Inequalities for C-S seminorms and Lieb functions
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 \textit{R. Bhatia} and \textit{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. \textit{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.
scientific article; zbMATH DE number 1331318
scientific article