Don't unitary operators have the same domain and codomain?

Question

As far as I know, an unitary operator $U$ is defined to be an operator on an Hilbert space such that its adjoint is its inverse. However, when dealing with the unitary equivalence, unitary operators are defined on domain that is different from the codomain. Then, unitary operators can have domains and codomains that are different from each other? I am very curious...

Answer 1

Nothing in the definition of a unitary operator requires the domain and codomain to be the same in order to make sense. If $E$ and $F$ are Hilbert spaces and $U:E\to F$ is a bounded linear map, it has an adjoint $U^*:F\to E$, defined by the formula $\langle Ux,y\rangle_F=\langle x,U^*y\rangle_E$ for all $x\in E$ and $y\in F$. We then say $U$ is unitary if $U$ and $U^*$ are inverses.

Answer 2

If the requirement was that $A=A^{-1}$, then naturally you would expect the domain and codomain of $A$ to be the same, because the domain of $A^{-1}$ is the codomain of $A$.

Now the domain of $A^\star$ and that of $A^{-1}$ are naturally both the same in any case, namely the codomain of $A$. Therefore the equality $A^\star =A^{-1}$ does not come with any implicit requirement regarding domains.

Answer 3

The Wikipedia article uses the right words, but restricts (unnecessarily) to a single Hilbert space.

A unitary is a linear surjective isometry between two Hilbert spaces. Since this is equivalent to being linear, surjective, and preserve the inner product, it turns out that unitaries are precisely the isomorphisms of Hilbert spaces.