Daugavet centers are separably determined

Author T. Ivashyna
t.bosenko@mail.ru
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
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Pages 66-70
Volume 40
Issue 1
Year 2013
Journal Matematychni Studii
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