Graduate Hilbert-adjoint operator vs self-adjoint operator

[SemM]

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:

https://en.wikipedia.org/wiki/Hermitian_adjoint

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

 

[jedishrfu]

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.

 

[SemM]

Sounds good!