Talk:Fredholm alternative

Non-zero lambda
Perhaps one should mention, in the statement of Fredholm's alternative, that lambda is non-zero. 208.164.49.125 03:34, 24 April 2006‎

adjoint
shouldnt we mention the orthagonality between the kernel of the adjoint and the range of operator. 169.237.30.124 03:29, 19 November 2006‎

Alternative... to what?
To what is the Fredholm alternative an alternative? —Ben FrantzDale 15:50, 20 December 2006 (UTC)


 * not quite sure, but i don't think the article's claim that Fredholm alternative was due to Fredholm is right. Fredholm considered integral operators and it was F. Riesz who developed the general spectral theory of compact operators and, IIRC, formulated the Fredholm alternative. Mct mht 12:36, 8 April 2007 (UTC)


 * The theorem states that either option 1 is true or option 2 is true. One must be true and both cannot be true. This is the alternative, because we alternate between the two options. Italo Tasso (talk) 20:54, 1 May 2015 (UTC)

clean up
i would suggest that the "linear algebra" and "correspondence" section be deleted. sure, a linear map on a finite dimensional space is injective iff it's surjective. so every number in the spectrum is an eigenvalue. the Fredholm alternative says something similar about compact operators. but to say there's some correspondence with the finite dimensional case would be pushing it, IMHO. for one thing, the spectrum of a compact operator on an infinite dim Banach space must accumulate at 0. (of course the Fredholm alternative hinges crucially on the compactness assumption. one can mention this connection with the finite dimensional case, where this analytic aspect is completely surpressed.) Mct mht 15:14, 8 April 2007 (UTC)


 * There is definitely something messy there. One of the linear algebra formulations is about the kernel being nontrivial, but the other formulation is more about the cokernel (or perhaps rather the orthogonal complement of the range) being nontrivial. For a square matrix they're equivalent, but is that equivalence trivial in comparison to the theorem that is being discussed? 130.243.94.123 (talk) 13:15, 4 December 2023 (UTC)

Almost-Fredholm?
Anything smart that can be said about the case of "almost-resonances" or "small denominators", where there are small but finite $$\varepsilon>0$$ with
 * $$\int _a^b \left| \lambda \varphi(x) - \int_a^b K(x,y) \varphi(y) \,dy\right|^p dx <\varepsilon \int _a^b \left| \varphi(x)\right|^p dx .$$

That is, there are some phi that are "almost" eigenfunctions, but not quite. If one could find such phi for any epsilon, then, by completeness or compactness, a sequence of such phi would converge to an actual eigenvalue. So, the interesting case is where there's a minimum epsilon that acts as a "band-gap" or "mass gap", bounding away from an actual eigen-solution. As I recall, Poincare called this the "small denominator problem" when analyzing the perturbative series for the three-body problem. In general, such "almost resonances" lead to thermalization (mixing and ergodicity) in wave mechanics (e.g. the Fermi–Pasta–Ulam–Tsingou problem). So my question is "where is this covered in wikipedia"? (It's also got something to do with solitons, i.e. the Lax pairs, but I can't quite figure out how it all fits together. Are the Lax pairs somehow specifying the small-epsilon that prevents an almost-resonance from becoming a true resonance? How does that work?) 67.198.37.16 (talk) 00:08, 13 January 2024 (UTC)