Linear operator on Hilbert space with empty spectrum

In summary, The conversation is about constructing an example of a linear operator on Hilbert space with empty spectrum. It is stated that it is easy to construct an example with empty point-spectrum, but not with empty spectrum. It is also mentioned that any bounded operator on Hilbert space would have a non-empty spectrum, so an unbounded example is needed. There is a discussion about the definition of an operator on Hilbert space and the definition of spectrum. The book "Methods of Modern Mathematical Physics" is referenced as a potential resource for constructing an example.
  • #1
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Homework Statement


Much as the title says, I need to construct an example of a linear operator on Hilbert space with empty spectrum. I can very easily construct an example with empty point-spectrum (e.g. the right-shift operator on l_2), but this has very far from empty spectrum.

If I recall correctly, any such operator which is also bounded will have non-empty spectrum, so it falls to construct an unbounded example. However, I'm just not really sure how to get going on this. Could anyone help me? Thanks in advance! :)
 
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  • #2
What is your definition of "an operator on Hilbert space"? Must the domain be dense? What else? What is your definition of the "spectrum"? The devil is in these details.

Reed, "Methods of Modern Mathematical Physics", I. Functional analysis:
 

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  • #3
arkajad said:
What is your definition of "an operator on Hilbert space"? Must the domain be dense? What else? What is your definition of the "spectrum"? The devil is in these details.

Reed, "Methods of Modern Mathematical Physics", I. Functional analysis:

My definition of a Hilbert space is standard, i.e. a real or complex inner product space which is complete under the norm defined by the inner product. The spectrum of an operator T is, for me, the set of points 'p' in the complex plane for which T-pI is not invertible (I the identity map). I have never really looked at operators which are not on the whole space before - so though it was never explicitly stated, I believe the operator should have domain the entire Hilbert space (whichever Hilbert space it may be) - is such a thing possible? The wording of the question makes it seem to me like we aren't expected to state such an operator but merely to provide a means to construct one - though of course, if you can state one explicitly that's definitely better!

I am nowhere near a library which has any decent math texts at the moment unfortunately, but as soon as I get near one again I will be sure to track it down, thankyou!
 
  • #4
Alright I am attaching the relevant page. Part of the example (that T2 is closed) is left as an exercise. But this example may help you to construct your own example.
 

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1. What is a linear operator on Hilbert space with empty spectrum?

A linear operator on Hilbert space with empty spectrum is a type of mathematical operator that is used in functional analysis and quantum mechanics. It acts on a Hilbert space, which is a mathematical concept that represents a complete vector space, and has the property that its spectrum is empty, meaning it has no eigenvalues.

2. How is an empty spectrum different from a non-empty spectrum?

An empty spectrum of a linear operator means that it has no eigenvalues, while a non-empty spectrum means that it has at least one eigenvalue. An eigenvalue is a scalar that, when multiplied by a vector, produces the same vector but with a different magnitude. In other words, an empty spectrum indicates that the operator has no distinct states or energy levels.

3. What are the applications of a linear operator on Hilbert space with empty spectrum?

Linear operators on Hilbert space with empty spectrum have various applications in mathematics and physics. In functional analysis, they are used to study the properties of Hilbert spaces, and in quantum mechanics, they are used to describe the behavior of quantum systems. They are also used in other branches of mathematics, such as differential equations and mathematical physics.

4. How is the spectrum of a linear operator on Hilbert space with empty spectrum determined?

The spectrum of a linear operator on Hilbert space with empty spectrum can be determined using various mathematical techniques, such as the spectral theorem and the Gelfand-Mazur theorem. These theorems provide methods for finding the spectrum of an operator, which can be either empty or non-empty depending on the properties of the operator and the underlying Hilbert space.

5. Why is the concept of an empty spectrum important in mathematics and physics?

The concept of an empty spectrum is important in mathematics and physics because it helps us understand the behavior of linear operators and their corresponding Hilbert spaces. It also has practical applications in various fields, such as quantum mechanics and functional analysis. Additionally, the study of empty spectra can lead to a better understanding of the mathematical structures underlying physical phenomena.

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