What is the difference between isometric and unitary operators on a Hilbert space?

Solution 1:

An isometric operator on a (complex) Hilbert space is a linear operator that preserves distances. That is, $T$ is an isometry if (by definition) $\|Tx-Ty\|=\|x-y\|$ for all $x$ and $y$ in the space. By linearity, this is equivalent to $\|Tx\|=\|x\|$ for all $x$. Because of the definition of the norm in terms of the inner product and the definition of adjoint operators, this is equivalent to $\langle T^*Tx,x\rangle=\langle x,x\rangle$ for all $x$. This implies that $T^*T=I$. Conversely, if $T^*T=I$, you can show that $T$ is an isometry (this direction is easier).

A unitary operator $U$ does indeed satisfy $U^*U=I$, and therefore in particular is an isometry. However, unitary operators must also be surjective (by definition), and are therefore isometric and invertible. They are the isometric isomorphisms on Hilbert space. One way to characterize them algebraically is to say that $U$ is a unitary if $U^*U=UU^*=I$.

On infinite dimensional Hilbert spaces (unlike in finite dimensional cases), there are always nonunitary isometries. For example, on $\ell^2$, the operator sending $(a_0,a_1,a_2,a_3,\ldots)$ to $(0,a_0,a_1,a_2,\ldots)$ is a nonunitary isometry.

I'm not sure what you mean by "isomorphic". One notion of equivalence of linear transformations is similarity; but a surjective operator is never similar to a nonsurjective operator. A stronger notion is unitary equivalence, i.e., similarity induced by a unitary transformation (since these are the isometric isomorphisms of Hilbert space), which again cannot happen between a nonunitary isometry and a unitary operator (or between any nonunitary operator and a unitary operator).

Solution 2:

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.