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Inequalities for certain powers of positive definite matrices. (English) Zbl 1426.15028
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
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##### References:
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