Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Hilber space and linear bounded operator

  1. Feb 5, 2008 #1
    Let H be a Hilbert space and A: H-> H be a Linear Bounded Operator. Show that A can be written as A=B+C where B and C are Linear Bounded Operators and B is self-adjoint and C is skew.

    This is suppose to be an easy question but i'm not sure where to start.
    I know that self-adjoint is (B*=B) and skew is (C*=-C) but can someone show this?
     
  2. jcsd
  3. Feb 5, 2008 #2

    morphism

    User Avatar
    Science Advisor
    Homework Helper

    Hint: A+A* is self-adjoint. Can you find a similar expression that's skew? Then what?
     
  4. Feb 5, 2008 #3
    A-A* is skew
    so A = B+B* + C-C*?
     
  5. Feb 5, 2008 #4

    morphism

    User Avatar
    Science Advisor
    Homework Helper

    What are B and C?
     
  6. Feb 5, 2008 #5

    HallsofIvy

    User Avatar
    Science Advisor


    Okay, you know that A+ A* is self-adjoint and that A- A* is skew. Your second sentence is non-sense because you have not defined B and C. A is the only operator you have! Your answer must be entirely in terms of A.
     
  7. Feb 5, 2008 #6
    ohh, so B can be rewritten as A+A*, and C can be rewritten as A-A*?
     
  8. Feb 5, 2008 #7
    in the question, doesn't it say that B and C are linear bounded operators? i thought this meant i could write the whole B+B* thing
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook