Results 1 to 2 of 2

Math Help - frames of a hilbert space

  1. #1
    Newbie
    Joined
    Apr 2011
    Posts
    1

    frames of a hilbert space

    Hi! I came across this problem while I was reading a supplementary material for our course on hilbert spaces. Can anyone help me with this? Thank you!

    given an orthonormal basis { \phi_{n}} of a hilbert space H and a one-to-one and onto continuous linear transformation T from H to H, prove that {T \phi_{n}} is a frame for H.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by krooooochan View Post
    Hi! I came across this problem while I was reading a supplementary material for our course on hilbert spaces. Can anyone help me with this? Thank you!

    given an orthonormal basis { \phi_{n}} of a hilbert space H and a one-to-one and onto continuous linear transformation T from H to H, prove that {T \phi_{n}} is a frame for H.
    For x in H, \sum_n|\langle x,T\phi_n\rangle|^2 = \sum_n|\langle T^*x,\phi_n\rangle|^2 = \|T^*x\|^2 \leqslant \|T\|^2\|x\|^2. That gives you one of the two inequalities that you need for a frame. The other one comes from the Banach inversion theorem, which says that T (and therefore also T*) must have a bounded inverse. It follows that \|x\| = \|(T^*)^{-1}T^*x\| \leqslant \|(T^*)^{-1}\|\|T^*x\| and hence \|T^*x\|^2 \geqslant  \|T^{-1}\|^{-2}\|x\|^2.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Banach space & Hilbert space
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: May 24th 2011, 02:06 PM
  2. Hilbert Space
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: October 25th 2009, 10:18 AM
  3. Hilbert Space
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: October 14th 2009, 02:39 PM
  4. Inverse of Mapping from Hilbert Space to Hilbert Space exists
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: June 2nd 2009, 09:15 PM
  5. Hilbert Space
    Posted in the Calculus Forum
    Replies: 1
    Last Post: January 25th 2009, 12:11 PM

Search Tags


/mathhelpforum @mathhelpforum