Can unitary operators on hilbert space behaive like rotations?

[cyeus]

Homework Statement

unitary operators on hilbert space

Homework Equations

is there a unitary operator on a (finite or infinite) Hilbert space so that cU(x)=y, for some
constant (real or complex), where x and y are fixed non-zero elements in H ?

The Attempt at a Solution

I know the answer in R^2, it is enough to consider U a suitable rotation so that U(x)
be a point on the straight line Ry={ry; r ε R}, and then there is a scaler r in R so that
rU(x)=y. I guess this is true for R^n too.

 

[voko]

You would need to define "rotation" in a Hilbert space. In 2D Euclidean space it is a one-parameter linear transformation. Scaling is also a one-parameter linear transformation. Together that is two parameters, which is just enough to describe any element in 2D. This, of course, would not work in a higher-dimensional space, not even Euclidean 3D.