Talk:Positive operator (Hilbert space)

Additional mention of self adjointness in second paragraph
While self adjointness is mentioned in the introduction, the paragraph "Positive and positive definite operators" begins like a new definition, also introduces a new variable, but doesn't mention self adjointness, which is necessary for the stated equivalences.

Bobrummel (talk) 17:55, 10 December 2020 (UTC)

you should really put a new positive that will tell you about grade 6 stuff because ots hard to figure out this stuff

Multiple notions of "Positive Operator"
The section about positive operators on ordered vector spaces does not really belong into this article. And even if it should be included, it must be pointed out that it refers to a different notion of "positive operator" that has few in common with positive definiteness. In particular, one should abstain from calling the Laplace operator and the Banach limit positive operators in a single sentence, and I still wonder what is meant by an "absolute value operator" (the only things I could think of here are nonlinear, unless it refers to the fact that the absolute value of a self-adjoint operator is positive; but this shouldn't be called "absolute value operator" anyway).

I am not sure whether the right action would be to delete the second part of the section on "Positive and positive definite operators". Anyone with more experience regarding Wikipedia's netiquette is welcome to help me out.

194.95.184.127 (talk) 13:26, 29 September 2011 (UTC)

This is in fact a serious problem. These two notions are incompatible with each other when the same ordered vector space is used as the inner product space in one case and the ordered banach space in the other. Consider the space R2 with the usual inner product, the cone of the positive y axis and the projector A onto the line y=x. Clearly this projector is a positive operator in the sense introduced in the introduction and the section "Positive and positive definite operators". Further a=(0,1) is a positive vector in R2 under this order, however Aa = (1,1) is not a positive element of R2 with this order. Thus these definitions are unrelated and incompatible. In keeping with rule 3 I will delete this section. The current situation is very confusing. Any attempt to reinstate the removed section should preferably include some text to highlight this inconsistency to the reader, or a correction to the argument laid out here.

C Chippyprice (talk) 12:30, 4 June 2019 (UTC)

Positive element of what?
The term "positive element" is meaningless without further qualification - i.e. positive element of a partially ordered vector space, or of a C*-algebra (which is, BTW, partially ordered, so it's a special case). So I don't think there should be a page with this name at all. Or maybe we should redirect that to "partially ordered vector space". On the other hand, positive operators in Hilbert space certainly do deserve an article on them. The C*-algebra case could be treated there as a generalization.

213.57.178.241 (talk) 20:05, 21 March 2014 (UTC)

Cauchy-Schwarz inequality
I don't think this proof is self-contained. The article defines an inner product using $$A+\epsilon.1$$ and then applies Cauchy-Schwarz to it. But the Cauchy-Schwarz inequality only applies to inner products that are symmetric (or conjugate-symmetric), and we haven't yet proved that this condition applies. But the next section proves that it's conjugate-symmetric in the complex case. Maybe we should swap the 2 sections? David.knipe (talk) 20:40, 2 July 2022 (UTC)

Inconsistent terminology
The article contains this sentence:

"A natural ordering of self-adjoint operators arises from the definition of positive operators."

But nowhere in the article has the term "positive operator" been defined.

The terms "positive semi-definite" and "positive definite" have been defined.

I hope someone knowledgeable about this subject can introduce the term "positive operator" into the article and define it clearly for readers (since this term is certainly in use). 2601:200:C000:1A0:52D:8748:7384:FB5A (talk) 20:13, 26 October 2022 (UTC)

Some changes
I have made some changes in response to the comments above.

Mike Stone (talk) 19:29, 1 January 2024 (UTC)

Feedback from New Page Review process
I left the following feedback for the creator/future reviewers while reviewing this article: Ref 1,5,8 and 13 and have Harv errors as the Palmer 1977 doesn't exist. Please add it.

 scope_creep Talk  17:59, 12 January 2024 (UTC)