Another version of Maher's inequality (Q1884382)

For a separable, infinite-dimensional Hilbert space \(H\), let \(L(H)\) denote the \(C^*\)-algebra of bounded linear operators on~\(H\). For two \(n\)-tuples of operators \(A=(A_1,\ldots,A_n)\), \(B=(B_1,\ldots,B_n)\) in \(L(H)\), let \(\Delta_{A,B}\) denote the following elementary operator on \(L(H)\): \[ X\mapsto\sum_{i=1}^nA_iXB_i-X. \] Fix \(T\in\ker\Delta_{A,B}\cap C_p\), where \(C_p\) is the \(p\)-th Schatten class and let \(F_p\) be defined by \[ F_p(X)=\| T-\Delta_{A,B}(X)\| _p^p. \] The author obtains several results on global minimizers and critical points for~\(F_p\). These results make essential use of the results by \textit{A.~Turnšek} in [Linear Algebra Appl. 317, 207--216 (2000; Zbl 1084.47510)].