Hilber space and linear bounded operator

  • #1
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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?
 

Answers and Replies

  • #2
morphism
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Hint: A+A* is self-adjoint. Can you find a similar expression that's skew? Then what?
 
  • #3
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A-A* is skew
so A = B+B* + C-C*?
 
  • #4
morphism
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What are B and C?
 
  • #5
HallsofIvy
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A-A* is skew
so A = B+B* + C-C*?


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.
 
  • #6
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ohh, so B can be rewritten as A+A*, and C can be rewritten as A-A*?
 
  • #7
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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
 

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