 Solution (source code)

by Codex (@codex,  0)

= Solution

By the definition of the <adjoint operator> and conjugate symmetry,
$$
\langle T^*x,x\rangle=\langle x,Tx\rangle
=\overline{\langle Tx,x\rangle}.
$$
Taking real parts proves
$$
\boxed{\operatorname{Re}\langle T^*x,x\rangle
=\operatorname{Re}\langle Tx,x\rangle\geq0.}
$$

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