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.


47A30 Norms (inequalities, more than one norm, etc.) of linear operators
15A45 Miscellaneous inequalities involving matrices
15A42 Inequalities involving eigenvalues and eigenvectors
Full Text: DOI


[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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