Abstract
Let A and Z be n-by- n matrices. Suppose A ⩾ 0 (positive semi-definite) and Z > 0 with extremal eigenvalues a and b. Then, for given scalars s, t > 0, there exist unitary matrices U and V such that 2 ab a + b U | ZA s + t | U ∗ ⩽ | A s ZA t | ⩽ a + b 2 ab V | ZA s + t | V ∗ . These are sharp inequalities for singular values. More generally, for monotone pairs of positive operators A and B, there exist unitaries U and V such that 2 ab a + b U | ZAB | U ∗ ⩽ | AZB | ⩽ a + b 2 ab V | ZAB | V ∗ .
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