On the Convexity of the Heinz Means |
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| Authors: | Fuad Kittaneh |
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| Affiliation: | (1) Graduate School of Information Sciences, Tohoku University, 980-8579 Aoba-ku, Sendai, Japan;(2) Graduate School of Mathematics, Kyushu University, 812-8581 Higashi-ku, Fukuoka, Japan |
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| 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 | | | ·| | | ,{leftvert leftvert leftvert cdot rightvert rightvert rightvert ,} | 2| | | A1/2XB1/2| | | ￡ | | | AvXB1-v+A1-vXBv| | | ￡ | | | AX+XB| | |.2leftvert leftvert leftvert A^{1/2}XB^{1/2}rightvert rightvert rightvert leq leftvert leftvert leftvert A^{v}XB^{1-v}+A^{1-v}XB^{v}rightvert rightvert rightvert leq leftvert leftvert leftvert AX+XBrightvert rightvert rightvert. |
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