Unitary Transformation Problem

1. Jun 30, 2009

Sam_Goldberg

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

Hi guys, I'm working on a question in Tom M Apostol's second calculus book, on page 141, number 5. It is: Prove that if T (an operator on a vector space V) is linear and norm-preserving, then T is unitary.

2. Relevant equations

Okay, a transformation T is linear if it satisfies (x and y are vectors, c is a complex scalar):

T(x + y) = T(x) + T(y)
T(cx) = cT(x)

A norm-preserving transformation satisfies for all x (these are norms and inner products):

|T(x)| = |x|
Equivalently, (T(x), T(x)) = (x, x)

Finally, T is unitary if for all x and y:

(T(x), T(y)) = (x, y)

Note that this is a complex vector space, so, in particular:

(x , y) = (y, x)*

where * is the complex conjugate. This fact is the key to my difficulties in this problem.

3. The attempt at a solution

Here goes:

(T(x), T(y))
= (T(x + y), T(y)) - (T(y), T(y))
= (T(x + y), T(y)) - (y , y)
= (T(x + y), T(x + y)) - (T(x), T(x)) - (T(y), T(x)) - (y, y)
= (x + y, x + y) - (x, x) - (y, y) - (T(y), T(x))
= (x, y) + (y, x) - (T(y), T(x))

Therefore: (T(x), T(y)) + (T(y), T(x)) = (x, y) + (y, x)

Now, due to the complex conjugate rule for the inner product, I can only deduce from this equation that:

(T(x), T(y)) + (T(x), T(y))* = (x, y) + (x, y)*

So, I have only proven that the real parts of (T(x), T(y)) and (x, y) are equal, not the imaginary parts. I would appreciate any inputs on how to prove this too. Thanks.

2. Jun 30, 2009

g_edgar

Try it again with the inner product (T(x), T(iy)), or some other combination with an i in there.

3. Jul 1, 2009

Sam_Goldberg

You are absolutely right. If we replace in my final equation x by ix, we get:

i(T(x), T(y)) - i(T(x), T(y))* = i(x, y) - i(x, y)*

which clearly shows that the imaginary parts of (T(x), T(y)) and (x, y) are equal. Thank you very much.