If $A$ and $B$ are two positive definite matrices such that $\|A\| \leq \|B\|$ for every unitarily invariant norm $\| \cdot \|$, and $U$ is an $n\times k$ matrix with adjoint $V$ such that $VU = I_k$, then can we establish some relation between $\operatorname{trace}(VAU)$ and $\operatorname{trace}(VBU)$?
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3$\begingroup$ No. Your assumption satisfied if $A,B$ are unitarily equivalent, so for example we can take two diagonal matrices with the diagonal entries reshuffled. Then you won't be able to control $a_{11}$, say, in terms of $b_{11}$. $\endgroup$– Christian RemlingCommented Aug 3, 2017 at 18:14
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As Christian said, no. Your condition ``$A$ and $B$ are two positive definite matrices such that $\|A\| \leq \|B\|$ for every unitarily invariant norm $\| \cdot \|$" is equivalent to "$\max\limits_{VU = I_k}\operatorname{trace}(VAU)\le \max\limits_{VU = I_k} \operatorname{trace}(VBU)$, $k=1, \ldots, n$" by Fan's dominance theorem.
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1$\begingroup$ This also follows directly, because $\max \textrm{tr}\ldots = \sigma_1+\ldots +\sigma_k$ is a unitarily invariant norm. $\endgroup$ Commented Aug 7, 2017 at 0:08