Explicitly, this means the following.
A unitary operator on a Hilbert space is a bounded linear operator that satisfies
where is the Hilbert space adjoint? of and is the identity operator. This property is equivalent to saying that the range of is dense and that preserves the inner product on the Hilbert space. An operator is unitary if and only if .
Unitary operators are the isomorphisms of Hilbert spaces since they preserve the basic structure of the space, e.g. the topology. The group of all unitary operators from a given Hilbert space to itself is sometimes called the unitary group or Hilbert group of and is denoted Hilb().
If a basis for a finite dimensional Hilbert space is chosen, the defnition of unitary operator reduces to that of unitary matrix.
A unitary matrix is an matrix with complex entries that satisfies the condition
This is equivalent to saying that both the rows and the columns of form an orthonormal basis in with respect to the respective inner product. is also a normal matrix? whose eigenvalues lie on the unit circle.
The notation used here for the adjoint, , is commonly used in linear algebraic circles (as is ). In quantum mechanics, is exclusively used for the adjoint while is interpreted as the same thing as , i.e. the complex conjugate.