One-parameter versions of the Heinz inequality

The celebrated Heinz inequality ([6]) states

$\displaystyle \vert\vert H^{\theta}XK^{1-\theta}+H^{1-\theta}XK^{\theta}\vert\vert\leq\vert\vert HX+XK\vert\vert
\quad (0 \leq \theta \leq 1),

where $ H,K,X$ are Hilbert space operators with $ H,K \geq 0$. This estimate actually remains valid for arbitrary unitarily invariant norms $ \vert\vert\vert\cdot\vert\vert\vert$, and in the recent years such norm inequalities for various operator means have been investigated by several authors. For example, the following arithmetic-logarithmic-geometric mean inequality ([7]) is known:

$\displaystyle \vert\vert\vert H^{1/2}XK^{1/2}\vert\vert\vert \leq \vert\vert\ve...
\leq \frac{1}{2}\vert\vert\vert HX+XK\vert\vert\vert.

Firstly, the general theory on operator means worked out in [8,9]) is reviewed, which is built upon the theory of Stieltjes double integral transformations ([4]). It enables us to associate operator means to certain scalar means and moreover to establish norm comparison results as above in a unified fashion. Secondly, as applications several norm inequalities involving quantities such as

$\displaystyle \vert\vert\vert H^{\frac{1+\alpha}{2}}XK^{\frac{1-\alpha}{2}}+
\quad (\alpha \in [0,1] \ $   and$\displaystyle \ x \in {\mathbf R})

are reported.


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