It is commonly to define an antilinear or antiunitary operator $\Theta$ acting on the ket state $|\alpha\rangle$ such as $$\Theta|\alpha\rangle.$$
It is commonly avoid to pursue a definition an antilinear or antiunitary operator $\Theta$ acting on the bra state $\langle \beta|$ such as $$\langle \beta| \Theta.$$ See Modern Quantum Mechanics (2nd Edition) Chap 4.4 by J. J. Sakurai, Jim J. Napolitano commenting on this issue below in the image file.
My question is that could we in fact define antilinear or antiunitary operator $\Theta$ acting on the ket state $|\alpha\rangle$ and on the bra state $\langle \beta|$ consistently simultaneously? If you can or if you can not, how does it reveal the (1) antilinear or antiunitary operator, (2) inner product $\langle \beta|\alpha\rangle$ constraint, and (3) structure of Hilbert space?