What is your definition of unitary operator?
I'm going to assume that it's also an isometry (otherwise take and and .
) Let be an isometric isomorphism then (by the polarization identity) preserves the interior product, and so if is an orthonormal basis for then is also an orthonormal set in . Take since is a basis we have such that then and (Since T, being an isometry, belongs to ). Since is onto, we have that is a basis for .
) Let be an orthonormal basis in and one for . Let be such that for all and if . It follows from Bessel's inequality that is well defined and it's clearly linear and one-one. If we define (again by Bessel) and by definition and so is onto.
If then then for all and we have (Note that this previous argument proves injectivity):
and so is an isometry.
Since and coincide on a l.i set, they are equal on the whole linear span and (assuming which I think is a necessary hypothesis) hence they agree on the closure of the linear span which is . So and this finishes the proof.