I thought that I knew what a unitary transform is, until I started thinking about it.
(2^n-ons are hypercomplex numbers that are related via the 2^n-on construction. Including n=3 the 2^n-on construction gives the same numbers as the Cayley-Dickson construction. From there the 2^n-ons are "nicer".)

I know the following:

· Unitary transformations are special kinds of normal transformations.

· In Hilbert space unitary transformations have eigenvectors that together form an orthogonal base of the Hilbert space.

· The eigenvalues are unit size numbers.

· They leave inner products of vectors untouched.

· The adjoint transformation equals the inverse of the original.

· Unitary transformations move vectors around in Hilbert space, but not their own eigenvectors.

· They move subspaces around in Hilbert space, but their eigenvectors stay put.

· A sequence of unitary transforms can move a subspace over a significant distance.

· A Fourier transform is a special kind of unitary transformation.

Now my questions:

1.    Are the eigenvalues involved in the transfer of vectors?

2.    If the Hilbert space is defined over the 2^n-ons, can the eigenvalues be 2^m-ons with m > n? (2^m-ons resemble 2^n-ons in their lower n dimensions)

3.    What makes a unitary transform a Fourier transform?

4.    Can the action of any unitary transform be represented as a trail of infinitesimal unitary transforms?

5.    A subspace can be redefined by replacing the vectors that span this subspace. Can this redefinition be seen as (part of) a unitary transformation?

6.    Can replacements be done freely? (Inertia learns that there exists a reaction in case of a disordered replacement! So, what is well ordered?)