On an analogue of \(ABA\) when the operator variables \(A\) and \(B\) are convex functionals (Q2341418)

Let \(A\) be a positive operator on a Hilbert space \(\mathscr{H}\). The quadratic convex functional \(f_A\) associated to \(A\) is defined on \(\mathscr{H}\) by \(f_A(x)=\frac{1}{2}\langle Ax,x\rangle\). Assume that \(A\mapsto\Phi(A)\) is a mapping on the set of all positive operators on a Hilbert space \(\mathscr{H}\). There has been some research in which \(\Phi\) is considered on the set of convex functionals on \(\mathscr{H}\). In particular, the equation \[ \Phi(f_A)=f_{\Phi(A)} \tag{1} \] has been examined for some mappings \(\Phi\). \textit{J. I. Fujii} [Sci. Math. Jpn. 57, 351--363 (2003; Zbl 1062.47026)] proved the equation ({1}) with \(\Phi(A,B)=A:B=\left(A^{-1}+B^{-1}\right)^{-1}\) by showing that \(f_{A:B}=f_A:f_B\). In this paper, the author first defines a functional transform \(\Lambda(f,g)\) for every convex function \(f: \mathscr{H}\to\mathbb{R}\cup\{\infty\}\) and every function \(g: \mathscr{H}\to\mathbb{R}\cup\{\infty\}\) and gives its properties. Then, the author considers the mapping \(\Phi(A,B)=AB^{-1}A\) and proves that, if \(A\) and \(B\) are positive operators and \(B\) is invertible, then \(f_{AB^{-1}A}=\Lambda(f_A,f_B)\).