Antilinear map

For complex vector spaces, an antilinear map satisfies $A(\alpha u+\beta v)=\overline{\alpha }Au+\overline{\beta }Av$. It is linear over the real numbers and conjugates complex scalar multiplication. This differs from two conjugate linear operators, which are related by similarity rather than conjugate-linearity.

An endomorphism $J$ that is an antilinear map with $J^2=-I$ equips a complex vector space with a quaternionic structure: complex scalar multiplication supplies the quaternion generator $i$ and $J$ supplies $j$, satisfying $Ji=-iJ$. In finite dimensions the complex dimension must be even. Indeed, there is no $J$-invariant complex line: $Jv=cv$ for $v\ne 0$ would imply $-v=|c{|}^{2}v$. Consequently $v,Jv$ span a $J$-invariant complex two-plane; passing to its quotient vector space and repeating proves evenness.

In complex dimension four the invariant complex two-planes are quaternionic lines, forming the quaternionic projective space ${\mathbb{HP}}^1\cong S^4$. This is the Euclidean reality structure on twistor space used in twistor theory.

An antiunitary operator between complex Hilbert spaces is a surjective antilinear map satisfying the displayed inner product identity. It preserves norms but conjugates scalars. In a chosen orthonormal basis it is $UK$, with $U$ a unitary operator and $K$ coefficientwise complex conjugation. Conversely any such $UK$ is antiunitary. Antilinearity by itself does not preserve norms.