On the Convexity of the Heinz Means

Abstract

Let A, B, and X be operators on a complex separable Hilbert space such that A and B are positive, and let 0 ≤ v ≤ 1. The Heinz inequalities assert that for every unitarily invariant norm ${\left\vert \left\vert \left\vert \cdot \right\vert \right\vert \right\vert ,}$

$$2\left\vert \left\vert \left\vert A^{1/2}XB^{1/2}\right\vert \right\vert \right\vert \leq \left\vert \left\vert \left\vert A^{v}XB^{1-v}+A^{1-v}XB^{v}\right\vert \right\vert \right\vert \leq \left\vert \left\vert \left\vert AX+XB\right\vert \right\vert \right\vert.$$

Using the convexity of the function ${f(v)=\left\vert \left\vert \left\vert A^{v}XB^{1-v}+A^{1-v}XB^{v} \right\vert \right\vert \right\vert }$ on [0, 1], we obtain several refinements of these norm inequalities and we investigate their equality conditions.