Bellman inequality for Hilbert space operators

TL;DR

This paper develops operator versions of Bellman's inequality within Hilbert spaces, providing new inequalities involving positive linear maps and contractions, which extend classical scalar inequalities to the operator setting.

Contribution

The paper introduces novel operator inequalities related to Bellman's inequality, extending scalar inequalities to the framework of Hilbert space operators with positive linear maps.

Findings

Established operator Bellman inequalities for contractions.

Extended classical inequalities to the operator setting.

Provided conditions for inequalities involving positive linear maps.

Abstract

We establish some operator versions of Bellman's inequality. In particular, we prove that if $\Phi: \mathbb{B}(\mathscr{H}) \to \mathbb{B}(\mathscr{K})$ is a unital positive linear map, $A,B \in \mathbb{B}(\mathscr{H})$ are contractions, $p>1$ and $0 \leq \lambda \leq 1$, then {eqnarray*} \big(\Phi(I_\mathscr{H}-A\nabla_{\lambda}B)\big)^{1/p}\ge\Phi\big((I_\mathscr{H}-A)^{1/p}\nabla_{\lambda}(I_\mathscr{H}-B)^{1/p}\big). {eqnarray*}