## A matrix subadditivity inequality for $$f(A + B)$$ and $$f(A) + f(B)$$.(English)Zbl 1123.15013

Authors’ abstract: T. Ando and X. Zhan [Math. Ann. 315, No. 4, 771–780 (1999; Zbl 0941.47004)] proved a subadditivity inequality for operator concave functions. We extend it to all concave functions: Given positive semidefinile matrices $$A,B$$ and a non-negative concave function $$f$$ on $$[0,\infty)$$, $\|f(A+B)\|\leq\|f(A)+ f(B)\|$ for all symmetric norms (in particular for all Schatten $$p$$-norms). The case $$f(t)=\sqrt t$$ is connected to some block-matrix inequalities, for instance the operator norm inequality $\left\|\left( \begin{matrix} A & X^*\\ X& B\end{matrix}\right)\right\|_\infty \leq\max\{\| |A|+ |X|\|_\infty;\||B|+|X^*|\|_\infty\}$ for any partitioned Hermitian matrix.

### MSC:

 15A45 Miscellaneous inequalities involving matrices 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory 47A30 Norms (inequalities, more than one norm, etc.) of linear operators

Zbl 0941.47004
Full Text:

### References:

 [1] Ando, T.; Zhan, X., Norm inequalities related to operator monotone functions, Math. ann., 315, 771-780, (1999) · Zbl 0941.47004 [2] J.S. Aujla, J.-C. Bourin, Eigenvalue inequalities for convex and log-convex functions, Linear Algebra Appl., in press, doi:10.1016/j.laa.2006.02.027. · Zbl 1132.15015 [3] Aujla, J.S.; Silva, F.C., Weak majorization inequalities and convex functions, Linear algebra appl., 369, (2003) · Zbl 1031.47007 [4] Bhatia, R.; Kittaneh, F., Norm inequalities for positive operators, Lett. math. phys., 43, 225-231, (1998) · Zbl 0912.47005 [5] Bourin, J.-C., A concavity inequality for symmetric norms, Linear algebra appl., 413, 212-217, (2006) · Zbl 1092.47009 [6] Bourin, J.-C., Hermitian operators and convex functions, J. inequal. pure appl. math., 6, 5, (2005), Article 139 [7] Kosem, T., Inequalities between $$\parallel f(A + B) \parallel$$ and $$\parallel f(A) + f(B) \parallel$$, Linear algebra appl., 418, 153-160, (2006) · Zbl 1105.15016 [8] Uchiyama, M., Subadditivity of eigenvalue sums, Proc. amer. math. soc., 134, 1405-1412, (2006) · Zbl 1089.47010 [9] Zhan, X., Matrix inequalities, Lnm, vol. 1790, (2002), Springer Berlin
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.