Alrimawi, Fadi; Hirzallah, Omar; Kittaneh, Fuad Inequalities for certain powers of positive definite matrices. (English) Zbl 1426.15028 Math. Inequal. Appl. 22, No. 3, 791-801 (2019). Summary: Let \(A,B\), and \(X\) be \(n \times n\) matrices such that \(A,B\) are positive definite and \(X\) is Hermitian. If \(a\) and \(b\) are real numbers such that \(0< a \leqslant s_n(A)\) and \(0 < b \leqslant s_n(B)\), then it is shown, among other inequalities, that \[|||A^bX+ XB^a||| \geqslant (1 + \min(a^2,b^2))|||X|||\] for every unitarily unitarily invariant norm. MSC: 15A45 Miscellaneous inequalities involving matrices 15A18 Eigenvalues, singular values, and eigenvectors 15A42 Inequalities involving eigenvalues and eigenvectors 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory Keywords:positive definite matrix; Hermitian matrix; singular value; unitarily invariant norm; inequality PDF BibTeX XML Cite \textit{F. Alrimawi} et al., Math. Inequal. Appl. 22, No. 3, 791--801 (2019; Zbl 1426.15028) Full Text: DOI References: [1] A. ABU-AS’AD ANDO. HIRZALLAH, Some inequalities for powers of positive definite matrices, Math. Ineq. Appl. 20 (2017), 13-27. · Zbl 1362.15016 [2] R. BHATIA, Matrix Analysis, Springer-Verlag, New York, 1997. [3] D. S. MITRINOVIC´, Analytic Inequalities, Springer-Verlag, 1970. [4] J. R. RINGROSE, Compact Non-Self-Adjoint Operators, Van Nostrand Reinhold Co., 1971. · Zbl 0223.47012 [5] B. SIMON, Trace Ideals and their Applications, Cambridge University Press, 1979. · Zbl 0423.47001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.