Example of dense non surjective operator

In summary, a dense non surjective operator is a linear operator in functional analysis that maps elements of one vector space to another, with a dense but incomplete image. It is significant in the study of vector spaces and their relationship with the operator. One example is the differentiation operator on continuously differentiable functions. It differs from a dense surjective operator, which has a dense and complete image. Applications of dense non surjective operators include the study of differential equations, partial differential equations, spectral theory, integral transforms, and boundary value problems.
  • #1
ak416
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Hi, can anyone give me an example of a bounded operator on a Hilbert space that has dense range but is not surjective? (Preferably on a separable Hilbert space)

Im pretty sure such an operator exists since the open mapping theorem requires surjectivity and not just dense range, but its just bothering me that I can't find an example.
 
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  • #2
How about \ell_1 (all absolutely summable sequences) being mapped into L_2[0, 2\pi) ( all complex valued square integrable functions on [0, 2\pi) ) by

(a_n) -> \sum a_n \exp( \sqrt{-1} n x)
 

1. What is a dense non surjective operator?

A dense non surjective operator is a type of linear operator in functional analysis that maps elements of one vector space to another. It is considered "dense" because the image of the operator is a dense subset of the codomain, and "non surjective" because the operator does not cover the entire codomain.

2. What is the significance of a non surjective operator?

A non surjective operator is important in functional analysis because it allows for the study of vector spaces that are not fully covered by the operator. This can provide insight into the structure and properties of the vector space and its relationship with the operator.

3. Can you give an example of a dense non surjective operator?

One example of a dense non surjective operator is the differentiation operator on the space of continuously differentiable functions. While the operator maps continuously differentiable functions to continuous functions, it does not cover the entire space of continuous functions.

4. How is a dense non surjective operator different from a dense surjective operator?

A dense non surjective operator does not cover the entire codomain, while a dense surjective operator does cover the entire codomain. In other words, a dense non surjective operator has a dense but incomplete image, while a dense surjective operator has a dense and complete image.

5. What are some applications of dense non surjective operators?

Dense non surjective operators have various applications in functional analysis, such as in the study of differential equations, partial differential equations, and spectral theory. They are also used in the construction of certain types of integral transforms and in the analysis of boundary value problems.

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