Matrix Hölder-McCarthy inequality via matrix geometric means
This paper examines a number of matrix inequalities which generalize well-known scalar inequalities and uses matrix geometric means to give unified proofs. If \(A\) and \(B\) are \(n\times n\) positive definite matrices then the matrix weighted geometric mean is defined by \(A\natural_{p}B:=A^{1/2}(A^{-1/2}BA^{-1/2})^{p}A^{1/2}\) for \(p\in\lbrack0,1]\). The definition can be extended in certain cases to semidefinite matrices by continuity and by using Moore-Penrose inverses to a larger range of \(p\) although \(\natural_{p}\) may not be a matrix mean in the sense of Kubo-Ando when \(p\notin\lbrack0,1]\). The authors prove the following version of the Hölder-McCarthy inequality (Theorem 1): if \(A\) is positive semidefinite and \(X\) is an \(n\times k\) matrix then \(X^{\ast}A^{p}X\leq(X^{\ast}X)\natural_{p}(X^{\ast}AX)\) for all \(p\in\lbrack0,1]\) and the reverse inequality holds for \(p\in\lbrack1,2]\). Moreover if \(A\) is positive definite then the reverse inequality also holds for \(p\in\lbrack-1,0]\). On the other hand, suppose in addition that the eigenvalues of \(A\) lie in an interval \([m,M]\) where \(0 < m < M\). Then a matrix version of the Kantorovich inequality holds (Theorem 3): \(K(h,p)(X^{\ast}X)\natural_{p}(X^{\ast}AX)\leq X^{\ast}A^{p}X\leq(X^{\ast}X)\natural_{p}(X^{\ast}AX)\) for all \(p\in\lbrack0,1],\) where \(K(h,p)\) in the Kantorovich constant which depends only on \(h:=M/m\) and \(p\). Again the reverse inequality holds when \(p\in\lbrack-1,0]\cup\lbrack1,2]\). The paper explores related inequalities and gives examples to show that some matrix generalizations of these inequalities fail.
scientific article; zbMATH DE number 7225053