On Strang's fifth bound for eigenvalues of the Jordan product of two self-adjoint operators on a finite-dimensional unitary space (Q1123375)

Let A, B be selfadjoint operators on a finite dimensional unitary space. Let \(\alpha_ 1,\beta_ 1\) and \(\alpha_ n,\beta_ n\) be respectively the largest and the smallest eigenvalues of A and B. In 1962 \textit{W. G. Strang} found [Am. Math. Monthly 69, 37-40 (1962; Zbl 0101.254)] that the best lower bound on the lowest eigenvalue and the best upper bound on the highest eigenvalue of the operator \(AB+BA\) are contained in the set of five numbers \(\{2\alpha_ 1\beta_ 1,2\alpha_ 1\beta_ n,2\alpha_ n\beta_ 1,2\alpha_ n\beta_ n,\gamma \}\), where the formula for \(\gamma\) given by him was complicated. In the paper the fifth element in the above set is studied in detail. An alternative somewhat simpler expression for \(\gamma\) is given and it is explained when \(\gamma\) is the upper or the lower bound. The considerations are based on the algebraic version of the Mohr circles theorem, a proof of which is also presented.