# Unitary operators

1. May 9, 2007

### Raven2816

1. The problem statement, all variables and given/known data

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

2. Relevant equations

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

3. 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 dont understand why ALL of them are on
the unit circle. (i understand the operator, but spectrums are confusing to me.

thanks

2. May 9, 2007

### Jimmy Snyder

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

$Ux = \lambda x$

3. May 9, 2007

### Raven2816

hmmm, i know that, so i have Ux = Lx and L is 1....so then Ux = x...? i'm just getting lost

4. May 9, 2007

### matt grime

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.

5. May 9, 2007

### Raven2816

heh, sorry about that. a unitary operator in a Hilbert space is what i'm working with

6. May 10, 2007

### Jimmy Snyder

L may not be 1, and you don't know what L is. But what is the length of Ux?

7. May 10, 2007

### matt grime

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>=.....?

8. May 10, 2007

### Raven2816

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?

9. May 10, 2007

### Jimmy Snyder

L is a number. There is a formula for pulling a number multiplier out of an inner product.

10. May 10, 2007

### Raven2816

a formula? isn't <Lx, Lx> = ||Lx||^2?
and i know that L<x, y> = <Lx, y> ....

11. May 10, 2007

### Jimmy Snyder

What about <x,Ly>?
What about <Lx,Lx>?

12. May 10, 2007

### Jimmy Snyder

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. May 10, 2007

### Raven2816

ahhh i see what you mean! thanks!

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