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
$$\vert y \rangle \langle T^*x \vert?$$
$\begingroup$
$\endgroup$
2
-
3$\begingroup$ Replace $T^*$ for $T$ and your sipppsition is correct. $\endgroup$– Valter MorettiAug 15, 2018 at 18:33
-
$\begingroup$ Related. $\endgroup$– Cosmas ZachosAug 15, 2018 at 19:00
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
2
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 $|y\rangle \langle Tx|$.
-
1$\begingroup$ It might also be helpful to note that if $T|x\rangle = |Tx\rangle$ then $\langle Tx| = \langle x|T^†$ $\endgroup$ Aug 15, 2018 at 19:31
-
$\begingroup$ But why does that property hold true in the first place? $\endgroup$– asd11Mar 16, 2019 at 11:31