DE for an arbitrary into isomorphism $J: \, L_1 \to L_1$ instead of the identity $I$.
Keywords	classical Daugavet Equation, compact operators $T$, wide classes of operators, inequalities
DOI	doi:10.30970/ms.20.1.75-84
Reference	1. Abramovich Yu. New classes of spaces on which compact operators satisfy the Daugavet equation // J. Operator Theory. – 1991. – V.25. – P.331–345. 2. Alspach D. E. Small into isomorphisms on $L_p$ spaces // Ill. J. Math. -- 1983. -- V.27, No 2. -- P.300--314. 3. Benyamini Y., Lin P.-K. An operator on $L^p$ without best compact approximation // Israel J. Math. -- 1985. -- V.51, No 4. -- P.298--304. 4. Даугавет И. К. Об одном свойстве вполне непрерывных операторов в пространстве $C$ // Успехи мат. наук. -- 1963. -- Т.18, № 5. -- С.157--158. 5. Dor L. E. On projections in $L^1$ // Ann. Math. -- 1975. -- V.102, No 3. -- P.463--474. 6. Franchetti C. Lower bounds for the norms of projections with small kernels // Bull. Austral. Math. Soc. – 1992. – V.45. – P.507–511. 7. Franchetti C., Semenov E. M. A Hilbert space characterization among function spaces // Anal. Math. – 1995. – V.21. – P.85–93. 8. Godefroy G., Kalton N. J., Li D. Operators between subspaces and quotients of $L^1$ // Indiana Univ. Math. J. -- 2000. -- V.9, No 1. -- P.245--286. 9. Holub J. R. A property of weakly compact operators on $C([0,1])$ // Proc. Amer. Math. Soc. -- 1986. -- V.97. -- P.396--398. 10. Holub J. R. Daugavet's equation and operators on $L^1(\mu)$ // Proc. Amer. Math. Soc. -- 1987. -- V.100, No~2. -- P.295--300. 11. Kadets V. M. Some remarks concerning the Daugavet equation // Quaestiones Math. – 1996. – V.19. – P.225–235. 12. Kadets V. M., Popov M. M. On the Liapunov convexity theorem with applications to sign-embeddings // Укр. мат. ж. – 1992. – V.44, No 9. – P.1192–1200. 13. Кадец В. М., Попов М. М. Свойство Даугавета для узких операторов в богатых под\-про\-странствах пространств $C[0,1]$ и $L_1[0,1]$ // Алгебра и анализ. -- 1996. -- Т.8, вып.4. -- С.43--62. 14. Kadets V.\ M., Popov M. M. Some stability theorems about narrow operators on $L_1$ and $C(K)$ // Mat. fiz., anal., geom. -- 2003. -- V.10, No 1. -- P.49--60. 15. Kadets V. M., Shvidkoy R. V. The Daugavet property for pairs of Banach spaces // Mat. fiz., anal., geom. – 1999. – V.6, No 3/4. – P.253–263. 16. Kadets V. M., Shvidkoy R. V., Sirotkin G. G., Werner D. Banach spaces with the Daugavet property // Trans. Amer. Math. Soc. – 2000. – V.352. No 2. – P.855–873. 17. Kadets V. M., Shvidkoy R. V., Werner D. Narrow operators and rich subspaces of Banach spaces with the Daugavet property // Stud. Math. – 2001. – V.147, No 3. – P.269–298. 18. Kalton N. J. The endomorphisms of $L_p \, (0 \leq p \leq 1)$ // Indiana Univ. Math. J. -- 1978. -- V.27, No 3. -- P.353--381. 19. Kalton N. J., Randrianantoanina B. Surjective isometries of rearrangement-invariant spaces // Quart. J. Math. Oxford. – 1994. – V.45. – P.301–327. 20. Kalton N. J., Peck N. T., Roberts J. W. An $F$-space sampler. -- Cambridge etc.: Cambridge Univ. Press, 1984. -- 154 pp. 21. Kwapień S. On the form of a linear operator in the space of all measurable functions // Bull. Acad. Polon. Sci. – 1973. – V.21. – P.951–954. 22. Lindenstrauss J., Tzafriri L. Classical Banach Spaces I. – Berlin etc.: Springer-Verlag, 1977. – 188 pp. 23. Lindenstrauss J., Tzafriri L. Classical Banach Spaces II. – Berlin etc.: Springer-Verlag, 1979. – 243 pp. 24. Лозановский Г. Я. О почти интегральных операторах в $KB$-пространствах // Вестн. Ленингр. ун-та. Сер. мат. мех. астроном. -- 1966. -- Т.21, № 7. -- С.35--44. 25. Oikhberg T. The Daugavet property of $C^*$-algebras and non-commutative $L_p$-spaces // Positivity. -- 2002. -- V.6. -- P.59--73. 26. Oikhberg T. Spaces of operators, the $\psi$-Daugavet property, and numerical indices // Preprint. 27. Plichko A. M., Popov M. M. Symmetric function spaces on atomless probability spaces // Dissertationes Mathematicae. – 1990. – V.306. – P.1–85. 28. Попов М. М. О нормах проекторов в $L_p(\mu)$ с ``малыми" ядрами // Функц. анализ и его прил. -- 1987. -- Т.21, вып. 2. -- С.84--85. 29. Popov M. M., Randrianantoanina B. A pseudo-Daugavet property for narrow projections in Lorentz spaces // Ill. J. Math. – 2003. (to appear). 30. Randrianantoanina B. Contractive projections in nonatomic function spaces // Proc. Amer. Math. Soc. – 1995. – V.123. – P.1747–1750. 31. Randrianantoanina B. A Daugavet type property in real rearrangement invariant function spaces // Preprint. 32. Shvidkoy R. Geometric aspects of the Daugavet property // J.Funct. Anal. – 2000. – V.176, No 2. – P.198–212. 33. Shvidkoy R. The largest linear space of operators satisfying the Daugavet equation in $L_1$ // Proc. Amer. Math. Soc. -- 2001. -- V.130, No 3. -- P.773--777. 34. Shvidkoy R. Operators and integrals in Banach spaces. – Dissertation, Univ. of Missouri-Columbia, 2001, 125 pp. 35. Wojtaszczyk P. Some remarks on the Daugavet equation // Proc. Amer. Math. Soc. – 1992. – V.115. – P.1047–1052.
Pages	75-84
Volume	20
Issue	1
Year	2003
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Daugavet type inequalities for narrow operators in the space $L_1$