Antiunitary operator

In mathematics, an antiunitary transformation, is a bijective antilinear map
between two complex Hilbert spaces such that 
for all formula_3 and formula_4 in formula_5, where the horizontal bar represents the complex conjugate. If additionally one has formula_6 then U is called an antiunitary operator.
Antiunitary operators are important in quantum theory because they are used to represent certain symmetries, such as time reversal. Their fundamental importance in quantum physics is further demonstrated by Wigner's theorem. 
Invariance transformations.
In quantum mechanics, the invariance transformations of complex Hilbert space formula_7 leave the absolute value of scalar product invariant:
for all formula_3 and formula_4 in formula_11.
Due to Wigner's theorem these transformations fall into two categories, they can be unitary or antiunitary. 
Geometric Interpretation.
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.
Decomposition of an antiunitary operator into a direct sum of elementary Wigner antiunitaries.
An antiunitary operator on a finite-dimensional space may be decomposed as a direct sum of elementary Wigner antiunitaries formula_29, formula_30. The operator formula_31 is just simple complex conjugation on formula_32
For formula_34, the operator formula_29 acts on two-dimensional complex Hilbert space. It is defined by 
Note that for formula_34
so such formula_29 may not be further decomposed into 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.