For a surjective bounded linear operator, does $U^{*}U = I_X$ imply $UU^{*} = I_Y$?

Question

Let X, Y be Hilbert spaces and $U\in\mathcal{B}(X,Y)$ be a surjective bounded linear operator between the two spaces. Given that the adjoint $U^{*}\in\mathcal{B}(Y,X)$ is such that $U^{*} U = I_X$ does it follow that $U U^{*} = I_Y$?

Answer 1

Yes, every surjective isometry is unitary.

One way to see this is to note that $UU^\ast$ is the projection onto the range of $U$, which is $Y$ in your case.

Answer 2

Since $U^*U=1$, $U$ is injective. If $U$ is also surjective, then $U$ is invertible by the Open Mapping Theorem. Hence $U^*=U^{-1}$, and $UU^*=1$. Note that this is general for Banach spaces $X$,$Y$.