TITLE: What is the difference between isometric and unitary operators on a Hilbert space?
QUESTION [37 upvotes]: It seems that both isometric and unitary operators on a Hilbert space have the following property:
$U^*U = I$ ($U$ is an operator and $I$ is an identity operator, $^*$ is a binary operation.) 
What is the difference between isometry and unitary? Which one is more general,
or are they the same? Are they isomorphic?

REPLY [5 votes]: In finite dimensions, there is a straightforward characterisation of isometries and unitaries in terms of their matrix representations.
The basic observation is that $U^*U=I$ means that the columns of (the matrix representation of) $U$ are orthonormal, while $UU^*=I$ means that the rows of $U$ are orthonormal.
A unitary $U$ is a matrix whose columns (equivalently, rows) form an orthonormal basis. This is equivalent to both its columns and rows being orthonormal.
An isometry, on the other hand, only requires that the columns are orthonormal, but not that they form a basis.
In summary, the distinction between the two objects can be stated as follows: an isometry is a matrix whose columns are orthonormal, while a unitary is squared matrix whose columns are orthonormal.
For completeness: one can also talk about partial isometries, which are matrices such that $U^* U$ is an orthogonal projection, rather than being the identity. This is equivalent to asking for $U$ to be an isometry on the orthogonal complement of its ker.