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Eigenvalue inequalities for differences of means of Hilbert space operators. (English) Zbl 1248.47019
Given bounded linear operators $$A,B$$ on a Hilbert space such that $$A\geq B>O$$ and $$A-B$$ is compact, and a scalar $$\mu\in (0,1)$$, the authors consider the $$\mu$$-weighted arithmetic (resp., geometric) mean $$A\nabla _{\mu} B = (1-\mu)A+\mu B$$ (resp., $$A\sharp _{\mu}B=A^{1/2}(A^{-1/2}BA^{-1/2})^{\mu}A^{1/2}$$) and the $$\mu$$-weighted Heinz mean $$H_{\mu}(A,B)=1/2(A\sharp _{\mu}B +A\sharp _{1-\mu}B)$$. The estimates for the singular values of the differences $$A\nabla _{\mu} B -A\sharp _{\mu}B$$, $$(A\nabla _{\mu} B)A^{-1}(A \nabla _{\mu} B) -A\sharp _{2 \mu}B$$, $$A\nabla _{\mu}(BA^{-1}B)-A\sharp _{2 \mu}B$$, $$B\nabla _{\mu}(AB^{-1}A)-B\sharp _{2 \mu}A$$ and $$\frac{A+B}{2} - H_{\mu}(A,B)$$ are expressed in terms of the singular values of the operators $$A^{-1/2}(A-B)^{2}A^{-1/2}$$ or $$B^{-1/2}(A-B)^{2}B^{-1/2}$$. Equality conditions for the corresponding inequalities are also obtained.

##### MSC:
 47A64 Operator means involving linear operators, shorted linear operators, etc.
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##### References:
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