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Generalizing a result of Babenko and Pichugov, it is shown that if      T T       is a weakly compact operator on      L 1 ( μ  ) {L^1}(\mu )      , where      μ  \mu       is a      σ  \sigma      -finite nonatomic measure, then      ‖ I + T ‖ = 1 + ‖ T ‖ \left \| {I + T} \right \| = 1 + \left \| T \right \|      . A characterization of all operators      T T       on      L 1 ( μ  ) {L^1}(\mu )       having this property is also given.

Daugavet’s equation and operators on 𝐿¹(𝜇)