Inequalities for C-S seminorms and Lieb functions. (English) Zbl 0932.15015
Let \(M_n\) denote the space of \(n\times n\) complex matrices. A seminorm \(\|\cdot \|\) is called a Cauchy-Schwarz (C-S) seminorm provided \(\|A^\ast A\|=\|AA^\ast\|\) for all \(A\in M_n\) and \(\|A\|\leq \|B\|\) whenever \(A,B\) and \(B-A\) are positive semidefinite. The authors prove that any nontrivial C-S seminorm is unitarily invariant. This allows many known inequalities that pertain to unitarily invariant norms to be extended to C-S seminorms. For example, the authors prove the following generalization of a theorem of R. Bhatia and C. Davis [ibid. 223-224, 119-129 (1995; Zbl 0824.47006)]: Let \(A,B,X\in M_n\) be given with \(A,B\) positive semidefinite. Then
\[
\||AXB|^r \|\leq \||A^pX|^r \|^{1/p} \||XB^q|^r \|^{1/q}
\]
for every C-S seminorm \(\|\cdot\|\) on \(M_n\), all \(r>0\) and all conjugate indices \(p,q\). The authors also discuss applications of the ideas to the Lieb class [cf. E. H. Lieb, Adv. Math. 20, 174-178 (1976; Zbl 0324.15013)] which, as they show, can be viewed as a generalization of C-S seminorms.
Reviewer: Joseph Lakey (Las Cruces)
MSC:
| 15A60 | Norms of matrices, numerical range, applications of functional analysis to matrix theory |
| 15A45 | Miscellaneous inequalities involving matrices |
References:
| [1] | Bhatia, R., (Matrix Analysis (1997), Springer: Springer New York) |
| [2] | Bhatia, R., Perturbation inequalities for the absolute value map in norm ideals of operators, J. Operator Theory, 19, 129-136 (1988) · Zbl 0698.47029 |
| [3] | Bhatia, R.; Davis, C., A Cauchy-Schwarz inequality for operators with applications, Linear Algebra Appl., 223/224, 119-129 (1995) · Zbl 0824.47006 |
| [4] | Horn, R. A.; Johnson, C. R., (Topics in Matrix Analysis (1991), Cambridge University Press: Cambridge University Press New York) · Zbl 0729.15001 |
| [5] | Horn, R. A.; Mathias, R., An analog of the Cauchy-Schwarz inequality for Hadamard products and unitarily invariant norms, SIAM J. Matrix Anal. Appl., 11, 481-498 (1990) · Zbl 0722.15019 |
| [6] | Horn, R. A.; Mathias, R., Cauchy-Schwarz inequalities associated with positive semidefinite matrices, Linear Algebra Appl., 142, 63-82 (1990) · Zbl 0714.15012 |
| [7] | Kittaneh, F., Norm inequalities for fractional powers of positive operators, Lett. Math. Phys., 27, 279-285 (1993) · Zbl 0895.47003 |
| [8] | Lieb, E. H., Inequalities for some operator and matrix functions, Advances in Math., 20, 174-178 (1976) · Zbl 0324.15013 |
| [9] | Magnus, J. R., A representation theorem for (tr\(A^p})^{1/p} \), Linear Algebra Appl., 95, 127-134 (1987) · Zbl 0632.15011 |
| [10] | Thompson, R. C., Convex and concave functions of singular values of matrix sums, Pacific J. Math., 66 (1976) · Zbl 0361.15014 |
| [11] | Wimmer, H. K., Extremal problems for Hölder norms of matrices and realizations of linear systems, SIAM J. Matrix Anal. Appl., 9, 314-322 (1988) · Zbl 0653.15018 |
| [12] | Zhang, F., Another proof of a singular value inequality concerning Hadamard products of matrices, Linear and Multilinear Algebra, 22, 307-311 (1988) · Zbl 0641.15008 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
