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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

Citations:

Zbl 0412.47013
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Full Text: DOI Euclid

References:

[1] R. Bhatia, Matrix analysis, Grad. Texts in Math. 169, Springer-Verlag, New York, 1997.
[2] R. Bhatia, Positive definite matrices, Princeton Ser. Appl. Math., Princeton University Press, Princeton, NJ, 2007. · Zbl 1125.15300
[3] W. F. Donoghue, Jr., Monotone matrix functions and analytic continuation, Springer-Verlag, New York-Heidelberg, 1974. · Zbl 0278.30004
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[5] F. Kubo and T. Ando, Means of positive linear operators, Math. Ann. 246(1980), 205-224. · Zbl 0412.47013
[6] M. Uchiyama, A converse of Loewner-Heinz inequality, geometric mean and spectral order, Proc. Edinb. Math. Soc. (2) 57 (2014), 565-571. · Zbl 1293.47016
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