Spaces of operators, the \(\psi\)-Daugavet property, and numerical indices
Let \(\psi: [0, +\infty]\to[1, +\infty]\) be a strictly increasing function. A Banach space \(X\) is said to have the \(\psi\)-Daugavet property if \(\| I + T\| \geq \psi(\| T\| )\) for every compact operator \(T: X \to X\). This property has been studied extensively for \(\psi(t)=1+t\), but for general \(\psi\) the only results are in \(L_p\) [see \textit{Y. Benyamini} and \textit{P. K. Lin}, Isr. J. Math. 51, 298--304 (1985; Zbl 0596.47020); \textit{K. Boyko} and \textit{V. Kadets}, Visn. Khark. Univ., Ser. Mat. Prykl. Mat. Mekh. 645, No. 54, 22--29 (2004; Zbl 1071.46014), for non-commutative \(L_p\)-spaces, see \textit{T. Oikhberg}, Positivity~6, No. 1, 59--73 (2002; Zbl 1004.46040)]. The author shows that if \(1< p < \infty\) and \(X \subset L(\ell_p)\) is a subspace containing all compact operators, then there is a constant \(c_p>0\) such that \(X\) has the \(\psi\)-Daugavet property with \(\psi(t)=(1+c_p t^q)^{1/q}\), where \(q = \max\{2, p\}\). For \(p=2\), a better estimate is given with \(\psi(t)=1+t/(8\sqrt2)\). Spaces of numerical index~1 are studied as well. It is shown that for every real \(n\)-dimensional space \(X\) with numerical index~1, either \(X\) itself or \(X^*\) contains an isometric copy of \(\ell_1^m\) with \(m\geq c\sqrt n\), where \(c>0\) is an absolute constant.
scientific article; zbMATH DE number 5070782
scientific article