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

Popov, M. M.

Arch. Math. 90, No. 6, 537-544 (2008).

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.

Reviewer: Vladimir Kadets (Kharkov)

MSC:

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.)

Keywords:

Daugavet equation; narrow operator

Citations:

Zbl 0715.46011; Zbl 1056.46013