I Antilinear Operators

[hokhani]

TL;DR

The correctness of an equation for antilinear operators

An antilinear operator $\hat{A}$ can be considered as, $\hat{A}=\hat{L}\hat{K}$, where $\hat{L}$ is a linear operator and $\hat{K} c=c^*$ ($c$ is a complex number). In the Eq. (26) of the text https://bohr.physics.berkeley.edu/classes/221/notes/timerev.pdf the equality $(\langle \phi |\hat{A})|\psi \rangle=[ \langle \phi|(\hat{A}|\psi \rangle)]^*$ is given but I think this equation is not correct within a minus sign. For example, in the Hilbert space of spin up and down, having $\hat{L}=\hat{\sigma_y}$ and $|\psi\rangle=\psi_1 |+\rangle +\psi_2 |-\rangle$ and $|\phi\rangle=\phi_1 |+\rangle +\phi_2 |-\rangle$ we have: $\langle \phi | (\hat{A} |\psi\rangle)=-i\phi_1^* \psi_2^*+i\phi_2^*\psi_1^*$ and $(\langle \phi|\hat{A})|\psi \rangle=i\phi_2 \psi_1 -i\phi_1 \psi_2$ which gives $(\langle \phi |\hat{A})|\psi \rangle=-[ \langle \phi|(\hat{A}|\psi \rangle)]^*$. I appreciate any help.