Daugavet type inequalities for operators on L ^p–spaces

Abstract

Let T be a regular operator from L ^p → L ^p. 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 L ¹–spaces. The main result of this paper (Theorem 9) is a converse of the above result. Let T be a regular linear operator on L _p and denote by T _A the operator T○χ_A. 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 $T \bot I$.