Talk:Compact operator on Hilbert space

Untitled

This article is a misunderstanding. It is pointless to include mathematical proofs of properties of compact operators in an article addressed for a general reader. A reference to a book would be enough.

:i am going to dispense with politeness here. who says this is "addressed for a general reader"? how "general" is general? if general is meant as in the general public, then no, this is not addressed to the general reader. a general reader who can't read this is supposed to be able to read a book? i believe the level of the article is comparable to introductory books on the subject. since you didn't make it clear, far as i see, there are two possibilities here,

:#if you can read, say, bounded operator, or uniform boundedness principle, or any other similar article, and find this article completely incomprehensible, then i would agree that certainly changes are needed. if that's the case, ignore 2. below and let's hear your suggestions. :#otherwise, how about directing your attention to stuff that you are actually competent enough to read and not waste your, and possibly other people's, time with comments like that? Cracking (chemistry) seems like Greek to me. if i left a comment similar to yours on its talk page, it certainly would be unhelpful and very stupid of me.

Mct mht 14:15, 30 August 2006 (UTC)

to anon: this is a very late follow up but if you still around, sorry for the tone in my reply above. it was way too harsh. i apologize. Mct mht 22:20, 13 December 2006 (UTC)

let me apologize again. there's really no excuse for my reaction there. Mct mht 04:29, 3 April 2007 (UTC)

A mistake?

The restriction of an operator on a closed invariant subspace is self-adjoint? All operators are invariant on the whole space H trivially.

That's definitely not true. Where does the article make that statement? Mct mht (talk) 03:29, 11 April 2013 (UTC)

Answer to "Question to writer"?

According to Theorem 6.6.1 in Functional Analysis: Applications in Mechanics and Inverse Problems by Lebedev, Vorovich, and Gladwell, separability is not required for the if and only if statement to be true. The Google books link to this book is: [[1]]

I hope this helps resolve the question. (The article's mainpage still contains the question; it should be changed as soon as someone more familiar with Hilbert spaces checks this fact.)

Owlcatowl (talk) 04:34, 23 February 2008 (UTC)

Please work on the readibility of the article

I've taken an entire year of Analysis at the graduate level and have a hard time understanding a lot of this. For instance:

The family of compact operators is a norm-closed, two-sided, *-ideal in L(H). Consequently, a compact operator T cannot have a bounded inverse if H is infinite dimensional. If ST = TS = I, then the identity operator would be compact, a contradiction.

The first sentence needs to be unpacked. If nothing else, there should at least be links to each of:

• norm-closed
• two-sided
• *-ideal
• L(H)

only the last of which I've been formally exposed to in my education. (at least in the way its named here)

--watson (talk) 09:44, 25 December 2008 (UTC)

say what it is useful for

it is badly written, just say what it is useful for, what are the main concepts we should think to, when, and why. give the fundamentals and historical examples, tell the difference between a compact operator and a bounded operator, and to which norm we should be thinking ($L_{1}$ or $L_{2}$ or $L_{\infty }$). So my advice would be : delete all these articles, and write a big article with all the definitions and concepts on how to extend finite dimensional linear operators : norms, inner product, orthogonality, continuity, convergence, boundedness, compactness, spectrum, singularities, exponential and holomorphic functional calculus, etc.. and stop using the word compact ! as it is the main purpose of the article to explain it !!!! 78.227.78.135 (talk) 01:47, 26 January 2016 (UTC)

Regarding "citation needed"

When reading this article I came across the passage "If a sequence of bounded operators $S_{n}\to S$ in the strong operator topology and $T$ is compact, then $S_{n}T$ converges to $ST$ in norm[citation needed]". We showed this in our 2018 LAMA article "The C-Numerical Range in Infinite Dimensions" (Lemma 3.2.(c)). Due to its recency, would the above article be a fitting reference here? I'm not utterly surprised that this result is known already but I personally am not aware of any book which lists it. Frederik vE (talk) 11:52, 6 September 2018 (UTC)

References

Is this statement correct? (problem with countability)

A priori, the statement "If X and Y are Hilbert spaces (...), then T : X → Y is compact if and only if it is continuous when viewed as a map from X with the weak topology to Y (with the norm topology)" (in this article) cannot be inferred from the quoted theorem in Zhu's book, because X with the weak topology (WT) is not first countable when X isn't finite-dimensional (if I'm right) UKe-CH (talk) 23:56, 29 March 2023 (UTC)

Moreover in Zhu's book he says in section 1.2. "In this book, we will only deal with separable Hilbert spaces" and this restriction is repeated at the beginning of section 1.3. in which one finds his theorem 1.14 used as source / reference in the part of this article containing the sentence I am quoting here. And said theorem 1.14 might be false when X is not separable. Indeed, in Zhu's second part of his proof (where the 'if' is treated) he seems to implicitly use that the closed unit ball B_1 of H with the WT is metrizable - which is easy to prove for H separable and is probably false when H isn't separable. This happens where he uses the compactness of B_1 to extract a weakly convergent sequence from a bounded sequence. BTW: that B_1 with the WT is metrizable doesn't mean that the same is true for the whole space H. I read in the article Weak topology things that seem to show that when H is infinite-dimensional, its WT is indeed non-metrizable ...--UKe-CH (talk) 22:40, 2 April 2023 (UTC)

My corrections in § Some general properties

I made two changes, each to correct one of the (probable) errors which I critized above (§ of this talk page immediately before this one) UKe-CH (talk) 19:58, 4 April 2023 (UTC)

weak-to-norm continuity

The proof of the spectral theorem contains the statement "Also, the compactness of T means (see above) that T : X with the weak topology → X with the norm topology is continuous."

I believe this is incorrect. Being weak-to-norm continuous requires a stronger condition than compactness: T would have to be finite rank. [2]

Kenneth.harris (talk) 17:03, 7 November 2023 (UTC)