A unitary representation of a group 
on a complex Hilbert space 
is a group
homomorphism 
into its group of unitary
transformations. Thus 
and each 
preserves the inner product.
For example, the left regular representation acts on 
by
 |
The norm of each representing operator is 1 when 
.
For a discrete group there is no additional continuity restriction. A uniformly bounded representation need not already be unitary. Unitarizability asks whether a common bounded invertible change of coordinates can make every representing operator unitary.
Factors.