Around Choi inequalities for positive linear maps
Copyright policy )Around Choi inequalities for positive linear maps
Let \(\Phi\) be a unital positive linear map between two matrix algebras \({\mathcal A}\) and \({\mathcal B}\) and let \(A\in {\mathcal A}\) be positive. \textit{J.-C.\thinspace Bourin} and \textit{E.\,Ricard} [Linear Algebra Appl.\ 433, No.\,3, 499--510 (2010; Zbl 1208.15019)] showed that, if \(0 \leq p \leq q\), then \(|\Phi(A^p)\Phi(A^q)| \leq \Phi(A^{p+q})\). \textit{M.-D.\thinspace Choi} [Ill.\ J.\ Math.\ 18, 565--574 (1974; Zbl 0293.46043)] proved that, if \(1\leq p \leq 2\), then \(\Phi(A)^p\leq\Phi(A^p)\). Using some ideas of the first mentioned paper, the author interpolates the inequalities above by establishing that, if \(0\leq p\leq q\) and \(\frac{q}{q+p} \leq r\leq \frac{2q}{q+p}\), then \(|\Phi(A^p)^r\Phi(A^q)^r|\leq \Phi(A^{(p+q)r})\). He also presents an asymmetric variant of \(\Phi(A)^{-1}\leq \Phi(A^{-1})\).
Numerical Analysis, Algebra and Number Theory, Order preserving operator inequality, unital positive linear map, Kadison inequality, General theory of \(C^*\)-algebras, Discrete Mathematics and Combinatorics, Linear operator inequalities, Geometry and Topology, Unital positive linear map, Choi inequality, order preserving operator inequality
Numerical Analysis, Algebra and Number Theory, Order preserving operator inequality, unital positive linear map, Kadison inequality, General theory of \(C^*\)-algebras, Discrete Mathematics and Combinatorics, Linear operator inequalities, Geometry and Topology, Unital positive linear map, Choi inequality, order preserving operator inequality
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