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Lieb-Thirring trace inequalities and a question of Bourin. (English) Zbl 1287.47011

In the study about matrix subadditivity inequalities, J.-C. Bourin [Int. J. Math. 20, No. 6, 679–691 (2009; Zbl 1181.15030)] asked the following question for any unitarily invariant norm \(|||\cdot |||\).
{Bourin’s question}: Given positive, bounded, linear operators \(A,B\) and \(p,q>0\), is it true that \[ ||| A^pB^q + B^pA^q|||\leq |||A^{p+q}+B^{p+q}|||? \]
Without loss of generality, one can take \(0 \leq p \leq 1\) and \(q =1-p\), by absorbing \((p+q)\)-th powers into \(A\) and \(B\). Then the conjecture can be rewritten as follows: for \(A,B>0\) and \(t\in[0,1]\), is it true that \[ ||| A^tB^{1-t} + B^tA^{1-t}|||\leq |||A+B|||?\tag{1} \]
This inequality is related to the well-known Heinz inequality, which states that, for any positive, bounded, linear operators \(A,B\) and \(0\leq t \leq 1\), \[ ||| A^tB^{1-t} + A^{1-t}B^t|||\leq |||A+B|||. \]
In order to solve Bourin’s question by using the Heinz inequality, the authors propose the following conjecture.
{Hayajneh-Kittaneh’s conjecture:} Let \(A,B >0\) and \(0\leq t \leq 1\). Then, for any unitarily invariant norm \(|||\cdot |||\), \[ ||| A^tB^{1-t} + A^{1-t}B^t|||\leq ||| A^tB^{1-t} + B^tA^{1-t}|||. \tag{2} \]
In the particular case of the Hilbert-Schmidt norm (\(||T||_2=(\text{tr} T^*T)^{1/2}\)), the inequality (2) is equivalent to a Lieb-Thirring type trace inequality.
In the present article, the authors give a partial answer to Bourin’s question, for the Hilbert-Schmidt norm and particular values of \(t\). Recently, R. Bhatia [J. Math. Phys. 55, No. 1, 013509 (2014; Zbl 1288.15022)] proved that a more general result is valid, more precisely: Theorem. Let \(A,B\) be positive definite matrices and \(t \in [1/4,3/4]\), then \[ || A^tB^{1-t} + B^tA^{1-t}||_2\leq || A^tB^{1-t} + B^tA^{1-t}||_2. \]
Finally, the reviewer wants to mention that Bottazzi et al. [T. Bottazzi, R. Elencwajg, G. Larotonda and A. Varela, “Inequalities related to Bourin and Heinz means with a complex parameter”, arXiv:1403.7472] have extended the result obtained by Bhatia to complex values of the parameter \(t=z\) in the strip \(\{z\in \mathbb{C}: \text{Re}(z)\in [1/4;3/4]\}\). They also showed a counterexample to Bourin’s question by exhibiting a pair of positive matrices such that inequality (1) does not hold for the uniform norm.

MSC:

47A30 Norms (inequalities, more than one norm, etc.) of linear operators
15A45 Miscellaneous inequalities involving matrices
15A42 Inequalities involving eigenvalues and eigenvectors
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[1] DOI: 10.1007/BF01045887 · Zbl 0705.47020
[2] Audenaert K. M. R., Int. J. Inform. Sys. Sci. 4 pp 78– (2008)
[3] DOI: 10.1016/j.laa.2009.01.021 · Zbl 1168.15014
[4] DOI: 10.1007/978-1-4612-0653-8
[5] DOI: 10.1137/0611018 · Zbl 0704.47014
[6] DOI: 10.1142/S0129167X09005509 · Zbl 1181.15030
[7] E. Lieb and W. Thirring, Studies in Mathematical Physics, edited by E. Lieb, B. Simon, and A. Wightman (Princeton University, 1976), pp. 301–302.
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