Inequality between powers of positive semidefinite matrices
The order relation \(A \geq B\) between two \(n \times n\) Hermitian matrices \(A,B\) is defined to mean that \(A-B\) is positive semidefinite. Hence \(A \geq 0\) means that \(A\) is positive semidefinite. Given \(p > 1\) and \(A \geq 0\), let \({\mathcal A}_{(p)}\) be the class of all positive semidefinite matrices \(S\) for which \(0 \leq B \leq S\) implies \(B^ p \leq A^ p\). Similarly let \({\mathcal A}_{[p]}\) be the class of all positive semidefinite matrices \(S\) for which \(S \leq B\) implies \(A^ p \leq B^ p\). In this paper the authors investigate some properties of the classes \({\mathcal A}_{(p)}\) and \({\mathcal A}_{[p]}\). Given \(p > 1\) and \(\lambda > 0\), let \(f_{p, \lambda} (t) = p \lambda t^ p/((p - 1) t^ p + \lambda^ p)\) on \([0, \infty)\), and let \(g_{p, \lambda} (t) = (\lambda^ p t^ p + p - 1)/p^ \lambda\) on \([0, \infty)\). They show that \(f_{p, \lambda} (A) \in {\mathcal A}_{(p)}\) and \(g_{p, \lambda} (A) \in {\mathcal A}_{[p]}\) for any \(\lambda > 0\), and that \(A \in {\mathcal A}_{(p)}\) if and only if \(A\) is a scalar multiple of an orthoprojection, and \(A \in {\mathcal A}_{[p]}\) if and only if \(A\) is a scalar. From these results they derive some corollaries containing the properties on maximum element of \({\mathcal A}_{(p)}\) and on minimum element of \({\mathcal A}_{[p]}\). Finally they prove that for \(p > 1\) and \(\lambda > 0\), the function \(f_{p, \lambda} (t)\) is the (pointwise) maximum in the class of nonnegative functions \(f(t)\) on \([0, \infty)\) such that \(f(\lambda) = \lambda\) and \(f(A) \in {\mathcal A}_{(p)}\) for \(A \geq 0\) of every order.
inequality between powers of positive semidefinite matrices
scientific article; zbMATH DE number 665777
scientific article