Some operator Bellman type inequalities

Abstract

In this paper, we employ the Mond–Pečarić method to establish some reverses of the operator Bellman inequality under certain conditions. In particular, we show δIℋ+∑j=1nωjΦj((Iℋ−Aj)p)≥(∑j=1nωjΦj(Iℋ−Aj))p, where Aj(1≤j≤n) are self-adjoint contraction operators with 0≤mIℋ≤Aj≤MIℋ, Φj are unital positive linear maps on B(ℋ), ωj∈R+(1≤j≤n) such that ∑j=1nωj=1, δ=(1−p)(1p(1−m)p−(1−M)pM−m)pp−1+(1−M)(1−m)p−(1−m)(1−M)pM−m and 0<p<1. We also present some refinements of the operator Bellman inequality.