Proving Maximal Symmetric Extension of a Symmetric Operator

Thread starter Nusc
Start date Mar 6, 2009
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In summary, a maximal symmetric extension of a symmetric operator is the largest possible self-adjoint extension of the operator, preserving its symmetry and defined on the largest possible domain. It is important to prove this extension in order to ensure a unique and consistent solution, as well as to allow for its use in various mathematical contexts. The most commonly used method for proving the maximal symmetric extension is the Friedrichs extension theorem, but other methods such as von Neumann and Krein extensions can also be used. The maximal symmetric extension is closely related to spectral theory, allowing for a complete set of eigenfunctions and eigenvalues to analyze the operator's properties. However, there can be limitations and challenges in proving this extension, such as the operator not being self

Mar 6, 2009

Nusc

A symmetric operator has a maximal symmetric extension.

How do you prove this?

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Mar 6, 2009

dvs

Zorn it!

What is a maximal symmetric extension of a symmetric operator?

A maximal symmetric extension of a symmetric operator is the largest possible self-adjoint extension of the operator. In other words, it is the extension that preserves the symmetry and is defined on the largest possible domain.

Why is it important to prove the maximal symmetric extension of a symmetric operator?

Proving the maximal symmetric extension of a symmetric operator is important because it ensures that the operator is well-defined and has a unique and consistent solution. It also allows for the operator to be used in a wider range of mathematical contexts and applications.

What methods are used to prove the maximal symmetric extension of a symmetric operator?

The most commonly used method for proving the maximal symmetric extension of a symmetric operator is the Friedrichs extension theorem. Other methods include the von Neumann and Krein extensions.

How is the maximal symmetric extension of a symmetric operator related to the spectral theory?

The maximal symmetric extension of a symmetric operator is closely related to the spectral theory, as it allows for the operator to have a complete set of eigenfunctions and eigenvalues. This makes it possible to analyze the operator's properties and behavior in a more precise and comprehensive manner.

Are there any limitations or challenges in proving the maximal symmetric extension of a symmetric operator?

Yes, there can be limitations and challenges in proving the maximal symmetric extension of a symmetric operator. These can include the operator not being self-adjoint, which would require more complex methods for determining the maximal symmetric extension. Additionally, the operator may have a singular or unbounded domain, which can also complicate the proof process.