What, exactly, has Louis de Branges proved about the Riemann Hypothesis? I know this is a dangerous topic which could attract many cranks and nutters, but:
According to Wikipedia [and probably his own website, but I have a hard time seeing exactly what he's claiming] Louis de Branges has claimed, numerous times, to have proved the Riemann Hypothesis; but clearly few people believe him. His website is:
http://www.math.purdue.edu/~branges/site/Papers
but I find his papers difficult to follow. However, whether or not you believe him, his arguments presumably should prove something, even if not the full RH.
So, my question is:
Are there any theorems related to the Riemann Hypothesis and similar problems, arising from his work, which have been fully accepted by the mathematical community and published (or at least submitted)?
 A: The paper by Conrey and Li "A note on some positivity conditions related to zeta and L-functions"
https://arxiv.org/abs/math/9812166
discusses some of the problems with de Branges's argument. They describe a (correct) theorem about entire functions due to de Branges, which has a corollary that certain positivity conditions would imply the Riemann hypothesis. However Conrey and Li show that these positivity conditions are not satisfied in the case of the Riemann hypothesis.
So the answer is that de Branges has proved theorems in this area that are accepted, and his work on the Riemann hypothesis has been checked and found to contain a serious gap. (At least the version of several years ago has a gap; I think he may have produced updated versions, but at some point people lose interest in checking every new version.)
Update: there is a more recent  paper by Lagarias discussing de Branges's work. Lagarias, Jeffrey C., Hilbert spaces of entire functions and Dirichlet $L$-functions, Cartier, Pierre (ed.) et al., Frontiers in number theory, physics, and geometry I. On random matrices, zeta functions, and dynamical systems. Papers from the meeting, Les Houches, France, March 9–21, 2003. Berlin: Springer (ISBN 978-3-540-23189-9/hbk). 365-377 (2006). ZBL1121.11057, MR2261101. Author's website and Wayback Machine.
A: In the years since this question was first asked and answered, there have been some new developments (namely, 80+ pages of commentary by Eric Kvaalen, and a large number of new versions of de Branges' paper).
Starting with the Wiki (which provides a decent account of events), I followed a link to Eric Kvaalen's "Commentary on work of Louis de  Branges" from 2016, a 70-page paper, available only in the very unorthodox format of PNG files. The relevant versions of de Branges' paper can be found here: https://web.archive.org/web/20160701000000*/http://www.math.purdue.edu/~branges/proof-riemann.pdf, from 2013-2016 (past 2016 the links are dead). I will type out Kvaalen's conclusion (from page 70), regarding I think this version of de Branges' paper (archived on Nov. 17 2015):

In this wide-ranging paper, de Branges has enunciated many true statements but also quite a few false statements or statements with no proof offered. Having analyzed the first 59 of his 90 pages, I will stop at this point because the later portions of what I have analyzed do not provide a good foundation for the rest. Besides which, de Branges has now replaced this version with a new version (dated November 23, 2015).

Linked in the aforementioned Kvaalen webpage is a slightly later paper (also 2016), "The application of the de Branges method to prove the Riemann Hypothesis" regarding a much earlier version of de Branges' paper from 2006. The versions from 2004-2008 can be found here
http://web.archive.org/web/20080901000000*/http://www.math.purdue.edu/~branges/riemannzeta.pdf (past 2009 the links are dead).
(Looking only at the first page, there seems to be at least 5 eras of de Branges' paper: 2004 where he began with a section on "Locally compact skew-fields"; May 2005 where he began with a section on "Hypercommplex analysis"; Sep. 2005-2008 where he began with a definition of a de Branges space; 2013-2014 where he began with a section titled "The Inverse Problem for the Vibrating String"; and 2015 onward where he begins with a section titled "Generalization of the Gamma Function". And that's only looking at obvious differences on the first page…)
Anyways, I retype the introduction and some conclusions that Kvaalen had about the 2006 version:

… in 2005 [de Branges] wrote a 41-page paper called "A Proof of the Riemann Hypothesis", the last version of which is dated July 2006. Around 2007 he started on a somewhat different line of attack, and he removed the 2006 paper from his website. His own opinion on his older argument has wavered. As recently as June 2014 he told me that the older argument was correct, but in conversations in May 2017 he was unable to recall exactly what he had meant in a key sentence, and thought that there was not a proof in the paper.
I have analyzed a recent version of his current work [above mentioned "Commentary …"] and found it wanting. But in this paper I would like to go back to his earlier work. The 2006 paper is hard to follow and does not explain all his reasoning. It presents three theorems, but does not even show how they can be applied to the Riemann Hypothesis. I will show how to prove the Riemann Hypothesis using Theorem 3 in it, if that theorem is true.
Unfortunately, the last sentence of his attempted proof of Theorem 3 is the key step which remains unexplained. There are also errors in the proof of Theorem 1 which I do not go into here. Theorem 1 may not be true, but here may be a form which is true that covers the specific case needed.

And from Section 5 "Conclusion regarding the Riemann zeta function":

I have demonstrated how it would be possible to apply the 2006 draft by de Branges to prove the Riemann Hypothesis, assuming Theorem 3 to be true. But the last sentence in de Branges's attempt to prove Theorem 3, “The computation of adjoints denies $w-ih$ as a zero of $E'(z)$ distinct from $\overline w$ or as a double zero equal to $\overline w$”, is problematic. It is not clear what exactly he meant, and whether the meaning is true. There is also doubt about the truth of Theorem 1. My analysis of the proof (which I am not yet ready to put on line) finds errors. However, de Branges has suggested to me that a weaker form would be true and sufficient. Theorem 1 is not actually mentioned in the rest of the paper, but there is one place where it seems to be used.

The version found on his webpage, which I presume is the most current one, seems to date from 2017: https://www.math.purdue.edu/~branges/proof-riemann-2017-04.pdf. His webpage also has his most recent attempts on the invariant subspace problem (105 pages), and a handwritten 13-page scan on the "Banach measure problem". I haven't seen any discussion on these papers, which although sad, seems justifiable based on the above complaints.
