On an Analogue of $ABA$ when the Operator Variables $A$ and $B$ are Convex Functionals

Abstract

The operator-product $ABA$ appears in many mathematical contexts, such as in algebraic Riccati equation, in operator entropy and in operator-mean theory. The purpose of the present paper is to investigate a reasonable analogue of $ABA$ when the positive linear operators $A$ and $B$ are convex functionals. As consequence, the square of a convex functional extending $A^2$ is provided as well.