Let be a linear operator on a finite-dimensional unitary space such that, for every the following implication holds:
Prove that then there exists a scalar and a unitary operator (where ) such that .
I'm aware of the following facts:
if an operator is unitary, and , then for every we have
Also, on a unitary space the following is true for every :
And finally, if is a unitary operator, then if is an orthonormal basis for , then is also an orthonormal basis for .
But so far, I've had no luck in employing these to solve this problem, so would greatly appreciate your help!