Hilber space and linear bounded operator

In summary, given a Hilbert space H and a Linear Bounded Operator A: H-> H, it can be shown that A can be written as A=B+C, where B and C are also Linear Bounded Operators. B is self-adjoint (B*=B) and C is skew (C*=-C). This can be proven by utilizing the fact that A+A* is self-adjoint and A-A* is skew, and then expressing B and C in terms of A. Thus, B+B* and A-A* can be written as linear bounded operators, satisfying the requirements for B and C.
  • #1
braindead101
162
0
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?
 
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  • #2
Hint: A+A* is self-adjoint. Can you find a similar expression that's skew? Then what?
 
  • #3
A-A* is skew
so A = B+B* + C-C*?
 
  • #4
What are B and C?
 
  • #5
braindead101 said:
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
ohh, so B can be rewritten as A+A*, and C can be rewritten as A-A*?
 
  • #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
 

FAQ: Hilber space and linear bounded operator

1. What is a Hilbert space?

A Hilbert space is a mathematical concept that refers to a complete vector space with an inner product defined on it. It is named after the German mathematician David Hilbert and is widely used in functional analysis, quantum mechanics, and other areas of mathematics.

2. What is the significance of a Hilbert space?

Hilbert spaces are important in mathematics because they provide a framework for studying functions and vectors in a way that is similar to how we study numbers using real or complex numbers. They also have many applications in physics, engineering, and other fields.

3. What is a linear bounded operator?

A linear bounded operator is a mathematical concept that refers to a function that maps one Hilbert space to another in a linear manner. It is called "bounded" because it has a finite or bounded range, which means that it does not map elements of the first Hilbert space to elements of the second space with infinite or unbounded magnitude.

4. How are Hilbert spaces and linear bounded operators related?

Hilbert spaces and linear bounded operators are closely related because linear bounded operators are defined on Hilbert spaces. They are used to model linear transformations between Hilbert spaces and play a crucial role in functional analysis, which is the study of infinite-dimensional vector spaces.

5. What are some examples of linear bounded operators?

Some common examples of linear bounded operators include differentiation and integration operators, as well as the Fourier transform. In quantum mechanics, observables such as position and momentum are also represented as linear bounded operators on Hilbert spaces.

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