## Order of operators determined by operator mean.(English)Zbl 1326.47017

A real continuous function $$f$$ defined on an open interval $$J\subset \mathbb R$$ is said to be operator monotone (in short, $$f\in\mathbb{P}(J)$$), if $$A\leq B$$ implies $$f(A)\leq f(B)$$ for any self-adjoint matrices $$A, B$$. The notable special monotone function in order theory is order embedding (function for which $$A\leq B$$ if and only if $$f(A)\leq f(B)$$). In this paper, the authors focus their study on $$J=(0,\infty)$$ and study a binary operation $X\sigma Y=X^{\frac{1}{2}}f(X^{-\frac{1}{2}}YX^{-\frac{1}{2}})X^{\frac{1}{2}},$ where $$X, Y$$ are positive definite. The notion was introduced by F. Kubo and T. Ando [Math. Ann. 246, 205–224 (1980; Zbl 0412.47013)]. Given $$f\in\mathbb{P}(0, \infty)$$, $$0\leq A\leq B$$ implies $$Y\sigma(tA+X)\leq Y\sigma(tB+X)$$ for any $$t\geq 0$$ and positive definite $$X, Y$$. Consequently, the operator monotone function $$f(t)$$ is an order embedding for sufficiently small $$t>0$$ if and only if $Y\sigma(tA + X)\leq Y\sigma(tB + X) \text{ for sufficiently small } t>0 \tag{$$\ast$$}$ implies $$A\leq B$$. Furthermore, they prove that, if $$X=cY$$ for some $$c>0$$ or $$f(t)=\frac{at+b}{ct+d}$$ with $$ad-bc > 0$$ and $$cd\geq 0$$, then ({$$\ast$$}) implies $$A\leq B$$ (see Corollary 4.2, Corollary 4.4). If $$X$$ is not a positive scalar multiple of $$Y$$ and $$f(t)$$ does not have the form $$\frac{at+b}{ct+d}$$, then there exist $$A\geq 0,B\geq 0$$, and positive definite matrices $$X, Y$$ such that $$A\nleq B$$ and ({$$\ast$$}) holds (see Theorem 4.6). Combining these facts, the authors prove that ({$$\ast$$}) implies $$A\leq B$$ if and only if $$X$$ is a positive scalar multiple of $$Y$$ or the operator monotone function $$f$$ associated with $$\sigma$$ has the form $f(t)=\frac{at+b}{ct+d}, \quad a, b, c, d\in\mathbb{R},\;ad-bc>0,\;cd\geq 0.$

### MSC:

 47A63 Linear operator inequalities 15A39 Linear inequalities of matrices

Zbl 0412.47013
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### References:

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