On the Symmetry of Heinz Means and Equality Cases in Symmetrically Normed Ideals

The Heinz mean is well known for its symmetry. This property naturally extends to operators through the function f(κ)=|||AκXB1−κ+A1−κXBκ|||,0≤κ≤1. Since this function is convex and symmetric, it reaches its minimum at κ=12 and its maximum at the endpoints κ=0 and κ=1. In this paper, we exploit this symmetry together with the strict convexity of symmetrically normed ideals to characterize the equality cases in the Heinz inequality and related Young-type inequalities. Most of our results do not require the operators A and B to be invertible. For the cases that do require invertibility, such as those involving the logarithmic mean, we state this assumption explicitly. We also show new connections between equality in operator means, the relation AX=XB, and the Sylvester equation. Finally, for the trace-class ideal, we fully describe when the maximum and minimum values of f(κ) are equal, using trace properties and Birkhoff–James orthogonality.