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Let $T$ be a linear operator, then we can consider the rank-one operator

$$\vert Tx \rangle \langle y \vert.$$

I am wondering what is its adjoint operator, is it

$$\vert y \rangle \langle T^*x \vert?$$

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    $\begingroup$ Replace $T^*$ for $T$ and your sipppsition is correct. $\endgroup$ Aug 15 '18 at 18:33
  • $\begingroup$ Related. $\endgroup$ Aug 15 '18 at 19:00
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For any two vectors $|v\rangle$ and $|w\rangle$, the adjoint of $|w\rangle\langle v|$ is $|v\rangle\langle w|$.

So, the adjoint of $|Tx\rangle \langle y|$ is $|y\rangle \langle Tx|$.

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    $\begingroup$ It might also be helpful to note that if $T|x\rangle = |Tx\rangle$ then $\langle Tx| = \langle x|T^†$ $\endgroup$ Aug 15 '18 at 19:31
  • $\begingroup$ But why does that property hold true in the first place? $\endgroup$
    – asd11
    Mar 16 '19 at 11:31

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