Talk:Spectrum of Self-Adjoint Bounded Linear Operator is Real and Closed

We seem to have chosen the term "Hermitian" for self-adjoint bounded linear operators. Caliburn (talk) 12:56, 24 February 2023 (UTC)

Can you elaborate, please? I cannot get your point. --Usagiop (talk) 00:21, 25 February 2023 (UTC)

I think what Caliburn means is that we can simplify the wordage of this and its title by referring to "Hermitian operator" instead of "self-adjoint bounded linear operator", which would indeed improve reality. --prime mover (talk) 08:58, 25 February 2023 (UTC)

OK, Caliburn, do you also want to call self-adjoint densely-defined linear operators densely-defined Hamiltonian operators? --Usagiop (talk) 11:03, 25 February 2023 (UTC)

The question needs to be asked as to whether we need to implement a definition as specific as this. Any problem with densely-defined Hermitian operator? Unless we have a whole body of work on such a hybrid definition, it's probably not a quality move to define every combination of properties on its own page. We do it on occasion but unless there is a specific difference in definition between a densely-defined Hermitian operator and a Hermitian operator which happens to be densely-defined, I wouldn't go about creating a separate page for it. There is a danger of increasing confusion. --prime mover (talk) 19:12, 25 February 2023 (UTC)

I would prefer we call it all self-adjoint. But since Hermitian is what came first, Hermitian is what it should be from here. Wikipedia also calls bounded self-adjoint operators "Hermitian" and general self-adjoint operators to be self-adjoint, so it's not unusual. Caliburn (talk) 12:25, 25 February 2023 (UTC)

Did you suggest to call a self-adjoint operator a Hamiltonian operator, only if it is bounded? In that case, once you call it self-adjoint, does it imply that it is unbounded and densely-defined? I did not get your idea, really. --Usagiop (talk) 20:46, 25 February 2023 (UTC)

I am not sure where "Hamiltonian" comes from - I think "Hamiltonian" is something else in physics. Let's call general densely-defined linear operators self-adjoint, and if they are bounded and globally defined we will call them Hermitian. Caliburn (talk) 20:49, 25 February 2023 (UTC)

Sorry, I always meant Hermitian, not Hamiltonian. OK, you mean Hermitian = self-adjoint + bounded. --Usagiop (talk) 21:11, 25 February 2023 (UTC)

One idiosyncrasy of $\mathsf{Pr} \infty \mathsf{fWiki}$ is that we use a named definition or proof in preference to a non-named (perhaps more descriptive) name. Hence Hermitian Operator takes precedence over self-adjoint. I'm actually not too concerned, as long as we are as usual ruigorously consistent. Would be embarrassing to have one set of results about Hermitian operators overlapping another whole set of results about self-adjoint etc. operators without any direct acknowledgement that they are the same thing. This sort of thing has happened where 2 different people have posted up work in the same area over different names. Gets messy merging them unless you catch it early enough. --prime mover (talk) 19:03, 25 February 2023 (UTC)

Oh I only knew about that for results. Makes sense that we also like it for names too. It's no big deal, something to keep an eye on. Caliburn (talk) 19:34, 25 February 2023 (UTC)