One-parameter versions of the Heinz inequality

The celebrated Heinz inequality ([6]) states

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

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:

$$\vert\vert\vert H^{1/2}XK^{1/2}\vert\vert\vert \leq \vert\vert\ve... ...K^{1-x}dx\vert\vert\vert \leq \frac{1}{2}\vert\vert\vert HX+XK\vert\vert\vert.$$

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

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

are reported.

Bibliography

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T. Ando, Matrix Young inequalities, Oper. Theory Adv. Appl., 75 (1995), 33-38.

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R. Bhatia and C. Davis, More matrix forms of the arithmetic geometric mean inequality, SIAM J. Matrix Anal. Appl., 14 (1993), 132-136.

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R. Bhatia and K. R. Parthasarathy, Positive definite functions and operator inequalities, Bull. London Math. Soc., 32 (2000), No 2, 214-228.

4

M. Sh. Birman and M. Z. Solomyak, Stieltjes double operator integrals, Dokl. Akad. Nauk SSSR, 165 (1965), 1223-1226 (Russian); Soviet Math. Dokl., 6 (1965), 1567-1571.

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U. Haagerup, On Schur multipliers in ${\mathcal C}_1$, unpublished hand-written notes (1980).

6

E. Heinz, Beiträge zur Störungstheorie der Spektralzerlegung, Math. Ann., 123 (1951), 415-438.

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F. Hiai and H. Kosaki, Comparison of various means for operators, J. Funct. Anal., 163 (1999), 300-323.

8

F. Hiai and H. Kosaki, Means for matrices and comparison of their norms, Indiana Univ. Math. J., 48 (1999), 899-936.

9

F. Hiai and H. Kosaki, Means of Hilbert space operators, to appear as LNM, Springer.

10

H. Kosaki, Arithmetic-geometric mean and related inequalities for operators, J. Funct. Anal., 156 (1998), 429-451.

11

A. McIntosh, Heinz inequalities and perturbation of spectral families, Macqaurie Mathematical Reports, 79-0006, 1979.

12

V. V. Peller, Hankel operators and differentiability properties of functions of self-adjoint (unitary) operators, LOMI Preprints E-1-84, USSR Academy of Sciences Steklov Mathematical Institute Leningrad Department, 1984.

13

X. Zhan, Inequalities for unitarily invariant norms, SIAM J. Matrix Anal. Appl., 20 (1998), 466-470.