# zbMATH — the first resource for mathematics

Norm inequalities involving convex and concave functions of operators. (English) Zbl 07089815
Summary: Let $$A_1,\dots,A_n$$ be bounded linear operators on a complex separable Hilbert space $$\mathbb{H}$$ and let $$\alpha_1,\dots,\alpha_n$$ be positive real numbers such that $$\sum^n_{j=1}\alpha_j A_j=0$$ and $$\sum^n_{j=1} \alpha_j=1$$. Among other results, it is shown that
(a)
If $$f$$ is a non-negative function on $$[0,\infty)$$ such that $$f(0)=0$$ and $$g(t)=f(\sqrt{t})$$ is convex, then for every unitarily invariant norm, $\left|\left|\left| \sum\limits^n_{j=1} \alpha_j f(|A_j|) \right|\right|\right| \geq \left|\left|\left| f\left(\sqrt{\frac{\alpha_\ell}{1-\alpha_\ell}}|A_\ell|\right) + \sum\limits_{j,k\in S_\ell} f\left(\sqrt{\frac{\alpha_j \alpha_k}{2(1-\alpha_\ell)}} |A_j - A_k|\right)\right|\right|\right|$ for $$\ell=1,\dots,n$$.
(b)
If $$f$$ is a non-negative function on $$[0,\infty)$$ such that $$g(t)=f(\sqrt{t})$$ is concave, then for every unitarily invariant norm, $\left|\left|\left| \sum\limits^n_{j=1} \alpha_j f(|A_j|) \right|\right|\right| \leq \left|\left|\left| f\left(\sqrt{\frac{\alpha_\ell}{1-\alpha_\ell}}|A_\ell|\right) + \sum\limits_{j,k\in S_\ell} f\left(\sqrt{\frac{\alpha_j \alpha_k}{2(1-\alpha_\ell)}} |A_j - A_k|\right)\right|\right|\right|$ for $$\ell=1$$. Here $$S_\ell =\{1,\dots,n\}\setminus \{\ell\}$$.

##### MSC:
 47A30 Norms (inequalities, more than one norm, etc.) of linear operators 47B10 Linear operators belonging to operator ideals (nuclear, $$p$$-summing, in the Schatten-von Neumann classes, etc.) 47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.) 46B20 Geometry and structure of normed linear spaces
Full Text:
##### References:
 [1] Fack, T.; Kosaki, H., Generalized s-numbers of *** -measurable operators, Pac J Math, 123, 269-300, (1986) · Zbl 0617.46063 [2] Simon, B., Trace ideals and their applications, (1979), Cambridge: Cambridge University Press, Cambridge · Zbl 0423.47001 [3] Audenaert, KMR; Kittaneh, F., Problems and conjectures in matrix and operator inequalities, Banach Center Publ, 112, 15-31, (2017) · Zbl 1381.15008 [4] Bhatia, R.; Holbrook, J., On the Clarkson-McCarthy inequalities, Math Ann, 281, 7-12, (1988) · Zbl 0618.47008 [5] Bhatia, R.; Kittaneh, F., Clarkson inequalities with several operators, Bull London Math Soc, 36, 820-832, (2004) · Zbl 1071.47011 [6] Conde, C.; Moslehian, MS, Norm inequalities related to p-Schatten class, Linear Algebra Appl, 498, 441-449, (2016) · Zbl 1341.47011 [7] Hirzallah, O.; Kittaneh, F., Non-commutative Clarkson inequalities for unitarily invariant norms, Pac J Math, 202, 363-369, (2002) · Zbl 1054.47011 [8] Hirzallah, O.; Kittaneh, F., Non-commutative Clarkson inequalities for n-tuples of operators, Integral Equ Oper Theory, 60, 369-379, (2008) · Zbl 1155.47013 [9] Kissin, E., On Clarkson-McCarthy inequalities for n-tuples of operators, Proc Amer Math Soc, 135, 2483-2495, (2007) · Zbl 1140.47005 [10] Gumus, IH; Hirzallah, O.; Kittaneh, F., Estimates for the real and imaginary parts of the eigenvalues of matrices and applications, Linear Multilinear Algebra, 64, 2431-2445, (2016) · Zbl 1361.15021 [11] Horn, RA; Johnson, CR, Matrix analysis, (1985), Cambridge: Cambridge University Press, Cambridge [12] Gumus, IH; Hirzallah, O.; Kittaneh, F., Eigenvalue localization for complex matrices, Electron J Linear Algebra, 27, 892-906, (2014) · Zbl 1326.15015 [13] Aujla, J.; Silva, F., Weak majorization inequalities and convex functions, Linear Algebra Appl, 369, 217-233, (2003) · Zbl 1031.47007 [14] Hirzallah, O.; Kittaneh, F., Norm inequalities for weighted power means of operators, Linear Algebra Appl, 314, 181-193, (2002) · Zbl 1017.47003 [15] Uchiyama, M., Subadditivity of eigenvalue sums, Proc Amer Math Soc, 134, 1405-1412, (2006) · Zbl 1089.47010 [16] Bourin, J-C; Lee, E-Y, Unitary orbits of Hermitian operators with convex or concave functions, Bull London Math Soc, 44, 1085-1102, (2012) · Zbl 1255.15028 [17] Kosem, T., Inequalities between f(A+B) and f(A)+f(B), Linear Algebra Appl, 418, 153-160, (2006) · Zbl 1105.15016 [18] Bourin, J-C; Uchiyama, M., A matrix subadditivity inequality for f(A+B) and f(A)+f(B), Linear Algebra Appl, 423, 512-518, (2007) · Zbl 1123.15013 [19] Bhatia, R., Matrix analysis, (1997), New York (NY): Springer, New York (NY) [20] Gohberg, IC; Krein, MG, Introduction to the theory of linear nonselfadjoint operators, 18, (1969), Providence (RI): AMS, Providence (RI)
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.