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Let $A,B\in\mathbb{C}^{p\times p}$ s.t. $A$ and $B$ are contractions and $I_p-AA^*$ and $BB^*$ both are positive semidefinite. I want to show that the matrix $$B(I_p-\sqrt{I_p-AA^*}\sqrt{I_p-AA^*}^*)B^*$$ is positive semidefinite. Are the conditions enough to show this? I'd much appreciate any help!

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    $\begingroup$ Note that $\sqrt{I_p-AA^*}$ is itself Hermitian and positive definite. So, your matrix is simply $$ B(I - (I - AA^*))B^* = B(AA^*)B^* $$ And yes; this matrix is necessarily positive semidefinite. $\endgroup$ – Omnomnomnom Jun 3 '18 at 20:04
  • $\begingroup$ Of course! That resolves it. $\endgroup$ – Vorhang Jun 3 '18 at 20:11

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