Congruences of the plane form two distinct classes. The first conserves the orientation and is generated by translations and rotations. The second does not conserve the orientation and is obtained from the first class by applying a reflection. On the complex plane these two classes correspond (up to translation) to unitaries and antiunitaries, respectively.
holds for all elements of the Hilbert space and an antiunitary .
When is antiunitary then is unitary. This follows from
For unitary operator the operator , where is complex conjugation (with respect to some orthogonal basis), is antiunitary. The reverse is also true, for antiunitary the operator is unitary.
For antiunitary the definition of the adjoint operator is changed to compensate the complex conjugation, becoming
The adjoint of an antiunitary is also antiunitary and (This is not to be confused with the definition of unitary operators, as the antiunitary operator is not complex linear.)
An antiunitary operator on a finite-dimensional space may be decomposed as a direct sum of elementary Wigner antiunitaries , . The operator is just simple complex conjugation on
For , the operator acts on two-dimensional complex Hilbert space. It is defined by
Note that for
so such may not be further decomposed into 's, which square to the identity map.
Note that the above decomposition of antiunitary operators contrasts with the spectral decomposition of unitary operators. In particular, a unitary operator on a complex Hilbert space may be decomposed into a direct sum of unitaries acting on 1-dimensional complex spaces (eigenspaces), but an antiunitary operator may only be decomposed into a direct sum of elementary operators on 1- and 2-dimensional complex spaces.
Wigner, E. "Phenomenological Distinction between Unitary and Antiunitary Symmetry Operators", Journal of Mathematical Physics Vol 1, no 5, 1960, pp.414–416