1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Symmetric operator

  1. Feb 28, 2007 #1
    1. The problem statement, all variables and given/known data
    Let A be a linear operator on a Hilbert space X. Suppose that D(A) = X,
    and that (Ax, y) = (x, Ay) for all x, y in H. Show that A is bounded.

    3. The attempt at a solution

    I've tried to prove it by using the fact that if A is continuous at
    a point x implies that A is bounded.

    Suppose that x_n converges to x.
    || Ax_n - Ax ||^2 = || A(x_n - x) ||^2 = ( A(x_n - x), A(x_n - x) ) =
    (x_n - x, A^2 (x_n - x) ) <= ||x_n - x || || A^2(x_n - x) ||

    But i don't think I can conclude from this that because ||x_n - x|| -> 0, this expression goes to zero, since || A^2(x_n - x) || may blow up.. Or doesn't it?

    Please help me :)
  2. jcsd
  3. Feb 28, 2007 #2


    User Avatar
    Science Advisor
    Homework Helper

    D(A)=domain of A? I don't see how you can prove this. Self adjointness doesn't imply boundedness. You must be assuming A is also continuous? Then there is not much to prove. Better check the question.
    Last edited: Feb 28, 2007
  4. Mar 1, 2007 #3


    User Avatar
    Science Advisor
    Homework Helper

    In a Hilbert space bounded <=> continuous. If A is symmetric and everywhere defined, it's equal to its adjoint, hence self-adjoint hence closed. By the closed graph theorem, it is also continuous, end proof.

    The theorem you needed to prove is called "The Hellinger-Toeplitz theorem" and proves that unbounded symmetric operators cannot be defined on all [itex] \mathcal{H} [/itex].
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Symmetric operator
  1. Symmetric Polynomial (Replies: 2)

  2. Symmetric Equation (Replies: 1)

  3. Symmetric Group (Replies: 28)