nLab Fredholm operator



Functional analysis

Index theory




A continuous linear operator F:B 1B 2F \colon B_1\to B_2 between Banach spaces is Fredholm if it has finite dimensional kernel and finite dimensional cokernel.


The difference between the dimensions of the kernel and the cokernel of a Fredholm operator FF is called its index (the Fredholm index)

indFdim(kerF)dim(cokerF)=dim(kerF)codim(imF). ind F \coloneqq dim (ker F) - dim (coker F) = dim (ker F) - codim (im F) \,.

A standard equivalent characterization of Fredholm operators is the following:


A parametrix of a bounded linear operator F: 1 2F \colon \mathcal{H}_1 \to \mathcal{H}_2 is a reverse operator P: 2 1P \colon \mathcal{H}_2 \to \mathcal{H}_1 which is an “inverse up to compact operators”, i.e. such that FPid 2F \circ P - id_{\mathcal{H}_2} and PFid 1P \circ F - id_{\mathcal{H}_1} are both compact operators.


A bounded linear operator F:B 1B 2 F \colon B_1\to B_2 between Banach spaces is Fredholm, def. precisely it is has a parametrix, def. .




The image (range) of a Fredholm operator is closed.


The subspace Fred(B 1,B 2)B(B 1,B 2)Fred(B_1,B_2)\subset B(B_1,B_2) of Fredholm operators in the space of bounded linear operators with the norm topology is open.

In other words, given a Fredholm operator FF, there exists ϵ>0\epsilon\gt 0 such that every bounded linear operator GG satisfying GF<ϵ\| G-F\|\lt \epsilon is Fredholm. Fredholm operators on a fixed separable Hilbert space H=B 1=B 2H = B_1 = B_2 form a semigroup with respect to the composition and the index is a morphism of semigroups: indFG=indF+indGind F G = ind F + ind G.


The space FredFred of all Fredholm operators on an (infinite dimensional) separable Hilbert space is a model for the classifying space of degree-0 topological K-theory.



Fredholm operators generalize to Fredholm complexes. A finite chain complex

0C 0d 0C 1d 1C 2C n0 0 \to C_0 \stackrel{d_0}\to C_1\stackrel{d_1}\to C_2 \ldots C_n\to 0

of Banach spaces and bounded operators is said to be a Fredholm complex if the images d id_i are closed and the chain homology of the complex is finite dimensional. The Euler characteristic (the alternative sum of the dimensions of the homology groups) is then called the index of the Fredholm complex. Each Fredholm operator can be considered as a Fredholm complex concentrated at zero. Each Fredholm complex produces a Fredholm operator from the sum of the even- to the sum of the odd-numbered spaces in the complex.

One can consider Fredholm almost complexes, where d id i1d_i \circ d_{i-1} is not zero but the image of that operator is compact. Out of every Fredholm almost complex one can make a complex by correcting the differentials by compact perturbation terms. Elliptic complexes give examples of Fredholm complexes.


Textbook accounts:

Discussion of the space of Fredholm operators as the classifying space for topological K-theory:

  • Michael Atiyah, Appendix of: K-theory, Harvard Lecture 1964 (notes by D. W. Anderson), Benjamin 1967 (pdf, pdf)

See also:

  • Wikipedia, Fredholm operator

  • A. S. Mishchenko, Векторные расслоения и их применения (Vector bundles and their applications), Nauka, Moscow, 1984. 208 pp. MR87f:55010

  • S. Rempel, B-W. Schulze, Index theory of elliptic boundary problems, Akademie–Verlag, Berlin, 1982.

  • Lars Hörmander, The analysis of linear partial differential operators. III. Pseudo-differential operators, 1994, reprinted 2007.

  • Pietro Aiena, Fredholm and local spectral theory, with applications to multipliers, book page

  • Otgonbayar Uuye, A simple proof of the Fredholm Alternative, arxiv/1011.2933

  • Alexander Grothendieck, La théorie de Fredholm, Bulletin de la Société Mathématique de France 84 (1956), p. 319-384, numdam

  • Marina Prokhorova, Spectral Sections, arXiv:2008.04672.

  • Marina Prokhorova, Spaces of unbounded Fredholm operators. I. Homotopy equivalences, arXiv:2110.14359.

  • Marina Prokhorova, The continuity properties of discrete-spectrum families of Fredholm operators, arXiv:2201.09869.

  • Marina Prokhorova, From graph to Riesz continuity, arXiv:2202.03337.

For Fredholm complexes, see

  • Graeme Segal, Fredholm complexes, Quarterly Journal of Mathematics 21:4 (1970), 385–402. doi.
category: analysis

Last revised on March 12, 2023 at 04:32:03. See the history of this page for a list of all contributions to it.