Daugavet type inequalities for operators on Lp–spaces

Abstract

Let T be a regular operator from Lp → Lp. Then \(T \bot I{\text{ implies that }}\left\| {I \pm T} \right\|_r \geqslant (1 + \left\| T \right\|_r^p )^{\frac{1}{p}} \), 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○χA. Then \(\left\| {I_A \pm T_A } \right\|_r \geqslant (1 + \left\| {T_A } \right\|_r^p )^{\frac{1}{p}} \) for all A with μ(A)>0 if and only if \(T \bot I\).