Show that an operator on a Hilbert Space is unitary iff is a complete orthonormal set whenever is.

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- Oct 13th 2009, 07:32 PMproblemHilbert Space
Show that an operator on a Hilbert Space is unitary iff is a complete orthonormal set whenever is.

- Oct 13th 2009, 07:37 PMJose27
What is your definition of unitary operator?

- Oct 13th 2009, 09:17 PMproblem
Unitary here means an isomorphism in Hilbert space(linear,one to one and onto)

- Oct 14th 2009, 01:39 PMJose27
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.