I have a Real Hilbert space $H$ and an operator $\beta$ over $H$ satisfyng $\beta^{\ast}=-\beta$ and $\beta^2=-1$ (this easily implies $\beta$ would be unitary).
Now I consider the complixification of $H$ call it $H_{\mathbb{C}}$ and extend $\beta$ by linearity to $H_{\mathbb{C}}$ and call the extention $\beta_{\mathbb{C}}$. So my question is the following ¿Is $\beta_{\mathbb{C}}$ instead unitary? ¿or satisfies the aforementiones equalities?
Many thanks in advance!