Daugavet type inequalities for operators on \(L^p\)--spaces (Q1412297)

In this paper, the author studies a generalization of the Daugavet equation for linear operators. In particular, he studies conditions on a linear operator \(T\) on \(L^p\) to have the inequality \(\| I+T\|\geq (1+ \| T\|^p)^{1/p}\) and obtains that this condition holds for disjointness preserving operators \(T\) disjoint with \(I\). He also studies a similar kind of inequalities for regular operators where the usual norm \(\|\cdot\|\) is substituted by the regular operator norm \(\|\cdot \|_r\), and for this kind of operator he proves a converse result in the following sense: If \(T_A\) denotes the operator \(T\circ \chi_A\), then \(\| I+T_A\|_r\geq (1+ \| T_A\|^p_r)^{1/p}\) for every measurable set \(A\) such that \(\mu(A)>0\) if and only if \(T\bot I\).