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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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 $|y\rangle \langle Tx|$.