I have a small question of sorts, regarding a certain algebraic manipulation.

I know for operators $S$ and $T$ that $\langle (S-T)v,v \rangle\geq0$. So then \begin{align} \langle (S-T)v,v \rangle\geq0&\iff\langle Sv-Tv,v \rangle\geq0\\ &\iff\langle Sv,v \rangle-\langle Tv,v \rangle\geq0\\ &\iff\langle v,S^*v \rangle-\langle Tv,v \rangle\geq0\\ &\iff\overline{\langle S^*v,v \rangle}-\langle Tv,v \rangle\geq0\\ &\iff\overline{\langle S^*v,v \rangle}\geq\langle Tv,v \rangle. \end{align} Inequalities are only applicable in this way between real numbers. Hence, I know $\overline{\langle S^*v,v \rangle}$ must be real and equal to $\langle S^*v,v \rangle$. Is this a correct conclusion to come to? It seems fishy to me but I'm not sure, and I don't know why it would be wrong if so.

Thank you for any help!

Edit: $v\in V$ where $V$ is a complex finite-dimensional vector space, and $S$ and $T$ are both $V \to V$.

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    $\begingroup$ what else do you know about $S$ and $T$? it is conceivable that $\langle Sv,v \rangle = a_1 + bi$ and $\langle Tv,v \rangle = a_2 + bi$ with $a_1 \geq a_2$ for $b \in \mathbb R$ so while it's true that $a_1 + bi - (a_2 + bi) \geq 0$ it does not necessarily follow that $\langle Sv,v \rangle =a_1 + bi \geq (a_2 + bi) = \langle Tv,v \rangle $ $\endgroup$ – user8675309 Feb 23 at 19:17

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