Unitary Operators: Why is Spectrum on Unit Circle?

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Homework Statement



why is the spectrum of the unitary operator the unit circle?

Homework Equations



i know that U^(-1)=U* and i know this makes U normal
i also know that normal means UU*=U*U


The Attempt at a Solution



i know that from spectral theory there is some lambda in the spectrum
such that abs(lambda)=1, but i don't understand why ALL of them are on
the unit circle. (i understand the operator, but spectrums are confusing to me.

thanks
 
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If x is an eigenvector of U, and \lambda is its eigenvalue, then what is the length of Ux?

Ux = \lambda x
 
hmmm, i know that, so i have Ux = Lx and L is 1...so then Ux = x...? I'm just getting lost
 
Just so we're happy: you're only talkiong about operators on finite dimensional spaces, right? Because, in general, spectrum and 'set of eigenvalues' are not the same thing.
 
heh, sorry about that. a unitary operator in a Hilbert space is what I'm working with
 
Raven2816 said:
hmmm, i know that, so i have Ux = Lx and L is 1...so then Ux = x...? I'm just getting lost
L may not be 1, and you don't know what L is. But what is the length of Ux?
 
Let x be an e-vector of U with e-value L as above.What do we know?

<x,x>=<Ux,Ux>

because U is unitary. If you don't see that then consider the intermediate steps:

<x,x>=<Ix,x>=<U*Ux,x>=<Ux,Ux>

Note we've just used the unitariness of U. So now we've got to use the fact that x is an e-vector

<x,x>=<Ux,Ux>=<Lx,Lx>=...?
 
and <Lx, Lx> is the inner product of an e-vector with its e-value...so do i get one? or am i using L=1?
 
Raven2816 said:
and <Lx, Lx> is the inner product of an e-vector with its e-value...so do i get one? or am i using L=1?
L is a number. There is a formula for pulling a number multiplier out of an inner product.
 
  • #10
a formula? isn't <Lx, Lx> = ||Lx||^2?
and i know that L<x, y> = <Lx, y> ...
 
  • #11
Raven2816 said:
i know that L<x, y> = <Lx, y> ...
What about <x,Ly>?
What about <Lx,Lx>?
 
  • #12
Raven, could you satisfy my curiosity? Are you taking a course, or reading a book on your own? What is the name and level of the course or the name of the book?
 
  • #13
ahhh i see what you mean! thanks!
 
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