Trace inequalities for operations of matrix powers by aid of block matrices (Q2907984)

The main result of this paper is a trace inequality which depends on the determinant for positive semidefinite matrices \(A\) and \(B\), namely NEWLINE\[NEWLINE n(\det A \det B)^{m/n}\leq \mathrm{tr}(AB)^m ,NEWLINE\]NEWLINE where \(m \in \mathbb Z\). The authors give a matrix trace inequality for a positive semi-definite block matrix \(\biggl(\begin{matrix} A & B\\ B^{*}& C \end{matrix}\biggr)\) given by NEWLINE\[NEWLINE [\mathrm{tr}(|B|{}^p )]{}^2 \leq \mathrm{tr}(A^p ) \mathrm{tr}(C^p ),NEWLINE\]NEWLINE where \(p\in \mathbb Z^{+}\). The applications of the above inequality improve some results on traces of Hadamard products, ordinary products and sums of positive semidefinite matrices given; see [\textit{B. Y. Wang} and \textit{F. Z. Zhang}, SIAM J. Matrix Anal. Appl. 16, No. 4, 1173--1183 (1995; Zbl 0855.15009); \textit{F. M. Dannan}, JIPAM, J. Inequal. Pure Appl. Math. 2, No. 3, Paper No. 34, 7 p. (2001; Zbl 0998.15028); \textit{X. M. Yang} et al., J. Math. Anal. Appl. 263, No. 1, 327--331 (2001; Zbl 0995.15015); \textit{X. J. Yang}, J. Math. Anal. Appl. 250, No. 1, 372--374 (2000; Zbl 0974.15017)]. The authors also deduce some natual consequences.