An exact Daugavet type inequality for small into isomorphisms in \(L_{1}\). (English) Zbl 1155.46004

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 A.M.Plichko and M.M.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.


46B04 Isometric theory of Banach spaces
47B38 Linear operators on function spaces (general)
46B20 Geometry and structure of normed linear spaces
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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