Abstract

We establish some operator versions of Bellman’s inequality. In particular, we prove that if Φ : B(H ) → B(K ) is a unital positive linear map, A, B ∈ B(H ) are contractions, p > 1 and 0  λ  1, then  Φ(IH − A∇λB)  1/p  Φ  (IH − A)1/p∇λ(IH − B)1/p  .