An exact Daugavet type inequality for small into isomorphisms in \(L_{1}\)
The Daugavet property of a Banach space \(X\) means that \(\| I + T\| = 1+ \| T\| \) for every weakly compact operator \(T : X \rightarrow X\). For \(X = L_{1}[0,1]\), the above equation in fact is true for a much wider class of operators, including the narrow operators in the sense of \textit{A.\,M.\thinspace Plichko} and \textit{M.\,M.\thinspace Popov} [Diss.\ Math.\ 306 (1990; Zbl 0715.46011)]. Developing his previous results from [Mat.\ Stud.\ 20, No.\,1, 75--84 (2003; Zbl 1056.46013)], the author shows that for every into isomorphism \(J : L_{1}[0,1] \rightarrow L_{1}[0,1]\) and for every narrow operator \(T\), \[ \| J + T \| \geq \| T \| + \| J \| \left(\tfrac{2}{d} - 1\right), \] where \(d = \| J \| \| J^{-1}\| \). It is shown that this estimate is exact.
scientific article; zbMATH DE number 5303970