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Unitary Transformation Problem

  1. Jun 30, 2009 #1
    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. jcsd
  3. Jun 30, 2009 #2
    Try it again with the inner product (T(x), T(iy)), or some other combination with an i in there.
     
  4. Jul 1, 2009 #3
    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.
     
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