Daugavet type inequalities for operators on Lp-spaces

Let T be a regular operator from Lp\to Lp. Then T\perp I implies that ||I+ or -T||r\ge (1+||T||rp) {1/p}, where ||T||r denotes the regular norm of T, i.e., ||T||r=|| |T| || where |T| denotes the modulus operator of a regular operator T. For p=1 every bounded linear operator is regular and ||T||=||T||r, so that the above inequality generalizes the Daugavet equation for operators on L1-spaces. The main result of this paper (Theorem 9) is a converse of the above result. Let T be a regular linear operator on Lp and denote by TA the operator T\circ[chi]A. Then ||IA+ or -TA||r\ge (1+||TA||rp)1/p for all A with [mu](A)>0 if and only if T\perp I. [PUBLICATION ABSTRACT]