Daugavet centers are separably determined

Author T. Ivashyna
V.N. Karazin Kharkiv National University

Abstract A linear bounded operator $G$ acting from a Banach space $X$ into a Banach space $Y$ is a Daugavet center if every linear bounded rank-$1$ operator $T\colon X \to Y$ fulfills $\|G+T\|=\|G\|+\|T\|$. We prove that $G \colon X \to Y$ is a~Daugavet center if and only if for every separable subspaces $X_1\subset X$ and $Y_1\subset Y$ there exist separable subspaces $X_2\subset X$ and $Y_2\subset Y$ such that $X_1\subset X_2$, $Y_1\subset Y_2$, $G(X_2)\subset Y_2$ and the restriction $G|_{X_2} \colon X_2 \to Y_2$ of $G$ is a Daugavet center. We apply this fact to study the set of $G$-narrow operators.
Keywords Daugavet center; Daugavet property; narrow operator
1. A. Aviles, V. Kadets, M. Martin, J. Meri, V. Shepelska, Slicely countably determined Banach spaces, C. R., Math., Acad. Sci. Paris, 347 (2009), 12771280.

2. T. Bosenko, Strong Daugavet operators and narrow operators with respect to Daugavet centers, Visn. Khark. Univ., Ser. Mat. Prykl. Mat. Mekh., 931 (2010), 62, 519.

3. T. Bosenko, V. Kadets, Daugavet centers, Zh. Mat. Fiz. Anal. Geom., 6 (2010), 1, 320.

4. I. Daugavet, On a property of completely continuous operators in the space C, Uspekhi Mat. Nauk, 18 (1963), 5, 157158. (in Russian)

5. V. Kadets, R. Shvidkoy, G. Sirotkin, D. Werner, Banach spaces with the Daugavet property, Trans. Amer. Math. Soc., 352 (2000), 2, 855873.

6. V. Kadets, R. Shvidkoy, D. Werner, Narrow operators and rich subspaces of Banach spaces with the Daugavet property, Stud. Math., 147 (2001), 3, 269298.

7. G. Lozanovskii, On almost integral operators in KB-spaces, Vestnik Leningrad Univ. Mat. Mekh. Astr., 21 (1966), 3544. (in Russian)

8. D. Werner, The Daugavet equation for operators on function spaces, J. Funct. Anal., 143 (1997), 117128.

9. P.Wojtaszczyk, Some remarks on the Daugavet equation, Proc. Amer. Math. Soc., 115 (1992), 10471052.
Pages 66-70
Volume 40
Issue 1
Year 2013
Journal Matematychni Studii
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