Unitarily invariant norm inequalities for positive semidefinite matrices (Q2666918)

Let \(M_n(\mathbb{C})\) denote the space of all \(n\times n\) complex matrices. \textit{F. Kittaneh} [J. Funct. Anal. 250, No. 1, 132--143 (2007; Zbl 1131.47009)] proved that if \(A, B, X \in M_n(\mathbb{C})\) such that \(A, B\) are positive semidefinite, then \[ \|| AX-XB |\| \le \Vert X\Vert~\|| A \oplus B |\|, \] where \(\|| \cdot |\|\) denotes the unitarily invariant norm on \(M_n(\mathbb{C})\). This result is an improvement of the inequality by \textit{X. Zhan} [SIAM J. Matrix Anal. Appl. 22, No. 3, 819--823 (2000; Zbl 0985.15016)]. In this paper, the authors continue to improve the above inequality and show the following commutator inequality, for \(X, Y \in M_n(\mathbb{C})\): \[ \lVert XY-YX\rVert \le \Vert Y \Vert \Vert X \Vert + \frac{1}{2} \Vert X^*Y - YX^*\Vert. \] Additionally, \textit{O. Hirzallah} and \textit{F. Kittaneh} [Linear Algebra Appl. 432, No. 5, 1322--1336 (2010; Zbl 1188.47018)] proved another interesting singular values' inequality, that is, \[ s_j(X+Y) \le 2s_j(X \oplus Y), \] for \(X, Y \in M_n(\mathbb{C})\) and \(j = 1, \dots, n.\) In this paper, the authors generalize this inequality. Furthermore, they prove an inequality for nonnegative concave functions of \(A^{1/2}XB^{1/2}+B^{1/2}YA^{1/2}\).