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• SemM
In summary, self-adjointness and Hermitian adjointness are two different properties that can be possessed by operators in a normed Hilbert space. While Hermitian adjointness is a property of the relationship between two operators, self-adjointness is a property of the action of an operator on a mapping. Both properties do not necessarily occur at the same time and can be compared and discussed further in other threads for a better understanding.
SemM
Gold Member
Hi, while reading a comment by Dr Du, I looked up the definition of Hilbert adjoint operator, and it appears as the same as Hermitian operator:

This is ok, as it implies that ##T^{*}T=TT^{*}##, however, it appears that self-adjointness is different?

Please correct me if this is wrong. And it looks to me that a self-adjoint operator is defined self adjoint if and only if it satisfies the rule of the inner product in a normed Hilbert space, where the conjugate transpose of the matrix elements is equal to the matrix.

So one can say that the first is a property which defines a particular symmetric aspect of the relationship of two operators, T and ##T^{*}##, while the latter defines a symmetric aspect of the action (or operation) of either of the operators separately, T or ##T^{* }## on a mapping - and that both properties (the former and the latter) not need to occur at the same time?

I am aware that much of this can be answered by looking at other threads, but this question compares these two critical properties, which beginners like me can misconceive, and thus may contribute to increase impact of the forum as a resource.

Thanks

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In the meantime, why not look at those threads so you’ll be able to discuss it better once someone actually replies here?

Let’s be proactive and not inactive.

Sounds good!

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## 1. What is the difference between a Hilbert-adjoint operator and a self-adjoint operator?

A Hilbert-adjoint operator is a linear operator on a Hilbert space that has a unique adjoint, while a self-adjoint operator is a special case of a Hilbert-adjoint operator where the operator is equal to its own adjoint.

## 2. How do you determine if an operator is Hilbert-adjoint or self-adjoint?

To determine if an operator is Hilbert-adjoint, you must first verify that it is a linear operator on a Hilbert space. You can then check if it has a unique adjoint by using the definition of the adjoint and verifying that it satisfies the necessary properties. To determine if an operator is self-adjoint, you only need to check if the operator is equal to its own adjoint.

## 3. What are the properties of a Hilbert-adjoint operator?

A Hilbert-adjoint operator has the property of being bounded, which means that it maps bounded sets to bounded sets. It also has the property of being closed, which means that the limit of any convergent sequence in the operator's domain is also in the operator's domain. Additionally, a Hilbert-adjoint operator has a unique adjoint and satisfies the property of being Hermitian.

## 4. Can a self-adjoint operator always be diagonalized?

Yes, a self-adjoint operator can always be diagonalized. This is because self-adjoint operators are unitarily equivalent to multiplication operators on a Hilbert space, which have a diagonal representation.

## 5. How are Hilbert-adjoint and self-adjoint operators used in quantum mechanics?

Hilbert-adjoint and self-adjoint operators are important in quantum mechanics because they represent physical observables, such as position, momentum, and energy. These operators have special properties that allow them to accurately describe the behavior of quantum systems and make predictions about their behavior.

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