The largest linear space of operators satisfying the Daugavet equation in \( L_{1}\) (Q2759027)

Let \((\Omega,{\mathcal X},\mu)\) be a measure space withou atoms of infinite measure, let \({\mathcal X}^+= \{A\in{\mathcal X}:\mu(A)> 0\}\) and, for \(A\in{\mathcal X}^+\), let NEWLINE\[NEWLINE{\mathcal X}^+_A= \{B: B\subset A, B\in{\mathcal X}^+\}.NEWLINE\]NEWLINE If \(A\in{\mathcal X}^+\), and \(T\) is a bounded linear operator in \(L^1(\Omega)\), let \(T_A\) denote the restriction of \(T\) to \(A\). Then the results of this paper identify the largest linear space which contains bounded linear operators of the form \(T\) satisfyingNEWLINENEWLINENEWLINE(1) \(\|\text{Id}_A+ T_A\|= 1+\|T_A\|\), for any set \(A\) in \({\mathcal X}\).NEWLINENEWLINENEWLINEIf the set \({\mathcal M}\) consists of bounded linear operators of the form \(T\) in \(L^1(\Omega)\) which satisfyNEWLINENEWLINENEWLINE(2) for every \(\delta> 0\) and \(A\in{\mathcal X}^+\), there is a set \(B\) in \({\mathcal X}^+_A\) with \(\mu(B)< \infty\) and \(\|\chi_B T(\chi_B/\mu(B))\|< \delta\),NEWLINENEWLINENEWLINEthen the main theorem states that each bounded linear operator \(T\) in \(L^1(\Omega)\) which satisfies (1) is in \({\mathcal M}\), and \({\mathcal M}\) is a closed linear space consisting of such operators.NEWLINENEWLINENEWLINEThe classes of operators which satisfy (1) have been considered by various authors in earlier papers including, in particular, \textit{I. K. Daugavet} [Usp. Mat. Nauk. 18, No. 5(113), 157-158 (1963; Zbl 0138.38603)].